Implementing Beyond Crossroads with the Right Stuff Rob Kimball Project Director The 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) 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.

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

Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004 (p. 27) CUPM Curricular Guide 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, 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? Content Methods Technology Assessment

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 are altered ?

How do you know when learning has taken place ? What the Best College Teachers Do Ken 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.”

How do you know when learning has taken place ? 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 less important than understanding the underlying concepts. Curriculum Reform and the First Two Years (Curriculum Foundations Project)

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 philosophy will require a shift in the way many math teachers teach as well as in what they expect of their students.

Best Practices Understanding over Memorizing

Best Practices Integration over Isolation

Best Practices Depth over Breadth

Best Practices Application over Recognition

Promoting Promising Practices RIGHT STUFF MODULE 0

Implementing the Rule of Four Module 0 This project is sponsored, in part, by a grant from the National Science Foundation: NSF DUE 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 document The 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.

Graphical Representation What is an appropriate function to model the data? M P G

A Model – Discuss why it is necessary to use so many decimal places in the coefficients. Does the model make sense? Explain.

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 process Implementing Evaluating Documenting Redefining Planning

Teaching should be a reflective process Implementing Evaluating Documenting Redefining Planning Mentoring

Teaching should be a reflective process Peer Review

Teaching should be a reflective process Research

Teaching should be a reflective process Professional Development

Teachers who use promising practices: Believe knowledge is constructed – not received.

Teachers who use promising practices: Lead students to mental discourse and then help them resolve the issues: Elicit – Confront – Resolve – Assess

Teachers who use promising practices: 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.

Teachers who use promising practices: 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