Rethinking Precollege Math:

A Personal Odyssey

Michael C. BurkeCollege of San Mateo

TMP-RPM Summer Math InstituteSleeping Lady Mountain Retreat, Leavenworth, WA

August 23, 2010

burke@smccd.edu

I’ve gone to find myself.

If I come home before I return,keep me here.

*Seen on the T-shirt of a kid standing on The Charles Bridge in Prague

July 14, 2006

What are the major problems or issues our students, our

citizens, our country will need to resolve in the future?

• global warming • social justice

• capital punishment • energy

• nuclear proliferation • economic growth

• water • world population

Does mathematics have anything to say about issues such as these?

What contributions are we making, as a profession, to provide our

students with the tools they will need to address issues such as these?

Citizenship and Quantitative Literacy

“My thesis today is that by virtue of our training, mathematicians have distinctive habits of mind that can enhance public discussion of public issues. More importantly, we have a professional obligation to move beyond the boundaries of our own discipline to bring our special skills of analysis and clarification to bear on important public policy discussions.”1

1. “On Being a Mathematical Citizen: The Natural NExT Step” by Lynn Arthur Steen in Focus Magazine, October, 2007

AAC&U: The Essential Learning Outcomes

• Knowledge of Human Cultures and the Physical and Natural World

- Through study in the sciences and mathematics, social sciences, humanities, histories, languages and the arts

Focused by engagement with big questions, both contemporary and enduring

• Intellectual and Practical Skills, including - Inquiry and analysis - Critical and creative thinking

- Written and oral communication - Quantitative literacy

- Information literacy - Teamwork and problem solving

Practiced extensively, across the curriculum in the context of progressively more challenging problems,

projects, and standards of performance

• Integrative Learning, including

- Synthesis and advanced accomplishment across general and specialized studies

Demonstrated through the application of knowledge, skills, and responsibilities to new settings and

complex problems

Question:

What stands in our way? What is it that prevents us from engaging

questions like these in our mathematics classes?

Answer:

Our curriculum.

Overview

1. Introduction

2. Overview

3. What do we teach?

4. Why do we teach what we teach?

5. How well does this serve our students?

6. What should we use as an organizing principle?

7. What goals should we have for our students?

8. What are the implications for earlier coursework (backward design)?

9. Examples

10. Final Thoughts

What do we teach?

For precollege mathematics, we typically teach a sequence ofcourses culminating in Intermediate Algebra:

• Arithmetic• Pre-Algebra• Elementary Algebra• Intermediate Algebra

The purpose of each course is to prepare for the next course.

Our students cannot take a college level mathematics course until

they have completed Intermediate Algebra.

What do we teach?

Topics from two courses:

Elementary Algebra Intermediate Algebra

absolute valuerational expressionscomplex fractionssimplifying radicalsrationalizing denominators

What do we teach?

Definition: lim x a f(x) = L

if for every number > 0 there is a number > 0 such that

if |x – a| < then |f(x) – L| <

What do we teach?

Topics from two courses:

Elementary Algebra Intermediate Algebra

absolute value absolute valuerational expressions rational expressionscomplex fractions complex fractionssimplifying radicals simplifying radicalsrationalizing denominators rationalizing

denominators

What do we teach?

Topics from two courses:

Elementary Algebra Intermediate Algebra

absolute value absolute valuerational expressions rational expressionscomplex fractions complex fractionssimplifying radicals simplifying radicalsrationalizing denominators rationalizing

denominatorsfractional exponentsGaussian eliminationCramer’s Rulevertex form of a parabolaconic sections

What do we teach?

What do these topics have in common?

• they are abstract

• they involve intricate manipulation

• they require great skill with arithmetic

• they are taught in a procedural way

• for our students, they are devoid of context or meaning

Why do we teach what we teach?

• when each of us began teaching, we inherited a curriculum

Why do we teach what we teach?

• when each of us began teaching, we inherited a curriculum

• the curriculum grows organically -- the result is a curriculum that is “a mile wide and an inch deep”

Why do we teach what we teach?

• when each of us began teaching, we inherited a curriculum

• the curriculum grows organically -- the result is a curriculum that is “a mile wide and an inch deep”

• the big reason: the entire mathematics curriculum is aimed at Calculus – that is, preparation for Calculus serves as an organizing principle for the entire curriculum

How well does this serve our students?

• our students arrive with woeful preparation (see Stigler)

• our success rate for the precollege sequence is dismal

• mathematics has become an insurmountable barrier for far too many of our students

• our students have seen (and often failed at) this material before

• our students arrive at college with excitement – they are ready to explore new ideas – we give them the same old thing

• and finally, most of our students are not headed for calculus

This is not news to us. We know all of this …

and so we tinker around the edges.

• We look for other approaches to this material.

• We arrange for extra support for our students to help them learn this material.

• We use publisher supplied, computer generated homework with elaborate online help, and 24 hour hotlines.

• We add a counseling component to our classes …

• or college readiness …

• or summer bridge.

None of these ideas are bad, but …

Today, let’s think the unthinkable.

• Why are the topics in our curriculum so important?We did not design the curriculum, after all.

• Of what use are these topics to our (our) students?What do our students truly need to learn about mathematics?

• Perhaps it’s time to seriously examine what we teach; are we teaching the right stuff?

Questions

• Should we dispense with Calculus as the organizing principle of our curriculum?

• If so, what should we use as an organizing principle?

• What implications would such a decision have for the precollege curriculum?

Just for today, let’s answer the first question in the affirmative,and see where that leads us.

What should we use as an organizing principle?

My answer:

The treatment and interpretation of data about the world.

The goal is to develop, in our students, the tools they need to understand the world.

