THE DEVELOPMENT

OF TEACHING AND

LEARNING

MATHEMATICS

GLOBALLY AND

LOCALLY Theories and Principles

Servino, Edel A.

Bron, Irene B.

Delatado, Mary Joy

Ibo, Jessica

Penero, Mary Jonabelle

Villareal, Antonette

Mathematics Education

- In contemporary education, mathematics education is the practice of teaching and

learning mathematics, along with the associated scholarly research.

- Researchers in mathematics education are primarily concerned with the tools, methods

and approaches that facilitate practice or the study of practice.

History

- Elementary mathematics was part of the education system in most ancient civilisations,

including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most

cases, a formal education was only available to male children with a sufficiently high

status, wealth or caste.

- In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium

included the mathematical fields of arithmetic and geometry.

- The first mathematics textbooks to be written in English and French were published by

Robert Recorde, beginning with The Grounde of Artes in 1540.

- In the Renaissance, the academic status of mathematics declined, because it was strongly

associated with trade and commerce.

- This trend was somewhat reversed in the seventeenth century, with the University of

Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry

being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics

being established by the University of Cambridge in 1662.

- In the 18th and 19th centuries, the industrial revolution led to an enormous increase in

urban populations. Basic numeracy skills, such as the ability to tell the time, count money

and carry out simple arithmetic, became essential in this new urban lifestyle.

- By the twentieth century, mathematics was part of the core curriculum in all developed

countries.

Objectives

At different times and in different cultures and countries, mathematics education has attempted

to achieve a variety of different objectives. These objectives have included:

• The teaching of basic numeracy skills to all pupils

• The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid

geometry, trigonometry) to most pupils, to equip them to follow a trade or craft

• The teaching of abstract mathematical concepts (such as set and function) at an early age

• The teaching of selected areas of mathematics (such as Euclidean geometry) as an

example of an axiomatic system and a model of deductive reasoning

• The teaching of selected areas of mathematics (such as calculus) as an example of the

intellectual achievements of the modern world

• The teaching of advanced mathematics to those pupils who wish to follow a career in

Science, Technology, Engineering, and Mathematics (STEM) fields.

• The teaching of heuristics and other problem-solving strategies to solve non-routine

problems.

Methods

• Conventional approach - the gradual and systematic guiding through the hierarchy of

mathematical notions, ideas and techniques. Starts with arithmetic and is followed by

Euclidean geometry and elementary algebra taught concurrently.

• Classical education - the teaching of mathematics within the quadrivium, part of the

classical education curriculum of the Middle Ages, which was typically based on Euclid's

Elements taught as a paradigm of deductive reasoning.

• Rote learning - the teaching of mathematical results, definitions and concepts by

repetition and memorization typically without meaning or supported by mathematical

reasoning. A derisory term is drill and kill. In traditional education, rote learning is used

to teach multiplication tables, definitions, formulas, and other aspects of mathematics.

• Exercises - the reinforcement of mathematical skills by completing large numbers of

exercises of a similar type, such as adding vulgar fractions or solving quadratic equations.

• Problem solving - the cultivation of mathematical ingenuity, creativity and heuristic

thinking by setting students open-ended, unusual, and sometimes unsolved problems.

Problem solving is used as a means to build new mathematical knowledge, typically by

building on students' prior understandings.

• New Math - a method of teaching mathematics which focuses on abstract concepts such

as set theory, functions and bases other than ten.

• Historical method - teaching the development of mathematics within an historical, social

and cultural context. Provides more human interest than the conventional approach.

• Standards-based mathematics - a vision for pre-college mathematics education in the

US and Canada, focused on deepening student understanding of mathematical ideas and

procedures, and formalized by the National Council of Teachers of Mathematics which

created the Principles and Standards for School Mathematics.

1. Transfer of Learning

- Transfer of learning deals with transferring one's knowledge and skills from one problem-

solving situation to another. You need to know about transfer of learning in order to help

increase the transfer of learning that you and your students achieve.

- For example, suppose that when you were a child and learning to tie your shoes, all of

your shoes had brown, cotton shoelaces. You mastered tying brown, cotton shoelaces.

Then you got new shoes. The new shoes were a little bigger, and they had white, nylon

shoe laces. The chances are that you had no trouble in transferring your shoe-tying skills

to the new larger shoes with the different shoelaces.

Near Transfer

- A nearly similar problem or task is automatically solved with little or no conscious

thought.

In recent years, Salomon & Perkins (1988) developed the low-road/high-road theory on transfer

of learning and proven to be a more fruitful theory.

Low-road transfer

- refers to developing some knowledge/skill to a high level of automaticity. It usually

requires a great deal of practice in varying settings. Shoe tying, keyboarding, steering a

car, and single-digit arithmetic facts are examples of areas in which such automaticity can

be achieved and is quite useful.

High-road transfer

- involves: cognitive understanding; purposeful and conscious analysis; mindfulness; and

application of strategies that cut across disciplines.

- In high-road transfer, there is deliberate mindful abstraction of an idea that can transfer,

and then conscious and deliberate application of the idea when faced by a problem where

the idea may be useful.

2. General Learning Theory

Benjamin Bloom

- is probably best known for his 1956 "taxonomy."

- he also did seminal work on student learning through different methods such as tutoring, peer

tutoring, mastery learning, and so on.

Knowledge: Observe and recall information knowledge of dates, events, places

• know major ideas, mastery of basic subject matter

verbs:

• list, define, tell, describe, identify, show, label, collect, examine, tabulate, quote, name,

who, when, where

Comprehension

• understand information grasp meaning translate knowledge to a new context interpret

facts, compare, contrast order, group, infer causes predict consequences

verbs:

• summarize, describe, interpret, contrast, predict, associate, distinguish, estimate,

differentiate, discuss, extend

Application

• use information, use methods, concepts, theories in new situations solve problems; use

required skills or knowledge

verbs:

• apply, demonstrate, calculate, complete, illustrate, show, solve, examine, modify, relate,

change, classify, experiment, discover

Analysis

• see patterns, organize the parts, recognize hidden meanings, identify components

verbs:

• analyze, separate, order, explain, connect, classify, arrange,divide, compare, select,

explain, infer

Synthesis

• use old ideas to create new ones, generalize from given facts relate knowledge from

several areas, predict, draw conclusions

verbs:

• combine, integrate, modify, rearrange, substitute, plan, create, design, invent,

• what is it?, compose, formulate, prepare, generalize, rewrite

Evaluation

• compare/discriminate between ideas, assess value of theories, make choices based on

argument, verify value of evidence, recognize subjectivity

verbs:

• assess, decide, rank, grade, test, measure, recommend, convince, select, judge, explain,

discriminate, support, conclude, compare

3. Situated Learning

“To Situate”- means to involve other learners, the environment, and the activities to

create meaning

Situated Learning

- Learning is a function of the activity, context and culture in which it occurs.

- Knowledge and skills are learned in the contexts that reflect how knowledge is obtained and

applied in everyday situations.

- A means for relating subject matter to the needs and concerns of learners.

