commons.hostos.cuny.edu...mathematics teaching-research journal on-line a peer-reviewed scholarly...

Mathematics Teaching-Research Journal On-Line

A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)

Bronislaw Czarnocha (Hostos Community College) Vrunda Prabhu (Bronx Community College) City University of New York

Volume 2 Issue 1 Date September 2007

Readers are free to copy, display, and distribute this article, as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal On-Line, it is distributed for non-commercial purposes only, and no alteration or transformation is made in the work. All other uses must be approved by the author(s) or MT-RJoL. MT-RJoL is published jointly by the Bronx Colleges of the City University of New York.

Editorial With this issue of the Mathematics Teaching-Research Journal, we begin the theme of democratization of mathematics and science education. Over the next several issues, our articles, your thoughts and the discussion will create the climate to arrive at a common shared understanding of what it means to have a democratic access to mathematics education. The word democratic includes equal access and existence of a level playing field. Do these two phrases also then include, creating the conditions necessary to address the learning needs of the students? Section 1 of the current issue addresses different aspects of the process of democratization. In Section 2 we bring mathematics classroom innovations from around the world and in Section 3 we present two reflective pieces upon our teaching work. Editorial News Our journal is now funded jointly by the National Science Foundation and CUNY's Office of Undergraduate Education. A new project undertaken by the teaching-research methodology - VISUALIZE- A TR-NYC project is aimed at creating a base, where students combine their tactile and visual sense to guide themselves in visualizing the geometry they create out of making paper models of polyhedra. The project is conducted jointly with the Siedlce group of teachers and mathematicians in Poland. The nets for the polyhedra, largely a creation of Waclaw Zawadowski and Krystoff Mostowski, are entertaining students of all ages in mathematics classes and outside.

Mathematics Teaching-Research Journal On-Line A peer-reviewed scholarly journal Editors: Anne Rothstein (Lehman College)




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Acres of Diamonds in New York’s High Stakes Testing Environment;

Empowering Educators via the Internet Steve Watsoni

The Jefferson Math Project as a Prototype

The Jefferson Math Project (JMAP) is grounded in the New York commencement level

high school mathematics curriculum. It is a non-profit initiative of two New York City Teaching

Fellows, who both began their teaching careers at Thomas Jefferson High School in Brooklyn in

2003. As we initially worked to develop our own subject matter and teaching expertise in

mathematics, we began developing and refining a database and other resources for exploring and

analyzing historical assessment practices in the Regents Math A/B curriculum. We soon found

that we were looking at high stakes testing through a pragmatic new lens that facilitated the

alignment of instruction with state assessment practices, and we also found that other teachers

wanted the resources we were developing. We began giving away these resources through

teacher training programs at Thomas Jefferson High School, Brooklyn College and City College.

In March 2005, we presented JMAP at a symposium celebrating the fifth anniversary of the New

York City Teaching Fellows, after which 2,500 copies of the JMAP 611 CD-ROM were created





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by the New York City Department of Education (NYCDOE) and distributed to mathematics

teachers throughout the City in September 2005.

JMAP also established a website (www.jmap.org) for mathematics teachers in March

2005, and provides lesson plans and teaching resources on all aspects of the New York Math

A/B curriculum, as well as the new curriculum currently being implemented. These materials

include over three gigabytes of JMAP books, workbooks, worksheets, solutions, grids, graphs,

exams, etc., which are available in MS Word, Adobe pdf, and other electronic formats. JMAP

materials are intended for the use of teachers in the classroom, but students and researchers are

welcome to use JMAP’s resources as well. Several thousand visitors, mostly from the City and

State of New York, regularly download lesson plans and teaching resources from the JMAP

website. During Regents week in June 2007, the website averaged over 5,000 visitors each day.

JMAP is listed as a resource for teachers by the Association of Mathematics Teachers of

New York State, Math for America, the Drexel Math Forum, the New York Math Exchange, the

Home School Math Network, NYLearns, and many other groups interested in mathematics

education in New York State. JMAP is not affiliated with any publisher, school, or other

corporate entity, though its co-founders are both high school math teachers employed by the

NYCDOE and both have personal affiliations with either universities or publishers. As such,





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JMAP provides a prototype for a relatively low-cost lesson plan and resource management

system with high potential for influencing teaching praxis.

A Vision of Technology Empowered Educators

The following diagram provides a high level overview of the major elements of a lesson

plan and resource management system based on the JMAP model.





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This vision for a new system involves user friendly, readily available software, which seamlessly

integrates the following features:

database management of static and dynamic resources;

word processing; and





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desktop publishing.

Commercially available, integrated suites of software, such as Microsoft’s Office Suite and

Adobe Acrobat/Reader, provide all required functionality in single integrated packages, and can

serve as the central core of such a lesson plan and resource management system. A fundamental

assumption behind this design specification is that teachers are more likely to invest the time and

intellectual effort necessary for systems mastery when the lesson plan and resource management

system is free of additional costs and grounded in pre-existing, user-friendly software with which

the teacher is already familiar. If such a public domain lesson plan management system is

created, curriculum specialists/website coordinators could then focus on the development and

dissemination of databases and teaching resources that are consistent with a sponsoring

organization’s topical and/or pedagogical agenda. Wikipedia style, refereed websites for the

development and dissemination of model lesson plans in editable formats could be used to

identify and disseminate best teaching practices.

The incremental costs of such an initiative would be limited to salaries or stipends for the

curriculum specialists/website coordinator(s) plus the relatively low costs of maintaining a

website. These curriculum specialists/website coordinators could be inside or outside the official

bureaucracy of public education. (JMAP is an example of curriculum specialists/website





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coordinators who are outside the official bureaucracy.) These databases and resources, once

developed, are easily shared with large numbers of teachers (and students), as evidenced by the

JMAP statistics. To the extent that the lesson plans and resources thus developed and delivered

via Internet are value adding, they will be sought out and used by teachers, thereby constituting a

powerful means of communicating and influencing the implemented, as opposed to the stated,

curriculum.

Existing Prototypes of Lesson Plan Management Systems and Their Weaknesses

Many textbook publishers produce some form of technology-based lesson plan and

resource management systems on CD-ROMs, which typically accompany the teacher’s edition of

the textbook. Examples include:

Prentice Hall’s Resource Pro and Worksheet Builder;

Saxon’s Test Generator; and

McGraw Hill/Glencoe’s Teacher Works.

The above referenced programs are examples of proprietary programs belonging to “for-profit”

publishers. Their contents are typically copyrighted, non-editable, and relatively unfriendly to

users who wish to modify or adapt them to fit specific individual and/or classroom needs.

Almost all resources in these programs are presented in difficult or impossible to manipulate





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formats, presumably and understandably to protect the proprietary interests of the publishers.

Any teacher seeking to adapt or change such fixed-format resources for the benefit of a specific

individual or classroom faces intentional technological impediments imposed by the publishers

as well as potential copyright infringement issues.

Teachers frequently draw resources for a single lesson from several sources. Proprietary

lesson plan management systems are typically not designed to facilitate the development of

lesson plans from more than a single source, and are limited to that which is provided by the

textbook publisher. This severely restricts teacher creativity in meeting the needs of students.

Although these resource and lesson plan management systems are promoted by textbook

publishers as being user friendly, this writer can attest to numerous obstacles encountered while

adding over 1500 Regents Math A and Math B questions to the databases, and in formatting and

printing hundreds of resources from the enhanced databases. Such obstacles are simply

overwhelming to the majority of teachers, thus preventing the realization of the potential of these

database management systems. These obstacles are overcome through the JMAP prototype, in

which responsibility for systems mastery, database management, and desktop publishing of

ready-to-use, high quality teaching resources is concentrated in individuals who are motivated

and competent in the associated technologies and academic content areas.





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JMAP as a Prototype of a Lesson Plan Management System in the Public Domain

As a co-founder of JMAP, this writer attributes JMAP’s success to the integration of: 1)

highly relevant databases of public domain resources (i.e. previously administered Regents

questions); 2) highly relevant databases of dynamic proprietary resources that have, in essence,

been allowed entry into the public domain through the generosity of their proprietary owners; 3)

associated lesson plans and teaching resources in editable electronic formats; 4) Adobe portable

document format (pdf) supplements; and 5) desktop publishing capabilities. All of the above

resources are intended to provide individual teachers with the ability to quickly and easily access

and adapt teaching resources for specific classrooms and individuals.

The major emphasis of JMAP is teacher empowerment through free resources with

minimal emphasis on securing private property rights or serving economic self-interests. This

orientation toward the public good without corresponding economic interest promotes open

sharing of educational resources and is essential to JMAP’s vision of a viable lesson plan and

resource management system.

The Need for An Improved Public Domain Technology Platform

While JMAP provides an effective prototype for the empowerment of teachers and the

creation and distribution of high quality teaching resources, an improved public domain





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technology platform would overcome current obstacles and present additional educational

opportunities. JMAP currently uses proprietary database management and desktop publishing

software that is no longer being updated and improved. As time passes, the limitations of the

JMAP technology platform will become more obvious. A new, public domain technology

platform could facilitate improvements in JMAP as well as new initiatives based on the JMAP

prototype in other academic areas. Furthermore, a public domain technology platform, which

might conceivably take the form of enhanced templates and “wrap-around” programs for MS

Access, could be developed at relatively low cost and widely disseminated via the Internet.

Facilitating the Attainment of Educational Goals in Several Areas

As mentioned previously, the envisioned lesson plan and resource management system

will facilitate the attainment of educational goals in several areas, including: 1) teacher

preparation and subject matter awareness; 2) aligning classroom instruction with state mandated

assessments; 3) adapting lesson plans and learning materials to differentiated student and

classroom needs; and 4) reducing new teacher stress and turnover. These advantages are further

discussed in the following paragraphs.

Improving the Subject Matter Expertise of Teachers





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The development and distribution of databases of Regents and related teaching resources

promotes awareness and understanding of the intended curriculum. This is important because

awareness and understanding of the intended curriculum is useful in the training and professional

development of teachers. Research has shown that other factors being equal, people who know

more math make better mathematics teachers than people who know less math (Darling-

Hammond, Holtzman, & Gatlin, 2005) (Decker, Mayer, & Glazerman, 2004) (Hill, Rowan, &

Ball, 2005). These findings are arguably applicable for all academic subject matter areas.

Aligning Instruction with Assessment

Historical assessment practices are important for understanding current and future

assessment practices, even when assessment practices are changing. Despite appearances to the

contrary, the curriculum changes slowly and incrementally. Accordingly, historical assessment

practices will continue to inform teachers and students with respect to future assessment

practices. With respect to mathematics, the new high school curriculum currently being

implemented throughout New York shares much of its topical content with the current Math A/B

curriculum. The differences are not so much in the area of what is taught, but rather in when a

particular topic is taught and when it is assessed. This is not surprising, as there is evidence that





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the topics taught in mathematics have converged into a single common curriculum throughout

the developed countries of the world (Baker & Letendre, 2005).

There exists an inevitable dualism between the achievement of goals as measured by high

stakes testing and the roles of teachers in preparing students for examinations. This dualism

between Regents examinations and instructional practices was noted by Dr. John E. Bradley,

principal of Albany High School, approximately 130 years ago when he commented that "The

salutary influence of the primary examinations in stimulating both teachers and pupils to

thoroughness in the acquisition of the elementary branches suggested the extension of the system

to academic studies.” The occasion of Dr. Bradley’s remarks was the extension of Regents

examinations from the grade schools into the high schools of New York. (New York State

Education Department).

The NCTM assessment principle argues the need for aligning assessment and

instructional practices and asserts that

Good assessment can enhance students' learning in several ways. First, the tasks

used in an assessment can convey a message to students about what kinds of

mathematical knowledge and performance are valued. That message can in turn

influence the decisions students make—for example, whether or where to apply





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effort in studying. Thus, it is important that assessment tasks be worthy of

students' time and attention. Activities that are consistent with (and sometimes the

same as) the activities used in instruction should be included….Assessment

should reflect the mathematics that all students need to know and be able to do,

and it should focus on students' understanding as well as their procedural skills.

Teachers need to have a clear sense of what is to be taught and learned, and

assessment should be aligned with their instructional goals (NCTM, 2000).

Numerous other studies suggest that alignment of curriculum and assessment practices is a

practical concern of educators (Firestone, Schorr, & Monfils, 2004)( (Hamilton, 2003). JMAP’s

position is that teacher awareness and understanding of historical patterns in assessment practices

is an effective means through which awareness and understanding of state learning standards is

acquired. Accordingly, curriculum alignment is good for both students and teachers.

Adapting Instructional Resources for Diverse Student Populations

Good teachers adapt their lessons and teaching resources to the needs and skill levels of

their students (Dewey, 1915/2001) (Fosnot, 2005). Inherent within this idea is the premise that

teachers add value to the generalized curriculum associated with textbooks and commercially

available teaching resources. Teachers have long used scissors and glue, mimeograph machines





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and photocopiers, and pencil and paper to create and adapt teaching resources for their classes

and individual students. As computers and technology have increasingly asserted a presence in

our daily lives, a new generation of teachers has emerged that is willing and able to use the

Internet and, to a lesser extent, database programs and the desktop publishing capabilities of

personal computers, to replace scissors, glue, pencils and mimeograph machines. This new

generation of teachers will arguably be more effective in adapting lesson plans and teaching

resources to the diverse needs of their students if they are provided with Internet resources and

technology that integrates rich collections of resources in user-friendly formats. The New York

City Department of Education is among the most diversified public school systems in the world

and there is no single lesson plan or textbook that can ever meet the needs of all students and all

teachers. Thus, it seems reasonable that efforts be made to empower classroom teachers with

high quality, manipulable resources accompanied by desktop publishing capabilities. JMAP is a

prototype for such a system, and the JMAP prototype for teacher empowerment can easily be

extended to other academic subjects.

Reducing Stress Levels and Burnout Rates in New Teachers





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New teachers face challenges in many critical areas, including but not limited to: 1)

developing subject matter expertise; 2) developing pedagogical expertise and classroom

management skills; and 3) developing several hundred lesson plans during their first years of

teaching, when they are arguably least capable of producing good lesson plans. Providing

adequate resources and facilities to support teachers in instructional practice is a recommended

practice for preventing and/or reducing teacher burnout (Wood & McCarthy, 2002). Although

textbooks and teaching resources are readily available from numerous sources, most of these

resources have serious limitations for New York’s teachers. These limitations include: 1) they

are not grounded in the New York State learning standards, assessment practices, or urban school

environments; 2) they are in formats that are not easily manipulable; and 3) they are disjointed

and non-integrated from both topical and pedagogical perspectives. A comprehensive lesson

plan and resource management system for each academic subject area could reduce the workload

and stress levels of new teachers by giving them starter lesson plans in manipulable formats that

can be adapted to a wide variety of classroom needs and pedagogical beliefs. Each lesson plan

would be accompanied by a complete history of how the topic has previously been assessed on

Regents examinations, and by additional teaching resources suitable for the topic of the lesson.





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By “lessening the load” on new teachers, such a lesson plan management system could

conceivably influence both teacher retention and quality of instruction.

An Opportunity Within Our Reach

Russell Conwell, founder of Temple University in Philadelphia, was renowned for a

series of lectures, sermons, and a book, all entitled “Acres of Diamonds,” in which he told the

story of a Persian farmer named Ali Hafed (Conwell, 1915). Ali searched the world over for

diamonds, never to learn that the farm he left as he began his quest would become the site of one

of the world’s great diamond mines. The morals of the story are: 1) sometimes the best

opportunities are in your own backyard; and 2) one should always look at the present situation

when looking for a better opportunity. Such is the case with education in New York City.

All Regents examinations pass into the public domain immediately upon publication and

administration, and this fact represents a diamond in our own back yard. These historical

documents provide an excellent teaching resource that teachers can own and do with as they

please. In this time of high stakes testing for all in New York, Regents questions constitute the

best available representation of the intended curriculum embodied in the state’s learning

standards for any given subject area, and as such, they have added significance for both students

and educators. They are the gold standard for understanding what the state of New York thinks





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a student should know and understand, and they constitute a body of knowledge that can and

should be at or near the center of any curriculum in our state.

Summary

JMAP has demonstrated the need for and efficacy of technology based lesson plan and

resource management systems that empower teachers in the areas of subject matter expertise and

adapting lesson plans to diverse student populations. The JMAP approach offers potential for

reduced stress and turnover amongst new teachers as well as better alignment of instructions with

learning standards and assessment practices. Furthermore, the JMAP approach is a low cost

form of professional development with significant buy-in from teachers, and it can be replicated

and expanded beyond its current focus in mathematics. An opportunity exists for a sea-change in

teacher empowerment and educational publishing at a grass-roots level with minimal costs.

Additional Information and Questions

Feedback from educators, researchers, and others interested in the use of Internet-based

technologies for the empowerment of teachers and the improvement of teaching practice in

public education is encouraged. The opinions and vision expressed in this article are those of the

writer. Questions and comments concerning this article and its contents may be directed to its

author: The Jefferson Math Project is accessible at www.jmap.org.





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REFERENCES

Baker, D.P., & Letendre, G.K. (2005). National Differences, Global Similarities: World Culture

and the Future of Schooling (Chapter 10). Palo Alto, CA: Stanford University Press.

Conwell, R.H. (1915). Acres of Diamonds. United States: Harper and Brothers.

Darling-Hammond, L., Holtzman, D.J., & Gatlin, S.J. (2005). Does teacher preparation matter?

Evidence about teacher certification, Teach for America, and teacher effectiveness (13,

42). Education Policy Analysis Archives.

Decker, P., Mayer, D.P., & Glazerman, S. (2004). The effects of Teach for America on students;

Evidence from a national evaluation. Princeton, N.J.: Mathematica.

Dewey, J. (2001). The School and Society & The Child and the Curriculum. Mineola, NY: Dover

Publications, Inc. (Original work published 1915)

Firestone, W.A., Schorr, R.Y., & Monfils, L.A. (Eds.). (2004). The Ambiguity of Teaching to the

Test: Standards, Assessment, and Educational Reform. Chapters 2, 3, 4. Hillsdale, N.J.:

Lawrence Erlbaum.

Fosnot, C.T.,. (Ed.). (2005). Constructivism: Theories, Perspectives, and Practice. New York

and London: Teachers College Press.





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Hamilton, L. (2003). Assessment as a Policy Tool. In R.E. Floden (Ed.), Review of Research in

Education (Vol. 27, pp. 25-68). Washington, D.C.: American Educational Research

Association.

Hill, H.C., Rowan, B., & Ball, D.L. (Summer 2005). Effects of teachers’ mathematical

knowledge for teaching on student achievement. American Educational Research

Journal, 42, no. 2, 371-406.

NCTM (Ed.). (2000). Principles and Standards for School Mathematics. Reston, VA: National

Council of Teachers of Mathematics.

New York State Education Department. (November 24, 1987). History of Regents Examinations

1865 to 1987. In University of the State of New York State Education Department

Website. Retrieved May 20, 2006, from

http://www.emsc.nysed.gov/osa/hsinfogen/hsinfogenarch/rehistory.htm:

Wood, T., & McCarthy, C. (2002, Dec). Understanding and Preventing Teacher Burnout. ERIC

Digest. [Data File]. Washington DC: ERIC Clearinghouse on Teaching and Teacher

Education. Available from http://www.ericdigests.org/2004-1/burnout.htm

i STEVE WATSON is a doctoral student in Urban Education at the Graduate Center of the City University of New York. He also teaches high school mathematics at the International High School @ Prospect Heights in Brooklyn, and he is an adjunct faculty member of the mathematics





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education department at Brooklyn College, where most of his students are New York City Teaching Fellows. Steve's general research interests are in mathematics education and the use of technology for teacher empowerment and new teacher orientation programs. After receiving an M.S. Ed. from Purdue University in 1973, Steve spent over twenty years as an executive with American General Corporation, a subsidiary of AIG, where he held positions including Senior Vice President for Government and Industry Relations, Senior Vice President and Chief Administrative Officer, and board member of two insurance companies. He joined the New York City Teaching Fellows in 2003 and co-founded the Jefferson Math Project with Steve Sibol during his first year of teaching. E-mail author.



Volume 2 Issue 1 Date September 2007 Gender Differences

1


Running head: GENDER DIFFERENCES

Gender Differences in Mathematics Departments at Colleges and Universities Across the

United States: Towards an Inclusive Environment

by

Alexander Rolón

Assistant Professor

Northampton Community College

3835 Green Pond Road

Bethlehem, PA 18020

610.861.4163

[email protected]

Gender Differences 2

Gender Differences in Mathematics Departments at Colleges and Universities Across the

United States: Towards an Inclusive Environment

Introduction

When one thinks of great mathematicians, women do not come to mind.

Perhaps it is because the development of mathematics, as we know it today, was

controlled by men. For example, Sir Isaac Newton, Carl Friedrich Gauss, and Gottfried

Leibniz were pioneers in the development of calculus. Women were not permitted to

become intellectuals in this sophisticated and complex subject. Mathematics departments

across the United States are predominantly male dominated. This disproportionate

interest is ingrained back in the elementary (Pérez, 2000) and middle school grades

(Hanson, 1992). Research (Eccles, 1986; Nichols, 1989; Zaslavasky, 1994) suggests that

females are not encouraged to enroll in higher level mathematics and science curricula

due to the lack of self identification, family discouragement and societal prejudices.

There is a need for more female representation in the field of mathematics and science.

One way to explain this discrepancy is rooted in Social Identity Theory. Women,

like men, challenge themselves but academically there exists a gap in the challenges they


take as it relates to mathematics and science. Research further supports gender

differentiation when it comes to making mistakes in mathematics performance. Males

tend to find a scapegoat for their failures, while females often blame themselves for not

preparing adequately (Beal, 2000). When males perform well they attribute it to their

proficiency in the subject matter as if it were expected; by contrast women’s success is

highly praised to the extent of the effort or time they spent studying the material. The

lack of attribution to ability for success and the tendency to attribute failure to a lack of

ability promotes low mathematical self-esteem; hence creating a negative association or

bias toward other mathematics courses, allowing males to enroll in higher level

mathematics curricula. Males have a sense of superiority over females in mathematics.

