teaching and learning algebra in the 21...
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Teaching and Learning Algebra
in the 21st Century
Why, Who, How, and WhatVickie L. Inge
[email protected] of Virginia
School of Continuing and Professional StudiesOffice of Mathematics Outreach
Teaching and Learning Algebra in the 21st Century
Why, Who, How , and What
The Needs of the Workplace Almost 90% of new jobs require more than a high
school level of literacy and math skills
The mathematics students need to learn today is not the same mathematics that their parents and grandparents needed to learn…
All young Americans must learn to think mathematically, and they must think mathematically to learn.
NRC, Adding It Up, 2001
Among the Fastest‐Growing Occupations Through 2016Require Advanced Math
Computer software engineers Network systems and data communications analysts Computer systems analysts Financial analysts and personal financial advisors Database administrators Physician assistants and physical therapists
Source: Dohm, A. & Shniper, L. (2007, November). Employment outlook: 2006–16. Washington, DC: U.S. Department of Labor, Bureau of Labor Statistics. (p. 895)
Many “Blue Collar” Jobs that Pay Well Require Strong Math Skills
The testing company ACT found that electricians, construction workers, upholsterers, and plumbers and concluded need the same math skills that are necessary for success in freshman college courses.
The Association of General Contractors says electricians, pipe fitters, sheet metal workers, and draftsman all need courses like algebra, geometry, and trigonometry to be successful on the job.
Sources: ACT, Inc. (2006). Ready for College or Ready for Work: Same or Different? Iowa City, IA: Author. Association of General Contractors of New Hampshire website, www.agcnh.org/public/workforce_dev/employment_labor/job_opportunities.asp
Virginia Community Colleges 2004‐
52 % of entering students enrolled in at least one developmental course
Only 36 % who began in developmental math courses enrolled in college‐level mathematics courses within 4 years while 62% enrolled in college level English courses within four years
The Turning Point: Developmental Education in Virginia’s Community Colleges (2009)
Virginia’s College and Career Readiness Initiative
Initiative is designed to:
Ensure that college‐ready standards in reading, writing and mathematics are taught in each Virginia high school classroom.
Strengthen students’ preparation for college and the work force before leaving high school.
Virginia’s College and Career Readiness Initiative
Estimates suggest that in the next decade, nearly eight in 10 job openings will require post secondary education or training.
Governor McDonnell called for an increase of 100,000 additional associate and bachelor's degrees over the next 15 years.
Teaching and Learning Algebra in the 21st Century
Why, Who, What, How
Math Literacy and Civil Rights
“ Today I want to argue, the most urgent social issue affecting poor people and people of color is economic access. In today’s world, economic access and full citizenship depend crucially on math and science literacy.”
Robert MosesRadical Equations:
Math Literacy and Civil Rights2001
Equity Issues —Every Child Every Day
Opportunity to learn High quality curriculum Highly effective teachers High expectations and support
One thing I notice is that in the high schools most of the assessment is multiple choice so students have little experience in using their algebra in any non-standard ways.
This is because the low performing schools have set up units of study so students can continue on in the course and "restart" a unit that they have failed and retake exams on units they have failed.
So they have this whole system of multiple choice tests for each unit of study.
Needless to say multiple choice testing prepares one for little except more multiple choice tests.
Dr. Joanne Rossi Becker
Performance in High Poverty High School Districts
Partnership for 21st Century Skills
Learning and Innovation Skills
Learning and innovation skills increasingly are being recognized as the skills that separate students who are prepared for increasingly complex life and work environments in the 21st century, and those who are not.
A focus on creativity, critical thinking, communication and collaboration is essential to prepare students for the future.
Teaching and Learning Algebra in the 21st Century
Why, Who, How, What
Research on Transferability
“The students who had learned through a problem‐solving approach did not regard the mathematics of school and the world as different and talked with ease about their use of the school‐learned math in their jobs and lives.”Boaler, J. (2008). What’s math got to do with it?: Helping children learn to love their least favorite
subjectand why it is important for America.
Boaler & Staples (2008)
The success of students in the high‐achieving school was due in part to the high cognitive demand of the curriculum and the teachers’ ability to maintain the level of demand during enactment through questioning.