- Learning is essentially a matter of creating meaning from the real activities of daily living.

Social interaction

- Critical component of situated learning.

- Learners become involved in a “community of practice” which embodies certain

beliefs and behaviors to be acquired.

Four Major Premises Guiding the Development of Classroom Activities (Anderson, Reder

and Simon, 1996)

1. Learning is grounded in the actions of everyday situations;

2. Knowledge is acquired situationally and transfers only to similar situations;

3. Learning is the result of a social process, encompassing ways of thinking, perceiving, problem

solving and interacting in addition to declarative and procedural knowledge; and

4. Learning is not separated from the world of action but exists in robust, complex, social

environment made up of actors, actions and situations.

Situated learning uses cooperative and participative teaching methods as the means of acquiring

knowledge.

Knowledge is obtained by the processes described (Lave, 1997) as Way In and Practice.

Way in – a period of observation in which a learner watches a master and makes a first

attempt at solving a problem.

Practice- is refining and perfecting the use of acquired knowledge.

4. Constructivism

Constructivism

- Programs aim to have students construct their own knowledge through their own process

of reasoning.

Teachers

- Pose their problems and encourage students to think deeply about possible solutions.

- Promote making connections to other ideas within mathematics and other disciplines.

- Ask students to furnish proof/ explanations for their work.

- Use different representations of mathematical ideas to foster students’ greater

understanding.

Students

Are expected to:

- Solve problems

- Apply mathematics to real-world situations; and

- Expand on what they already know

Constructivist approach to thinking about mathematics education.

1. People are born with innate ability to deal with small integers and to make comparative

estimates of larger numbers.

2. The human brain has components that can adapt to learning and using mathematics.

3. Humans vary considerably in their innate mathematical abilities or intelligence.

4. The mathematical environments that children grow up vary tremendously.

5. thus, when we combine nature and nurture, by the time children enter kindergarten, they

have tremendously varying levels of mathematical knowledge, skills, and interests.

6. Even though we offer a somewhat standardized curriculum to young students, that actual

curriculum, instruction, assessment, engagement of intrinsic and extrinsic motivation, and

so on varies considerably.

7. thus, there are huge differences among the mathematical knowledge and skill levels of

students at any particular grade level or in any particular math course

8. thus, mathematics curriculum, instruction, and assessment needs to appropriately take

into consideration these differences.

Educational Trends:

1. The transition of the teachers’ role from “sage on the stage” to “guide on the side”

2. Teaching “higher order” skills such as problem-solving, reasoning, and reflection

3. Enabling learners to learn how to learn;

4. More open-ended evaluation of learning outcomes;

5. And, of course, cooperative and collaborative learning skills

5. Cooperative Learning

• is an approach to organizing classroom activities into academic and social learning

experiences. Students must work in groups to complete tasks collectively.

5 Elements

Positive interdependence

• Students must fully participate and put forth effort within their group

• Each group member has a task/role/responsibility therefore must believe that they are

responsible for their learning and that of their group

Face-to-Face Promotive Interaction

• Each group member has a task/role/responsibility therefore must believe that they are

responsible for their learning and that of their group

• Students explain to one another what they have or are learning and assist one another

with understanding and completion of assignments

Individual Accountability

• Each student must demonstrate master of the content being studied

• Each student is accountable for their learning and work, therefore eliminating “social

loafing”

Social Skills

• Social skills that must be taught in order for successful cooperative learning to occur

• Skills include effective communication, interpersonal and group skills

– Leadership

– Decision-making

– Trust-building

– Communication

– Conflict-management skills

Group Processing

• Every so often groups must assess their effectiveness and decide how it can be improved

Challenges

• Need to prepare extra materials for class use

• Fear of the loss of content coverage

• Do not trusts students to acquire knowledge by themselves

• Lacks of familiarity with cooperative learning methods

• Students lack the skills to work in group

VARIOUS MATHEMATICIANS AND EDUCATORS THAT CONTRIBUTED TO

THE DEVELOPMENT OF TEACHING AND LEARNING MATHEMATICS

GLOBALLY AND LOCALLY

Mathematics Educators

1. Euclid (fl. 300 BC), Ancient Greek, author of The Elements

2. Tatyana Alexeyevna Afanasyeva (1876–1964), Dutch/Russian mathematician who

advocated the use of visual aids and examples for introductory courses in geometry for high

school students [16]

3. Robert Lee Moore (1882–1974), American mathematician, originator of the Moore method

4. George Pólya (1887–1985), Hungarian mathematician, author of How to Solve It

5. Georges Cuisenaire (1891–1976), Belgian primary school teacher who invented Cuisenaire

rods

6. Hans Freudenthal (1905–1990), Dutch mathematician who had a profound impact on Dutch

education and founded the Freudenthal Institute for Science and Mathematics Education in

1971

7. Toru Kumon (1914–1995), Japanese, originator of the Kumon method, based on mastery

through exercise

8. Pierre van Hiele and Dina van Hiele-Geldof, Dutch educators (1930s - 1950s) who

proposed a theory of how children learn geometry (1957), which eventually became very

influential worldwide

9. Robert Parris Moses (1935-), founder of the nationwide US Algebra project

10. Robert & Ellen Kaplan (about 1930/40s-), authors of Nothing That Is, The Art of the

Infinite: The Pleasures of Mathematics, and Chances Are: Adventures in Probability (by

Michael Kaplan and Ellen Kaplan).

10 Famous Filipino Mathematicians and Physicists

The names of the 10 famous Filipino Mathematicians are listed randomly below. Let’s

get to know them.

1. Raymundo Favila: He was elected as Academician in 1979. He is one of the people who

initiated mathematics in the Philippines. He had extensive contributions to the progression of

mathematics in the country.

2. Amador Muriel: This mathematician was known due to his significant works and

contributions to theoretical physics. He has made new kinetic equation which is essential for

discovering problems on a statistical method that is non-equilibrium.

3. Bienvenido F. Nebres: Dr. Nebres contributed much to the development of higher

mathematics teaching in the nation being the president of Mathematical Society of the

Philippines for years. He has successfully published 15 documents about pure mathematics and

mathematics education.

4. Tito A. Mijares: This doctor performs studies in relation to multi-variety hypothesis and

analysis. These were published in the Annals of Mathematical Statistics, a global journal.

5. Gregorio Y. Zara: He became famous for his two-way television telephone and the Zara

Effect or the electrical kinetic resistance.

6. Casimiro del Rosario: He was honored with the Presidential Award in 1965 for his

excellent works in physics, astronomy, and meteorology. His workings on soft x-rays made him

well-known.

7. Dr. Melecio S. Magno: He made researches on the absorption and fluorescence

spectroscopy of rare-earth crystals and how gravitation is affected by typhoons.

8. Apolinario D. Nazarea: He played important roles to the theories on biophysics and

recombinant biotechnology. His own conceptual framework on the structure of RNA/DNA

investigation is also included.

9. Eduardo Padlan: He was elected as Academician in 2003. He has significant work on

humanized antibodies which have possible applications in the healing of different diseases

including cancer.