Women who continue in mathematics courses usually do so in the presence of a

group with which they can identify. In addition to these groups, there are national and

local associations like The Association for Women in Mathematics and Women in Math,

among others. The creation of groups or cohorts helps facilitate and overcome some of

the sexist behaviors that exist in the mathematics community. Often times, the women in

these groups support each other and become successful. For this, and many other reasons,

group identification is essential to the success in future enrollment of women in

mathematics curricula.

Individuals get much of their self-esteem from interactions with their peers. The

case for females in mathematics is no exception. Without group identification their

psychic immunity decreases, and so does their ability to perform well. It is imperative

that we as professionals in the field think more inclusively about female capabilities to


perform well as it relates to mathematics; hence recognizing that women are also part of

the realm of mathematics.

Enhancing Women in Mathematics Through History

Although the history of mathematics is male-centered, there were a few women

who were influential and contributed a great deal to the development of mathematics.

Maria Agnesi, often times referred to as the first woman professor of mathematics on a

university faculty. Sonya Kovalevsky impressed the French Académie des Sciences

judges, and unanimously voted her concealed paper, On the Rotation of a Solid Body

about a Fixed Point, the winner of the famous Prix Bordin. Because of the exceptional

merit of the work, the monetary value of the prize was raised from 3,000 to 5,000 francs,

a considerably sum of money at the time. Amelie Emmy Noether, who was a professor at

Bryn Mawr College in Philadelphia, PA and known for her work on the Theory of

Invariants. These women, as well as others who are not mentioned, were prodigies in

their field of expertise. A history of mathematics course could emphasize the

contributions of such prolific females, although it is not guaranteed to increase

awareness. Mathematics, as a field, is male dominated. Incorporating some biographical

vignettes of females into mathematics curricula will help shift the center and help to

communicate the message that women are also “good at math.”

Many women feel that graduating with a mathematics degree is equivalent to

winning a gold medal in the Olympics. They feel a great deal of accomplishment because

of their limited support from their male counterparts. Sexism governs the unwritten,

nonmathematical rules of this field. Women often times are told they are not good at


math, that they are better at nurturing or in careers where creativity and intuition prevail

(Eccles, 1986). Therefore they begin to believe such prejudices and act on them.

Counselors, teachers, peers, or even parents who are not “good at math” discourage

women to explore other than mathematical opportunities. But once they have

accomplished success in mathematics, they want to share with other women. They feel it

is important to break the stereotypes and hence educate others and reduce the sexism that

exists in mathematics and science. Universities across the country need to have a more

mathematics diverse faculty. By having such diversity within the department, perhaps

they would attract more females in graduate programs. Furthermore, recognizing this

deficiency would create for a more inviting, less intimidating atmosphere for women.

Social Identity Theory suggests that there are some positive intergroup biases

when it comes to women succeeding in mathematics. Women value their educational

experience in fields like mathematics and science more so than males. Their success is

attributed to their preparation and perseverance. They feel as if they are equal to men in

the field. Elva Treviño Hart, a Chicano author, who holds a B.S. in Mathematics and an

M.S. in Computer Engineering from Stanford University, once said that being a math

major gave her the opportunity to express herself without being prejudiced about her

gender or her ethnicity in a field where she was “twice a minority”. She and similar

women were able to excel because they looked past the sexism and prejudices that this

field brings, and because they were determined to succeed regardless of the myriad of

obstacles presented to them. Many women don’t want to deal with the pressures and

often change careers or go into related fields where mathematics is needed, yet not

essential. Moreover, they view themselves as low achievers and feel as if they were more


inferior to males. This feeling develops a negative association and belief that males are

better than females as it relates to mathematics competence and achievement.

Improving Intergroup Bias

How then do we go about creating an atmosphere of tolerance, respect, and

equality? The exploration of the following three activities incorporates methods to

improve intergroup bias: 1) recategorization, 2) decategorization and 3) mutual

differentiation. Each of these methods is equally important and they bear no order of

significance.

Recategorization

Many universities hold lecture series or a mathematics symposia sponsored by

mathematics departments, where distinguished mathematicians are invited to present their

latest research. A suggestion is that the faculty work together to feature more women in

this series or symposium. There are many women mathematicians to choose. For

instance Elva Treviño Hart, previously mentioned, is a good example of

multiculturalism as well gender issues in mathematics. Below you will read other

contemporary female mathematicians who are qualified for such lectures.

Margaret A. M. Murray who wrote a book in 2001 titled: Women Becoming

Mathematicians: Creating a Professional Identity in Post-World War II America, that

looks at the lives and careers of thirty-six of the approximately two hundred women who

earned Ph.D.s in mathematics in the United States from 1940 to 1959 an era when

American mathematical research enjoyed an unprecedented expansion, fueled by the

technological successes of World War II and the postwar boom in federal funding for

education in the basic sciences. Nevertheless women's share of doctorates earned in


mathematics in the United States reached an all-time low. Murray explains: "…the book

examines the development of mathematical identity across the life span, from childhood

through adulthood and into retirement. It focuses on the process by which women, who

are actively involved in the mathematical community, come to ‘know themselves’ as

mathematicians. The women's stories are instructive precisely because they do not

conform to a set pattern; compelled to improvise, the women mathematicians of the

1940s and 1950s followed diverse paths in their struggle to construct a professional

identity in postwar America.”

Lai-Sang Young a pioneer in the field of topology. Her research is on continuous

flows on compact 2-manifolds. She has been the keynote speaker at different

mathematical association throughout the United States and Europe as well as her home

country of China.

Linda Goldway Keen who was born and raised in Bronx NY. Her research

involves studying the interplay between the analytic and geometric aspects of classifying

Riemann surfaces. Dr. Goldway Keen is one of the few mathematicians to have studied

this branch of mathematics.

Lenore Blum has a story of perseverance. She always dreamed on attending

Massachusetts Institute of Technology both because it was an excellent place for her to

study and also because her husband was there, but she was not accepted. She was

discouraged and was not set on taking a degree in mathematics but had other interests so

she enrolled in the Department of Architecture at Carnegie Institute of Technology in

Pittsburgh. She still wanted to attend MIT and had made a number of unsuccessful

applications to there but at last she made a successful one and began to study there while


completing her first degree at Simmons College. She was awarded her B.S. from

Simmons in 1963 and continued working towards her doctorate at MIT.

These are great examples of contemporary female mathematicians who would

make excellent speakers at the lecture series or symposium throughout mathematics

departments at colleges and universities in the United States, with the end goal of

reducing biases in this field. The experience of both groups working together would be

sufficient to agree upon the fact that they are all professional mathematicians, regardless

of gender differences.

Decategorization

The male professors, after attending the lecture series or symposium, will discuss

with the lecturer how her research could be incorporated into their classes or their own

research. The purpose of this activity is to encourage more research and to have the

female mathematician as the focal point of the lecture or discussion. It is important to

have male mathematicians praise their female counterpart, with the objective of de-

emphasizing that mathematics is a male dominated field. Making these females

mathematicians the key person and expert in the field will confirm their role in the

mathematics world; hence, breaking the stereotypes that are associated with their

performance in mathematics.

Mutual Differentiation

Explaining how history has shaped the way we learn mathematics today would be

interesting to research to put into practice. The faculty will explore ways in which

females learn mathematics as opposed to men. Pedagogical practices will also be

explored to enhance the ways females perceive mathematics. Two of the key questions to


answer are: Is a hands-on curriculum embedded in their programs and/or classes? Are

there different delivery methods rather than all lecture classes? The latter will explain

spatial ability and its influence in learning mathematics. Understanding how females

think is fundamental to the creation of a less biased environment. They would then

discuss how these approaches can benefit men as well. As a suggestion, a lesson attached

in Appendix A can be used as a collaborative group work where students will be aware of

female mathematicians. Mathematics reform calls for a more inclusive curricula where

creative minds are generated.

Summary

Inclusion of women in mathematics programs as well as an increase of the

number of female faculty at colleges and universities will yield an environment that is

tolerant and conducive to learning. However, there are many obstacles that this change

may possibly bring. Resistance on the faculty due to the lack of women input on this

issue is a possible downfall. Bringing females into the scheme of things may spark some

negativism creating tension among the male faculty. Yet this can be a constructive,

learning experience for all. Social Identity Theory explains how clusters among women

will be encouraged and expected. However, conveying the reality of sexist behaviors in

mathematics may put this issue into perspective. It is important to understand women’s

role and thoughts about sexism in the mathematics field in general as well as their

feelings about being a minority in the department.

Initiating discussion about gender differentiation and sexism at mathematics

departments among American institutions of higher learning, will make certain of one

thing: communication of ideas that otherwise would have been overlooked will be


occurring. I am not an expert in sexism in the mathematics field; however, I am aware

that it exists. This is a step closer towards a more inclusive way of thinking. Being aware

of the biases should be the foremost, intricate step in understanding our own biases and

prejudices. Do we as teachers call on males more often than females? And when we call

on females, do we give the same wait-time for them to respond as we would for males? If

we can go beyond the basics and think about our own personal experiences, we can be at

a much better place in our professional lives.

We need to take initiative, be a pioneer, and be active participants in reducing

gender biases in the mathematics field. We must understand that we are all

mathematicians, regardless of sex or any other characteristics that come with each and

everyone of us; by doing so we build a better rapport within the mathematics community

and a more inclusive atmosphere.

In conclusion, in order to create an atmosphere of respect and tolerance one must

have knowledge, awareness, and skills when it comes to female competence in

mathematics. Knowing what to do when females are in our classes becomes an intricate

part of our professional growth and their interest in mathematics as a career; awareness of

the fact that sexism exists in today’s society will help with our own peace of mind to

accept; and we gain skills to be sensitive to everyone, especially women who have been

underrepresented in mathematics for a long time.


References

Beal, C. (2000). Gaining confidence in mathematics: Instructional technology for girls.

Paper presented at the International Conference on Mathematics/Science

Education & Technology, Sand Diego, CA.

Eccles, J. (1986). Gender-roles and women’s achievement. Educational Research, 15(6),

15-19.

Hanson, K. (1992). Teaching mathematics effectively and equitably to females.

NY: Eric Clearinghouse on Urban Education.

Nichols, R. (1989). Gender and mathematics contests. Arithmetic Teacher, 41(5), 238-

243.

Pérez, C. (2000). Equity in standard-based elementary mathematics classrooms.

Retrieved July 10, 2007, from http://wge.terc.edu/equity.html.

Zaslavsky, C. (1994). A mind is a terrible thing to waste!: Gender, race, ethnicity, and

class. In Fear of math: How to get over it and get on with your life (pp. 69-98).

New Brunswick, NJ: Rutgers University Press.


Appendix A

Lesson Plan #1: A Historical Perspective

Objective: To enhance student awareness of women mathematicians in education.

Grade Level: High School

Activity: Students will be paired and they will research a female mathematician of their

choice, with instructor approval. They will then create an article for a newspaper with a

short biography as well as one of her mathematical innovation. The article must be era

appropriate. The students will then have to put together a live interview of this person to

be presented in the class. The purpose is to emphasize the mathematical achievement(s)

of this person. (In the case where two male students are working together, they will need

to come up with a creative way to convey her contributions to mathematics.) The

instructor will help the students understand the mathematical concepts of these

mathematicians.

Assessment: A rubric will be developed for both the written article and the interview.

The students will have access to this rubric while they are researching and preparing their

presentations.

Technology: The students are encouraged to use power point presentations or other

technology to enhance their presentations.

Looking back: Individually, the students will write a journal entry reflecting on this

activity. In it they should explain what they learned, how valuable it was as well as their

reactions to the activity.





1

Collaborative research relationships in Urban Science Classrooms1

Rowhea Elmesky and Anita Abraham Washington University in St. Louis & Philadelphia School District

INTRODUCTION

Anita (Teacher Researcher): As a science teacher at City High School, a school where a nearby university

conducted ongoing research in science classrooms, I had seen university researchers walking down the

halls, in classrooms and also in the principal’s office. Most of the teachers were suspicious about the

university researchers. They tried to avoid them, were apprehensive about being interviewed by them, and

afraid that they might accidentally say something that might put them in ‘trouble.’ In those days, I wasn’t

sure what the ongoing research was about, and I didn’t make any effort to know either.

Things started to change when our vice principal, Dr. Al, a former science department head, asked me to

join the Masters in Chemistry Education (MCE) program offered at the same nearby university. At the

same time, Dr. Kenneth Tobin, the main university researcher from the Graduate School of Education,

asked me if I would be interested in joining the research group already working at City High School. He

further explained to me that, as a part of the research team, university researchers would have access to my

classroom and I also would be participating in the research as a teacher-researcher. As a regular classroom





2

teacher, I didn’t consider myself a researcher and didn’t know what qualifications were expected for a

researcher. Moreover I wasn’t comfortable letting a university researcher into my classroom. I was worried

that, if things went out of control, those events would become the focus of their research findings. Sensing

my uneasiness, Dr. Tobin told me about the university researcher whom he had in mind for my classroom,

and assured me that I would be comfortable working with her.

When I shared this information with one of my coworkers, Ms. Cloud, a 30-year veteran teacher, her

reactions were negative, mainly because in her opinion educational researchers always concluded their

findings without any input from the classroom teacher or students. However, I anticipated that my situation

would be different because I would act as a teacher-researcher and my students would also become a part

of the research team as student-researchers. Hence, I agreed to be a part of the university team, excited that

my voice and my students’ voices would also be heard during the research process.

Rowhea (University Researcher): When I began working with Anita, I sensed her nervousness. Perhaps

she thought that I was there to criticize her as a teacher. For the purposes of the research study, I brought in

a video camera to record the class events several times a week, and the camera seemed to reinforce the

apprehension. Early on, I worked to assure Anita that I was present in her class to assist her in conducting

HER research. I attempted to position myself as an assistant – someone who would be a co-researcher on a

study that she would design – around questions that were of interest to her and would help her to improve

1 The research in this paper was supported in part by the National Science Foundation under Grant No. REC-0107022. Any opinions, findings, and conclusions or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation.





3

self-selected aspects of her teaching. It took a while for us to establish a comfortable rhythm of trust and

for Anita to feel confident that I wanted to collaborate rather than dominate.

Nisha (student participant): I learned a lot from research. We sit in groups and talk about class an[d] stuff, I

never thought about the other kids and how they feel. I learned how Ms. A [Anita] cares about us. She

taught us to help other people in class. I get good grades. Class is just a big group of helpers for everybody.

(Journal entry, June 2002)

Maria (student researcher): We told Ms. Morris [the English teacher] about the research in your [Anita’s]

class and how we talk about what we like and what we don’t and all. She liked it. She said that she might

try it.

In this article, we describe some aspects of the research process in which a teacher researcher

(Anita), a university researcher (Rowhea), student researchers, and student research participants

collaborated on a study during 2002 in Anita’s 11th grade Chemistry class and her supplementary

laboratory at City High School. Conducted within a critical ethnographical framework, the study

focused on the teaching and learning of science in an urban high school where the majority of

students are African Americans from low socioeconomic backgrounds. The authors met through

our mutual involvement with an NSF-funded grant that invoked a model of collaborative

research (utilizing a ‘research with’ rather than ‘research on’ methodology) where teams were





4

created that consisted of two teacher-researchers from each participating urban school, at least

two student-researchers from each focal class, and university researchers. Thus we chose to

begin this paper with a series of reflections through multiple voices, to illuminate some of the

challenges and benefits associated with engaging in this collaborative model of research.

CONTROL OVER OR COLLABORATION WITH?

Anita: I have learned that model of ‘control over’ does not afford respect in urban school settings.

‘Collaboration with’ has allowed me to communicate care and build the understandings of my students. In

order to be a successful science teacher, I encouraged my students to give me feedback on my teaching

practices. Based on their suggestions and through negotiations, I tried to develop a learning community that

is built on respect and trust. As a teacher-researcher, I encouraged my students to make connections

between their experiences from their lifeworlds and science. Students brought new insights to common

concepts and raised questions that I had never considered. Now, three years following the completion of

this study, I am a teacher who is constantly evaluating and transforming my classroom practices in an

ongoing effort to create a learning community based on trust, collaboration, and shared responsibilities.

(Reflections, 2007)

Urban schools, such as City High where Anita taught, are marked by inequalities – visible in

school staffing, funding, courses offered, and the resources available. The schools are often

oppressive and hence become grounds for struggle – the teachers and/or administration against

students who are labeled as “resistant” or “unmotivated,” for example. This paper provides a

model for incorporating a critical ethnographic methodology and cogenerative dialogues as tools





5

for shifting school dynamics from control over to collaboration with – where participatory

critique is encouraged such that ongoing structural transformation in the classroom occurs and

schooling becomes a less oppressive experience and a more rewarding experience for both the

students and their teachers.

CRITICAL RESEARCH AS A TOOL FOR DAILY CLASSROOM CHANGE

When Barton (2001) discusses critical ethnography, she describes the research process as a

“dialectical theory- and practice-building process in which practice and research shape each

other in an endless cycle” (p. 907). Thus, critical ethnography calls for identifying the problems

and asks for transformation by connecting theory and practice. This dialectical relationship

between practice, theory, and research triggers local transformation of the structure by providing

tools for all participants to act in new ways as the findings from the research constantly inform

participants of their practices and vice versa. Critical ethnography also asks to increase the

agency of the participants who can draw strength from the research findings. Thus the research

process and associated findings can become a catalyst for growth and transformation (Seiler &

Elmesky, 2005).





6

Students as Researchers

Educational research that involves students as researchers “provides a way to obtain their

perspectives on what is salient in terms of school, teaching, learning, and myriad other issues”

(Tobin, 2006, p. 27). Student-researchers should be included in salient ways in the research so

that their perspectives on what is occurring in the school or neighborhood fields and ‘why’ can

emerge (Elmesky & Tobin, 2005). Student-researchers are empowered as they contribute

significantly to identifying patterns of coherence (as well as contradictions) within their

classrooms, in relation to the teaching and learning they experience.

In our study, student researchers engaged in activities such as the review and analysis of

videotapes, interviewing each other and fellow classmates, transcribing such interviews, writing

reflective journal entries, and developing video ethnographies that captured salient aspects of

their lifeworlds outside of school. Weekly, the researchers ate lunch together, during which time

they watched videotapes from class time and from within the laboratory. They were asked to

identify video vignettes of salient events that were taking place, and these video vignettes then

became central focal points for discussion. In addition, a selection of video vignettes was shared

with students who were participants within a captured video clip, in order to obtain their

perspectives and preserve and privilege their voices.





7

Cogenerative Dialogues

Out of the many research methods that were engaged in this study, we highlight the role that

cogenerative dialogue can play in catalyzing deeper understandings and ongoing change in a

classroom. Cogenerative dialogue (LaVan, 2004; LaVan & Beers, 2005; Roth & Tobin, 2001;

Wassel, 2004) can be understood as a conversion between co-participants about shared events

and experiences. As Tobin, Zurbano, Ford, and Carambo describe (2003), “such conversions

might focus on participation, access and appropriation of resources and the co-occurrence of

given patterns of coherence and associated contradictions” (p. 55). During cogenerative

dialogues, every participant should have the opportunity to speak and be heard while also

listening with interest and respect to the perspectives of the others who are involved. Moreover,

participants can discuss collective responsibility, participants’ responsibility, curriculum,

experiences, and power relationships. A critical characteristic of cogenerative dialogues is that

the outcomes of these interactions can and should be immediately applied to improve the

teaching and learning of science in a science classroom.

Cogenerative dialogues can take many different forms and involve different participants at

different times. For instance, some are held after class time while others occur during class time.

Martin (2004) classifies cogenerative dialogues as formal, informal and classroom huddles.

Classroom huddles occurred between us (the authors) on a regular basis in between, during, and





8

after each chemistry class. In these ‘huddles,’ we briefly discussed that day’s lesson, classroom

activities, how the materials would be distributed during the laboratory activity, and logistical

research details, such as where the video camera should be placed to best capture salient events

in the classroom. These classroom huddles were short, focused and occurred by moving to

physically stand in close proximity to each other; they enabled to evaluate and revaluate the

lesson before, during and after implementing it and to develop immediate suggestions and

implement ongoing modifications with the help of the university researcher. When a greater

amount of time was available right after the chemistry laboratory activity, informal cogenerative

dialogues occurred between us where we were able to sit down and discuss an individual

student’s performance or progress, laboratory group dynamics, and other salient events that

occurred during a particular laboratory activity.

During this research study, cogenerative dialogues occurred in different fields with different

participants. Formal cogenerative dialogues occurred about once a week between Rowhea and

student-researchers during their lunch time. Anita tried to attend most of these sessions but, at

times, her other responsibilities as a teacher prevented her from doing so. In these meetings,

Rowhea introduced the student-researchers to some basic concepts in sociocultural theory, and

we discussed classroom practices, especially Anita’s practices as a teacher, curriculum, and

laboratory activities, and came up with suggestions for improving our learning environment. In





9

order to help dissipate the power hierarchies existing, we sat wherever we felt comfortable, and

we constantly reminded the student-researchers that their interpretations and suggestions were

not going to affect their grades. We also watched video vignettes from previous classes, and

discussed concerns and suggestions. It is evident in the following entry from one student

research’s (Deidre’s) journal that she felt comfortable to be critical of Anita’s practices as a

teacher. She urged Anita to allow the students to access resources on their own to light the

Bunsen Burner.

I think Mrs. Abraham should trust us and plus the burner, She gotta go to group to group, lightning it and its gonna take a long time and we wanna do our lab real quick and by her keep goin to group to group she just need to give us like some matches or a lighter so we can [light] burner our own? Burner is easy to use. (Journal entry, February 2002)

These types of reflections were useful in helping us to identify practices that afforded and

truncated students’ performance within the laboratory setting. However, providing feedback in

writing lacks the interactive components of cogenerative dialogue and the possibilities to ask for

clarification. For example, in the following cogenerative dialogue, Rowhea was able to seek

clarification when a student researcher (Maria) advised that Anita needed to be more patient with

the students in the laboratory.

