Implementing Beyond Crossroads with the Right Stuff

Implementing Beyond Crossroads with the

Right Stuff

Rob Kimball

Project DirectorThe Right Stuff: Appropriate Mathematics for All Students (NSF)

Re-focusing College Algebra

an example of implementing the continuous improvement

cycle with the involvement of many stakeholders

The Right Stuff – what is the message ?

Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004 (p. 27)

http://www.maa.org/CUPM/curr_guide.html

CUPM Curricular Guide

Unfortunately, there is often a serious

mismatch between the original rationale

for a college algebra requirement and the

actual needs of the students who take the

course.

< www.maa.org/CUPM >

http://www.maa.org/CUPM

http://www.maa.org/CUPM




A critically important task . . .

is to clarify the rationale for the requirements, determine the needs of the students who take college algebra, and ensure that the department’s courses are aligned with these findings.




Offer suitable courses . . . designed to

• Engage students in a meaningful and positive intellectual experience;

• Increase quantitative and logical reasoning abilities needed for informed citizenship / workplace;

• Strengthen quantitative and mathematical abilities that will be useful to students in other disciplines;

• Improve every student’s ability to communicate quantitative ideas orally and in writing;

• Encourage students to take at least one additional course in the mathematical sciences.

Arnold Packer, from a conference on college algebra at West Point, 2002

http://www.maa.org/t_and_l/college_algebra.html

•Look at a typical description:

–This course is a modern introduction to the nature of mathematics as a logical system. The structure of the number system is developed axiomatically and extended by logical reasoning to cover essential algebraic topics: algebraic expressions, functions, and theory of equations.

•Who decided that "algebraic expressions, functions, and theory of equations" is essential, and if so, essential to whom or for what?

The course covers the following topics: Radicals, Complex Numbers, Quadratic Equations, Absolute Value and Polynomial Functions, Equations, Synthetic Division, the Remainder, Factor, and Rational and Conjugate Root Theorems, Linear-Quadratic and Quadratic-Quadratic Systems, Determinants and Cramer's Rule, and Systems of Linear Inequalities.

. . .

That is a long list of topics; yet, it is only half the topics listed in a typical college algebra syllabus.

How much can students learn in one or two days on a topic and what will they remember?

If a student passes the course, what, from the course, will they remember

and be able to use at a later date?

For too many students it looks like – and is – a painful experience that

they would prefer to skip.

What will make the student’s mathematical experience

more meaningful, enjoyable, and useful?

ContentMethods

TechnologyAssessment

How do you know when learning has taken place ?

• When students can get the correct answer to a problem ?

• When students can get a satisfactory grade on a test ?

• When students feel good about the course ?

• When changes in the student’s way of thinking and/or habits of mind arealtered ?


What the Best College Teachers DoKen Bain, Harvard Univ Press

“…the best teachers believe that learning involves both personal and intellectual

development and that neither the ability to think nor the qualities of being a mature human being are immutable. People can change, and those changes – not just the accumulation of information – represent

true learning.”


When changes in the student’s way of thinking and/or habits of mind are altered.

What changes do we want to cause/observe

in our students?

What should students be able to do?

•Create, analyze, and interpret basic mathematical models from informal problem statements;•Argue that the models constructed are reasonable; and•Use the models to provide insight into the original problem.

Algorithmic skills are much lessimportant than understanding

the underlying concepts.Curriculum Reform and the First Two Years (Curriculum Foundations Project)

http://www.maa.org/cupm/crafty/cf_project.html

What are the big ideas in your College Algebra course ?

• Solving quadratic equations• Solving exponential equations• Simplifying rational expressions• Simplifying logarithmic expressions• Finding solutions to systems of equations• Interpreting and using data shown in tables, graphs,

or formulas• Understanding variables and relationships between

variables• Understanding rate of change• Strengthening their problem solving skills• Using mathematics to support conclusions

It is safe to say

that these changes in that these changes in philosophy will require a philosophy will require a

shift in the way many math shift in the way many math teachers teach as well as teachers teach as well as

in what they expect of in what they expect of their students.their students.

Best Practices

UnderstandingUnderstandingoverover

MemorizingMemorizing

Best Practices

IntegrationIntegrationoverover

IsolationIsolation

Best Practices

DepthDepthover over

BreadthBreadth

Best Practices

ApplicationApplicationoverover

RecognitionRecognition

Promoting Promising Practices

RIGHT STUFFMODULE 0

Implementing the Rule of Four

Module 0This project is sponsored, in part, by a

grant from the National Science Foundation: NSF DUE 06 32883.

Any opinions, findings, and conclusions or recommendations

expressed in this material are those of the author(s) and do not

necessarily reflect the views of the National Science Foundation.

The Rule of Four

Have participants read the documentThe Rule of FourThe Rule of Four

Discuss the rule of four and the 12 pathways and what they mean.

Discuss the final figure, showing the shift to contextually based problems that motivate the mathematics.

Example 3: Gas Mileage

Have students read the scenario and discuss their experiences regarding situations similar to the data.

Discuss the trends in the data and how the data can be represented graphically.

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35 50 65 80

Ave Speed

M P

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Graphical Representation

What is an appropriate function to model the data?

M P

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A Model – Discuss why it is necessary to use so many decimal

places in the coefficients.

Does the model make sense? Explain.

y = 9E-05x3 - 0.0149x2 + 0.7029x + 16.364

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Questions

When Papa drives, he averages 77 mph. When Grandma drives, she averages 60 mph on the stretch down US 64 East.

How much money do they save (one way) when Grandma drives? (Let gasoline be $4 per gallon.)

How much time do they save (one way) when Papa drives?

Teaching should be a reflective processTeaching should be a reflective process

Implementing

Evaluating

Planning

Documenting

Redefining


Implementing

Evaluating

Planning

Documenting

Redefining

Mentoring


Implementing

Evaluating

Planning

Documenting

Redefining

Peer Review


Implementing

Evaluating

Planning

Documenting

Redefining

Research


Implementing

Evaluating

Planning

Documenting

Redefining

Professional Development

Teachers who use promising practices:

• Believe knowledge is constructed – not received.


• Lead students to mental discourse and then help them resolve the issues: Elicit – Confront – Resolve – Assess


• Ask the right questions until students learn to ask them themselves. If students can’t learn to judge the quality of their own work – they really haven’t learned.


• Create a natural critical learning environment.

Finally:

Mental Models Change Slowly

Next, on the Right Stuff:

Hurricanes – modeling the force of the wind

Building a collaborative environment for teachers

Classroom and Course Assessment

Documents

Implementing Beyond Crossroads with the Right Stuff