10. Jose A. Marasigan: He is a multi-awarded professor here and abroad. He has received

awards like Young Mathematician Grant of the International Mathematical Union (IMU) to the

International Congress of Mathematicians (Finland) and the Outstanding Young Scientist Award

from the National Academy of Science and Technology. He also initiated the Program of

Excellence in Mathematics for mathematically gifted high school students in Ateneo de Manila.

They are the pillars of mathematics in the Philippines. You can try searching for more

information about them through a Filipino/Pilipino dictionary.

References:

http://www.denznet.com/amazing/10-famous-filipino-mathematicians/

http://www.ejmste.com/v3n1/EJMSTEv3n1_Zakaria%26Iksan.pdf

http://en.wikipedia.org/wiki/Cooperative_learning

http://en.wikipedia.org/wiki/Mathematics_education

Contributions of Various Mathematicians and Educators to the Development of Teaching

and Learning Mathematics Locally:

The Mathematics Teachers Association of the Philippines (MTAP)

- An organization of Mathematics teachers working together to promote excellence in

Mathematics education.

- First organized in 1976 by Fr. Wallance G. Campbell, S.J. at the Ateneo de Manila

University

- MTAP has honed the mathematical skills of promising students through its Math

Competition.

-MTAP's other programs includes

Scholarship grants leading to Master of Science in Teaching for selected members

2.

intensive summer training programs for Math teachers

tutorial programs for students

Conduct of mastery/ inventory tests for teachers.

Mathematics Trainers’ Guild

-Dr. Simon L. Chua- the president of MTG. His primary task is to improve and provide

quality education and open training among mathematics teachers and students

Top 10 Filipino Mathematicians:

1. Raymundo Favila- He made a new kinetic equation and studied geometric inequalities

and differential equations with applications to stratifiable congruences, among other

things. Favila has also helped write algebra and trigonometry textbooks.

2. Dr. Melecio S. Magno- has researched rare-earth crystals, how typhoons affect

atmospheric ozone distribution, gravitation, radiation in the atmosphere, sky luminosity,

and the philosophy of science. He has co-written the physics textbook University Physics

used at the University of the Philippines.

3. Jose A. Marasigan- He was instrumental in establishing the Philippine Mathematical

Olympiad and developing the Program for Excellence in Mathematics. He designed a

program for high school students in Ateneo de Manila who are mathematically gifted.

4. Tito A. Mijares-He conducted studies on multi-variety hypothesis and analysis and his

results were published in the Annals of Mathematical Statistics. He manages the

statistical system in the Philippines as Executive Director of the National Census and

Statistics Office and Deputy Director-General of the National Economic Development

Authority.

5. Amador Muriel-He was one of the founders of the Quantum Theory of Turbulence and

has made other significant contributions in theoretical physics. He also studied stellar

dynamics and discovered that self-gravitation can make a system of structures in one

dimension.

6. Apolinario D. Nazarea- He also contributed to the design of synthetic vaccines. He was

elected as an Academician in 1990 mostly because of this work.

7. Bienvenido F. Nebres- He was one of the founders of the Consortium of Manila

universities that developed PhD programs in mathematics, chemistry, and physics. The

Consortium has become the center of a network of schools all over the Philippines

8. Eduardo Padlan-He was elected as an Academician in 2003, partially for his work on

antibodies that may have applications in the healing of certain diseases including cancer.

He has Ph. D in Biophysics and has 14 patents on the use of antibodies

9. Casimiro del Rosario-He has performed superior work in the fields of meteorology,

physics, and astronomy. He became well known for his work on soft X-rays. Other work

was done on radioactive radiation on Euglena, the different wavelengths of ultraviolet

light, and electrical discharges in a vacuum.

10. Gregorio Y. Zara- He designed a microscope that has a collapsible stage and helped on

the design of the Marex X-10 robot. He also invented an airplane engine that ran on

alcohol and contributed to new designs of producing solar energy.

Contributions of Mathematics Educators

The following are some of the people who have had a significant influence on the

teaching of mathematics at various periods in history.

1. Euclid of Alexandria

- was a Greek mathematician, often referred to as the "Father of Geometry". He was

active in Alexandria during the reign of Ptolemy I (323–283 BC).

- author of The Elements

Elements is one of his most influential works in the history of mathematics,

serving as the main textbook for teaching mathematics (especially geometry) from the t

ime of its publication until the late 19th or early 20th century. In the Elements, Euclid

deduced the principles of what is now called Euclidean geometry from a small set

of axioms.

- Euclid also wrote works onperspective, conic sections, spherical geometry, number

theory and rigor.

Other works

In addition to the Elements, at least five works of Euclid have survived to the present day. They

follow the same logical structure as Elements, with definitions and proved propositions.

• Data deals with the nature and implications of "given" information in geometrical

problems; the subject matter is closely related to the first four books of the Elements.

• On Divisions of Figures, which survives only partially in Arabic translation,

concerns the division of geometrical figures into two or more equal parts or into parts in

given ratios. It is similar to a third century AD work by Heron of Alexandria.

• Catoptrics, which concerns the mathematical theory of mirrors, particularly the

images formed in plane and spherical concave mirrors.

• Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite

similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.

• Optics is the earliest surviving Greek treatise on perspective.

In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which

emanate from the eye.

Other works are credibly attributed to Euclid, but have been lost.

• Conics was a work on conic sections that was later extended by Apollonius of

Perga into his famous work on the subject.

• Porisms might have been an outgrowth of Euclid's work with conic sections, but

the exact meaning of the title is controversial.

• Pseudaria, or Book of Fallacies, was an elementary text about errors

in reasoning.

• Surface Loci concerned either loci (sets of points) on surfaces or loci which were

themselves surfaces; under the latter interpretation, it has been hypothesized that the work

might have dealt with quadric surfaces.

• Several works on mechanics are attributed to Euclid by Arabic sources.

2. Tatyana Alexeyevna Afanasyeva

- Dutch/Russian mathematician who advocated the use of visual aids and examples for

introductory courses in geometry for high school students

- Tatyana collaborated closely with her husband, most famously on their classic review

of thestatistical mechanics of Boltzmann. She published many papers on various topics

such asrandomness and entropy, and teaching geometry to children.

3. Robert Lee Moore

- was an American mathematician, known for his work in general topology and

the Moore method of teaching university mathematics.

- originator of the Moore method

4. George Pólya

- Hungarian mathematician, author of How to Solve It

He was a professor of mathematics from 1914 to 1940 at ETH Zürich in Switzerland

and from 1940 to 1953 at Stanford University carrying on as Stanford Professor

Emeritus the rest of his life and career.

- He worked on a great variety of mathematical topics, including series, number

theory,mathematical analysis, geometry, algebra, combinatorics, and probability.

5. Georges Cuisenaire

- (1891–1976), Belgian primary school teacher who invented Cuisenaire rods

6. Hans Freudenthal

- was aDutch mathematician. He made substantial contributions to algebraic

topology and also took an interest in literature, philosophy, history and

mathematics education.

- nFreudenthal focused on elementary mathematics education.