Cognitive Demand of Classroom TasksKinds of thinking required:
Memorization Procedures without Connections
Requires little or no understanding of concepts or relationships.
Procedures with Connections Requires some understanding of the
“how” or “why” of the procedure. Doing Mathematics
Lower level
H igher level
Stein, Smith, Henningsen, & Silver, (2000)
Mathematical Tasks and the Cognitive Demand of the Task
MemorizationGiven y=2x+5 state the slope and the y‐intercept.
Procedures without ConnectionsGraph y= 2x + 5
“Too much of a focus on lower level tasks discourages student “involvement” in learning mathematics.”
Examples of Mathematical Tasks
Procedures with ConnectionsGiven y = 2x + 5 and y = 2x – 4 find the equation for the line that
is exactly half way between the two lines.Explain how you know your solution is correct. Is this the only
solution, how do you know?
Doing MathematicsDescribe a contextual situation that could be modeled with the
equation y = 2x + 5, then make a graph to represent the model. Explain how the situation, equation, and graph are interrelated.
“Higher level tasks or “Rich Tasks”, when wellimplemented, promote “involvement” in learning mathematics.”
What makes a “rich task” Martha was recarpeting her
bedroom, which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase?
Ms. Brown’s class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen to keep the rabbits.
If Ms. Brown’s students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?
How long would each of the sides of the pen be if they had only 16 feet of fencing?
How would you go about determining the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.
The Fencing Task Diagrams on Grid Paper
The Fencing Task Using a Table
Length Width Perimeter Area
1 11 24 11
2 10 24 20
3 9 24 27
4 8 24 32
5 7 24 35
6 6 24 36
7 5 24 35
8 4 24 32
x ?? ?? ??
The Fencing Task Using a Table
Length Width Perimeter Area
1 11 24 11
2 10 24 20
3 9 24 27
4 8 24 32
5 7 24 35
6 6 24 36
7 5 24 35
8 4 24 32
x 12-x 24 x(12-x) or 12x-x2
The Fencing Task Graph of Length and Area
0510152025303540
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Length
Area
y=x(12-x) or y=12x-x2
The Fencing Task Graphs
0510152025303540
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Length
Area
12w l
24p
212A l l
Multiple Representations
Similarities and Differences
Similarities• Both are “area” problems
• Both require prior knowledge of area
Differences• The amount of thinking
and reasoning required• The number of ways the
problem can be solved • Way in which the area
formula is used • The need to generalize• The range of ways to
enter the problem• The connections and
extensions that can be made with the problem
Get the Elephant Out of the Room
SOL
2009 Virginia Mathematics SOL
Teachers are encouraged to go beyond the standards and select instructional strategies and assessment methods appropriate for their students.
Page iv.
2009 Virginia Mathematics SOL
The content of the mathematics standards is intended to support the following five goals for students:
becoming mathematical problem solvers, communicating mathematically, reasoning mathematically, making mathematical connections, and using mathematical representations to model and interpret
practical situations.Page iv.
Teaching and Learning Algebra in the 21st Century
Why, Who, How, What
What is the right question?
Algebra for All?
Algebra by 8th Grade?
Or Something Else Entirely!!!!!!
34
Mathematical Proficiency For All
Must get beyond skills only focus and work toward developing reasoning and understanding in order to cultivate a productive disposition.
Proficiency is defined in terms of five interwoven strands, no one strand is more important than another.
Adding It Up: Helping Children Learn Mathematics, NRC ﴾2001﴿
Rigor vs. Complexity of ContentLe
vel o
f Diff
iculty
KnowledgeComprehensionApplication
AnalysisSynthesis
Evaluation To evaluate, make judgments based on criteria or standards, chec k based on data
To create, generate, design, produce
To analyze, differentiate, organize
To apply, execute with familiar tasks, implement with unfamiliar tasks
To understand, interpret, explain, exemplify
To remember , identify, recall
Bloom’s Taxonomy
Anderson, L. W., & Krathwohl, D. R. (2001). (Eds.). Taxonomy for learning, teaching, and assessing:
The National Mathematics Panel Final Report 2008Key Findings
Math knowledge is cumulative from Kindergarten through Algebra.