Maria: She is a nice teacher. She is all right. She explains how her culture is and everything. We ask her stuff. She explains why. She teaches but she still needs to be a little more patient with us also.





10

Row: Can you give me an example of it? Do you remember a time in lab that she got frustrated with you?

Maria: Um, I think our group was asking for something. She was doing something else and she got like real mad like “I WILL BE THERE IN ONE SECOND” and I understand that you [Anita] are only one person but we need help also.

Cogenerative dialogue provided a space for hearing the students’ voices and enabled Anita to

reflect on her practices and transform the structures to allow for greater student and teacher

empowe. Conversations like this one helped Anita perceive emotion as a central part of

instruction. Maria brought to our attention that the generation of positive emotional energy

(Collins, 2004) in the classroom (through patience) would encourage a positive atmosphere for

learning. Some of the other actions that Anita took to build positive emotional energy were

positive reinforcement (instead of being loud), avoiding argument, and complimenting students

for being successful in class and laboratory. More importantly, through the research process,

Anita was increasingly conscious of regularly reflecting upon her practices.

Anita: Everyday I tried to spend a couple of minutes reflecting on my actions, and at times asking the following question to myself, “If I were a student, would I want me as a teacher?”

In addition to providing critical remarks, the students also shared their perceptions of

teaching practices that they considered successful, during cogenerative dialogues. The following

cogenerative dialogue excerpt was in relation to Anita’s practice of allowing the students to





11

select their own laboratory groups, contrary to other teachers. The students enjoyed this

autonomy. Evident in her comments, Deidre felt that working with familiar peers assisted the

process to proceed smoothly and in an enjoyable manner:

When we are in the lab and we have to pick who we are in the group with, and you work with people you are already familiar with. Some teachers just put you with anybody if you don’t like that person and you are not familiar with that person you are not going to work because you don’t know anything about them. So you work with your friends and like we have the lab [Rate of Reaction], and we had to mix the chemicals, look at the color change and time it for one second or two second. It was fun.

Cogenerative dialogues as windows into student practices. On some occasions, the video

being watched during cogenerative dialogues spurred conversations that encouraged the study

practices of different students’ practices and related aspects of the learning environment. For

example, Maria made the following comment as we watched a videotape of the students engaged

in a lab (Flame Test Laboratory Activity).

But at 11:07 [AM] we seem like we all were writing down our observation and getting along well. Look at Earl. Earl the type of person that doesn’t do any work. He the one that copy and stuff like that. But he not dumb! Earl ain't dumb! He smart he just don’t wanna do it, He don’t wanna comprehend. He don’t wanna seem like he smart.

Earl was considered to be a disruptive student by other teachers. As his seat was near the

door, he preferred to look outside the classroom than to focus inside. During classroom

instruction, instead of paying attention and writing notes, he usually put his head down. Anita

tended to ignore him because he was not threatening her classroom environment. When Maria





12

brought Earl to our attention, Anita went to the counselor’s office to learn about Earl’s living

situation. She was told that Earl and his sister were living in a foster home. Even though efforts

to reunite them with their mother were being made by the child protection services, they were

constantly being moved from placement to placement. With no permanent home, the situation

was very stressful. Every day Earl brought his toothbrush and other personal belongings to

school because he was not sure whether he would be moved to a new foster home by the end of

the day. Through cogenerative dialogues around video footage of the classroom, Anita obtained

new insights into why particular students may act in particular ways.

Cogenerative dialogues and collective responsibility. When the students were in the

laboratory, there were constant requests for Anita’s assistance. The students expected her to

assist groups and she did – continuously circling throughout the duration of the laboratory

activity from group to group. While Rowhea also assisted students who were in close proximity

of the video camera, the most responsibility, initially, seemed to be placed on Anita. This issue

of the lack of human resources in the lab was raised by a student researcher during one

cogenerative dialogue. As Maria explained, the historical experiences of the students (since they

had never participated in a science laboratory activity previously in ninth and tenth grades)

invoked a higher demand for help then Anita and I had expected.





13

Row Yea, I know what you mean. When I am in that lab, it seems like there are a million people [calling out], “Ms Abraham, Ms Abraham.”

Maria This is our first time for doing something. This is our first time being in the lab. It is our first time all this stuff. It is the first time. But I think she can get more help somewhere else too. She needs to find some more help.

Interestingly, the students began to take responsibility for their own and each other’s

practices. Students kept an eye on their group members and on other groups to make sure that

they were following procedures correctly. They often provided information by answering

questions, sharing procedures, talking through the process and modeling for each other. For

example, during laboratory activity on physical and chemical changes, one group wanted to

finish the laboratory activity quickly and decided to put the baking powder directly into the

vinegar without first wrapping the powder up inside a paper towel, as the procedure required

them to do. However, surprisingly, this didn’t go unnoticed by another group’s member who

reacted quickly by shouting, “Stevenson you wrong! Don’t take it out! You wrong.” Such

interactions indicate that the students themselves were acting as resources for others within the

laboratory, illustrating an emergent spirit of collective responsibility.

CONCLUSIONS





14

The potential to transform the classroom lies in knowing oneself as a teacher and/or learner, and that only by collectively seeking to expose and examine the structures associated with the process of teaching and learning can contradictions be resolved to afford greater agency for all classroom participants. (Martin, 2004, p. 203)

This critical research on the teaching and learning of science in Anita’s chemistry classroom

demonstrates the possibilities for improving the teaching and learning of science. Cogenerative

dialogues between teachers and students can act as a tool to fill the gap between a teacher and

her students. However, due to their busy roster, teachers can barely find time to engage in

cogenerative dialogues with their students. We suggest that administrators provide support to

teachers who would like to engage in cogenerative dialogues with their students.

School and classroom structures and available resources impact the teaching and learning of

science in a classroom. Because structures can be changed to afford the learning of students in

the classroom and because teachers’ practices are also a part of the structure, we propose that

teachers should regularly reflect with students, through cogenerative dialogues, on their practices

to identify those that are successful in the classroom setting. When teachers identify successful

practices, they can reinforce them to achieve their own as well as their students’ goals.

Although educational research findings are always utilized to improve teaching and learning

in a classroom, the reality is that traditional research dynamics do not afford the immediate

participants of a study with opportunities to reap the benefits; rather the implications of the





15

research findings are for future classrooms. A research ‘with’ methodology empowers students

and teachers during the research process. That is, the model of critical research we discuss in this

article introduces a view of educational research that utilizes research as a tool that is

immediately effective and designed to encourage a sense of empowerment. In this manner, teams

of teacher and student researchers become integrated and natural parts of a classroom routine

where the learning environment is characterized by an openness to examining practices and

taking responsibility for one’s own actions.

REFERENCES

Elmesky, R., & Tobin, K. (2005). Expanding our understandings of urban science education by

expanding the roles of students as researchers. Journal of Research on Science Teaching, 42,

807-828.

Barton, A. (2001). Science education in urban settings: Seeking new ways of praxis through

critical ethnography. Journal of Research in Science Teaching, 38, 899-917.

Collins, R. (2004). Interaction ritual chains. Princeton, NJ: Princeton University Press.

LaVan, S. K. (2004). Cogenerating fluency in urban science classrooms. Unpublished doctoral

thesis, University of Pennsylvania, Philadelphia.





16

LaVan, S.K., & Beers, J. (2005). The role of cogenerative dialogue in learning to teach and

transforming learning environments. In K. Tobin, R. Elmesky, & G. Seiler (Eds.), Improving

urban science education: New roles for teachers, students, and researchers (pp. 147-163).

New York: Rowman & Littlefield Publishers, Inc.

Martin, S. (2004). The cultural and social dimensions of successful teaching and learning in an

urban classroom. Unpublished doctoral thesis, Curtin University of Technology, Perth, WA.

Roth, W.-M., & Tobin, K. (2001). The implications of coteaching/cogenerative dialogue for

teacher evaluation: Learning from multiple perspectives of everyday practice. Journal of

Personnel Evaluation in Education, 15, 7-29.

Seiler, G., & Elmesky, R. (2005). The who, what, where, and how of our urban ethnographic

research. In K. Tobin, R. Elmesky, & G. Seiler (Eds.), Improving urban science education:

New roles for teachers, students, and researchers (pp. 1-19). New York: Rowman &

Littlefield Publishers, Inc.

Tobin, K. (2006). Qualitative research in classrooms: Pushing the boundaries of theory and

methodology. In K. Tobin and J. Kincheloe (Eds.), Doing educational research – A

handbook (pp. 15-58). Rotterdam, The Netherlands: Sense Publishers.

Tobin, K., Zurbano, R., Ford, A., & Carambo, C. (2003). Learning to teach through coteaching

and cogenerative dialogue. Cybernetics & Human Knowing, 10, 51-73.





17

Wassell, B. (2004). On becoming an urban teacher: Exploring agency through the journey from

student to first year practitioner. Unpublished doctoral thesis, University of Pennsylvania,

Philadelphia.






A Pragmatic Mathematics: A new skills-for-life mathematics course addressing the

NSACS’s revelation of the dismal quantitative literacy of America’s college graduates

Dr. Elie Feder Kingsborough Community College - CUNY

Department of Mathematics and Computer Science [email protected]; [email protected]

1. INTRODUCTION

U.S. colleges are failing in their responsibility of training students in the practical

mathematical skills necessary to successfully enter society. This is the conclusion that was reached by

The American Institutes for Research’s new study examining the literacy of U.S. college students

(American Institutes for Research [AIR], 2006). “The National Survey of American College Students

[NSACS],” is based upon a sample of 1,827 graduating students from randomly selected 2-year and 4-

year, public and private, universities and colleges across the United States. According to Stephane

Baldi, the NSACS’s director at the American Institutes for Research, the study is intended to be used as

a tool to help college and university administrators identify specific academic areas where students

have literacy gaps that need to be rectified. The study reveals that students struggle most with

quantitative literacy, which the NSASC defines as follows:

The knowledge and skills required to perform quantitative literacy tasks, that is, to identify

and perform computations, either alone or sequentially, using numbers embedded in printed






materials. Quantitative examples include balancing a checkbook, figuring out a tip,

completing an order form, or determining the amount of interest on a loan from an

advertisement.

The study concludes that approximately twenty percent of U.S. college graduates completing

four year degrees – and thirty percent earning two year degrees – have only basic or below basic

quantitative literacy skills. This means they are unable to estimate if their car has enough gas to get to

the next gas station, or calculate the total cost of ordering office supplies. The results indicate

shortcomings in the educational system’s preparation of students to meet the mathematical challenges

of the real world.

As these shortcomings have considerable effects upon American society, the results of this

study indicate a crisis which must be addressed. “Many situations bring people into contact with

mathematics, including buying products, conducting business, producing products, managing people

and technology, using science and technology” (Arney, 1999). The rate of growth of mathematically

based occupations is about twice that for all other occupations (National Research Council, 1990).

Almost 40 percent of the workforce does not have sufficient quantitative literacy for jobs that pay more

than $26,900, on average. Additionally, close to two-thirds of new jobs will require quantitative skills

typical of those who currently have some college or bachelor’s degree. America cannot remain a first-

rate economic power with a population that has second-rate mathematical literacy. Additionally, if

educators cannot fulfill their economic responsibility to help our youth and adults achieve quantitative

literacy, they will also fail in their cultural and political missions to create good neighbors and good

citizens (Carnevale and Desrochers, 2003).

In response to the findings of the study, a new problem-based method of teaching basic

mathematical skills to those students who have difficulty in applying mathematics to their everyday






lives is presented. A course is designed which is not truly a mathematics course, but can be construed

as a skills-for-life course which involves mathematics. Mathematics is not taught as a subject, but as a

tool to be used on a day-to-day basis. This change in attitude impacts what is taught, what is not taught,

and most significantly, how it is taught. These changes signify an appropriate response to the challenge

presented by the findings of the NSACS.

The paper is organized as follows. Section 2 discusses the notion of quantitative literacy, and

surveys the literature and studies regarding the emphasis on quantitative literacy in the American

mathematical educational system. Section 3 relates the successful Israeli model of improving the

quantitative literacy of its weaker students while simultaneously maintaining strong standards for its

advanced students. Section 4 elucidates the new pragmatic mathematic1 course designed to improve

America’s quantitative literacy, illustrates its teaching method and content, and compares it to

traditional mathematics courses. Section 5 is devoted to addressing various oppositions towards the

implementation of this new course. Section 6 provides some concluding remarks.

2. BACKGROUND ON QUANTITATIVE LITERACY AND ITS OPPOSITION

2.1. The notion of quantitative literacy. The concept of numeracy, or quantitative literacy, emerged

in the Crowther report (1959), where “numerate” is defined as “a word to represent the mirror image of

literacy … an understanding of the scientific approach to the study of phenomena - observation,

hypothesis, experiment, verification [- and] the need in the modern world to think quantitatively.” The

landmark report, A Nation at Risk (U.S. Department of Education, 1983), calls for higher standards for

all students in mathematics, as well as curricula that would teach students to “apply mathematics in

1 Throughout this work, the term pragmatic mathematic will be used specifically to refer to the new course developed in this paper.






everyday situations”. Carnevale and Desrochers (2003) underscore that “most Americans seem to have

taken too little, too much, or the wrong kind of mathematics”. They recommend that “to fully exploit

mathematics as a practical tool for daily work and living, mathematics needs to be taught in a more

applied fashion.” In Mathematics and Democracy: The Case for Quantitative Literacy (Steen, 2001),

the Quantitative Literacy Design Team remarks: “Typical numeracy challenges involve real data and

uncertain procedures, but require primarily elementary mathematics. In contrast, typical school

mathematics problems involve simplified numbers and straightforward procedures, but require

sophisticated abstract concepts.”

In their insightful discussion of quantitative (in their terms, mathematical) literacy, Amit and

Fried (2002) comment that most educators struggle to supply the term with a precise definition which

enables one to determine its applicability to particular students. Despite this difficulty, they contend

that “at the heart of this notion lies students’ openness to mathematics”, rather than their mastery of

particular skills. Since it is unavoidable that students will confront mathematics in their lives, educators

must ensure that these encounters do not cripple them with fear. They conclude that “the

mathematically literate society is, thus, one characterized by a sense of ease, of feeling at home, with

mathematical ideas and mathematically presented information.” A similar attitude is presented by

Briggs, Sullivan and Handelsman (2004) who comment as follows:

Providing liberal arts students with a worthwhile experience in a quantitative literacy course

requires overcoming significant psychological obstacles. Students who take such courses often

are victims of previous mathematics courses and instructors. As a result, they harbor genuine

fears of mathematics, they have lost confidence in their quantitative skills, and they have little

belief that mathematics might be of use in their future. A successful quantitative literacy






course cannot subject students to more of the same experiences they have had in previous

mathematics courses.

Although some of the characterizations of quantitative literacy found in the

literature seem to differ, the difference is largely one of perspective more than of

substance. Many authors focus on the practical manifestations of quantitative

literacy– the possession of practical mathematical skills; while others define the

underlying cause of quantitative literacy – openness to and understanding of

mathematics. However one precisely defines quantitative literacy, almost all agree

about its necessity in today’s world. The urgent need for effective quantitative literacy

courses and programs in American colleges is expressed in recent reports

commissioned by the Mathematical Association of America [MAA] (Sons, 1995),

The American Mathematical Association of Two-Year Colleges [AMATYC] (Cohen,

1995), and the College Board (Steen, 1997). In 2001, the case for quantitative literacy

reappeared in a report that has inspired a new dialogue on the subject (Steen, 2001). A

recently released report, Beyond Crossroads II, extends the dialogue on quantitative

literacy by underscoring the interdisciplinary nature of quantitative literacy and

making a “call to coordinate across the disciplines to create a curriculum that

effectively supports quantitative literacy in our colleges (Blair, 2006).”






2.2. Related studies. The findings of the NSACS provide evidence for the shortcomings of American

college’s mathematical education. This problem is not new to the American mathematical educational

system. It simply provides additional evidence for the prevalence of a problem which was previously

recognized as a serious flaw in the American educational system. As evidence of this problem, The

Program for International Student Assessment [PISA] indicates that while U.S. high school students

match their peers in other nations when it comes to mathematical skills, this is not the case regarding

practical mathematical skills in which they ranked 24th out of 29 industrialized nations (NCES, 2004a).

Furthermore, The Trends in International Mathematics and Science Study [TIMSS] finds that

American eighth-grade students rank 15th internationally in mathematical achievement (NCES, 2004b).

In a similar vein, The National Assessment of Educational Progress [NAEP], billed as “the nation’s

report card”, reveals that only 36% of fourth graders, and 30% of eighth graders, have reached a level

of proficiency in mathematics (NCES, 2004c). These findings, which were reported prior to the

NSASC, indicated that the nation’s educational techniques must be improved to better prepare students,

throughout K-12, for the mathematical challenges that life presents. One would naturally assume that

this problem which is prevalent throughout K-12 would not vanish in American colleges. NSACS

confirms this assumption and demands that mathematics education reform be extended to American

colleges as well.

In response to the findings of weak mathematical performance of American

students throughout K-12, many educators have supported the implementation of a

Standards-Based Mathematics Curriculum in American schools. “The NCTM

Standards”, developed by the National Council of Teachers of Mathematics (NCTM,

1989, 1991, 1995, 2000), shift the focus of mathematics education from

memorization, rote learning, and application of facts and procedures, to the






development of conceptual understanding and reasoning. The NCTM Standards are

based on a set of core beliefs about mathematics as a body of knowledge and about

the learning processes that effectively promote mathematical understanding and

literacy (Goldsmith and Mark, 1999).

The need for reform in collegiate mathematics education has also been

documented in several national reports, and change in this area has begun. Specific

recommendations for curriculum change in two-year colleges are made in Curriculum

in Flux (Davis, 1989). A new undergraduate curriculum is put forth in Reshaping

College Mathematics (Steen, 1989). Everybody Counts (National Research Council,

1989) calls for detailed changes in mathematics education starting from kindergarten

all the way to graduate school. Additionally, Moving beyond Myths (National

Research Council, 1991) proposes major changes in undergraduate mathematics

education. The MAA’s Guidelines for Programs and Departments in Undergraduate

Mathematical Sciences (1993) recommends that every college graduate should be able

“to analyze, discuss, and use quantitative information; to develop a reasonable level of

facility in mathematical problem solving; to understand connections between

mathematics and other disciplines; and to use these skills as an adequate base of life-

long learning.” Significant work has also been done regarding calculus educational

reform in American colleges [Crocker (1990), Ross (1994), Tucker and Leitzel

(1994)].






The shortcomings revealed by the NSACS are specifically regarding students of two-year

colleges and the lower division of four-year colleges. Mathematics education of these students is

referred to as “introductory college mathematics” by the AMATYC’s Crossroads in Mathematics:

Standards for Introductory College Mathematics before Calculus (Cohen, 1995). Crossroads develops

standards for introductory college mathematics education with the following two goals in mind: “to

improve mathematics education at two-year colleges and at the lower division of four-year colleges and

universities and to encourage more students to study mathematics.” One theme in these standards is

that “the mathematics that students study should be meaningful and relevant” and that the problems

presented should “provide a context as well as a purpose for learning new skills, concepts and theories.

A similar sentiment is described by Haver and Turbeville (1995) in their formulation of the goals of a

mathematics course designed for nonscience majors. They explain:

The goals of the course are to develop, as fully as possible, the mathematical and quantitative

capabilities of the students; to enable them to understand a variety of applications of

mathematics; to prepare them to think logically in subsequent courses and situations in which

mathematics occurs; and to increase their confidence in their ability to reason mathematically.

In their recent report Beyond Crossroads (Blair, 2006), the AMATYC presents a renewed

vision for introductory college mathematics education by providing new Implementation Standards,

which “focus on student learning and the learning environment, assessment of student learning,

curriculum and program development, instruction, and professionalism.” These standards are designed

to “clarify issues, interpret, and translate research to bring standards-based mathematics instruction into

practice” in an attempt to reform American introductory mathematics education.






Many other conferences and articles have echoed the sentiments that the American

educational system must reform the way it teaches mathematics, and refocus on practical applications.

Crossroads suggests that “introductory mathematics courses hold the promise of opening new paths to

future learning and fulfilling careers to an often neglected segment of the student population” (Cohen,

1995). Despite numerous suggestions and implementations of reform in introductory college

mathematics education, the recent findings of NSACS reveal that these changes have not been as

widespread or as effective as would be desired. A large number of American college graduates are still

failing in their quantitative literacy and are not properly prepared for the mathematical challenges

which life presents. Hopefully, the adoption of the new Implementation Standards suggested in Beyond

Crossroads will help address these shortcomings.

2.3. Opposition to an educational system focused on quantitative literacy. Despite the strong and

widespread support for a shift of the American educational system towards quantitative literacy, there

is some tough opposition. Kaiser (1999) reveals that the educational method in Germany is more

focused on theoretical aspects of mathematics, while that of England is more focused on the pragmatic

side of mathematics. Thus, the impetus towards quantitative literacy can be seen as a push that America

follows England’s lead. However, Gardiner (2004) strongly cautions against such an approach and

furnishes evidence of the failings of the English educational system in equipping its students with basic

mathematical skills. He attributes this failing to the paradigm shift which occurred in England after the

publication of the Cockcroft report (Cockcroft, 1982), and advises educators to heed the warning of

Hyman Bass (quoted in (Steen, 2004)), who describes with uncanny accuracy what occurred in

England in the late 1980’s and 1990’s:






The main danger … is the impulse to convert a major part of the curriculum to this form of

instruction. The resulting loss of learning of general (abstract) principles may then deprive the

learner of the foundation necessary for recognizing how the same mathematics witnessed in

one context, in fact applies to many others.

Gardiner concludes that “mathematics and mathematics teaching are simply hard, and that

there is no “cheap alternative” to facing the fact that abstraction is a crucial part of elementary

mathematics - almost from the outset.” He cautions against making England’s mistake, and suggests

“that current abysmal levels of achievement indicate the need for hard work and incremental

improvement, rather than the launch of yet another bandwagon.” Gardiner’s points are insightful and

must be considered in any attempt at improving the American educational system. America must learn

from England’s mistakes and cannot afford to deprive its students of the true essence and beauty of

mathematics, and at the same time rob them of learning its basic skills. Simultaneously, the abysmal

levels of mathematical proficiency of America’s students must not be ignored, and its educational

methods must be improved.