- In the 1970s, his single-handed intervention prevented the Netherlands from

following the worldwide trend of "`new math"'.

- He was also a fervent critic of one of the first international school achievement

studies.

- In 1971 he founded the IOWO at Utrecht University, which after his death was

renamed Freudenthal Institute, the current Freudenthal institute for science and

mathematics education.

7. Toru Kumon

- was a Japanese mathematics educator

-In 1954, Kumon began to teach his oldest son, who was doing poorly in mathematics

in primary school, and developed what later became known as the Kumon method.

- This method involves repetition of key mathematics skills, such

as addition, subtraction, multiplication, and division, until mastery is reached. Students

then progress to studying the next mathematical topic. Kumon defined mastery as being

able to get an excellent score on the material in the time given, which is intended to benefit

students in all their studies. Kumon strongly emphasised the concepts of time andaccuracy.

- As a result of the method, other parents became interested in Kumon's ideas, and in

1956, the first Kumon Center was opened in Osaka,Japan.

- In 1958, Toru Kumon founded the Kumon Institute of Education, which set the

standards for the Kumon Centers that began to open around the world. The Institute

continues today to focus on individual study to help each student reach his or her full

potential. The underlying belief behind the Kumon Method is that, given the right kind

of materials and the right support, any child is capable of learning anything. At any

time, there are more than 4 million Kumon students worldwide, and since 1956, more

than 19 million students have enrolled in Kumon Centers worldwide.

8. Pierre van Hiele and Dina van Hiele-Geldof,

- Dutch educators (1930s - 1950s) who proposed a theory of how children learn

geometry (1957), which eventually became very influential worldwide

- Van Hiele model is a theory that describes how students learn geometry. The theory

originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van

Hiele (wife and husband) at Utrecht University, in the Netherlands.

9.Robert Parris Moses

- is an American, Harvard-trained educator who was a leader in the 1960s Civil Rights

Movement and later founded the nationwide U.S. Algebra project.

- In 1982 he received a MacArthur Fellowship, and used the money to create

the Algebra Project, a foundation devoted to improving minority education in math.

Moses taught math for a time at Lanier High School in Jackson, Mississippi, and used

the school as a laboratory schoolfor Algebra Project methods.

10.Robert & Ellen Kaplan

- (about 1930/40s-), authors of Nothing That Is, The Art of the Infinite: The Pleasures of

Mathematics, and Chances Are: Adventures in Probability (by Michael Kaplan and

Ellen Kaplan).

Filipino Mathematics Teachers and Their Contribution

The following people all taught mathematics at some stage in their lives, although they are better

known for other things:

Charles Lutwidge Dodgson

• was an English author, mathematician, logician, Anglican deacon and photographer.

Mathematical Findings and Work

Within the academic discipline of mathematics, Dodgson worked primarily in the fields of

geometry, matrix algebra, mathematical logic and recreational mathematics, producing

nearly a dozen books which he signed with his real name. Dodgson also developed new ideas in

the study of elections (e.g., Dodgson's method) and committees; some of this work was not

published until well after his death. He worked as a mathematics tutor at Oxford, an occupation

that gave him some financial security.

Mathematical works

• A Syllabus of Plane Algebraic Geometry(1860)

• The Fifth Book of Euclid Treated Algebraically(1858 and 1868)

• An Elementary Treatise on Determinants, With Their Application to Simultaneous Linear

Equations and Algebraic Equations

• Euclid and his Modern Rivals(1879), both literary and mathematical in style

• Symbolic Logic Part I

• Symbolic Logic Part II(published posthumously)

• The Alphabet Cipher(1868)

• The Game of Logic

• Some Popular Fallacies about Vivisection

• Curiosa Mathematica I(1888)

• Curiosa Mathematica II(1892)

• The Theory of Committees and Elections, collected, edited, analysed, and published in

1958, by Duncan Black

Thomas Andrew Lehrer

- born April 9, 1928) is an American singer-songwriter,satirist, pianist, and mathematician. He

haslectured on mathematics and musical theater. Lehrer is best known for the pithy, humorous

songs that he recorded in the 1950s and 1960s.

His work often parodies popular song forms, such as in "The Elements", where he sets the names

of the chemical elements to the tune of the "Major-General's Song" from Gilbert and

Sullivan's Pirates of Penzance. Lehrer's earlier work typically dealt with non-topical subject

matter and was noted for its black humor, seen in songs such as "Poisoning Pigeons in the Park".

In the 1960s, he produced a number of songs dealing with social and political issues of the day,

particularly when he wrote for the U.S. version of the television show That Was The Week That

Was.

Mathematical publications

The American Mathematical Society database lists Lehrer as co-author of two papers:

• RE Fagen & TA Lehrer, "Random walks with restraining barrier as applied to the biased

binary counter", Journal of the Society for Industrial Applied Mathematics, vol. 6, pp. 1–14

(March 1958) MR0094856

• T Austin, R Fagen, T Lehrer, W Penney, "The distribution of the number of locally

maximal elements in a random sample",Annals of Mathematical Statistics vol. 28,

pp. 786–790 (1957) MR0091251

Georg Joachim de Porris,

– also known as Rheticus (16 February 1514 – 4 December 1574), was a mathematician,

cartographer, navigational-instrument maker, medical practitioner, and teacher. He is perhaps

best known for his trigonometric tables and as Nicolaus Copernicus's sole pupil. He facilitated

the publication of his master's De revolutionibus orbium coelestium (On the Revolutions of the

Heavenly Spheres).

Trigonometry

For much of his life, Rheticus displayed a passion for the study of triangles, the branch of

mathematics now called trigonometry. In 1542 he had the trigonometric sections of Copernicus'

De revolutiobis published separately under the title De lateribus et angulis triangulorum (On the

Sides and Angles of Triangles). In 1551 Rheticus produced a tract titled Canon of the Science of

Triangles, the first publication of six-function trigonometric tables (although the word

trigonometry was not yet coined). This pamphlet was to be an introduction to Rheticus' greatest

work, a full set of tables to be used in angular astronomical measurements.

At his death, the Science of Triangles was still unfinished. However, paralleling his own

relationship with Copernicus, Rheticus had acquired a student who devoted himself to

completing his teacher's work.Valentin Otto oversaw the hand computation of approximately

100,000 ratios to at least ten decimal places. When completed in 1596, the volume,Opus

palatinum de triangulus, filled nearly 1,500 pages. Its tables were accurate enough to be used in

astronomical computation into the early twentieth century.

Works

• Narratio prima de libris revolutionum Copernici(1540)

• Tabula chorographica auff Preussen und etliche umbliegende lender(1541)

• De lateribus et angulis triangulorum(with Copernicus; 1542)

• Ephemerides novae(1550)

• Canon doctrinae triangulorum(1551)

MATH and LITERATURES

" ... Using mathematics to tell stories and using stories to explain mathematics are two sides of

the same coin. They join what should never have been separated: the scientist's and the artist's

ways of uncovering truths about the world." (Frucht, xii)

LITERATURE

- It stirs our imaginations and emotions, making ideas more enjoyable and memorable.

