Why Teaching Math Is So Hard

Dan Kennedy Baylor School

Teaching math is so hard because…

Mathematics is not something you learn.

Mathematics is something you do.

This makes mathematics different from virtually every other subject in the traditional school curriculum.

Here are some things that some people can do:

Juggle.

Waterski.

Play the violin.

Dunk a basketball.

Run a marathon.

Mathematics.

Other people cannot.

People never say, “I never could do English,” or “I never could do History.”

These are subjects you learn.

People do not readily excuse themselves for not learning.

And who defines what it means to “do math?”

Math teachers.This is a big difference between the ability to do mathematics and the ability to read!

Someone who can read this sentence knows how to read.

How about this sentence:

Ontogeny recapitulates phylogeny.

What does it mean to do mathematics?

2. 24 6 3. Solve for : ( 2)(2 3) 49.x x x 4. Find cos( /3).5. Find the product: 874539 374958.

6. Find 239121.

7. What is ?ie

1. 5 2

Do we teach students to do mathematics or to understand mathematics?

2 1 1

1 4 2

1 1 0

2 1

1 4

1 1

+ + +– – –

0 + 2 + 1 – (–4) – (–4) – 0 = 11

Compare this to:

So how do we justify teaching a meaningless computational trick that is ONLY good for computing 3-by-3 determinants?

It does not generalize to higher orders.

It does not even suggest anything important about how determinants work!

Teaching math is so hard because…

The people who must identify which students cannot do mathematics are the same people entrusted with teaching them.

Their math teachers!

A sobering thought:

There are people walking the streets of Buffalo right now who became convinced years ago that they could not “do math” -- because they could not “do” some things that we no longer even teach today!

For example, here is what we were doing 45 years ago:

Theorem: (b + c) + (–c) = b Statement Reason 1. b and c are real numbers Hypothesis 2. b + c is a real number Axiom of closure for addition 3. –c is a real number Axiom of additive inverses 4. (b + c) + (–c) = b + [c + (–c)] Associative axiom of addition 5. c + –c = 0 Axiom of additive inverses 6. b + [c + (–c)] = b + 0 Substitution principle 7. b + 0 = b Additive axiom of 0 8. b + [c + (–c)] = b Transitive property of equality 9. (b + c) + (–c) = b Transitive property of equality

Teaching mathematics is so hard because:

This is America.

“The problem is, this is such an unusual country.”

-- Jan de Lange, Director, Freeudenthal Institute for Teaching and Learning, Utrecht, The Netherlands

“My Ph.D. was in mathematics; by most standards, I was very 'well trained.' Nonetheless, the education that I received was in many ways impoverished.”

-- Dr. Alan Schoenfeld, Reflections on an Impoverished Education, from Mathematics and Democracy: The Case for Quantitative Literacy, NCED 2001

Teaching math is so hard because…

Our professional leaders cannot agree on what mathematics we ought to be teaching!

In 1983, the College Board published a small booklet called Academic Preparation for College. It listed the following “basic academic competencies” for mathematics…

• The ability to perform, with reasonable accuracy, the computations of addition, subtraction, multiplication, and division using natural numbers, fractions, decimals, and integers.

• The ability to make and use measurements in both traditional and metric units.

• The ability to use effectively the mathematics of:− integers, fractions, and decimals;− ratios, proportions, and percentages;− roots and powers;− algebra;− geometry

• The ability to make estimates and approximations, and to judge the reasonableness of a result.

• The ability to formulate and solve a problem in mathematical terms.

• The ability to select and use appropriate approaches and tools in solving problems (mental computation, trial and error, paper-and-pencil techniques, calculator, and computer

• The ability to use elementary concepts of probability and statistics.

Ironically, it was that very same year, 1983, that another document was published, destined to change the rules for high school academic preparation for years to come…

A Nation at Risk: The Imperative for Educational reform

From A Nation at Risk:

“If an unfriendly foreign power had attempted to impose on America the mediocre educational performance that exists today, we might well have viewed it as an act of war.”

The Math Wars:

Quantitative Literacy vs. Calculus Preparation

Theory vs. Applications

Rote vs. Constructivism

Tracking vs. Mainstreaming

Etc.