3. A SUCCESSFUL MODEL FOR IMPROVING QUANTITATIVE LITERACY

In the search for a middle ground between the democratization of mathematics with its risk of

“watering down” mathematics on the one hand, and the maintenance of high mathematical standards on

the other, Amit and Fried (2002) present the successful model of Israel’s reformulation of their

National Completion Examinations in Mathematics (NCEM) administered at the end of high school. In

the early 1990’s, educators realized that the high level of mathematics demanded by their NCEM’s

intimidated weaker students, and caused many to terminate their mathematical studies as early as ninth






grade. This presented a serious impediment to the nation’s aspirations towards high quantitative

literacy for its citizens. At the same time, however, they did not want to lower their standards and

history of high achievement for their advanced students. To solve this dilemma, in 1996 they created an

alternative for weaker students by dividing their basic NCEM into two parts. The basic part is designed

with the reasonable expectation that all students can pass. One of the foci of this part of the test is

mathematical questions involving common sense and everyday experience. The hope was that the

weaker students would be able to face their mathematical inadequacies and strive for a modicum of

success in their mathematical endeavors. Instead of insisting on a standard which these students could

not achieve, Israel devised a bifurcated system which enabled its weaker students to pursue a more

reasonable goal, and thereby continue their mathematical training. In order to ensure that the standards

for the higher level students were not compromised, the completion of the exam demanded passing the

more advanced section as well. Although this is the case, Amit and Fried comment that:

The hope is that the option of taking this part of the examination independently of the rest of

the examination will encourage students to continue to study mathematics earnestly until the

end of high school, that they will work through the Basic Questionnaire with success, and that

this sense of success will push them eventually to complete the whole NCEM. There is

already evidence that this hope is not futile.

The data collected in the years following this shift indicate the success of the model. The number of

students who took the basic exam increased - an indication that once the students were given the

opportunity to take an exam more suited to their level, they became less intimidated and rose to the

occasion. Interviews with teachers reflected that these students felt more motivated to continue

studying mathematics, and gained a sense of competency through passing the basic part of the exam.






Although the evidence of Amit and Fried is limited to high school educational reform, the

concept should hold true in college educational reform as well. Assuming this to be the case, the Israeli

model indicates a successful method of maintaining America’s high mathematical achievement, while

pursuing reforms aimed at increasing its quantitative literacy. Namely, educators must continue their

advanced, more theoretical methods of mathematics instruction for stronger students, but

simultaneously reach out to weaker students and provide instruction more suited to their level and

interests. With this goal in mind, Section 4 introduces a new mathematics course, designed specifically

for weaker students. This pragmatic mathematic course should not be offered to stronger students who

are capable of higher mathematical achievement. By restricting the focus on practical mathematics to

its weaker students, America will follow Israel’s successful model and avoid the catastrophe which

occurred when England “watered down” its mathematics education for strong and weak students alike.

4. THE PRAGMATIC MATHEMATIC APPROACH

This section focuses on how the lesson learned from the Israeli model should be implemented in

teaching weaker students in American colleges. It elucidates the particulars of the pragmatic

mathematic approach and differentiates it from the traditional approach.

4.1. The traditional approach. It is common wisdom that to prepare students for life, the applicability

of mathematics to everyday situations must be demonstrated (see Senge (2000), for instance).

However, it is often overlooked that the role of practical examples must be different when teaching

strong, as opposed to weak students. In a traditional mathematics course, taken by both strong and






weak students, a rigorous approach is taken2. The focus is on content knowledge; to teach the student

certain pieces of subject matter (Cohen, 1995).This subject matter is presented in a logical sequence,

determined by the intrinsic mathematical development of the ideas. After a brief introduction regarding

motivation for a given topic, the mathematical theory is taught in an abstract setting. The methods of

computation are derived from this theory and extended to real life applications, wherever possible. The

applications are essentially introduced as an afterthought to the fundamental ideas. This approach is

effective in teaching advanced students who comprehend the underlying mathematical theory, follow

the computational methods derived from this theory, and appreciate the practical applications enabled

by these methods. They are engaged each step of the way, are truly involved in the mathematical

process of gaining knowledge, and reap the full benefits of the education provided. They are not part of

the troubling statistic regarding quantitative illiteracy.

This method, however, is inappropriate for students lacking in quantitative literacy. The

traditional college algebra or precalculus courses, which are primarily designed to prepare students for

calculus, do not provide the breadth and applicability of mathematics needed by liberal arts students

(Sons, 1995). They are not interested in the theory, find the computational methods difficult, and are

consequently not prepared to comprehend the real life applications. Their apathy, coupled with the

intricacies of the material, obfuscates them. Due to their weakness in comprehending advanced

mathematics, they are robbed of the opportunity of acquiring basic mathematical skills well within their

capabilities. Unfortunately they proceed to fail a quantitative literacy test (such as NSACS) and, more

importantly, are not properly equipped with the fundamental degree of mathematical ability necessary

for life’s challenges. It must be remembered that “making mathematics relevant and meaningful is the

collective responsibility of faculty” (Cohen, 1995) and the results of NSACS reveal a failure in this

responsibility which must be addressed.

2 For a more thorough discussion of the traditional mathematics education approach, see Quirk (n.d.).






4.2. A direction revealed by NSACS. One of the five Implementation Standards of

Beyond Crossroads (Blair, 2006) is “Curriculum and Program Development”. It

suggests that:

Mathematics departments will develop, implement, evaluate, assess, and revise courses,

course sequences, and programs to help students attain a higher level of quantitative literacy

and achieve their academic and career goals.

In attempting to discover an appropriate method for implementing this standard and reaching out to

college students with poor quantitative literacy, one aspect of the NSACS stands out and provides

direction to educators. Namely, the study reveals that students who take classes which stress analytic

thinking and applying theories to practical problems, have a higher degree of quantitative literacy

(AIR, 2006). This correlation suggests the introduction of a pragmatic mathematic course which

reaches out to students who have no intrinsic interest in mathematics, but realize its necessity in the

modern world3. This course addresses the frequent question posed by students, “Why do we need to

learn this stuff?” The pragmatic approach better motivates the student to study mathematics. Instead of

immersing students in pure mathematics and its multifarious abstractions, this course only teaches the

mathematical skills necessary in the modern society, and therefore succeeds in conveying these skills to

the students4. By giving the students a level of comfort with mathematics and allowing them to realize

the power and usefulness of mathematics, they are assisted in overcoming their fear of mathematics

and improving their quantitative literacy. 3 Though the correlation does not prove that the cause of the increased quantitative literacy is these courses, it is nonetheless a correlation which suggests a direction for needed reform. 4 Being that modern society is rapidly changing, the mathematical skills necessary in modern society are also rapidly changing. Thus, while the approach of this course is fixed, the particulars must be adjusted by the instructor to the changing demands of society. This being said, for a sample of some mathematical skills necessary in today’s society, see Section 5.4.






4.3. The pragmatic mathematic approach: What is taught? The new method that the pragmatic

mathematic utilizes in teaching these students is exactly the opposite of the traditional sink-or-swim

approach. The traditional approach unrealistically attempts to prepare all students to become

mathematical experts. On the other hand, the pragmatic mathematic aspires to train weaker students in

practical mathematical areas. The practical examples do not complement the theory, but are the focus

of this course. It is not a pure mathematics class, but a preparatory class for life’s challenges. As such,

each lesson focuses on a problem which the students encounter in their daily lives. The syllabus is

designed to have approximately 40 practical problems which will serve as springboards to introduce

mathematical skills and methods. Instead of concocting unrealistic word problems to illustrate remote

applications of an abstract subject, this course focuses on real life problems which every student relates

to and appreciates. Rather than the example of Phil, the farmer, using the quadratic formula to help

plant his field, this course considers the example of Steve, the student, using fractions to determine if

he will make it to the next gas station. A teacher of this course will not be concerned about skipping

abstract topics whose mathematical significance may be great, but whose practical significance is

small, or nonexistent. These topics simply do not belong in a pragmatic mathematic course, but in a

true mathematics course. Just as Shakespeare is not taught to beginners in reading, mathematical

abstraction should not be taught to beginners in mathematics. Will these students comprehend and

appreciate the full picture of mathematics? Absolutely not! They will not understand the theory, nor

will they have a solid grasp on the rigor of the computational methods. An honest analysis leads to the

realization that this is not appropriate for these students, as is evidenced by their failure to grasp real

mathematics in the traditional educational system. In a pragmatic mathematic course, these students

derive from mathematics the practical tools they truly need. As was the case with Israel’s high school

students, these positive experiences allow the students to become comfortable with mathematics. They






will enjoy the pragmatic mathematic as its beneficial function is immediately apparent. This method

addresses the American educational system’s shortcomings in educating its weaker students, and will

find success in increasing the nation’s quantitative literacy through teaching these students only what is

truly necessary.

4.4. The pragmatic mathematic approach: How is it taught? Besides for impacting what material is

taught, the pragmatic mathematic approach provides a new method as to how the material should be

taught. This method is elucidated by explaining the basic approach towards each lesson in a pragmatic

mathematic course, and then by illustration through an example. Each lesson begins by engaging the

students with a practically motivated exercise. Once the students gain interest in the problem, the

instructor demonstrates that its solution demands mathematics, and introduces the skills necessary to

solve the problem. Instead of insisting on a rigorous mathematical solution, the most efficient method

of solving the problem is illustrated. Whenever possible, tricks or shortcuts are introduced to simplify

the solution. The method can be illustrated with a simple example. On a regular basis, students are

faced with the task of computing a tip. Assume that their restaurant bill totals $51.07, and they want to

give a 15% tip. The NSACS reveals that many college graduates are perplexed by such a task. Why is

this so? Are American students incapable of such a simple procedure? Certainly not! The explanation is

usually that they were never properly taught how to calculate it. They were instructed how to take

2.47% of 0.0456, and other complex percent problems. They have a vague recollection of moving the

decimal two spots to the left and multiplying, but have forgotten long ago how to carry out this multi-

step process. They were never shown how straightforward it is to take 15% of a number, especially

when precision is unnecessary. It is in this context that percents are introduced - with the simplest, most

practical problems, which can be solved by a shortcut. The instructor emphasizes the significance of

rounding in real-life problems. Students are taught to quickly compute 10%, half it to get 5%, and add






these to get the desired 15%. No student should have trouble applying this method, especially with

practice. How many mathematicians evaluate a 15% tip by doing long multiplication by 0.15? Most

employ some shortcut which they have discovered on their own. So why not teach this method to

students who cannot figure it out by themselves? Why burden them with multiplying $51.07 by 0.15

when there is a simpler and quicker alternative? Just as in a regular mathematics course tricks cannot

be allowed to substitute for real mathematics, so too in a pragmatic mathematic course, real

mathematics cannot be allowed to substitute for tricks, and cloud the path towards a practical solution.

The goal of this course is not to make its students into mathematicians, but to give them the tools and

confidence needed to apply mathematics to their everyday lives. Although not every lesson lends itself

to the simplicity involved in computing tips, this example illustrates the approach of this course and

should, therefore, serve as a model for other lessons.

4.5. Education regarding real-life institutions. The pragmatic mathematic course possesses another

feature which addresses the failings of college graduates in the NSACS, but is unrelated to their

mathematical education per se. Often times, students are relatively well equipped with mathematical

skills, but are ignorant regarding the real world institutions which invoke these skills. For instance,

computing interest payments on a mortgage or credit card involve basic mathematics. Yet, even the

greatest “math genius” would be unable to perform these tasks without knowing the concept of a

mortgage, or the meaning of APR. This is specifically applicable to students with poor quantitative

literacy who fear anything involving mathematics and, therefore, never acquaint themselves with these

basic financial phenomena. With this realization in mind, this course attempts to overcome the

students’ fear of mathematics and teaches mathematical skills together with their accompanying real

world knowledge. The concepts of interest, insurance, credit card fees, deposit slips, checkbooks,

investments, odds, and other such notions are elucidated. By familiarizing students with sufficient






applications of mathematics, they are assisted in overcoming their dread of mathematics, and become

more suited to handle future situations involving mathematical skills.

The success of the course will be increased by designing hands-on lessons. This can be

accomplished by distributing: checkbooks, deposit slips, credit card offers, cell phone plans, lottery

rules, odds for sporting events, food labels containing nutritional information, and any other material

which invokes mathematics. This arouses the interest of the students and allows them to realize, in a

concrete manner, the value of improved quantitative literacy.

4.6. Comparable courses. A similar, but more sophisticated, pragmatic course design is suggested by

Bernard L. Madison (2004). The University of Arkansas course, developed by Madison, is centered on

newspaper and magazine articles which can only be understood or critiqued by applying mathematical

skills. A comparison between his course and the pragmatic mathematic course indicates that the subject

matter and mathematics involved in Madison’s university course are more complex and are suited to

higher level students.

Another similar liberal arts mathematics course is offered at University of Colorado at Denver

by William L. Briggs. The course has three stated goals: (1) to strengthen and broaden students’

quantitative skills; (2) to restore students’ confidence in using those skills; and (3) to demonstrate the

immediate relevance and applicability of mathematics to students’ lives and careers. (Briggs et al.,

2004). From their experiences, the authors conclude that “if student engagement is secured early in the

course, it can change student attitudes favorably and lead to an effective learning experience.” It would

be a fruitful study to thoroughly compare the three courses (Madison’s, Briggs’s and the pragmatic

mathematic) in their content and degree of success.

5. ADDRESSING OBJECTIONS TO THE PRAGMATIC MATHEMATIC COURSE






This section discusses a number of objections which can be raised against the implementation

of the pragmatic mathematic course in American colleges.

5.1. The level of the course. One might raise the following objection to the pragmatic mathematic

course: “Is this truly a college mathematics course? After all, the mathematical topics covered are of an

elementary nature and should have been mastered before entering college. Students who have difficulty

with these skills should take remedial classes to rise up to college standards!” To this objection, two

responses are offered. Firstly, the NSACS revealed the sobering fact that thirty percent of graduates of

American two-year colleges are severely lacking in their practical mathematical skills. This is

compared to the zero percent of students who graduate without remediation or placing out of remedial

courses. Colleges are simply not succeeding in preparing their students for life. What good is a

mathematics education which prepares students to pass a test requiring numerous calculations, if they

stumble as soon as they encounter a real life mathematical problem? Is this truly a college mathematics

course? Although it is not a college course in the current educational system, that is precisely the

problem. It should be, as indicated by the results of the NSACS. Educators cannot ignore the findings

of this national study and blindly assume that the current approach is flawless. They must rise to the

challenges presented by the directors of the study and change the manner in which weaker students are

educated. The pragmatic mathematic course suggested in this paper rises to this new challenge.

Additionally, one cannot judge the level of a mathematics course merely by the technical

skills it involves. Any mathematics teacher is well aware that many students who have mastered the

requisite technical skills become dumbfounded by challenging word problems. This explains why

many students performed poorly in the NSACS test of practical mathematics skills, despite the fact that

all community college graduates have either passed or placed out of remediation. Mathematics cannot






be limited to the application of step-by-step algorithms, but must include understanding, thinking and

applying mathematics to new situations. This is arguably the most difficult and important part of

mathematics, and is precisely what the current educational system is failing to teach its students. Thus,

although the pragmatic mathematic course covers mathematical skills which are considered elementary

in their technical level, it teaches its students how to understand and apply these skills to new problems.

It therefore truly merits the status of a college-level mathematics course.

5.2. The scope of the course. Another objection which might be raised is that by restricting its lessons

to truly practical examples, this course unnecessarily limits the scope of its students’ mathematical

education. After all, by the end of the semester its students have only learned how to solve

approximately 40 practical problems. A more traditional approach, however, would provide students

with tools to handle a larger variety of problems. There are two responses to this objection. First,

although the lessons are centered about practical problems, these problems are invariably solved by

mathematical methods which can be generalized to other problems. These 40 problems train the

students to think mathematically and prepare them to solve other mathematical challenges. More

importantly, Amit and Klein (2002) note that the major hindrance to students’ advancement in

mathematics is not their weak intellectual faculties, but is their frightful attitude towards mathematics.

Because of their early failings in mathematics, they shy away from anything involving mathematics.

An effective method of overcoming this obstacle is through accustoming students to applying basic

mathematics, and helping them realize the usefulness of mathematics in making important decisions. A

course meeting the Crossroads standards is one in which “the students will have the opportunities to be

successful in doing meaningful mathematics that fosters self-confidence and persistence” (Cohen,

1995). When mathematics is presented as a concrete tool instead of an abstract pursuit, it becomes

demystified in their minds. By breaking through their inner resistances, educators can open up a world






of mathematics which was previously closed to these students. Once students become comfortable with

mathematics, they readily learn the basic skills they need. Thus, while the initial approach of this

course is limited to the practical, its objective extends much further. Hopefully, this course will provide

an effective method of reaching those students which the educational system has failed thus far, and

will enjoy the success of Israel’s model which reached out to its weaker students in the reformulation

of its NCEM’s.

5.3 Avoiding the “England Disaster”. Beside addressing the troubling results revealed in the recent

studies, this new course stands up to Gardiner’s objections as well. In order to avoid the “England

disaster” which resulted from shifting the focus of all mathematics education towards the practical, this

syllabus should only be used as a substitute for a “liberal arts mathematics” syllabus. In general,

students who take this course are those who have no plans of advancement in mathematics, but need to

satisfy a college requirement. These classes attract students who struggle with quantitative literacy.

Since the studies underscore the failings of the traditional methods for the lower twenty to thirty

percent of students, it is only these weaker students who must be targeted by a new approach.

However, this syllabus must not be implemented for stronger students who can, and must learn the true

rigor and theory involved in mathematics. The Israeli model, which found success through clearly

differentiating between its standards for stronger and weaker students, must be followed. Advanced

classes must continue in the traditional educational approach, keeping in mind that “numeracy and

mathematical literacy are desirable byproducts of school mathematics” (Gardiner, 2004). America will

thereby maintain the strengths of its stronger students, while simultaneously alleviating the weaknesses

of its weaker students.






5.4. The quantity of material. One may suggest that there are not enough practical mathematical

problems to provide material for an entire semester. In order to address this concern, numerous

examples are listed by topic. These examples are merely a fraction of the numerous problems which

students with poor quantitative literacy are confounded by on a daily basis. Paying attention to every

day experiences furnishes many similar examples. Teachers are encouraged to elicit examples

involving mathematical reasoning from the students’ daily routines. As the objective is to help the

students gain the mathematical skills which they require in their lives, they will direct the instructor to

the areas they find difficult. Additionally, such an approach is an effective means of fostering the

interest of the students.

5.4.1 Financial Topics. Comparing credit card offers; computing interest on a credit card

based upon APR; computing simple and compound interest on mortgages, loans and

investments; balancing checkbooks; deposit slips; comparing investments: stocks, bonds,

cash, mutual funds; determining profits/losses on investments; retirement plans; income taxes.

5.4.2. Consumer Topics. Comparing the value of two products in the grocery store;

comparing nutritional information on food products; comparing cell phone offers; comparing

prices for different types of gasoline: full vs. self and premium vs. regular; computing miles

per gallon of a car; determining if a car has enough gas to reach a gas station; metric system

conversions; computing tips; determining square yardage of a room to buy carpet; analysis of

insurance premiums using mathematical expectation; comparing insurance plans based upon

deductibles, percent coverage and out-of-pocket expenses; determining price of an item with a

given percent off sale; estimating and determining sales tax.






5.4.3. Recreational Topics. Understanding probability; methods of counting; analysis of

lotteries; odds for sporting events; examination of card and dice games; interpreting statistics,

bar graphs, line graphs, and circle graphs.

5.4.4. Miscellaneous Topics. Mixture problems; work problems; averages and grades;

greatest common divisor and least common multiple; speed/miles per hour; and scientific

notation.

6. CONCLUSION

Many studies have underscored the failing of America’s mathematics educational system in

imparting quantitative literacy to its students at all levels. The NSACS has provided new evidence of

the severity of this problem in the nation’s colleges. The pragmatic mathematic course is a new

suggestion to help remedy this problem. Educators at both two-year and four-year colleges are

encouraged to offer this course as a liberal arts mathematics course. The author of this paper requests to

be kept informed of any progress. Hopefully, this course will build a foundation for raising the

quantitative literacy of American college graduates and adequately preparing them for the challenges of

the modern world.

ACKNOWLEDGEMENTS

The author would like to thank Aron Zimmer and Rachel Sturm-Beiss for their help with this paper.






REFERENCES

American Institutes for Research (AIR), (2006). The National Survey of America’s College Students:

The Literacy of America’s College Students. Washington, DC: Author. Amit, M., & Fried, M.N. (2002). High-Stakes Assessment as a Tool for Promoting Mathematical

Literacy and the Democratization of Mathematics Education. Journal of Mathematical Behavior, 21, 499-514.

Arney, D.C. (1999). Undergraduate Mathematics for the Future: Modeling and Solving Problems,

Understanding the New Sciences. Retrieved January 14, 2007 from http://www.dean.usma.edu/math/activities/ilap/workshops/1999/files/arney1.pdf.

Blair, R. (Ed.). (2006). Beyond Crossroads: Implementing Mathematics Standards in the First Two

Years of College. Memphis, TN: American Mathematical Association of Two-Year Colleges (AMATYC).

Briggs, W., Handelsman, M.M. & Sullivan, N. (2004). Student Engagement in a Quantitative Lieracy

Course. The AMATYC Review, 26(1). Carenvale, A.P., & Desrochers, D.M. (2003). The Democratization of Mathematics. In B.L. Madison,

& L.A. Steen (Eds.), Quantitative Literacy: Why Numeracy Matters for Schools and Colleges, (pp. 21-31). Princeton, NJ: National Council on Education and the Disciplines.