-It enlivens what many people see as the isolating abstractness of mathematics (cf.

Midgley, 1-39).

-It also elicits expressions of feeling, increasing our insight about joys and frustrations in

studying math.

Different way of using math in literatures

(1) Call on math to illuminate a theory

(e.g., Dostoyevsky, and Tolstoy, and Austen);

(2) Be inspired by mathematical themes to create a work of art based on the themes

(e.g., Doxiadis, Growney, Lem,Reese, and Upson);

(3) Poke fun at typical experiences in learning math or at mathematicians

(e.g., Dodgson,Leacock and Russell);

(4) want to produce an educational work

(e.g., Enzensberger); or

(5) want to write theimagined life of an intriguing mathematician

(e.g., Petsinis).

Advantages of Literature in Teaching Mathematics

Provide a context or model for an activity with mathematical content.

Introduce manipulatives that will be used in varied ways (not necessarily as inthe story).

Inspire a creative mathematics experience for children.

Pose an interesting problem.

Prepare for a mathematics concept or skill.

Develop or explain a mathematics concept or skill.

Review a mathematics concept or skill.

Components of Mathematical content

1. Accuracy

2. Visual and Verbal Appeal

3. Connections

4. Audience

5. “Wow” Factor

Literary samples

The Symbolic Logic of Murder by John Reese

"... adjusts Boolean algebra , of an admittedly elementary order, to the requirements of popular

fiction." (Fadiman, Fantasia ..., 223.)

The solution to the murder depends on facility with negations, unions and intersections!

Young Archimedes by Aldous Huxley

- young hero combines a loving proficiency in music with an extraordinary ability in

math.

*As the story unfolds, we encounter both geometric and algebraic proofs of the Pythagorean

Theorem!

Star, Bright by Mark Clifton

two aspects:

o the problem of rearing a genius, and

o mathematical activities.

--- Star, a three-year old child, invents a Moebius strip and also figures out a way to teleport

herself into 4-dimensional space and to travel backwards and forwards in time.

Arcadia by Tom Stoppard

Although the 19 th century heroine (aged 13) of Arcadia fails to solve Fermat's Last

Theorem, she does anticipate the 20th century topics of chaos and iteration.

Proof by David Auburn

- a play about genius and love, considers the probability that a young woman could have

authored a path-breaking proof.

The Law by Robert Coates

-focuses on insights into human behavior, and the important role of statistics. It motivates

discussion of the meaning of the familiarly cited "law of averages," the various types of

averages,

The Brothers Karamazov

(an excerpt )

-it shows Dostoyevsky=s use of the new mathematical ideas to his philosophy.

War and Peace, another Russian novel by Tolstoy

*Tolstoy's theory is that history needs to be analyzed mathematically and statistically: not

as discrete incidents, but (in a reference to calculus) as a continual process.

Tolstoy also use Achilles and the Tortoise to show that history cannot be analyzed as a series of

discrete vignettes.

In addition, Tolstoy provides an example of the use of ratio and linear equations to clarify how

the disadvantaged (such as the Russians) can win battles against more advantaged (such as the

French) if they have enough spirit and energy.

Emma by Jane Austen

- it alludes to the ratio M/A, based on the 18th century philosophy of Francis Hutcheson,

who believed that the ratio measured "virtue, " where A is perfect virtue and M is attained

virtue.

The Extraordinary Hotel or the Thousand and First Journey of Ion the Quiet by Stanislaw Lem's

- the story goes on to many other possible scenarios, illuminating beautifully many

properties of infinite sets.

"A Mathematician's Nightmare“ by JoAnne Growney

- seems on the surface to be about decision-making in pricing and shopping, but it is an excellent

depiction for a student or lay reader of the Collatz Conjecture, a famous unsolved problem.

"My Dance is Mathematics,“

- poem about Emmy Noether

"If a woman's dance / is mathematics,/ must she dance alone?“

- The relationship to mathematics is usually seen in the content of the poem, but may also

be a matter of structure.

OTHER EXAMPLES

Sorting

Strega Nona by Tomie De Paola

Noodles by Sarah Weeks

Counting

How Many Snails? by Paul Giganti, Jr.

Who Took the Cookies from the Cookie Jar? by Bonnie Lass

Addition/Subtraction

Time

The Very Hungry Caterpillar by Eric Carle

The Grouchy Ladybug by Eric Carle

Ten Sly Piranhas by William Wise

Mouse Count by Ellen Stoll Walsh

Fractions

Eating Fractions by Bruce McMillan

Lunch with Cat and Dog by Rozanne Williams

Measurement

How Big is a Foot? by Rolf Myllar

Inch by Inch by Leo Lionni

Money

Bennie's Pennies by Pat Brisson

Research studies on the use of teaching and learning aids in math.

“Teaching and Learning Mathematics using Research”

-Dr.Terry Bergeson

Four key ingredients

• The students trying to learn mathematics

• The teachers trying to teach mathematics

• The content of mathematics and its organization into a curriculum

• The pedagogical models for presenting and experiencing this mathematical content

Advantages of Research in Math Education

• It can inform us.

• It can educate us.

• It can answer questions.

• It can prompt new questions.

• It can create reflection and discussion.

• It can challenge what we currently do as

educators

• It can clarify educational situations

• It can help make educational decisions and educational policy

• It can confuse situations

• It can focus on everything but your situation

• It can be hidden by its own publication style.

STUDIES CONDUCTED ON THE RESEARCH:

RESEARCH IN NUMBER SENSE

1. Number and Numeration

2. 2. Estimation

RESEARCH ON MEASUREMENT

1. Attributes and Dimensions

2. Approximation and Precision

O Difference between estimation and approximation

3. Systems and tools

O Measurements strategies

RESEARCH ON GEOMETRIC SENSE

O Define shape

O Characteristic of different shape

O 3-D environment

O Relationships/ Transformation

The research conducted implies that manipulative materials are good teaching aids in

teaching mathematics.

ROLES AND IMPACT OF USING MANIPULATIVES

O Increase mathematical achievement

O Students’ attitude towards mathematics are improved

O Help students understand mathematical concepts and processes

O Increase students” flexibility of thinking

O Tool to solve new mathematical problem

O Reduce students” anxiety

Note:

Manipulative need to be selected and used carefully.

Students do not discover or understand math concepts simply by manipulative concrete

materials.

Math teachers need assistance on selecting appropriate manipulative materials.

Mistaken beliefs about manipulative materials

-Jackson(1979)

1. Almost all manipulative can be used to teach any mathematical concept.

2. It simplify students’ learning of math.

3. Good math teaching always include manipulative.

4. The number of manipulative is positively correlated to the amount of learning that occur

5. There is a multipurpose manipulative

6. It is more useful in primary grades that in the upper grades.

7. It is more useful with low-ability students than high-ability students.

The use of concrete manipulative do not seem as effective in promoting algebraic

understanding.

Manipulative help students at all grade levels conceptualize geometric shapes and their

properties.

Suggestions in using Manipulative

1) Use it frequently and throughout the instructional program

2) It should be used in conjunction with other learning aids.