If this is our organizing principle, what would the curriculum look like?

Curriculum

The common approach is to ask:

“What mathematics do we want college graduates to know?”

I suggest a different approach:

“What types of problems do we want college graduates

(or educated citizens) to be able to solve?”

and only then,

“What mathematics must be learned in order to solve these

problems?”

I list three types of problems.

Curriculum: Problems

Estimation:

- What is the average price of a home in Seattle?- What proportion of Americans support believe that Obama is a Muslim?- What is the difference in annual income for those with and without a

college degree?

Decision making:

- Will the addition of stannous fluoride to toothpaste reduce the number of cavities in the population at large?- Is the level of lead in a local water supply within government standards, so that the water is safe to drink?- Does the use of the death penalty lower the murder rate?

Description of trends, projection into the future:

- What do we know about global carbon-dioxide levels, and what are the implications for global warming?- How long will it take for nuclear waste to decay to a safe level, and what are the implications for our use of nuclear power?- What are the trends at work in world population, and what are the implications for our future on the planet?

Curriculum: Mathematics Necessary to Solve These Problems

Statistics:

estimation -----> confidence intervalsdecision making -----> hypothesis testing

Trend Analysis:

description of trends, projection into the future, growth and decay -----> linear functions (linear growth/decay)-----> exponential functions (exponential growth/decay)

These suggest that it would be appropriate to design a curriculum that concludes

with two college level courses:

• a course in statistics• a course in the mathematics of trend analysis

Question

Suppose that we have a curriculum that concludes with two college

level courses:

• a course in statistics• a course in the mathematics of trend analysis

What precollege curriculum will prepare students for these two courses?

Before we begin to answer this question (very tentatively), let’s look at some examples.

Example 1 Extension of Unemployment Benefits(matters of scale, clarity of a graph)

In July, after weeks of disagreement, Congress passed an extension

of unemployment benefits for those whose unemployment compensation had run out.

The cost of this extension was $34 billion dollars.

Consider the graph on the following slide.

Example 1 Extension of Unemployment Benefits

Example 2 Ethanol (clarifying an issue, matters of scale)

Can ethanol be an important part of the solution to our energy problems?

Example 3 Unpaving Roads (matters of scale, close reading, anecdote vs. trend)

Krugman article

Paved and Confused

Example 3 Unpaving Roads

Precollege Curriculum – Very Tentative Thoughts

Can we do this in less than four semesters? I think so.

We must include:

some algebra, because algebra is the language of mathematics

experience with functionsthe rule of three (table, graph, formula)linear functions (certainly)quadratic functions (because not everything is linear)students must acquire some facility algebraic

manipulation

some geometry, because we live in a world that is Euclidean(at least locally)

Beyond that, I offer the following thoughts …

Precollege Curriculum – Very Tentative Thoughts

The precollege curriculum should offer extensive experience with

each of the following:

• isolated events (ie. anecdotal evidence) vs. long term trends

• problems of scale

• clarifying issues

• natural variation vs. overall trends

• political questions vs. empirical questions

• simple writing about what the data tells us

• graphing …

Precollege Curriculum – Very Tentative Thoughts: Graphing

• Why do we graph?

• What does a graph mean?

• Students need experience interpreting graphs of relationships between two real world quantities (not x and y)

• Students need experience reading graphs in both directions:

input ------> outputoutput ------> input

• students should build tables and construct graphs of relationships between two real world quantities (not x and y)

Final Thoughts: Impediments

Suppose we decide to reformulate our curriculum along these lines.

What sorts of difficulties will we encounter?

• tradition

• articulation

• the perceived necessity of coverage

It’s a political problem in our departments, on our campuses, at the

state level.

Final Thoughts: Reason for Hope

At this time, in this place … you are uniquely positioned:

• this project

• the Gates money

• the desire for change that motivates your group and the policy makers associated with your group

• the cachet that comes with the Gates money

• the excellent leadership that I am seeing here

Final Thoughts: The Payoff

Suppose we succeed at reformulating our curriculum along these lines. What is the payoff?

We get comments like these at the end of the course.

• Graphing has always been my mathematical Achilles heel. But with the papers we’ve written, I’ve come to appreciate graphs; they give such a clear visualization of data. I like mathematical modeling because it is open-ended; unlike most math, there is no right or wrong answer when interpreting the model’s projections.

Final Thoughts: The Payoff

• I think writing about math really helps you understand what you are doing. The integration of mathematical modeling and writing has taught me how to support my arguments logically using data. My writing is much more structured and thorough in addressing all questions.

• It is really useful to write a paper that applies math to real world situations. My papers are visually stronger when presenting an argument. My writing is becoming more detailed and developed. Writing papers like these has helped my understanding of mathematics.

• This course was the most important and most fun I’ve had in any college course. This course is as close to real life as any course I have taken.

I’m done.

Unless …

Burke’s Opinion on the Death Penalty

Do you know what Mr. Burke’s opinion of the death penalty is? If so, what is it? And why do you think you are describing his position?

• No idea

• No, I don’t. Maybe he is against it.

• I think Mr. Burke is for the death penalty.

• I did not know which side you were on.

• Not sure, but probably against it because he showed the data about there is not much correlation between murder rate and execution rate.

Burke’s Opinion on the Death Penalty

• not sure – think he is against it.

• I have no idea. He seems conservative, and based on that, I would guess he is for the death penalty. Then again, he would present statistics for and against the death penalty so he could be against it.

• I’m not sure of his opinion about the death penalty

• I believe Mr. Burke believes in the death penalty. He seems like the person who is very disciplined and takes very little bs.

• No. If I had to guess … nevermind, I can’t even guess.