Cockcroft, W.H. (1982). Mathematics Counts. Report of the Committee of Inquiry into the

Teaching of Mathematics in Schools. London: Her Majesty’s Stationery Office.

Cohen, D. (Ed.). (1995). Crossroads in Mathematics: Standards for Introductory College Mathematics before Calculus. Memphis, TN: American Mathematical Association of Two-Year Colleges (AMATYC).

Crocker, D. A. (1990). What has happened to calculus reform? The AMATYC Review, 12(1), 62-66.

Crowther Report (1959). 15 to 18: Report of the Central Advisory Council for Education (England). London: Her Majesty’s Stationery Office.

Davis, R.M. (Ed.). (1989). A curriculum in flux: Mathematics at two-year colleges (A report of the Joint Subcommittee on Mathematics Curriculum at Two-Year Colleges). Washington, DC: Mathematical Association of America (MAA).

Gardiner, T. (2004). What is Mathematical Literacy? UK: University of Birmingham. Retrieved February 9, 2006, from http://www.pims.math.ca/.






Goldsmith, L.T. & Mark, J. (1999). What is Standards-Based Mathematics Curriculum? The Constructivist Classroom, 57 (3), 40-44.

Haver, W. & Turbeville, G. (1995). An Appropriate Culminating Mathematics Course. The AMATYC

Review, 16(2), 45-50. Kaiser, G. (1999). Comparative Studies on Teaching Mathematics in England and Germany. In Kaiser,

G. & Luna, E. & Huntley, I. (Eds.), International Comparisons in Mathematics Education (pp. 140–150). London: Falmer Press.

Madison, B.L. (2004, December). To Build a Better Mathematics Course. All Things Academic, 5 (4).

Mathematical Association of America (MAA). (1993). Guidelines for programs and departments in undergraduate mathematical sciences. Washington, DC: Author.

National Center for Education Statistics (NCES). (2004a). International Outcomes of Learning in Mathematics Literacy and Problem Solving: PISA 2003 Results from the U.S. Perspective (NCES 2005-003). Washington, DC: U.S. Department of Education: Author.

National Center for Education Statistics (NCES). (2004b). Highlights from the Trends in International

Mathematics and Science Study (TIMSS) 2003 (NCES 2005-005). Washington, DC: U.S. Department of Education: Author.

National Center for Education Statistics (NCES). (2004c). The Nation’s Report Card: Mathematics

Highlights 2003 (NCES 2004-451). Washington, DC: U.S. Department of Education: Author. National Council of Teachers of Mathematics (NCTM). (1989). Curriculum and

evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (1991). Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (1995). Assessment standards

for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (NCTM). (2000). Principles and standards in school mathematics. Reston, VA: Author.

National Research Council. (1989). Everybody counts. Washington, DC: National Academy Press: Author.






National Research Council. (1990). A Challenge of Numbers. Washington, DC: National Academy Press: Author.

National Research Council. (1991). Moving beyond myths: Revitalizing undergraduate mathematics. Washington, DC: National Academy Press: Author.

Quirk, B. (n.d.). Traditional K-12 Math Education: Chapter 1 of Understanding the Original NCTM Standards, retrieved on January 14, 2007 from http://www.wgquirk.com/Genmath.html.

Ross, S. C. (Ed.). (1994, January). NSF Beat: $2.1 million in NSF calculus grants. UME Trends.

Senge, P. (2000). Schools That Learn: A Fifth Discipline Fieldbook for Educators, Parents, and Everyone who cares about Education. New York: Doubleday Currency.

Sons, L. R. (Chair). (1995). Quantitative reasoning for college graduates: A complement to the Standards. (Report of the CUPM Subcommittee on Quantitative Literacy). Washington, DC: Mathematical Association of America.

Steen, L. A. (Ed.). (1989). Reshaping college mathematics. Washington, DC: Mathematical Association of America.

Steen, L. (Ed.). (1997). Why Numbers Count: Qualitative Literacy for Tomorrow’s America, The College Board, New York.

Steen, L.A. (Ed.). (2001). Mathematics and Democracy: The Case for Quantitative Literacy. Princeton,

NJ: National Council on Education and the Disciplines. Steen, L.A. (2004). Achieving Quantitative Literacy. MAA Notes, 62. Washington D.C.: Mathematics

Association of America.

Tucker, A. C., & Leitzel, J. R. C. (1994). Assessing calculus reform efforts. Washington, DC: Mathematical Association of America.

U.S. Department of Education. The National Commission on Excellence in Education (1983). A Nation at Risk: The Imperative for Educational Reform, Washington, DC: U.S. Government Printing Office.






Using Alternative Numeral Systems in Teaching Mathematics

Farida Kachapova School of Computing

and Mathematical Sciences Auckland University of

Technology [email protected]

Murray Black School of Computing

and Mathematical Sciences Auckland University of

Technology [email protected]

Ilias Kachapov 13 Fairlands Ave

Waterview Auckland 1026, New

Zealand [email protected]

1. Introduction

The purpose of this article is to stimulate interest in number theory and history of

mathematics. Teachers can include parts of this article as additional topics in their

classroom teaching. On one hand, the origin of our decimal numbers and possible

numeral systems can make an exciting lesson and lead to quite advanced topics in

mathematics. On the other hand, this material does not require any specific knowledge

and can be explained even to primary students. So it can be used by high school

teachers to motivate students and expand their horizons.

A numeral system is a language where numbers are represented by symbols –

numerals. In the modern mathematics we use the decimal numeral system. It has base

10, which means that all numerals are made of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The

probable reason for using base 10 is that humans have 10 fingers and they used them

for counting. 10 is the base of the most common numeral system but it is not the only

possible one. Some native Americans used spaces between their fingers for counting,

so their numeral system had base 8.






The binary system (with base 2) has originated in China, was studied by G.Leibniz

in the 17th century and had useful applications in computer science in the modern

time. The hexadecimal system (with base 16) also has applications in computer

science because its base can be written as a power of two: 16 = 2 4.

The numeral systems with base 12 and base 60 were also used in the past. We still

have 12 hours in the clock and 12 months in the year. When we measure time we use

60 minutes in an hour and 60 seconds in a minute; we have 60 seconds in a degree as

angular measure.

In practice we mostly use the decimal numeral system. The alternative numeral

systems can be used by mathematics teachers to stimulate students’ interest in

mathematics and research. The students will realise that the decimal system is not the

only possible one and even not the best one, so they will learn to think “outside the

square”. The binary, senary and other alternative numeral systems can be explained

in simple terms at different levels starting from primary school. At the same time they

lead to some interesting topics in number theory and general algebra at the tertiary

level, such as modular arithmetic and Mersenne primes.

2. Senary Numeral System






Now we will take a closer look at the senary numeral system – the numeral system

with base 6.

Imagine aliens from another planet that have three fingers on each hand. They use 6

fingers for counting, so their numeral system has base 6 and all the numerals are

constructed from 6 digits: 0, 1, 2, 3, 4, 5. Besides each alien has 6 limbs: two arms,

two legs and two wings.

The aliens have the same arithmetic as we do but their arithmetic has a simpler

representation because it is expressed in the senary numeral system instead of our

decimal numeral system.

We will write numerals in the senary system with a subscript 6. So numbers 0, 1, 2,

3, 4, 5 are represented by numerals 06 , 16 , 26 , 36 , 46 , 56 respectively. Next number

6 is represented by 106 , 7 is represented by 116 , etc. Actually humans with 5 fingers

on each hand can show two-digit senary numerals if they use each hand to show a

digit from 0 (a fist with no fingers out) to 5 (all fingers out).

3. Arithmetic in the Senary System

Addition table:

16 + 16 = 26

26 + 16 = 36 26 + 26 = 46






36 + 16 = 46 36 + 26 = 56 36 + 36 = 106

46 + 16 = 56 46 + 26 = 106 46 + 36 = 116 46 + 46 = 126

56 + 16 = 106 56 + 26 = 116 56 + 36 = 126 56 + 46 = 136 56 + 56 = 146

Multiplication table:

26 × 16 = 26 36 × 16 = 36 46 × 16 = 46 56 × 16 = 056

26 × 26 = 46 36 × 26 = 106 46 × 26 = 126 56 × 26 = 146

26 × 36 = 106 36 × 36 = 136 46 × 36 = 206 56 × 36 = 236

26 × 46 = 126 36 × 46 = 206 46 × 46 = 246 56 × 46 = 326

26 × 56 = 146 36 × 56 = 236 46 × 56 = 326 56 × 56 = 416

26 ×106 = 206 36 ×106 = 306 46 ×106 = 406 56 ×106 = 506

The senary addition table is shorter than the decimal one but otherwise is not very

different.

The senary multiplication table is much easier to learn for the aliens’ children than

the decimal one for the humans’ children because it has several patterns in it:

1. Last digits 2, 4 and 0 make a cycle in the first and third columns.






2. Last digits 3 and 0 make a cycle in the second column.

3. In the fourth column the first digits are 0, 1, 2, 3, 4, 5 and the last digits are the

same in reverse order.

Notation: for digits an, …, a1, a0, the symbol an … a1 a0 will denote the numeral

in the senary system made of these digits (versus a product an ⋅ …⋅ a1 ⋅ a0).

As with any other base, there are algorithms for transferring numbers from senary

form to decimal form and vice versa.

Senary to Decimal. For a numeral an an-1… a1 a0 in the senary system its decimal

form is given by the formula an ⋅6n + an-1 ⋅6n-1 +…+ a1 ⋅ 61 + a0 ⋅ 60.

Examples:

12346 = 1×63 + 2×62 + 3×6 + 4 = 310,

543216 = 5×64 + 4×63 + 3×62 + 2×6 + 1 = 7465.

Decimal to Senary. To transform a natural number in decimal form to senary form

we keep dividing numbers by 6 and then write all remainders from right to left.

Example: 1244 ÷ 6 = 207 with remainder 2

207 ÷ 6 = 34 with remainder 3



So 1244 = 54326.






Since the senary multiplication table has several patterns, divisibility criteria are

easier in the senary system. To check divisibility of a number by 2, 3 or 6 we use only

its last digit: if the last digit is a, then the number can be written as 6b + a for some

integer b.

Suppose a natural number n ends on a digit a in the senary system.

Divisibility by 2 (3, 6). n is divisible by 2 (3, 6) if and only if a is divisible by 2 (3,

6).

We can write this in different terms.

Divisibility by 2. n is divisible by 2 if and only if a equals 0, 2 or 4.

Divisibility by 3. n is divisible by 3 if and only if a equals 0 or 3.

Divisibility by 6 = 10 6. n is divisible by 6 = 106 if and only if a equals 0.

Divisibility by 4. Suppose a natural number n has digits a and b as its last two

digits in the senary system. Then n is divisible by 4 if and only if the number ab is

divisible by 4.

Proof. The numeral ab denotes the number 6a + b. For some integer c,

n = 36c + 6a + b = 4 ⋅ 9 + 6a + b.

So n is divisible by 4 if and only if 6a + b is divisible by 4, which is if and only if

ab is divisible by 4.

Next divisibility criterion is proven similarly.






Divisibility by 9 = 13 6. Suppose a natural number n has digits a and b as its last

two digits in the senary system. Then n is divisible by 136 if and only if the number

ab is divisible by 136.

Divisibility by 5. A natural number n is divisible by 5 if and only if a sum of all its

digits in the senary system is divisible by 5.

Proof. The proof is easy if we use congruency modulo m. Two integers x and y

are said to be congruent modulo m if (x − y) is divisible by m; this is denoted by

x ≡ y (mod m).

Suppose n = ak … a1 a0. Then n = ak ⋅6k +…+ a1 ⋅ 6 + a0 .

Apparently 6 ≡ 1 (mod 5). For i = 1, 2,…, k by properties of the congruency,

6 i ≡ 1 (mod 5), ai 6 i ≡ ai (mod 5) and n ≡ ak +… + a1 + a0 (mod 5). Hence n is

divisible by 5 if and only if a0 + a1 +…+ ak is divisible by 5.

Next divisibility criterion is proven similarly.

Divisibility by 7 = 11 6. A natural number n = ak … a1 a0 is divisible by 116 if and

only if a sum of its signed digits a0 − a1 + a2 −… ak is divisible by 116.

4. Fractions in the Senary System






A fraction 0.c1c2 c3 … in the senary system is transformed to the following

decimal numeral: ...ccc

+++3

3

2

21

666.

Examples: 0.346 = 18

11

6

4

6

3

2=+ = 0.6111… ;

0.111…6 = 5

1

6

11

1

6

1

6

1

6

1

6

1

32=

!

"=+++ ... = 0.2.

The following example demonstrates how to transform a decimal numeral to senary

form by subsequent multiplication by 6.

Transforming 16

1 to senary form. 6×16

1 = 8

3 = 8

30 ; 6×

8

3 = 4

12

4

9= ;

6×4

1 = 2

11

2

3= ; 6×

2

1 = 3. We collect the whole parts of all results to get a senary

numeral: 0.0213. So 16

1 = 0.02136.

In the decimal system some fractions can be written as finite decimals and others as

infinite (recurring) decimals. For example, 2

1 = 0.5 and 25

1 = 0.04 are finite;

3

1 = 0.333… and 6

1 = 0.1666… are infinite. In general, a fraction k

1 can be written

as a finite decimal if and only if any prime factor of k is either 2 or 5. The reason is

that 2 and 5 are the only prime factors of 10.






For base 6 the only prime factors are 2 and 3. So a fraction k

1 has a finite

expression in the senary system if and only if any prime factor of k is either 2 or 3.

Since multiples of 3 occur more often than multiples of 5, the fractions with finite

expressions occur more often in the senary system than in the decimal system:

Senary Decimal

62

1 = 0.36 2

1 = 0.5

63

1 = 0.26 3

1 = 0.333…

64

1 = 0.136 4

1 = 0.25

65

1 = 0.111…6 5

1 = 0.2

610

1 = 0.16 6

1 = 0.1666…

611

1 = 0.0505…6 7

1 = 0.142857…

612

1 = 0.0436 8

1 = 0.125

613

1 = 0.046 9

1 = 0.111…






Base 60 seems even more practical for expressing fractions, since it has three prime

factors: 2, 3 and 5.






5. Prime Numbers and Perfect Numbers in the Senary System

If a number in the senary system ends on 0, 2 or 4, then it is divisible by 2. If a

number in the senary system ends on 0 or 3, then it is divisible by 3. So any prime

number in the senary system (except 2 and 3) ends on 1 or 5. These are the first

eleven prime numbers:

Decimal

5 7 11 13 17 19 23 29 31 37 41

Senary

56 116 156 216 256 316 356 456 516 1016 1056

We should emphasize that most properties of numbers (for example being a prime

number) are independent of the numeral system because numerals are only symbols

for expressing numbers. But some numeral systems express number properties in a

simpler form than others.

Perfect numbers are another class of numbers that have a simpler form in the senary

system. A perfect number is a natural number, which is a sum of its positive factors,

excluding itself. Here are a few first perfect numbers:

6 = 1+2+3,

28 = 1+2+4+7+14,






496 = 1+2+4+8+16+31+62+124+248,

8128,

33550336.

These are their senary forms:

6 = 106,

28 = 446,

496 = 21446,

8128 = 1013446,

33550336 = 31550333446.

One can see that every numeral on the right of this list, except 6, ends on 44. It can

be proven that in the senary system every perfect number (except 6) has 44 as its last

two digits.

Thus, the senary system would make a better symbolic base for dealing with

numbers in mathematics but due to historical reasons we use the decimal system and

it is too late to change it now.

References

Ore, O. (1988). Number theory and its history. New York: Courier Dover Publications






Lam, L.Y.& Ang, T.S. (2005). Fleeting footsteps: tracing the conception of arithmetic and

algebra in ancient China. River Edge, N.J.: World Scientific.

Kachapova, F. and Kachapov, I. (2005) Senary Numeral System. Workshop at the 9-th

conference of NZ Association of Mathematics Teachers. Retrieved October 1, 2005 from

NZAMT website http://www.nzamt.org.nz/nzamt9/ka/Senary%20System.ppt

Senary. (2007). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2007,

from http://en.wikipedia.org/w/index.php?title=Senary&oldid =121979068

Perfect number. (2007). In Wikipedia, The Free Encyclopedia. Retrieved April 22, 2007, from

http://en.wikipedia.org/w/index.php?title=Perfect_number&oldid=123382935






Effective Procedures in Teaching Mathematics

Farida Kachapova Auckland University of Technology

<[email protected]>

Murray Black Auckland University of Technology

<[email protected]> Ilias Kachapov

< [email protected]>

.

Introduction and Framework In our practice of teaching tertiary mathematics we want students to gain both

conceptual and procedural knowledge and develop their problem solving abilities. In

mathematics, procedural knowledge is the knowledge of symbolic representations,

algorithms and rules; conceptual knowledge is the knowledge of core concepts,

principles and their interrelations (Byrnes & Wasik, 1991). On the tertiary level, the

dominance of procedural mathematics is a characteristic of mathematics itself.

Mathematical theories are based on axioms and derivation rules, thus this knowledge

is highly procedural by nature: it must be derived from the fundamental definitions

and axioms by a finite sequence of logical steps (Tossavainen, 2006). Also

understanding a mathematical concept often does not provide the relevant procedural

knowledge, for example the definitions of a limit, a derivative or an integral are not

linked to the methods of their evaluation.

One approach to learning mathematics labeled the ‘dynamic action view’ (Byrnes

& Wasik, 1991), justifies the ‘traditional’ emphasis on procedural knowledge.






Another approach, the ‘simultaneous action view’, focuses on conceptual knowledge

using the assumption that student’s conceptual knowledge will necessarily increase

their procedural knowledge (Haapasalo & Kadijevich, 2000).

In constructivism, in particular in Piaget’s theory of cognitive development

(Piaget, 1985), conceptual knowledge and procedural knowledge are both integral

parts of the learning process. The ‘iterative model’ developed these views further

(Rittle-Johnson, Siegler, & Alibali, 2001). Their research shows the causal relations

between conceptual and procedural knowledge: concepts and procedures develop

iteratively reinforcing each other. Increased conceptual knowledge leads (through

training) to gains in procedural and problem solving abilities. Use of correct

procedures leads to improved conceptual understanding.

The authors of this paper share the iterative views. Therefore in our teaching

practice we look for effective methods to improve students’ procedural knowledge

that will in turn enhance their conceptual knowledge. Here we describe some non-

traditional methods that we used in courses in algebra, calculus and probability theory

at the Auckland University of Technology and the Moscow Technological University

for several years.






Substitution Method

The authors used the substitution method in teaching some topics of secondary

and tertiary mathematics, where other methods were traditionally applied. They

noticed that substitution simplifies learning and mastering some mathematical

techniques, especially for weaker students. The students who use substitution do not

have to guess; they just apply simple formulas, get answers faster and make fewer

mistakes on the way.

Completing the Square

To complete the square in a quadratic y = cbxax ++2 we introduce a new

variable a

bxt

2+= . Then the linear term in the quadratic cancels and at the end we

substitute x back. This can be justified as follows:

=+!=+"#

$%&

'!+"

#

$%&

'!=

!=

+=

=++= ca

batc

a

btb

a

bta

a

btx

a

bxt

cbxaxy422

2

22

2

2

2

a

bc

a

bxa

42

22

!+"#

$%&

'+= .






The new variable means the change to the set of coordinates where the

corresponding parabola has a simple form: symmetrical in the y-axis.

The following three examples illustrate this.

Example 1.

( ) ( ) =!=+!+!=!=

+==++ 27363

3

376

222ttt

tx

xtxx ( ) .x 23

2

!+

Example 2.

=!=+"#

$%&

'+!"

#

$%&

'+=

+=

!=

=+!8

123

4

55

4

52

4

5

4

5

3522

2

2ttt

tx

xt

xx

.x8

1

4

52

2

!"#

$%&

'!=

Solving Quadratic Equations

The same substitution a

bxt

2+= helps to solve quadratic equations in an easier

way than traditional factorising or quadratic formula.






Example 3.

0158122

=!! xx .

3

1

3

1

+=

!=

tx

xt

. 0153

18

3

112

2

=!"#

$%&

'+!"

#

$%&

'+ tt ,

03

4912

2=!t ,

6

7

36

492±== t,t . 1)

6

5

3

1

6

7

3

1

1!=+!=+= x,tx . 2)

2

3

3

1

6

7

3

1

2=+=+= x,tx .

Example 4.

0142

=++ xx . 2

2

!=

+=

tx

xt. ( ) ( ) 01242

2

=+!+! tt , 032

=!t ,

3±=t . 1) 3221

!!=!= x,tx . 2) 3222

+!=!= x,tx .

The factorising method is not very useful in Example 3 because the roots are

fractional, and does not work at all in Example 4 because the roots are irrational. The

quadratic formula works in all cases but it is harder to apply than the substitution

formula a

bxt

2+= .






Evaluating Limits of the Form 0

0 .

When we have to evaluate a limit of the form ( )xflimax!

with 0!a , it is often

useful to do the substitution axt != ; then we get 0!t . It is often easier to study

the behaviour of a function near point 0 than near other points. Here are a few

examples.

Example 5.

( ) ( )=

!=

+=

!=="

#

$%&

'=

!

!(( t

tlnlim

tx

xt

x

xlnlim

tx

1

5

5

0

0

5

6

05

( )1

1

0

!=!

!!

" t

tlnlimt

.

(Using the fact that: ( ))1

1

0

=+

! z

zlnlimz

.

Example 6.

=!

"#

$%&

'+

=

+=

!=

="#

$%&

'=

! (( t

tcos

lim

tx

xt

x

xcoslim

tx 2

2

2

2

0

0

2 0

2

)

)

)

)) t

tsinlimt 20 !

!

"=

2

1

2

1

0

==! t

tsinlimt

. (Using the fact that: )10

=! z

zsinlimz

.

Example 7.






( )( )

=+

+=

+=

!=="

#

$%&

'=

(( )

)

)

)

) 55

44

0

0

5

4

0 tsin

tsinlim

tx

xt

xsin

xsinlim

tx

=!" tsin

tsinlimt 5

4

0

5

4

5

5

4

4

5

4

00

!="!=## tsin

tlim

t

tsinlim

tt

. (Using the fact that: )10

=! z

zsinlimz

.