3) It should be used by students in a manner consistent with the mathematical content

4) used with learning activities that are exploratory and deductive in approach

5) Simplest and yet ideal

6) Used with activities that include that symbolic recording of results and ideas

Research study of material use in teaching and learning in mathematics

Abstract

The introduction of laptops in the teaching of mathematics and science in English under

the Teaching and Learning of Science and Mathematics in English Programme (Pengajaran dan

Pembelajaran Sains dan Matematik dalam Bahasa Inggeris, PPSMI) has been implemented by

the Ministry

Education since 2003. The preliminary observations found that teachers are not fully

utilising these facilities in their teaching. A survey was conducted to study the barriers preventing

the integration and adoption of information and communication technology (ICT) in teaching

mathematics. Six major barriers were identified: lack of time in the school schedule for projects

involving ICT, insufficient teacher training opportunities for ICT projects, inadequate technical

support for these projects, lack of knowledge about ways to integrate ICT to enhance the

curriculum, difficulty in integrating and using different ICT tools in a single lesson and

unavailability of resources at home for the students to access the necessary educational materials.

To overcome some of these barriers, this paper proposes an e-portal for teaching mathematics.

The e-portal consists of two modules: a resource repository and a lesson planner. The resource

repository is a collection of mathematical tools, a question bank and resources in digital form

that can be used for other teaching and learning mathematics. The lesson planner is a user

friendly tool that can integrate resources from the repository for lesson planning.

Digital Teaching Aids Make Mathematics Fun

"Students are increasingly living in two worlds: the world of the classroom and the real

world... and the two are growing farther apart," cautions Chronis Kynigos, a researcher at the

Research Academic Computer Technology Institute (RACTI) and director of the

Educational Technology Lab at the University of Athens.

Working in the EU-funded ReMath project, the team developed new teaching aids, in

the form of software tools known as Dynamic Digital Artefacts (DDAs), and a comprehensive

set of Pedagogical Plans for teachers to use within the guidelines of national education curricula.

A specific set of six Dynamic Digital Artefacts (DDAs) was designed and developed

during the ReMath Project. They have been selected in order to reasonably reflect the existing

diversity of representations provided by ICT tools

Examples of Program use inDDA’s

AlNuSet

- the building of a microworld consisting of an Algebraic Line and Algebraic manipulator

component for visual representation of geometrical and symbolic manipulation of number

sets,

MoPiX

- a tool for programming games and animations with equations,

MaLT

- an extension to the ‘Machine-lab’ authoring system for interactive virtual reality scenes

to include a mathematical scripting mechanism and a set of programmable and

mathematical controllers (such as variation tools and vectors) for manipulating virtual

objects, their properties and relations between them in small-scale 3d spaces,

Cruislet

- an extension to the ‘Cruiser’ G.I.S. and geographic space navigator to include a

mathematical scripting mechanism and custom mathematical user interface controls for

vector-driven navigation in 3d large-scale spaces.

A Study on the Use of ICT in Mathematics Teaching

Chong Chee Keong, Sharaf Horani & Jacob Daniel

Faculty of Information Technology

Multimedia University, 63100 Cyberjaya

Selangor Darul Ehsan, Malaysia

Abstract

The introduction of laptops in the teaching of mathematics and science in English under

the Teaching and Learning of Science and Mathematics in English Programme (Pengajaran dan

Pembelajaran Sains dan Matematik dalam Bahasa Inggeris, PPSMI) has been implemented by

the Ministry of Education since 2003. The preliminary observations found that teachers are not

fully utilisingthese facilities in their teaching.

A survey was conducted to study the barriers preventing the integration and adoption of

information and communication technology (ICT) in teaching mathematics.

Six major barriers were identified:

1. lack of time in the school schedule for projects involving ICT

2. insufficient teacher training opportunities for ICT projects,

3. inadequate technical support for these projects,

4. lack of knowledge about ways to integrate ICT to enhance the curriculum,

5. difficulty in integrating and using different ICT tools in a single lesson and;

6. unavailability of resources at home for the students to access the necessary educational

materials.

To overcome some of these barriers, this paper proposes an e-portal for teaching

mathematics. The e-portal consists of two modules: a resource repository and a lesson planner.

The resource repository is a collection of mathematical tools, a question bank and other

resources in digital form that can be used for teaching and learning mathematics. The lesson

planner is a user friendly tool that can integrate resources from the repository for lesson

planning.

METHODOLOGY

This research deployed a survey method to investigate the use of ICT and the barriers of

integrating ICT into the teaching of mathematics. The survey was carried out during a

mathematics in-service course conducted by the State Education Department. Before the

commencement of the survey, the respondents were given a briefing on the purpose of the

survey.

A total of 111 responses was received and they were analyzed using the SPSS statistical

package. A questionnaire was adapted from the Teacher Technology Survey by the American

Institute for Research (AIR, 1998).

The questionnaire was divided into seven areas:

(A) the teacher’s profile,

(B) how teachers use ICT,

(C) professional development activities,

(D) the teacher’s ICT experience,

(E) the level of use in ICT,

(F) the barriers faced by teachers and

(G) the proposed solution.

ORIGAMI

Origami is a Japanese compound word which means “paper folding”. It is used to

describe craft made from folded paper in Japan as well as pieces originating in other regions,

since so many people associate folded paper crafts with Japan in particular. Individual origami

pieces can vary widely in size and design, from simple folded boxes to ornate creatures made by

joining several different sheets of paper. Many young people learn origami in school, and some

people continue to practice this craft into adulthood.

The art of paper folding actually originated in China around the first century CE. The

Chinese referred to their folded paper crafts as zhe zhi, and monks brought the tradition with

them to Japan when they visited in the sixth century. The Japanese quickly took to paper folding

as a pastime, developing a number of traditional folds, shapes, and styles, many of which were

considered fortuitous for particular occasions or life events. The crane is a particularly famous

lucky origami shape.

Highlights in Origami History

100 AD

Paper-making originated in China by Ts'ai Lun, a servant of the Chinese emperor. The art of

paper folding began shortly after.

600 AD

Paper-making spread to Japan where origami really took off.

800-1100AD

Origami was introduced to the West (Spain) by the Moors who made geometric origami models.

1797

Hiden Senbazuro Orikata, the oldest origami book for amusement in the world is published.

Translated it means "The Secret of One Thousand Cranes Origami".

1845

Kan no mado (Window on Midwinter)-The first published collection of origami models which

included the frog base

1900

Origami spread to England and the United States

1935

Akira Yoshizawa developed his set of symbols used for origami instructions.

1960

Sadako and One Thousand Cranes was published by Eleanor Coerr and is linked with the

origami crane and the international peace movement.

2000

International Peace Project-An international project which is engaging communities in

collaborative activities to promote peace, non-violence and tolerance - A Million Paper

Cranes for Peace by the Year 2000!

Folding a single piece:

Start with a 1.5-inch square of paper:

Make a precise and creased fold lengthwise.

Dividing the square in half.

The actual purpose of this fold is just to give you a reference to make the next two folds.

Unfold the paper and lay it flat.