The goal of K – 7 mathematics instruction should be to build “algebraic thinking” and not just “arithmetic thinking”.
Algebra Proficiency sits on top of 8 years of math instruction, and every year matters.
The National Mathematics Panel Final Report 2008Key Findings
Our task is to prepare our students to be Proficient in Algebraand not be content with simply placing them into Algebra.
The Panel recommends that struggling students receive some explicit mathematics instruction regularly. Some of this time should be dedicated to ensuring that these students possess the foundational skills and conceptual knowledge for understanding the mathematics they are learning at their grade level.
Higher Order Thinking Skills ?
Arithmetic is doing something to numbers to get an answer.
Algebra is exploring the relationships between numbers.
Algebraic ThinkingA key to success “in algebra” is the development of algebraic thinking as a cohesive thread in the mathematics curriculum from prekindergarten through high school.
Cathy Seeley, Past President
National Council of Teachers of Mathematics
Developing Algebraic Reasoning in Elementary School
Rather than teaching algebra procedures to elementary school children, support them to develop ways of thinking about arithmetic that are more consistent with the ways that students have to think to learn algebra successfully.
Adapted from the work of Thomas Carpenter
Predict What Students Might Say
8 + 4 = + 5
Falkner, Levi, & Carpenter 1999
Percent of Students Offering Various Solutions to 8 + 4 = + 5
Response/Grade
7 12 17 12 & 17
1 and 2 5 58 13 8
3 and 4 9 49 25 10
5 and 6 2 76 21 2
Falkner, Levi, & Carpenter 1999
Algebraic Thinking Begins in Kindergarten8=8
8 = 2 + 62+6 = 6+ 22+6 = ____ + 5
159 + 24 = ____ + 3412 x 15 = 6 x ___
3x – 2 = 10 Relational Thinking
A web of ideas!
Blue dots represents of prior knowledge Lines represent connectionsRed dot represents a new ideaVan de Walle
How many pennies?
Multiplication with Connections
My Penny Jar
I was cleaning my closet and found an old jar that contained 5 pennies.
I decided to start saving some of the pennies I accumulate each day by putting 3 pennies in the jar daily.
My Penny JarHow many pennies will I have after 5 days, 12 days, 30 days?
How many pennies after any number of days?
My Penny JarHow many pennies will I have after 12 days?
How many pennies after any number of days?
My Penny JarHow many pennies after any number of days?
Y = 5 + (3+3+3….+3)
Y = 5 + 3x where x is the number of days pennies are put into the jar.
Algebraic Thinking
Where are we going with the pennies? Can you represent the relationship recursively? Explicitly?
What changes or what are the variables in the situation?
As one variable changes, what happens to the other?
How would you recognize the relationship in another context?
What teachers do in classrooms has
significant impact on student achievement
(knowledge).
Sanders & Rivers 1996, Kain, 1998; National Research Council, 2000
Public Agenda, 2000;Wenglinsky 2000, Whitehurst 2002; Hill, et al, 2005
K‐12 Focus and CoherenceInstructional Issues
How do we know that students are receiving good instruction in an “algebrafied” curriculum”?
How do we know that there is vertical coherence of “algebrafied” ideas?
How can we maximize teacher collaboration focused on student’s making sense of the “algebrafied” ideas which will impact readiness for Algebra?
What professional development do teachers need? How can we make it jobembedded?
Caveat
Generic professional development that focuses on teaching techniques without also focusing on content does not appear to be effective.
L. Desimone, A. Porter, B. Birman, M. Garet, K. Yoon 2001
Virginia’sCollege and Career Readiness Initiative
VDOE is developing new tests that will measure students’ mastery of the more rigorous content standards.
The new assessments will be administered online, and include technology‐enhanced items that require students to demonstrate content mastery in ways that were not possible with multiple‐choice tests.
The result of the standards revision, therefore, will be that students will have to demonstrate mastery of more rigorous content in order to pass the revised SOL tests.
Paradoxical Intention
Consider:
If we didn’t believe algebra was an important goal for ALL students, what would we do?
Algebra is the Challenge…
They are the pay-off!
Algebra is a Critical Key toBeing College and Career Ready