In the last example many students who apply the traditional technique give the

wrong answer 5

4 .

The substitution axt != always helps to find the limits of rational functions of

the indeterminate form 0

0 . With this substitution we can avoid factorising, which

involves trial and error process and can be sometimes quite difficult. Here are a few

examples.

Example 8.

=!=

+=="

#

$%&

'=

!+

!+

!( 1

1

0

0

12

23

2

2

1 tx

xt

xx

xxlimx

( ) ( )( ) ( )

=!!+!

!!+!

" 1112

2113

2

2

0 tt

ttlimt

3

5

32

53

32

53

02

2

0

=!

!=

!

!=

"" t

tlim

tt

ttlim

tt

.






Example 9.

=+=

!=="

#

$%&

'=

!+!

!+!( 2

2

0

0

1021143

674

23

23

2 tx

xt

xxx

xxxlimx

( ) ( ) ( )( ) ( ) ( )

=++

++=

!+++!+

!+++!+=

"" ttt

tttlim

ttt

tttlim

tt

23

23

023

23

0 43

32

1022121423

627242

3143

32

2

2

0

=++

++=

! tt

ttlimt

.

Example 10.

=+=

!=="

#

$%&

'=

!!!

!!+

( 1

1

0

0

13

4352

234

24

1 tx

xt

xxx

xxxlimx

( ) ( ) ( )( ) ( ) ( )

=!+!+!+

!+!+++=

" 11113

4131512

234

24

0 ttt

tttlimt

7

15

714113

151782

714113

151782

23

23

0234

234

0

=+++

+++=

+++

+++=

!! ttt

tttlim

tttt

ttttlim

tt

.






Obviously all these limits can be calculated by l’Hôpital’s rule. But the

substitution method gives students the chance to calculate these limits easily before

they have learned differentiation.

Integration

Substitution is widely used for integration. Here we will consider the integrals,

where substitution is not usually used but can be quite useful.

These integrals have the form ! ++ cbxax

dx

2 and dx

cbxax

BAx

! ++

+2

. So we do not

need partial fractions to integrate these particular types of rational functions. The idea

is the same as for the problem of completing the square: in each case the substitution

a

bxt

2+= is used.

The following three examples illustrate this.

Example 11.

=

+

=

+

=

=

!=

+=

=++ """

19

49

2

2

92

2

1

2

1

522 222

t

dt

t

dt

dtdx

tx

xt

xx

dx =

+!"

#$%

&'

13

29

2

2

t

dt

=+!!"

#$$%

&!"

#$%

&+=+!

"

#$%

&''=

((CxtanCttan

2

1

3

2

3

1

3

2

2

3

9

2 11C

xtan +!

"

#$%

& +'

3

12

3

1 1 .






Example 12.

=

!

!=

+!

=

=

+=

!=

=! """

25

15

1

5

15

5

1

5

1

52 222

t

dt

t

dt

dtdx

tx

xt

xx

dx

=+!

+=+

+

!!=

""""

#

$

%%%%

&

'

+

!!

(!= ) Ct

tlnC

t

t

lndt

tt15

15

2

1

5

1

5

1

2

1

5

1

1

5

1

1

2

5

5

1

Cx

xln +

!=

25

5

2

1 .

Example 13.

=+

!=

=

!=

+=

=++

!"" dtt

t

dtdx

tx

xt

dxxx

x

13

741

1

463

34

22

( )=

+!

+ "" dt

t

dtt

t

13

7

13

6

6

4

22

( ) ( ) =+!+= !Cttantln 3

3

713

3

2 12






( ) ( )( ) Cxtanxxln ++!++= !13

3

7463

3

2 12 .

The authors noticed that many students had difficulties with applying standard

techniques to the problems described above; but they can easily master and apply the

substitution technique. As the reader can see, the advantage of this method is that it is

algorithmic. So the students only have to remember the formula a

bxt

2+= and do

routine operations like expanding brackets and simplifying polynomials. The method

does not involve trial and error or guessing, unlike the ordinary techniques of

factorising and completing the square.

We invite readers to teach students to apply the substitution method to the classes

of mathematical problems described above, and see how this will enhance the

students’ learning.

Using Probability Trees: Making Procedures Meaningful

In basic courses in probability theory many secondary and tertiary teachers use

probability trees as a tool for teaching conditional probabilities, (e.g. Lipschutz &

Lipson, 2000, pp. 87-89). We put some thought and investigation into this practice. It

is easy for students to apply probability trees to simple problems. But often this

technique is not justified, so the students apply it without a clear understanding of its

meaning and limitations. Let us consider the following common example.






Example 14.

A test for a certain disease shows positive with probability 0.8 on an ill person and

with probability 0.05 on a healthy person. 10% of the population are affected by this

disease. If the test on a person shows positive what is the probability that this person

is ill?

In our lessons we explain that every person in this example represents an

outcome, or an elementary event. The sample space Ω (the set of all outcomes) can be

represented as a union of two events H 1 and H 2 (hypotheses):

H 1 = {a person is ill} and H 2 = {a person is healthy}.

Consider event A = {the test shows positive}. Construct a probability tree, where

the first level of branches represents hypotheses and the second level shows

conditional probabilities:

H1

H2

A∩H1

A∩H2

Ac∩H2

Ac∩H1 0.1

0.9

0.8

0.05






The students usually quickly learn to multiply probabilities along each branch and

then add some of the products:

P(A) = 0.1×0.8 + 0.9×0.05 = 0.125; ( )( )( )

6401250

80101

1.

.

..

AP

HAPA|HP =

!=

"= .

Common mistakes here are choosing wrong hypotheses and applying probability

trees to the problems, where this method is not appropriate. For example, some

students came up with such a probability tree for Example 14:

To avoid these mistakes, we justify the method of probability trees in the

following three steps.

1. The hypotheses H1, H2 ,…, Hn are disjoint events, which union contains all

outcomes.

2. Multiplication theorem: if P(H k) > 0, then P(A! H k) = P(H k) ⋅ P(A | H k),

k = 1, 2,…, n. It justifies multiplying probabilities along each branch.

positive result result

ill

ill

healthy

healthy

negative result result






3. Partition theorem: ( ) ( )!=

"=n

k

kHAPAP

1

. It justifies adding the products for

the branches corresponding to event A.

With these underlying principles the students will choose the correct hypotheses

and use the correct conditional probabilities in their probability tree. We call this

‘meaningful procedural knowledge’ because the procedure – probability trees – is

applied with understanding of the underlying mathematical concepts and theorems. It

is also useful to encourage the students to think about proofs of the multiplication and

partition theorems. One can be explained by the teacher and the other offered as an

exercise.

In past the authors taught probability trees as an algorithm. Now we teach them

together with the three justifying steps. We notice that our students more often get

correct solutions for this kind of problems than they used to in the past.

Conclusion

The described procedures were used for several years at the Auckland University

of Technology and the Moscow Technological University. Verbal responses from

students and their assessment results show effectiveness of these procedures. The

students who applied the suggested procedures are more successful than the ones

using traditional procedures, in technical manipulations as well as in learning the






relevant concepts. Applied to certain types of mathematical problems, these

procedures involve fewer technical details, are logical, relatively universal and

eliminate most of memorising. We would like to hear from the readers who used the

described procedures and we will appreciate their feedback on how this affected their

students’ learning.

References

Byrnes, J.P. & Wasik, B.A. (1991). Role of conceptual knowledge in mathematical procedural

learning. Developmental Psychology, 27(5), 777-786.

Haapasalo, L. & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation.

Journal für Mathematikdidaktik, 21(2), 139-157.

Lipschutz, S. & Lipson, M.L. (2000). Probability. (2nd ed.). New York: McGraw-Hill.

Piaget, J. (1985). The equilibrium of cognitive structures. Cambridge, MA: Harvard University Press.

Rittle-Johnson, B., Siegler, R.S., & Alibali, M.W. (2001). Developing conceptual understanding and

procedural skill in mathematics: an iterative process. Journal of Educational Psychology, 93(2),

346-362.

Tossavainen, T. (2006). Conceptualising procedural knowledge of mathematics – or the other way

around. Retrieved February 26, 2007, from website:

http://www.distans.hkr.se/rikskonf/Grupp%206/

Tossavainen.pdf





Counter-Examples and Paradoxes in Teaching Mathematical Statistics: A Case Study

Farida Kachapova Auckland University of Technology

<[email protected]>

Murray Black Auckland University of Technology

<[email protected]>

Sergiy Klymchuk Auckland University of Technology

<[email protected]>

Ilias Kachapov < [email protected]>

Introduction and Framework Counter-examples are a powerful and effective tool for scientists, researchers and

practitioners. They are good indicators showing that a suggested hypothesis or a chosen

direction of research is wrong. Before trying to prove a conjecture or a hypothesis it is

often worth to look for a possible counter-example. It can save lots of time and effort.

Creating examples and counter-examples is neither algorithmic nor procedural and requires

advanced thinking which is not often taught at school (Selden & Selden, 1998; Tall, 1991;

Tall, et al. 2001). Many students are used to concentrate on techniques, manipulations,

familiar procedures and do not pay much attention to concepts, conditions of theorems and

rules, reasoning and justifications. As Seldens argue, coming up with examples requires

different cognitive skills from carrying out algorithms – one needs to look at mathematical

objects in terms of their properties. To be asked for an example can be disconcerting.

Students have no prelearned algorithms to show the ‘correct way’ (Selden & Selden,

1998).

2

There are several publications on using counter-examples in teaching/learning of

mathematics, in particular calculus (Gelbaum and Olmstead, 1964; Peled & Zaslavski,

1997; Zaslavski & Ron, 1998; Bermudez, 2004; Gruenwald & Klymchuk, 2003;

Klymchuk, 2004 & 2005). There are three well-known books on counter-examples in

statistics at an advanced level (Stoyanov, 1997; Romano, 1986; Wise & Hall, 1993). But

we could not find any publication on using counter-examples in teaching/learning of first-

year probability and statistics. So we decided to apply counter-examples along with

paradoxes as a pedagogical strategy in our first-year probability course.

The main objective of the study was to check our assumptions on how effective the

usage of counter-examples is for deeper conceptual understanding, eliminating students’

misconceptions and developing creative learning environment in teaching/learning of first-

year probability course.

In this study, practice was selected as the basis for the research framework and, it was

decided ‘to follow conventional wisdom as understood by the people who are stakeholders

in the practice’ (Zevenbergen & Begg, 1999). The theoretical framework was based on

Piaget’s notion of cognitive conflict (Piaget, 1985). Some studies in mathematics education

at secondary level (Swan, 1993; Irwin, 1997) found conflict to be more effective than

direct instruction. ‘Provoking cognitive conflict to help students understand areas of

mathematics is often recommended’ (Irwin, 1997). Swedosh and Clark (1997) used

conflict in their intervention method to help undergraduate students to eliminate their

misconceptions. ‘The method essentially involved showing examples for which the

misconception could be seen to lead to a ridiculous conclusion, and, having established a

conflict in the minds of the students, the correct concept was taught’ (Swedosh and Clark,

1997). Another study by (Horiguchi & Hirashima, 2001) used a similar approach in

3

creating discovery learning environment in their mechanics classes. They showed counter-

examples to their students and considered them as a chance to learn from mistakes. They

claim that for counter-examples to be effective they ‘must be recognized to be meaningful

and acceptable and must be suggestive, to lead a learner to correct understanding’

(Horiguchi & Hirashima, 2001). Mason and Watson (2001) used a method of so-called

boundary examples, which suggested creating by students examples to correct statements,

theorems, techniques, and questions that satisfied their conditions. ‘When students come to

apply a theorem or technique, they often fail to check that the conditions for applying it are

satisfied. We conjecture that this is usually because they simply do not think of it, and this

is because they are not fluent in using appropriate terms, notations, properties, or do not

recognise the role of such conditions’ (Mason and Watson, 2001). In our study, often not

the lecturer but the students were asked to create and show counter-examples to incorrect

statements, so the students themselves established a conflict in their minds. The students

were actively involved in creative discovery learning that stimulated development of their

advanced statistical thinking.

The Study

The students from a first-year course ‘Probability Theory and Applications’ were given

mathematical statements and asked to create counter-examples to disprove these

statements. They had enough knowledge to do that. However, for most of the students this

kind of activity was absolutely new, very challenging and even created psychological

discomfort and conflict for a number of reasons. In the beginning some of the students

could not see the difference between “proving” that the statement is correct by an example

and disproving it by an example. It agrees with the following observation from Selden &

Selden (1998): ‘Students quite often fail to see a single counter-example as disproving a

4

conjecture. This can happen when a counter-example is perceived as “the only” one that

exists, rather than being seen as generic’. To illustrate the idea of disproving by a counter-

example it might be helpful to use non-mathematical examples first. For instance, it might

be discussed with students: What does it take to disprove the statements like ‘all

Scandinavians are blond’ or ‘there are no numbers such that when they are spelled they

contain the letter "a". Apart from the activity on using counter-examples the students were

also given some paradoxes and were asked to explain them.

In our study we did not use ‘pathological’ cases. All exercises given to the students

were within their knowledge and often were related to their common misconceptions.

Below are examples of the incorrect statements to be disproved by counter-examples

and the paradoxes to be explained that were discussed with the students.

Counter-Examples

Use counter-examples to disprove the following incorrect statements.

1) Pairwise independence of events implies their independence.

2) a) If events A and B are independent, then they are conditionally independent.

b) If events A and B are conditionally independent, then they are independent.

3) Uncorrelated random variables are independent.

a) Consider the case of discrete random variables.

b) Consider the case of continuous random variables.

4) Pairwise independence of random variables implies their mutual independence.

Paradoxes

We consider the following problems as paradoxes because the correct answer to each

of them contradicts intuition.

5

Galton’s paradox. (Grimmett & Stirzaker, 2004, p. 14).

You flip three fair coins. At least two results are alike (the same). There is 50-50

chance that the third one is a head or a tail. Therefore the probability that all three results

are alike equals 0.5. Do you agree?

Simpson’s paradox. (Grimmett & Stirzaker, 2004, p. 19).

A doctor has performed clinical trials to determine the relative efficacies of two drugs,

with the following results:

Table 1

Results of Drug Treatment

Women Men

Drug 1 Drug 2 Drug 1 Drug 2

Success 200 10 19 1000

Failure 1800 190 1 1000

Total 2000 200 20 2000

The success rate of Drug 1 is 219/2020 ≈ 0.108 and of Drug 2 is 1010/2200 ≈ 0.459, so

the overall success rate is greater for Drug 2.

Among women the success rates are:

200/2000 = 0.1 for Drug 1 and 10/200 = 0.05 for Drug 2.

Among men the success rates are:

19/20 = 0.95 for Drug 1 and 1000/2000 = 0.5 for Drug 2.

So the success rates are greater for Drug 1 when the proportions are calculated for men

and women separately.

Which drug is better?

6

Monty Hall paradox. (Grimmett & Stirzaker, 2004, p. 12).

Suppose you are in a game show, and you are given the choice of three doors of which

one contains a prize. The other two contain gag gifts like a goat or a donkey. You pick a

door, say, No. 1. The host (who knows what behind the doors) opens door 3, which has a

donkey. He then says to you, “Do you want to pick door 2?”. Is it to your advantage to

switch your choice? That is, will your probability of winning increase if you switch to door

2?

The intuition of many students tells them that switching the door does not change the

probability of winning. Actually this probability increases from 3

1 to 3

2 .

Prisoners’ paradox. (Grimmett & Stirzaker, 2004, p. 11). There are three prisoners, A,

B, and C. The warden tells them that two of them will be released and one will be

executed. But he is not permitted to reveal to any prisoner the fate of that prisoner.

A asks the warden to tell him the name of one of his cohort who will be released. The

warden obliges and says, “B will be released.” Assume that the warden tells the truth.

a) What are A’s and C’s respective probabilities of dying now?

b) If A could switch fates with C now, should he?

This paradox is similar to Monty Hall paradox. Contrary to what intuition tells us, the

conditional probabilities of dying are different for A and C (3

1 and 3

2 respectively).

St Petersburg’s paradox. (Grimmett & Stirzaker, 2004, p. 55).

In a game of chance, a player pays a fixed fee to enter, and then a fair coin is tossed

repeatedly until a head appears ending the game. If the first head appears after n tosses,

then the player gets $2 n.

a) What is the expected win of a player?

7

b) What is the “fair” entrance fee?

The answer for both questions is ∞. It seems that any high fee is worth paying to enter

this game, which contradicts common sense.

The Questionnaire

After several weeks of using such exercises in class the students were given the

following questionnaire to investigate their attitudes towards the usage of counter-

examples and paradoxes in learning/teaching.

Question 1.

Do you find counter-examples and paradoxes useful for understanding this course?

a) Yes Please give the reasons:

b) No Please give the reasons:

Question 2.

Do you feel confident using counter-examples and paradoxes?



Question 3.

Do you find this strategy effective?



Question 4.

Would you like this kind of activity to be a part of assessment?



8

Findings from the Questionnaire

The statistics from the questionnaire are presented in the following table.

Table 2

Summary of Findings from the Questionnaire

Number of Students

Question 1

Useful?

Question 2

Confident?

Question 3

Effective?

Question 4

Assessment?

Yes No Yes No Yes No Yes No

11

100%

10 1

91% 9%

7 4

64% 36%

11 0

100% 0%

4 7

36% 64%

The majority of the students found counter-examples and paradoxes useful for

understanding the course. The typical comments from those students were as follows:

- they are both entertaining and informative;

- they are helpful because we can look back at them when we do assignments;

- we go though reasoning of counter-examples and paradoxes that helps

understanding the course.

About 2/3 of the students (64%) felt confident using counter-examples and 36% did

not. The ones who answered ‘no’ to the question about confidence provided the following

comments:

- counter-examples are difficult;

- sometimes they are confusing;

- I need more practice with them.

9

All surveyed students found the method of counter-examples effective and provided

the following typical comments:

- it improves my understanding of probability and random variables;

- it builds my logical skills;

- it strengthens my thinking ability.

About 2/3 of the students (64%) did not like the idea of counter-examples being a part

of assessment. In their comments they wrote that they could cope only with simple

counter-examples in assessment or with counter-example problems only in home

assignments but not in class tests. To some extent this last result contradicts the responses

to questions 2 and 3, where many students indicated that they felt confident using counter-

examples and that they considered the method effective.

Conclusions and Recommendations

The statistical results of this study show positive attitudes of the students towards using

paradoxes and counter-examples as a pedagogical strategy in a first-year course in

probability and random variables. All students surveyed stated that the pedagogical

strategy was effective. The majority of the students (91%) stated that the strategy was

useful for understanding the course. Many students commented that this method helped

them improve their logical skills and critical thinking and made the learning environment

more creative and entertaining. Though most of the surveyed students did not encounter

counter-examples in past and often found them challenging, they also found them useful

and effective and wanted to practise more with such problems.

As with any other case study the question is: to which extent can the results of the

study be generalised? The question remains regardless of the number of students surveyed

in the study. It doesn’t matter whether there are 11 students or 20 students or 50 students in

10

a class – there is only one lecturer and one learning environment and the number of

students surveyed is drop in the ocean compared to the number of students in the world

studying the first-year university probability course. In addition in this particular class

many students were in year 2 and 3 of their studies, therefore they had a better

mathematical background than typical year 1 students. It makes the study a bit biased. So

the results of the study can be treated as an invitation for colleagues to try the suggested

strategy with their own students and see how it works with them. It definitely worked for

our students.

As the first step in introducing counter-examples the authors recommend that a lecturer

provides a paradox or a counter-example and asks the students to explain or justify it. Next

the students can be asked to create their own counter-examples for a given incorrect

statement. And finally, the lecturer can ask the students to decide whether a given

mathematical statement is correct, so the students have to come up with a proof to show

that the statement is true, or with a counter-example to show that the statement is wrong. In

a one-semester course we tried to lead the students through these three steps with a certain

amount of success. But we observed that many students needed a lot more practice to

succeed and feel more confident in this area.

Further Study

We would like to extend the study to measure the effectiveness of this pedagogical

strategy on the students’ exam performance on the questions that require good

understanding of concepts, not just manipulations and techniques. We plan to compare the

performance of two groups of students with similar backgrounds. In one group we will

extensively use counter-examples and paradoxes, with the other group being the control

11

group. Then we will use statistical methods to establish whether the difference is

significant or not.

References

Bermudez, C.G. (2004). Counterexamples in calculus teaching. Paper presented at the 10th International

Congress on Mathematics Education (ICME-10). Copenhagen, Denmark.

Gelbaum, B.R. & Olmstead, J.M.H. (1964). Counterexamples in Analysis. San Francisco: Holden-Day.

Grimmett, G. & Stirzaker, D. (2004). Probability and Random Processes. (3rd ed.). New York: Oxford

University Press.

Gruenwald, N. & Klymchuk, S. (2003). Using counter-examples in teaching calculus. The New Zealand

Mathematics Magazine. 40(2), 33-41.

Horiguchi, T. & Hirashima, T. (2001). The role of counterexamples in discovery learning environment:

Awareness of the chance for learning. Proceedings of the 1st International Workshop on Chance

Discovery (pp. 5-10). Matsue, Japan.

Irwin, K. (1997). What conflicts help students learn about decimals? Proceedings of the International

Conference of Mathematics Education Research Group of Australasia (pp. 247-254). Rotorua, New

Zealand.

Klymchuk, S. (2005). Counter-examples in teaching/learning of calculus: Students’ performance. The New

Zealand Mathematics Magazine. 42(1), 31-38.

Klymchuk, S. (2004). Counter-examples in calculus. New Zealand: Maths Press.

Mason, J. & Watson, A. (2001). Getting students to create boundary examples. MSOR Connections, 1(1),

9-11.

Peled, I. & Zaslavsky, O. (1997). Counter-examples that (only) prove and counter-examples that (also)

explain. Focus on Learning Problems in Mathematics. 19(3), 49-61.