Take the bottom-right corner of the paper and fold it into a triangle so that the left side of the

paper now lies on top of the second fold you made.

Leave that folded, spin the paper 180 degrees and make the same fold.

Now, take the bottom-right corner of the paper and make another needle nose-type fold.

Unfold the paper and lay it flat.

Take the bottom edge of the paper and fold it to the center crease. Then spin the paper 180 degrees and do the same.

That means bringing the fold that you just made to lie exactly on top of the second fold you

made.

Then rotate the paper 180 degrees and make the same fold.

Another "needlenose" type fold.

Now is the time to remake the second and third folds you made:

Fold the bottom point of the paper straight up to meet another vertex of the parallelogram,

Then rotate the paper 180 degrees and

repeat, producing this:

Now, take the bottom-left corner of the paper and fold it so that what was the left edge of the paper now lies on top of the top edge of the paper, producing a triangle, like this:

Rotate the paper 180 degrees and repeat. A parallelogram! Now, you must tuck in that large triangle fold into the paper. . Here is what I mean:

Then rotate the paper 180 degrees and tuck

in the other fold, resulting in:

Now flip the paper over and rotate it so

that it looks like this:

Now you need to give the paper a bend in the middle. You will end up with this:

THE BASIC UNIT

Making models:

1. The cube. The easiest to construct, it takes 6 pieces.

2. The stellated octahedron. Takes 12 pieces.

3. The icosahedron. (A stellated icosahedron.) Takes 30 pieces.

4. The stellated truncated icosahedron. Takes 270 pieces.

Model construction

A piece has two sharp corners and two pockets, which allow them to interlock.

Here is a third piece, placed over the first two:

Here are two pieces placed to illustrate

this:

And here they are locked together, corner in pocket:

And here the third piece is locked in:

There is a free corner and free pocket that can be locked together. Doing so necessitates forming

the three pieces into a three-dimension configuration that I call a peak:

REFERENCES:

http://nuwen.net/

library.thinkquest.org/5402/history.html

en.wikipedia.org/wiki/History_of_origami

Solving Quadratic Equations

Using Quadratic Formula

and

TI – 84 Plus

(Graphing Calculator)

THE QUADRATIC FORMULA

Entering a calculation

Use the Quadratic Formula to solve the quadratic equation

3x2

+ 5x + 2 = 0

1. Press 3 STO > ALPHA [A] (above MATH) to store the coefficient of the x2 term.

2. Press ALPHA [:]( above .). the colon allows you to enter more than one instruction

on a line

3. Press 5 STO > ALPHA [B] (above APPS) to store the coefficient of the X term.

Press ALPHA [:] to enter a new instruction on the same line. Press 2 STO>

ALPHA [C] (above PRGM) to store the constant.

3 → 𝐴: 5 → 𝐵: 2 → 𝐶

4. Press ENTER to store the values to the variables A, B, and C.

3 → 𝐴: 5 → 𝐵: 2 → 𝐶

5. Press ( ( ) (-) ALPHA [B] + 2nd

[√] ALPHA [B] x2 – 4 ALPHA [A] ALPHA [C] ) )

÷ ( 2 ALPHA [A] ) to enter the expression for one of the solutions for the quadratic

formula,

−𝒃 ± √𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

( -B+√ (B2 −4AC) )/(2A)

6. Press ENTER to find one solution for the equation 3x2

+ 5x + 2 = 0

( -B+√ (B2 −4AC) )/(2A)

-.6666666667

Converting to a Fraction

You can show a solution as a fraction

1. Press MATH to display the MATH menu

2. Press 1 to select 1:> Frac from the MATH menu.

When you press 1, Ans>Frac is displayed on the home screen. Ans is a variable that

contains the last calculated answer.

3. Press ENTER to convert the result to a fraction.

To save the keystrokes, you can recall the last expression you entered, and then edit it

for a new calculation.

4. Press 2nd

[ENTRY] (above ENTER) to recall the fraction conversion entry, and then

press 2nd

[ENTRY] again to recall the quadratic formula expression

−𝒃 + √𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

5. Press ^ to move the cursor onto the + sign in the formula. Press – to edit the quadratic

formula expression to become

−𝒃 + √𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

6. Press ENTER to find the other solution for the quadratic equation

3x2

+ 5x + 2 = 0

NeoCube

Neocube magnets

are small high-energy sphere magnet that allows you to create and recreate an endless

number of different patterns and shapes. Neocube magnets are very strong because are made of

neodymium iron boron material and it is pretty fun to play with it. Has 216 pieces of magnets.

It is not important where you will buy the NeoCube. If it will be in some country as

Canada, India, Mexico, South Africa, Australia, Hong Kong, UK, Ireland or in some city as

London, Delhi, Dublin. You can buy it from local retail stores or order it on an internet shop.

Bulk NeoCube you can buy from wholesaler or from factory. From different brands as

BuckyBalls, Nanodots, Zen Magnets you will get different packing of magnets. But the magnets

are always made in China. Original source of neodymium magnets.

Features of Neocube Magnets

Neocube is the future of puzzles.

Dual-brain hemisphere stimulation.

Gaming.

Stress relief.

Boredome busting.

Most common on the market are Neocube magnets made of neodymium N35. Nickel coating

with the diameter size 5mm.

neocube diameter size : 4.8mm, 5mm, 6mm, 7mm, 8mm

neocube colors : nickel, black, silver, gold, blue, red

neocube grade : N35, N35, N40, N42, N45, N48, N52

neocube coating : Ni-Cu-Ni, Ni-Cu-Ni-Cr / nickel, copper, chrome

Different size, color and grade of the material means also a different price.

Warnings

This product is not designed or intended for children under the age of fourteen.

This product contains small parts that may be harmful or fatal if swallowed.

Consult a doctor immediately if this occurs.

This product contains magnets. Magnets sticking together or becoming attached to a

metallic object inside the human body can cause serious or fatal injury. Seek immediate

medical help if the magnets are swallowed or inhaled.

The NeoCube or any of the spheres should never be put in the mouth, ears, nose, or any

other bodily orifice.

The strong magnets in the NeoCubeTM

can damage or destroy some electronic devices.

Therefore, it should never be put close to or directly in contact with electronic products.

Strong magnets can even damage electronic medical devices. Therefore the NeoCubeTM

should never be handled, used by, or brought near anyone with a pacemaker or other

electronic medical device.

Strong magnets can also damage or destroy information stored magnetically. Some

examples of these are: credit card strips, floppy disks and hard disks. Therefore the

NeoCubeTM

should not be put close to or directly in contact with any type of

magnetically stored data.

Never attempt to burn the NeoCubeTM

.

If the metallic coating around the spheres breaks down, discontinue use. This is

precautionary. The NdFeB material which is the magnetic material in the NeoCubeTM

is a

relatively new material, and long term effects of direct skin exposure are therefore

unknown, although there have been no studies which indicate that it is in any way

transdermally toxic.

This product is not intended to treat, diagnose or cure any diseases.

This product contains small balls.

.