Piaget, J. (1985). The Equilibrium of Cognitive Structures. Cambridge, MA: Harvard University Press.

Romano, J.P. (1993). Counterexamples in probability and statistics. Chapman & Hall/CRC.

Selden, A. & Selden, J. (1998). The role of examples in learning mathematics. The Mathematical Association

of America Online. Retrieved February 2, 2007, from website: www.maa.org/t_and l/sampler/rs_5.html

Stoyanov, J.M. (1997). Counterexamples in probability. (2nd ed.) England: Wiley.

12

Swan, M. (1993). Becoming numerate: Developing conceptual structures. In Willis (Ed), Being numerate:

What counts? (pp. 44-71). Hawthorne VIC: Australian Council for Educational Research.

Swedosh, P. & Clark, J. (1997). Mathematical misconceptions – can we eliminate them? Proceedings of the

International Conference of Mathematics Education Research Group of Australasia (pp. 492-499).

Rotorua, New Zealand.

Tall, D. (1991). The psychology of advanced mathematical thinking. In Tall (Ed), Advanced Mathematical

Thinking (pp. 3-21). Kluwer: Dordrecht.

Tall, D., Gray, E., Bin Ali, M., Crowley, L., DeMarois, P., McGowen, M., et al. (2001). Symbols and the

bifurcation between procedural and conceptual thinking. Canadian Journal of Science, Mathematics and

Technology Education, 1, 81–104.

Wise, J.L. & Hall, E.B. (1986). Counterexamples in probability and real analysis. New York: Oxford

University Press.

Zaslavsky, O. & Ron, G. (1998). Students’ understanding of the role of counter-examples. Proceedings of the

22nd Conference of the International Group for the Psychology of Mathematics Education. 1, 225-232.

Stellenbosch, South Africa.

Zevenbergen, R. & Begg, A. (1999). Theoretical framework in educational research. In Coll, R.K., et al

(Eds) SAME Papers (pp. 170-185). New Zealand.






Planning a route for a performing trip of a music team: A group project in 4th grade mathematics

Grażyna Pawłowska,

Szkoła Podstawowa Nr1 im. Gustawa Morcinka w Warszawie,Poland e-mail: [email protected]

Overview

I have used a project-based approach in two 4th grade classes at a grammar school. The project entailed planning the most efficient route for a music team. The group work method used was presented in NIM and Forum of Education (Pawlowska, 2001). This was the first experience of this kind for for my students, although they had already worked with the collaborative group method in the context of an integrated approach.

I wanted all pupils to understand precisely what is expected of them in similar situations, to be clear about the principles of the cooperative work and to find the task doable as a whole. I wanted the children – also those working slower or requiring help of a teacher or classmates – to experience satisfaction and success. In order not to discourage able pupils with too simple problems, each set of them contained separate, more difficult problems designed for the members of the team and its leader. In order to accommodate children, who after initial teaching still have problems with reading, the instruction was written in larger font and tasks were discussed in front of the blackboard. The aim of the class was to use counting by memory in the practical context of calculating the length of a car route on the basis of a map of Poland. The skill of using the map is very useful in life and I wanted children to also learn map skills. Observing them, I think that the problem posed in front of them was interesting, convincing and speaking to childrens‘ imagination.

As usual with group work, pupils had to take on certain roles and perform actvities connected with them. This simple approach engages pupils imagination and reinforces their emotional attitude toward their work. During the time of the proposed game, every group became a musical team preparing a performance tour. The route commenced in Frombork through Ostróda to Mława to Płock. There were three concerts on each day in each of the cities. In order to make the problem more real I suggested that each of the music teams played well but was not famous enought to have a manager. Consequently, each member of the team had to play and also plan the route of the travel. The work had to be divided: each member of the team was supposed to plan the route on a different interval of the trip and calculate its length.






The computational part of the work consisted of adding lengths of individual pieces read off the map. After finishing its calculations, each group had to glue coordinated pieces of the map onto a poster board indicating each of the travel intervals. The groups also included tables with the intervals and their calculated length, as well as the list of villages and towns passed along the road. The presentation was designed to enhance and diversify the work of the teams. Goals and Objectives The general aims of a class period were to: --Develop the competency of applying mental calculus to daily life problem solving, and in particular to problems encountered while using car road maps. -- Develop the competency to create a simple outline showing the problem situation. -- Develop the capacity for collaborative work. The proposed design of the class will also impact: --Competency of reading with understanding. --Competency of using tables as means for organization and presentation of information. The proposed problem is also useful for the realization of wider, interdisciplinary educational aims such as: --Familiarization with geographic regions contained in the road map. --Familiarization with the history and development of the passing communities. --Familiarlization with the scale of distance in Poland. --Introduction to the art of writing ads and announcements. --Organization of the musical program of the team. Among the detailed outcomes, students will be able to: -Plan the route of the trip from given initial site to the final site using the car map. -Make a sketch of the route. -Read the distances between different cities. -Complete the form describing the route in accordance with the template; -Do simple mental calculations applied to practical problem. -notice the relationships between different details of the map. -Synthisize details and make a useful chart or table.






Materials The design involves the following didactical materials in addition to a large map of Poland:

a) three different fragment maps,limited by the horizontal lines passing through the passing and final destinations (Fig.1)

b) instructions for the leader of the team c) instructions for the team’s members d) three forms for each of the members of the

team e) one form to collect information for the

leader of the team e) a handout with information about the cities

passed along the route f) large piece of the poster board to integrate

all the elements of the work.

Method

The teacher informs the class that each of the teams is responsible for plotting the performance route for a music team, scheduled to play three concerts in three different cities on one day. In the morning the teams leave Frombork where they stayed overnight and have to go to Ostróda, where they have the first concert at 10:00am in the gymnasium. The trip to Mława follows, where they have to play at 14:00 (2 p.m.) in the Cultural Center. Finally, for the evening they have engagement for 19:00 (7 p.m.) in the discoteque in Płock.

While presenting the outline of the game/activity, the teacher shows the essential cities on the map of Poland and briefly describes their history, development, significance for the region. Next, she shows the set of materials to be received by each of the groups. She discusses the distribution of tasks for the team. The teacher reads the instructions aloud and coordinates the forms with the map. Finally, she explains the sequential activities using the demonstration map.

Each member of the team is responsible for one interval of the trip and must design the route and compute its length. The activities are:

Frombork , miasto w województwie warmi_sko -mazurskim, nad Zalewem Wi_lanym, na skraju Równiny Warmi_skiej. 2,7 tys. mieszka_ców (2000). Port rybacki, przysta_ _eglugi pasa_erskiej, regularne po__czenia z Krynic_ Morsk_. Spe_nia funkcje miejscowo_ci wypocz ynkowej. Dobrze rozwini_te zaplecze noclegowe.

Historia Prawa miejskie nadano w 1310 osadzie, która rozwin__a si_ wokó_ warowni biskupów warmi_skich. Od 1466 w granicach Polski. Miejsce pracy i _mierci M. Kopernika, który mieszka_ we Fromborku w lat ach 1512 -1516 i 1522 -1543. W XVI i XVII w. rozwój handlu, powstanie portu i huty szk_a. W okresie rozbiorów pod panowaniem pruskim. W 1945, znacznie zniszczony, powróci_ do Polski.

Zamek we Fromborku, w_a_ciwie zespó_ katedralny wraz z umocnieniami obronnymi, za_o_ony na miejscu staropruskiego grodu. Umiejscowiony na wysokim wzgórzu. Ok. 1278 kapitu_a warmi_ska przenios_a tu swoj_ siedzib_ ze zniszczonego przez Prusów Braniewa. Wówczas wybudowano tu pierwszy ko_ció_, którego wygl_d nie jest znany, przypuszczalnie by_ on otoczony umocnieniami z drewna i ziemi. Istniej_cy dzisiaj ko_ció_ katedralny zbudowano w latach 1329 -1388. Wtedy te_ prawdopodobnie zacz_to wznoszenie murowanych umo cnie_

wzgórza. W latach 1510 -1543 z czteroletni_ przerw_ pracowa_ tu i mieszka_ Miko_aj Kopernik, b_d_cy kanonikiem warmi_skim. W 1626 wojska szwedzkie pod wodz_ Gustawa Adolfa zdoby_y i ograbi_y miasto i katedr_ ze wszystkich skarbów, biblioteka katedraln a i zbiory Kopernika zosta_y wywiezione do Szwecji.

Rys. 1. Collection of team’s materials

Szkic trasy

Kolejne miasta na trasie

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Fromborka

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Planowana godzina wyjazdu

z Fromborka

(godzina:minuty)

Czas przejazdu trasy przy

pr_dko_ci 60 km na godzin_

(godziny:minuty)

Planowana godzina

przyjazdu do Ostródy –

oblicz ! (godzina:minuty)

Potrzebna ilo__ paliwa przy

zu_yciu 10 litrów na 100 km

(litry)

Warto__ potrzebnego

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przejazd Frombork -Ostróda 7:00 I.

koncert w Ostródzie 10:00 10:30

przejazd: Ostróda -M_awa 11:00 II.

koncert w M_awie 14:00 14:30

przejazd: M_awa -P_ock 16:00 III>

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Matematyka GRA -IV, ©Paw_owscy`2002 KAPELA: Praca w grupach (dodawanie)

Instrukcja dla zespo_u i jego lidera.

Jeste_cie zespo_em muzycznym, wybieraj_ cym si_ na tras_ koncertow_. W sobot_ macie

zagra_ a_ trzy koncerty, wi_c musicie dok_adnie to zaplanowa_. Z samego rana musicie

pojecha_ swoim busem z Fromborka do Ostródy, gdzie przed po_udniem , o godzinie 10:00

gracie w miejscowym gimnazjum. Potem czeka was podró_ do M_awy , gdzie o godzinie

14:00 gracie w Domu Kultury. Na wieczór macie umówiony koncert w dyskotece w

P_ocku o godzinie 19:00 .

Wasz zespó_ gra bardzo _adnie, ale jeszcze nie zdobyli_cie s_awy, wi_c nie macie

mena_era. Ka_dy z muzyków musi nie tylko gra_, ale i planowa_ tras_ – tak jak pilot

rajdowy w rajdzie samochodowym . Lider zespo_u nadzoruje prac_ wszystkich i pomaga

tym, którzy sobie nie radz_.

Zadania dla lidera (zadania na szarym tle nie s_ obowi_zkowe) :

1. Sprawd_ , czy koledzy prawid_ow o policzyli d_ugo_ci tras.

2. Wpis z d_ugo_ci tras do tabelki zbiorczej (pola I. II. i III.)

3. Policz ca_kowit_ d_ugo__ sobotniej trasy (pole |IV. RAZEM: ).

4. Pomó_ kolegom zrozumie_ i wykona_ zadania dodatkowe.

5. Wpis z wyniki zada_ dodatkowych do tabelki zbiorczej.

6. Oblicz potrzebn_ ilo__ paliwa i jego koszt.

7. Rozplan uj rozmieszczenie mapek i tabelek na kartonie, i pomó _ kolegom je naklei_.

8. Nakle jcie razem karteczki z plakatami informuj_cymi o koncertach.

Matematyka GRA -IV, ©Paw_owscy`2002 KAPELA: Praca w grupach (dodawanie)

Instrukcja dla pilota trasy (zadania na szarym tle nie s_ obowi_zkowe):

1. Odszukaj na mapce swoje miast o pocz_tkowe i ko_cowe , i zaznacz tras_ przejazdu.

2. Naszkicuj tras_ w tabelce .

3. Wpisz do tabelki nazwy co najmniej 4 kolejnych miast mijanych na Twoim

odcinku trasiy (pola nr 1a., 2a., 3a., ....) . Cz___ pól mo_e pozosta_ pusta.

4. W polu obok ka_dego miasta, wpisz d_ugo__ drogi od miasta poprzedniego (pola nr

1b., 2b., 3b., ...) . Nie zapom nij o mie_cie ko_cowym, które ju_ jest wpisane!

5. W polu „RAZEM” tabelki podsumuj d_ugo__ ca_ej trasy.

6. Je_li potrafisz i masz czas, to wype_nij kolejne pola „Tabel i zada_ dodatkowych”.

Podpowied_: pr_dko__ 60 km na godzin_ to to samo co 1 km na minut_

7. Wytnij i naklej na karton swoj_ mapk_, dopasowuj_c j_ do mapek kolegów.

8. Naklej na karton sw oj_ tabel_ .

9. Wykonaj projekt plakatu informuj_cego o koncercie waszego zespo_u w mie_cie, do

którego prowadzi Twoja trasa. Podaj miejsce, dat_ i godzin_ .

Figure 1. Fragment of a map obtained by one of the groups/






-find the initial and final town on the map. - draw the route on the map. - make a draft of the route on the appropriate form. -wirte in at least 4 names of passing towns. -write in the distances between them. -calculate the sum of the distances for the total trip.

Team leaders have their own tasks. They are responsible for the work of the members of

the teams: they should check whether the members do their work correctly; offer suggestions; and assist where needed. They organize common activities including: naming the team; planning the composition of maps and forms on the final poster presentation; and creating the announcements. The leader also supervises filling out the final form, plans the course for the day and sums up the length of the whole route. The forms for the particular intervals of the route contain additional tasks with instructions and needed information. Understanding these instructions is part of the overall task, hence it’s not discussed by the teacher.

When all the members of the team end their activities, the team aprpoaches the construction of the final product on the poster board. Each team member cuts out their map and glues them to the board making sure all individual maps fit together. The team leader supervises the gluing and adds the handout about one of the concerts. The team continues to work on the announcements of their concerts. (The teacher demonstrated different possible arrangements of the final product using the magnetic pieces.) The lesson and project can be accomplished in about 45 minutes. After checking the group submissions it is important that the teacher display each groups work (Figure 2.).

Figure 2. Example of the work design






Outcomes The described design was realized in two 4th grades with19 and 18 pupils respectively. Pupils were informed about upcoming activity and about the car maps and their utilization one class in advance; the appointed leaders were good in mathematics. The class experienced real joy and satisfaction in both classes. I was available to help children but only after each question was directed back to the team leaders. I was reminded children about the necessity of mutual help, about the final product depending on the collaboration of every one in the team. I looked for students who needed help the most and was helping them in more difficult situations. The real difficulty was reading off real distances from the map. I had also encouraged children to organize their work better.

Figure 3. Working Group. In the background there is a blackboard with attachad materials..






Assessment and evaluation of knowledge I used the following rubric to assess pupils‘ work - the member of the team could get 5 points for basic activities, which - gave 2 pts for starting all activities - gave 2 pts for correct reading off the distances from the map - gave 1 point for correct sum of all the distances. Captain could also get 5 points for basic activities: -2 points for starting -1 point for correct writing in the data into the table; - 1 point for finding sum -1 point for good leadership of the team (averaging 4 points by team members). Both captain and the team members could get max 3 points for additional activities.

Figure 4. One of the work results.






The distribution of points for the tasks is presented Figure 5. The average was 4.05. More than 75% of students received 4 points which means that basic tasks were done correctly or at most with small corrections. The additionaL task was taken by 4 pupils in the class Ivd, including two team captains.

Histogram of teams‘ average results (Figure 6) shows that all teams, except one, averaged above 3.5 pts, and four out of nine teams averaged more than 4.5 pts. These data confirm the subjective

assessment that the lesson was able to motivate collaboration in the majority of the teams. Corrected and graded products were discussed a couple of days later. The classroom where the posters were produced was changed into a gallery of finished work. The exhibition evoked deep interest amongst the children. Every child received the grade for individual work in a team in agreement with the accepted criteria. Besides two best teams in each class received the special prize to be divided by the team members. Assessment of effectiveness

The value of the project based learning is in its effectiveness. At the same time it is difficult to assess objectively because of the individual characteristics of the pupils. One measure can be the sum of all activities undertaken by pupils. All activities are counted independently of the correctness of the result. There were 6 activities in the described lessons which can be objectively registered on the basis of the documentation. Those activities together with the percentage of activities undertaken is presented in Table 1. Activity 3 was

0.0

1.0

2.0

3.0

4.0

5.0

6.0

b1.1

b1.3

b2.1

b2.3

b3.1

b3.3

b4.1

b4.3

b5.1

b5.k

d1.2

d2.1

d2.3

d3.1

d3.3

d4.1

d4.3

Figure 5. Distributionof points

0.0

1.0

2.0

3.0

4.0

5.0

6.0

b1 b2 b3 b4 b5 d1 d2 d3 d4

Figure 6. Team Averages






not required accounting for the smaller percentage. The presented data suggest that all or almost all students intensively participated in the classroom work. This confirms my subjective assessment. Summary and discussion

The sequence of activities in grade 4 as well as the effectiveness demonstrates that the group work method worked in this case. The lesson achieved it’s outlined goals. The coefficient of difficulty was 0.8 which means that the problems were well chosen for the class. Even children from the class who usually work slower and are a less able team presented the final effect of their work.

An additional problem was done by few children so the coefficient of difficulty is not reliable as there was not much time for this problem. Pupils who had undertaken the problem demonstrated high levels of competency and work organization. More difficult problems are needed so that more able students experience have satisfaction with the work.

The class design can be easily adapted for use in mathematics, biology, language, history and music. Use of the music team performance task including characteristics of the geography, history, legend and curiosities of passing cities) worked in the case of mathematics. At the same time the children had more time for the actual task of mathematical computation of the best route. Expanding the range of subjects to be included would necessitate using additional class periods. Although for this project I didn’t plan collaboration with other teachers I think it would be interesting and useful to realize multidisciplinary project. Talking with my colleagues, I think such a collaboration is possible. References

1. G. Pawłowska. Zamawiamy sadzonki na klomby. Praca w grupach, klasa V. Nauczyciele i

Matematyka 35 (2000) 10. 2. G. Pawłowska. Płacimy podatki. Praca w grupach na temat procentów, klasa V. Nauczyciele

i Matematyka 36 (2000) 21. 3. G. Pawłowska. Witraż: Praca w grupach na przykładzie lekcji w geometrii w klasie piątej.

Nauczyciele i Matematyka 40 (2001) 21. 4. G. Pawłowska. Planujemy zakupy na wycieczkę: Praca w grupach, klasa IV. Część I. Forum

Edukacji 2/3 (2001) 70.






5. G. Pawłowska. Planujemy zakupy na wycieczkę: Praca w grupach, klasa IV. Część II. Forum Edukacji 4 (2001) 72.





Mathematical Competencies in the nascent state“on the basis of the Educational Project „In the Middle of the Way…“ Elżbieta Jaworska, Bożena Makulska-Dąbkowska, Elżbieta Ostawiczuk, Andzrej Wawrzyniak, Andrzej Werner1 Mathematics Teacher-Coaches, Department of Education, Office of the City Warszawa, Poland. A person taken by the new idea is a weirdo

until the idea is succesful. Mark Twain Education is the main determinant of the cultural and economical development of a society. Hence the strategic role of education should be high on the list of government’s priorities. Already during the seventies of the twentieth century, countries of the European Economical Community (OPEC) and now EU started to attach increasing importance to the formulation of the correct educational policy2. The educational priorities of EU contain increasing the quality of education at every level, development of vocational education, mutual recognition professional qualification by the member states, and the development of higher education. European pedagogical categories contain multidisciplinary approaches, equalization of educational opportunities, raising the quality through the reform of curriculum, stimulation of the creative, innovative approaches by the teachers. The idea of the vision is to raise a European as a citizen of the World.

1 Originally the authors of the project „In the Midde of the Way…“ was created by the teacher-coaches of mathematics – Andrzej Wawrzyniak and Andrzej Werner and full of enthusiasm teachers of mathematics, Elżbieta Jaworska and Elżbieta Ostaficzuk. In the Fall 2004, the team was augmented by Bożenę Makulską-Dąbrowską. Since then all five teachers constituted the group of teacher-coaches. At present Elżbieta Ostaficzuk works as aa consultant to the Małopolska Centrum of Teacher Development. The project „In the Middle of the Way …“ is the object of our formative concerns and interventions. 2 Tadeusiewicz, G. – Education in Europe PWN, Warszawa-Łodź, 1977.

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Polish society, although it continuously learns, is badly educated.3 Most probably it can’t be otherwise because independent Poland has been left with the results of the absence of concern for education from the previous years. Expenditures for education kept steadily below 4% of NGP, the level beneath which, according to UNESCO, one encounters the „educational death“ of the society, thus creating deep depression in Polish education. For the first time this low level was crossed in 1990. Unfortunately again, according to the World Bank. Poland had systematically lowered its educational spending towards 2.9% of NGP. The reasons for of the low conditions of teaching mathematics, the Queen of Science, are multidimensional. First of all since 1986, mathematics is not required for the final graduation exam from high school4. Many years of carelessness in the education of teachers and overloading the content of the curriculum causes the deepening of the crisis. This state of affairs requires a systemic approach of improvement. From the beginning of the nineties that is from the beginning of the deconstruction of previous reality, the difficult process of the transformation of the socio/economical system, the Polish school is in the state of continuous „tectonic movement“. The reform of education had entered the schools with big effort; its beginning are well taken by R. Nowakowski5: „Aims and expectations deduced from superficially understood meaning of life, understood as free movement around the garbage hill of the world had deformed the conception of education. As a result, teaching of mathematics not only doesn’t shape the rational man being able to distinguish the objective world from the illusion and being able to find a way of understanding phenomena in the broadest sense of word, but also is not able to reach assumed mistaken goals.“ These critcal words convey not only the factual state of affairs but also the emotions which are woken up by the issue of education reform. There are two worlds in the school millieu: the world of child/pupil and the world of the adult teacher. All have good intentions and enthusiasm. Children are convinced that they will be at least as succesful as their parents if not more. Adult teachers are convinced they will never reproduce the behavior of that „terrible mathematics teacher“ they had themselves. How to help them in

3 Kwieciński Z. – Education vis-à-vis hopes and dangers of the contemporary ssociety- presentation at the 3rd Polish Pedagogical Congress, Poznań, 21-13 Sept. 1998. 4 The exam might be restored by 2010, according to the last Minister of Education, Roman Giertych. 5 Nowakowski, R. – Dokad zmierzasz, edukacjo? O tendencjach I sytuacji nauczania matematyki (I nie tylko w niej). Wydawnictwo Naukowo Oswiatowe, Wrocław, 2005,p.5.

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their endeavor? Professor Zawadowski6 usually says:“Teach in agreement with medical principles – do not harm!“ In the contemporary reality of the school one needs to notice a pupil who doesn’t want to learn. It seems correct to say that the responsible for the education of teachers should accept that at present, pedagogical studies prepare the army (of teachers) for the war „that was“. That means that the teacher – constantly trained professionally- is in effect helpless in the classroom. Then he/she more often than not uses the freedom to do bad in the name of doing good7. One needs to notice the pupil who doesn’t want to learn in the present school reality; one needs to develop means and strategies of facilitating pupil’s interest, to impact the motivation. Convince the pupil that the basic goal of every individual is independence and self- assertion. The community of teachers is standing at present in front of such issues and attempts to solve them. The realization of the Project „In the Middle of the Way…“ became Warsaw’s laboratory of developing those two qualities of independence and self-assertion amongst teachers of mathematics, and therefore amongst their pupils. The idea of the test was born on the background of teachers‘ concern about the development of mathematical competencies by students of liceum and technicum of Warsaw. Mathematics teachers of Warsaw were carried by the creative, professional impulse - the authors of the test proposed the cycle of workshops to Warsaw teachers of mathematics whose aims were: - presenting the teachers with the advantages of didactic measurement in teachers systems of teaching; - preparing the content of the large scale test for students of 2nd grade of Warsaw high schools. The workshops discussed the issues of -three dimensions of the teaching content; -typology of written assignments; -statistical analysis of written assignments and results of testing; -graphical representation of the results, and percentiles; -interpretation and communication of the results. Before the problems for the test were chosen, a questionnaire was conducted 6 Prof. W. Zawadowski, known mathematician, founder of Stowarzyszenie Nauczycieli Polskich. 7 The teacher generally asseses the pupil negatively. Kwiatkowska „„Chaos i autonomia, czyli o wolności i przymusie w wychowaniu”, w listopadzie 2004, w Warszawie

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amongst the Warsaw teachers of mathematics concerning the conduct of mathematics education in their schools. The average number of hours of mathematics teaching per week was 8 hours in licea and 6 hours in technica. Using this information, the scope of material to be presented on the test was decided: numbers and sets, functions and their properties, linear, quadratic and trigonometric functions of acute angle, polynomials of higher degree, plane geometry. The tests „In the Middle of the Way…“ 2004, 2005, 2006, 2007 were designed to meet the Programatic Principles as well as the Examination Standards. The tasks with this content were grouped in three categories: FiW-functions and their properties, RiN – equations and Inequalities; GiT – geometry and Trigonometry. Problems are designed in the Basic level as well as in the Advanced level. In 2005, in order to enrich educational diagnosis, a multidimensional description of problem solving had been estanlished, where: A – is the analysis of the problem, that is using the mathematical language, understanding and creation of symbols, drawings, familiairty with mathematical terminology; M – choice of the method, that is meritoric correctness based on the familiarity with mathematical theory U-independence of learning; R – correctness of computations and transformations of equations. The level of mathematical competency was investigated with the help of the difficulty coefficient for the given competency in a diagnosed cohort of students: - larger than 0.75 – mastery of the competence by the cohort; - between 0.30 – 0.75 – the whole cohort needs to review given competence - less than 0.30 – the cohort needs to learn the competency anew. Several cycles of the test had shown that important element of teachers‘ knowledge about student competencies is the information about student errors and the frequency of their occurrence. The errors were coded in the following manner: - d – absence of familiarity with the definition of the concept; - w - absence of knowledge of properties of the object, rules of operation, a theorem (that is the sentence in the form If…then… - p - errors in computations, graphs, rounding off; - k - errors of the type d,w or p. Succesive editions of the Test In the Middle of the Way… were conducted to obtain precise measurements and independence of the results, Each of the editions of the test were piloted with cohorts of 100 students. While checking

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the tests, teachers were giving points: 1 – the mathematical activity done correctly 0 – the mathematical activity done wrongly x – mathematical activity not undertaken. Correct non-standard solutions get the maximal number of points. In 2004 there was 9,000 students taking the test, whose analysis allowed the teachers to formulate extensive diagnosis of their classes. The value of the difficulty coefficient for different categories:

Taxonomic category B Understanding of the concept C Application of knowledge in similar situations

D Application of knowledge in Problem

situations.

Difficulty Coefficient 0.64

0.31

0.37

It turned out that our pupils manifest certain type of functional limitation. They can apply their knowledge in situations directly described like tests, but can’t if the instruction is more general and does not contain suggestions about the mathematical domain involved or the method of solving. Equally troubling are results in geometry and trigonometry. Independently of the school, student knowledge in this areas is sporadic. The coefficient of difficulty of plane geometry problems was p=0.38, what translates itself into: -50% pupils doesn’t know how to solve the problem; -25% can solve it partially and 25% - completely. The dsitribution of points obtained by students in this problem suggests that they don’t know basic geometrical facts, don’t know how to solve problems which do not have any numerical data. We suggest the cause to be the absence of coherence in the program of geometry and small number of hours devoted to plane geometry. The coefficient of difficulty of trigonometric problems was p=0.49. That translates into -34% of students could not react at all to the problem -33% solved partially; -33% students solved it correctly. We think that these results are due to the fact that trigonometrical content appears late, and therefore students can retain

6

the acquired knowledge. There was 7,000 students participating in the 2005 edition of the test In the Middle of the Way…The difficulty coefficient reached by the test had following numbers

Taxonomic category B Understanding of the concept C Application of knowledge in similar situations

D Application of knowledge in Problem

situations.

Difficulty Coefficient 0.47

0.22

0.57

The analysis of results allows the teacher to reflect upon teacher’s system of teaching. Of special interest is the analysis of pupils‘ work from the point of view of new competencies being introduced in schools at present, called anchoring competencies. The analysis of the change in the mastery of A M U R competencies was made throughout different ediitions of the test. The comparison of difficulty coefficients:

Fig. 2 Comparison of difficulty coefficients of the test In the Middle of the Way…

7

0,44

0,45

0,38

0,49

0,37

0,22

0,32

0,39

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

F i W R i N G i T Zestaw

podtesty

wspó?czynnik

?atw

oœ

ci

2004

2005

The analysis of the comparisons suggest that there is a development in pupils‘ mastery of anchoring competencies. The questionnaire given to the teachers whose students participated in the test inform that 61% of them have adapted their methods of teaching to needs of the classrooms. Work on the project In the Middle of the Way… initiated the process of activization of professional development of mathematics teachers in Warsaw. More than 85% of teachers expressed interest in further development of professional knowledge conencted with the assessment of student mastery of mathematics. The Project In the Middle of the Way… is recognized by those Warsaw teachers who recognize the need to improve one’s own quality of work; who treat teaching profession as a necessary profession in contemporary society, who give high priority to the issue of learning assessment. May 2007






The Process of Doing Mathematics By

Rony Gouraige I have been actively involved for the last year as a mentor to several students under the New

York City Louis Stokes Alliance for Minority Participation program. This experience forced me

to think about the best approach to teaching undergraduates with a limited knowledge of

mathematics how to actually do mathematics. I hope that my ideas on this matter may prove

useful to someone who is acting as a mentor. I also hope to encourage teachers to become

mentors by highlighting one of the less obvious benefits of mentoring.

I began with four students in the summer of 2006. By the spring of 2007, two of my students

had moved on to schools outside of New York City. My students started with a background that

included three semesters of calculus and a course in linear algebra. From the outset, I had two

goals in mind. First, I wanted to teach my students some mathematics that would be useful no

matter what careers they decided to pursue. Second, I wanted my students to gain the experience

of actually doing mathematics, which in my mind meant that they should do some original

research.






With these aims in mind, I had to first of all decide what subject to study. My background is in

non-commutative algebra. For a variety of reasons, I decided that this area was not appropriate

for an undergraduate research project. There seems to me to be at least two approaches that one

could adopt. Choose a subject that requires no specialized knowledge, but which is rich in open

and challenging problems that could be solved without resorting to any complex machinery. On

the other hand, one could spend some time teaching the students some specialized knowledge,

and then give them problems that could be solved with the techniques that they had learned,

perhaps supplemented by further theory according to the circumstances. I rejected the first

approach because in my mind it was inconsistent with the first goal I stated above. To pursue the

second approach, I finally decided to teach my students some number theory. Number theory

has a rich history, is an active research area, and, most importantly for my immediate aims, has

motivated the development of much of the abstract machinery which pervades modern

mathematics

I spent the summer of 2006 teaching my students elementary number theory, roughly the

equivalent of a junior-senior level undergraduate course in number theory. However, I adopted

the point of view that in teaching number theory, I would make maximal use of the basic

structures of abstract algebra: Groups, rings, fields, and vector spaces. This was consistent with






my first goal. After the initial period of teaching my students elementary arithmetic, I had to

now focus on my second goal, which was to get my students to do original research. Not being a

professional number theorist, I was not conversant with the open problems, even the elementary

ones. Frankly, I did not want to simply assign problems that were already well-known, on the

assumption that such problems were already receiving plenty of attention. I decided to bring to

bear my experience in non-commutative algebra, and look for problems in arithmetic that had

analogues in algebra. Having done some work in the theory of central simple algebras, it was

natural that I would look for the matrix analogues of results in elementary number theory. This

is hardly a novel approach, but it turned out to be remarkably fruitful. For example, one of my

students is currently investigating the analogies between Euler’s totient (also called phi) function

and row equivalence of matrices while another is looking at the analogues of Fermat’s sum of

two squares theorem over modular rings of integers.

How does one initiate a student into the process of doing research? The first thing that I had to

overcome was the natural tendency of my students to turn to a textbook for the answer to a

problem that they could not solve. More fundamentally, I wanted to encourage my students to

embrace the unknown as an opportunity for discovery. My constant refrain is that at some point,






no book, or paper, or person will have the answer to the question that one is asking. This fear of

the unknown is primarily a psychological barrier, and it takes some time to overcome.

My next goal was to inculcate the notion that the most difficult part of doing mathematics is not

in proving theorems, but rather in coming up with theorems that are worthy of being proved. For

the novice in mathematics, the notion of proof and the techniques of proof seem to be of

paramount importance. This is not entirely wrong, but it risks a fundamental misunderstanding

of the nature of mathematics. To focus on proofs is to take the theorems to be proved as given.

But in the practice of doing research, discovering the theorem to be proved comes before one can

even begin to undertake the search for a proof.

How does one make new discoveries? I have spoken to other mathematicians about this topic,

and it seems that the process of discovery is as individualized as one’s fingerprints. So whatever

I write about this topic is necessarily biased and reflective of my own temperament and abilities.

I advise my students to pursue analogies and to examine many examples. The analogies one

pursues will change as one’s knowledge increases. However, I have been influenced by my

advisor, Ravindra Kulkarni, to think in terms of three fundamental categories in mathematics:

Number, space, and symmetry. The key is to appreciate the subtle and symbiotic relationship






between these categories. This is not the place to elaborate on these themes, but the idea is to

pursue analogies that follow from these basic categories.

The analysis of examples is of fundamental importance, and underappreciated by students. How

often have you observed that a student can state a theorem and repeat its proof verbatim, but is

completely lost when asked what the theorem implies about a basic example? A good example

gives meaning to a theorem. Better yet, the examination of many examples is a fruitful way to

make new discoveries. Of course, the discoveries that one makes will reflect one’s talents. But

the examination of many examples is a critical step in the process of discovery, and ought not to

be bypassed. This is inductive reasoning. Students find it very difficult at first because it takes a

lot of hard work. I encourage my students to engage in this process, and to persevere even if it

doesn’t yield immediate results. The process of doing research involves hard work and tenacity

as well as creativity, and I want my students to understand and accept this.

In closing, I want to comment about some of the benefits of mentoring for the mentor. I teach at

a community college, and as a result, have a heavy teaching load. The free time that I have for

research is precious. Why should I devote any of that free time to mentoring students? Each

teacher will have to find their own answer to this question, but my experience has taught me that






mentoring can be personally rewarding and at the same time can further one’s own research.

The personal rewards are obvious, and need no elaboration from me. I think that it is the benefit

to one’s research that tends to be overlooked when one is considering whether or not to become a

mentor. If one is actively engaged in research, why take the time to mentor someone who is

unprepared to understand or contribute to the work that you are doing? I would argue that

mentoring such a student forces you to look very carefully at basic questions. These questions

have been the stimulus for original research for millennia. The time that one spends

contemplating such questions can only benefit one’s research program. This benefit may seem

tenuous at best, but I am convinced that if a researcher brings to bear his or her creativity in the

examination of such problems, then they will discover new avenues of investigation that they

might not otherwise pursue or even be aware of.

R. Gouraige

Bronx Community College

[email protected]






Reflections of a Teacher Educator.

Harrie Broekman, Freudenthal Institute for Math and Science Education, University of Utrecht, The Netherlands.

If we expect a lot of effect of teachers researching their own practice, we as teacher trainers

have not only ‘to teach as we preach’, but also ‘research our own practice’.

Shortly after writing about my reading of an interesting article and writing about it 1 for some

colleagues, I realised that as a participant in a project for Teacher Researchers2, I had to look

more carefully at my own teaching. Not only what I’m doing in reality, but also the main ideas

behind my activities had to be ‘researched’, or at least reflected on in a critical way.

The following is part of the ongoing reflection about “what tasks to use in my teacher training

courses, why to use them and how to use them”.

The main background ideas for selection of and work on tasks in teacher education.

1 See appendix 2 A EU sponsored project ‘Professional Development of Teacher Researchers, called the Krygowska project.






1. Mathematics is a human activity; a lot of doing and reflecting on the doing as well on the

background of this doing. The strategy ‘think, share and compare’ supports this.

2. In Realistic Mathematics Education both the horizontal mathematisation (real contexts

used3) as well as vertical mathematisation (processing within the mathematical system4)

are important.

3. Teachers’ stories are an important tool for building a strong fundament for making

teachers’ tacit (hidden and not conscious) knowledge explicit.

4. Reflection – private, but also with the support of others – is useful and maybe even

necessary for learning.

5. Implicitly is the importance of a questioning style, about which Vrunda Prabhu5 wrote:

“The questioning style also has the hope that the enquiring attitude m the regular

classroom discourse become a part of students way of learning and creating their own

mathematics”

6. As a mathematics teacher educator it is my task to investigate the most effective methods

of improving learning in my pre-service and in-service courses.

7. As a trainer/educator of teacher educators it is also my task to investigate teaching and

learning processes in general; the selection of classroom tasks included. 3 Transforming a problem field into a mathematical problem 4 Going ‘deeper’ in the mathematical structures etc 5 Vrunda Prabhu, Independence of learning, MTRJ Online. V2;N1.






An example used as a starter to ‘open the minds of participating pre-service and in-service

teachers for aspects of teaching, worthwhile to write about’.

A short article, written by a teacher, was given to a group of in-service teachers to comment, and

– more specific - to ‘look for’ the aspects of her experience the teacher wrote about in this article

and what aspects she didn’t mention. I expected a lively discussion about what is interesting to

write about and what is interesting to read about, and that happened and made us all become

aware of our privileged focus points.

I named the activity: “learning from a human activity described in a teachers’ story”.

The teachers’ story

A primary teacher – Lonneke Boels – described in a Dutch teachers’ magazine how she added to

the textbook (paragraph about tables and graphs) the question: “bring to the class newspapers,

journals, telephone books, bus- and train time schedules, or other sources with tables and graphs

in them. The children had to select from their materials at least one graph, one table and one






diagram, and glue that on a A4 page. Next they had to circle one point in each figure and write

down what the meaning was of that specific point in the given context.”

The results were shared, compared. The teacher describes in the article what she observed as

learning outcomes for the children and – according to her reflection, important for her own

learning – what she didn’t expect, what surprised here and what she intended to do as a result of

that.

[Lonneke didn’t mention that she was working in a classroom culture that is supportive,

conversational and respectful; a community where ideas are valued (regardless of their

mathematical validity), where there is trust (no ridiculisation) and where risk-taking is

rewarded.]

Two illustrations of some important aspects of mathematics education.

Horizontal and vertical mathematising illustrated by two short stories.






Story A. My daughter had to go to the railway station every working day of the week. She had

the possibility to travel with bus 3 or bus 4, for both directions. Since she is not a routine person

she wanted to have different routes on each day. Is that possible?

After a group of teachers worked on this many of them came with the (correct) answer NO.

The next question I gave them was: can you show this with some kind of a visualisation? And

next: can you prove it?

These next questions were given because I wanted to involve the teachers in a discussion about

‘modes of communication’ and about the triple ‘ being sure’, ‘being able to convince a friend’,

and ‘ being able to convince a math teacher ‘ (proof?) The given ‘ realistic situation’ was used to

provoke different solutions. This time I did not expect a discussion about that aspect.

I was surprised by the results, both by the creative solutions as well as by the inability of some

participants to visualise or explain their thoughts.

This made us – the participants agree with my decision to use a second, more inner-mathematical

context, for further exploration.






Story B. Each participant of my in-service group is asked to choose 5 lattice points on a ‘square-

dotted’ paper with a coordinate system on it, and to find the midpoints of the connecting lines

between each pair of dots. After my question if anybody found a midpoint without whole

coordinates the answer was YES.

So, the next question that came up (asked by a participant) was: is it possible to find 5 points in

such a way that all of the midpoints have whole coordinates?

And the whole group started to think, share, compare and reflect on the process.

What didn’t happen was the spontaneous emerging of the question that could have made the

work on these two problems into a real “mathematics learning by horizontal and vertical

mathematising”. This question seemed to be in the Zone of Proximal Development of the

participants. So, they needed a teacher/coach to ask some question to help them to ‘learn’.

The best question I could think of was: ‘What is similar in the problems A and B?’

A bit later in the conversation I also was the person who asked the question ‘What presentation

can you use to show that?’

Reflection on the work.






Afterwards I realised that we missed in the conversation attention for the question: ‘how can we

help our students to start asking these kind of questions themselves and become independent of a

teacher?’ Next sessions with this group we have to find a moment to reflect on this question.

For me the most important part of the experience with these tasks (story 1 and 2) and the work

on it was the reflection of one of the participants.

“At the beginning I didn’t understand why you asked the second question, the one about

‘presentation’. But then I listened to the reactions of my colleagues and thought that is a bit crazy

that all of us tried to use ‘language’ (words) to ‘show’ the similarity in the problems A and B.

Only when you pushed us to think deeper about the second question we started to look for other

‘modes’ of communication. Maybe that is the reason we don’t stress enough the ‘visual’ ways of

describing in our own classrooms. We don’t use them easily ourselves”

These different modes of communication are – for sure – much more salient in contemporary

society (Kress&Jewitt6, 2003, p.1) and are for that reason stressed in the PISA tests as can be

seen in the main processes (8 in 2000, or 7 in 2003), called representation (number 6) and

mathematical communication (number 3).

6 Kress, G. and Jewitt, C. , 2003, Íntroduction’, in: Kress, G. and Jewitt, C. (eds) Multimodial literacy, New York, NY, Peter Lang, pp.1-18.






My personal learning.

As a result of the described experience - and some other experiences - I’m much more focused at

finding and using opportunities /tasks that can act as a challenge for participants in mathematics

teaching courses. Challenges to work on, to reflect on their own work in mathematics, but also to

reflect on the work of their students and to investigate their own role in the (mathematics)

learning of their students.

Appendix.

Harrie Broekman, Freudenthal Institute for Math and Science Education, Utrecht University, was

reading for you:

Ana Maria Lo Cicero, Yolanda De La Cruz, Karen C. Fuson.

Teaching and Learning Creatively: Using Children’s Narratives.

Teaching Children Mathematics, May 1999, pp.544-547.






Children’s Math Worlds project seeks to integrate student’s social, emotional and cultural

experiences into classroom mathematics.

“We build on the individual experiences, interests, and practical mathematics knowledge that

diverse children bring to our classrooms.”

There needs to be a balance between building on children’s knowledge and teaching within

the zone of proximal development (also called the ‘learning zone’).

The project uses a Vygotskian model for unfolding, formulating, and solving mathematics

problems from children’s experience. This model describes one way in which teachers build on

children’s prior knowledge about various situations to facilitate student’s construction of

understandings of formal mathematical concepts, symbolism, and problems. The unfolding

multiple narratives of different children’s experiences provide a framework that is co-constructed

by the teacher and children and within which teachers relate new mathematical ideas to children'’

lives. The ZPD -–learning zone – is what children can accomplish with assistance. The teacher

leads the children from a starting point to more advanced mathematical knowledge. This

knowledge includes being better at listening, explaining and helping one another understand;






learning more advanced, efficient and accurate solution methods; and learning mathematical

symbolism, language, and new ideas.

- Getting started: eliciting and using children’s stories.

- Understanding, listening and describing.

- Putting a story in mathematical terms.

- Problem solving, reflecting and explaining.

- The co-constructing process. The classroom conversation is co-constructed by all those

involved. The active participants in a conversation each direct the conversation in certain

ways. Each contribution stimulates thinking. Throughout the conversation, personal

meanings are continually being constructed and reconstructed in ways that are influenced by

the classroom process.

Conclusion

Listening to children, putting their stories in a mathematical context, using children’s labeled

mathematics drawings and number drawings, and eliciting explanations from children about how

they solved problems are powerful approaches. But these approaches need constant leadership by






the teacher so that children can progress in their knowledge of mathematical methods,

vocabulary, and understanding.

In Utrecht we say that the teacher is needed to foster and coach the horizontal

mathematising as well as the vertical mathematising. In classroom settings interaction plays

an important role.

Part of the teachers’ role is: giving students opportunities to share their ideas, opinions, and

questions (creating a classroom environment in which mathematical thinking is encouraged and

valued) The selection of tasks or learning situations/contexts as well as the teachers’ own

questions are an important ingredient in this teachers’ role.

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