Some Objects formed by neocubes:

NEOCUBE SHAPES, patterns - unique magnet gadget toy

Neocube Magnet Ball - 216 Neo Cube Magnet Ball - China Cybercube .

PAPER SPINNER

A type of manipulative that can be used to teach about chance and random

choices.

How to Make a Spinner?

Things you’ll need

Paper (printed)

Markers (optional)

Scissors (or just tear it)

Creativity (for markers)

Steps:

1. Get some printed paper. (it also works with loose leaf notebook paper)

2. Fold the piece of paper in half vertically.

3. Cut down the crease.

4. Fold the two large rectangles in half vertically, so that they become long and skinny.

5. Fold the bottom corner of each rectangle to the right, so that it forms a triangle shape.

6. Repeat at the top, except this time, make sure the triangles are facing left.

7. Put one of the triangles (it should have the little triangles) facing vertically upward.

8. Put the other rectangle horizontally, facing down in space between the two triangles on the

other rectangle.

9. Fold the bottom triangle to the center, then fold the left triangle to the center, overlapping

the one you just folded. With the top triangle, fold to the center also.

10. Fold the right triangle so it overlaps the top triangle and make sure it goes under the

bottom triangle.

HOW IT IS USED?

Color the ¼ part of the paper spinner by RED and the ¾ by BLUE. Spin the paper spinner and

find out what color will be on top when it stops.

FOLDABLES

o FOLDABLES

an artistic graphic organizer.

This Foldable project is used to help teachers analyze data, sort the strengths and

weaknesses of their students and determine the question levels from a TAKS-Released

Test so that they can make informed decisions about instruction.

Example of Foldables

TYPES OF FOLDABLE

o A POCKET BOOK FOLDABLE

1. Fold a piece of 8 ½” x 11” paper in half horizontally

2. Open the folded paper and fold one of the long sides up two inches to form a pocket.

3. Glue the outer edges and the center (on the valley/crease) of the two inch fold with a

small amount of glue.

o A LAYERED LOOK BOOK FOLDABLE

1. Stack four sheets of paper (8 ½” x 11”) together, placing each consecutive sheet

around ¾ of an inch higher than the sheet in front of it.

2. Bring the bottom of both sheets upwards and align the edges so that all of the layers or

tabs are the same distance apart.

3. When all of the tabs are equal distance apart, fold the papers and crease well.

4. Open the papers and glue them together along the valley/center fold.

o A JOURNAL RESPONSE THREE QUARTER BOOK FOLDABLE

1. Fold a piece of 8 ½” x 11” paper in half horizontally

2. Fold it in half again horizontally.

3. Unfold the paper (just once so that it is still folded in half) and cut up (along the edge

of the paper at the center where you can see the crease) to the mountain top

4. Open flat, lift the left-hand tab. Cut the tab off at the top fold line.

o A STUDENT INTEREST BOUND BOOK FOLDABLE

1. Fold two pieces of ¼ sheet paper (4 ¼” x 5 ½”) separately in half horizontally

2. Place the folds side-by-side allowing 1/16” between the mountain tops. Mark both

folds 1” from the outer edges.

3. On one of the folded sheets, cut-up from the top and bottom edge to the marked spot

on both sides.

4. On the second folded sheet, start at one of the marked spots and cut out the fold

between the two marks. Do not cut into the fold too deeply, only shave it off.

5. Take the “cut-up” sheet and burrito it.

6. Place the burrito through the “cut out” sheet and then open the burrito up.

7. Fold the bound pages in half to form a book.

o A TWO-TAB POINT OF VIEW BOOK FOLDABLE

1. Fold a piece of (4 ¼” x 5 ½”) paper in half horizontally

2. Fold it in half again horizontally

3. Unfold the paper (just once so that it is still folded in half) and cut up the valley (along

the edge of the paper at the center where you can see the crease) to the mountain top.

o A THREE-TAB BOOK FOLDABLE

1. Fold a piece of (8 ½ x 11”) paper in half vertically

2. With the paper horizontal and the fold up, , fold the right side toward the center, trying

to cover one half of the paper. (Make a mark here, but do not crease the paper.)

3. Fold the left side over the right side to make a book with three folds.

4. Open the folded book. Place your hands between the two thicknesses of paper and cut

up the two valleys on one side only. This will form three tabs.

o A TWO-TAB BOOK FOLDABLE Z

1. Fold a piece of (4 ¼” x 5 ½”) paper in half vertically

2. Fold it in half again horizontally

3. Unfold the paper (just once so that it is still folded in half) and cut up the valley (along

the edge of the paper at the center where you can see the crease) to the mountain top.

MANIPULATIVES

‘Multiplying two Binomials using Teaching Manipulative”

SQUARE BASE TABLE MANIPULATIVE

INSTRUMENT

S

= 3 = 𝑥2

= x

= 3 = 1

These instruments can be used in multiplying two binomials.

for example:

(x-1)(x+3)

(x+1)(x-3)

= x2-

3x+1x-3

= x2-2x-3

-2x x2

-3

+

-+

-

+

-+

-

Pentominoes as Math Manipulative

Definition

Use the 12 pentomino combinations to solve problems.

Is a geometric pattern which is the basis of a number tiling patterns and puzzles.

An arrangement of five identical squares in a plane, attached to one another edge to edge.

Is a polymino composed of five congruent squares, connected long their edges (which

sometimes is said to be an orthogonal connection).

How it is done or constructed?

Know that there are 12 pentominoes shape. They are named for the letters they represent:

F I L N P T U V W X Y Z. A pentomino is a shape composed of five congruent

squares connected by at least one side. Since there are twelve pentominoes made of five

squares each, pentomino puzzles are played on grids of 60 squares: 6 by 10, 5 by 12, 0r 3

by 20.

Make your grids. Sketch them on one sheet of graph paper. Cut them out, then trace them

on two card stock. Go over the lines with a permanent marker to make a boarder, then cut

the grids out and set aside.

Make your puzzle pieces. Sketch out one of each pentomino onto graph paper. Cut them

out and trace them onto card stock. Color the pentominoes. Try to use one color for each

piece if you have enough markers available. Otherwise, just make it as colorful as

possible. Then, cut out the pieces and set aside.

Construct your folder, which will contain your puzzle. Open the folder and staple the

zipper bag to one side of it. Your grids and pieces will be stored in the bag when you are

not playing with your puzzle. Put the paper clip on the other side. This will be used to

hold which ever grid you are playing on at that time.

Play pentominoes. Take a grid out of your bag and clip it onto the folder. Using your

pentomino pieces, fill the gried by leaving no empty spaces and overlapping no pieces.

Each grid size has several olutions, so enjoy fiding them all.

When to use?

Subject Tag: problem Solving involving geometry and Algebra graphing.

Pentominoes Shape

F I L N P T

U V W X

Y Z

Group1

Manipulatives and other

Instructional Materials

GROUP3

Carlos, Aiza A.

Francisco, Ma. Salome V.

Gonzales, Karen C.

Habana, Sarah Mae

Laguna, Jan Rea O.

Poche, Michille

Baylon, Kevin

Instruments for Mathematics Teaching