No matter what mathematics we choose to teach or how we choose to teach it, some people will believe that we have made the wrong choice. Some of them will say so. And they might be right.

Teaching mathematics is so hard because:

Everybody recognizes the importance of mathematics, even if they do not understand it.

There are only three R's:

Reading ‘Riting ‘Rithmetic

God help us; we are one of them.

If Johnny can read and Johnny can write, then the fate of Johnny will be determined by whether Johnny can do 'rithmetic.

The mathematics teacher is the Gatekeeper.

Whether Johnny becomes a wealthy CEO or a penniless beggar is entirely up to you. Have a nice day.

Teaching mathematics is so hard because:

Most people believe mathematics is constant over time.

Unlike any other science.

Sort of like religion.

Thou shalt factor.

Parents realize that physics, chemistry, biology, history, and geography are different today from when they were in school.

So they allow those subjects to change.

Not so mathematics.

The following things upset people:

Inability to do their child's homework.

Any appearance of the words NEW and MATH in the same sentence.

Calculators.

The assertion that mathematics is for everyone.

Some people seem to think that pre-college mathematics is timeless.

If it was important for our parents, how can it be unimportant today?

But technology has been rendering our parents’ mathematics obsolete for decades.

For example, consider log tables.

Here is a 1928 College Board mathematics achievement exam.

It looks a lot like today’s college placement tests.

But that is another talk.

And speaking of logarithms…

Teaching mathematics is so hard because:

Assessment is out of control.

NYSRegents

What should we assess?

•What we value

•Learning

I learned an important fact about classroom assessment when I began teaching AP courses:

It changes the entire classroom dynamic when the teacher honestly does not know what will be on the test.

The teacher has no other option but to teach the students how to think for themselves!

Why students don’t think on tests:

•Thinking takes time.

•Thinking is only necessary when you cannot do something “without thinking.”

•If you can do something without thinking, you can do it very well.

•Students who can do something very well have been well-prepared.

•Therefore, if you prepare them well, your students will proceed through your tests without thinking!

This is how the educational game is played:

•We show the students how to do math.

•We let them practice at it for a while.

•Then we give them a test to see how well they can mimic what we did.

The game is won and lost for BOTH of us on test day.

We must value correct mathematics more than we value correct answers.

We must let our students know it.

What we don't see can hurt our students.

Good algebra:

2

2

2

3( 1) 12

( 1) 4

( 1) 4 0

( 1 2)( 1 2) 0

( 3)( 1) 0

3 0 or 1 0

3 or 1

x

x

x

x x

x x

x x

x x

Bad algebra:

2

2

3( 1) 12

3 3

( 1) 9

1 3

1 or 3

1 and 3 don't work.

1 and 3

x

x

x

x x

x x

Teaching mathematics is so hard because:

Articulation is out of control.

High School College

Once upon a time there were 11 AP courses.

One of them was in mathematics.

Today there are 37 AP exams in 20 subject areas.

Three of them are in mathematics.

“Currently, the greatest growth in the high school curriculum is in courses that have traditionally been taught in colleges.

“The greatest growth in the college curriculum is in courses that have traditionally been taught in high schools.

“It is not clear that either institution is serving its clients very well.”

--Dr. Bernard Madison, Chair of the MAA Task Force on Articulation, 2002

0

50000

100000

150000

200000

250000

300000

350000

1950 1960 1970 1980 1990 2000 2010 2020

1955285 exams

196710,703 exams

198651,273 exams

1993101,945 exams

2003212,794 exams

2008276,004 exams

Unofficial 2009 point

What once was designed to be a program for validating college-level study of a few courses by a small percentage of exceptional students for the purposes of college placement

has gradually morphed into a program for validating a rigorous high-school curriculum in a wide variety of courses for the purposes of college admissions.

What effect is this AP scramble having on the students?

On the one hand, they are condensing or skipping foundational courses, so they are less prepared for advanced courses.

On the other hand, they are taking more advanced courses, assuring that their lack of preparation will be exposed!

The Race to Begin Algebra I

8th Grade… 7th Grade… 6th Grade…

College Mathematicians want our best high school students to have:

Deeper understanding

More proofs

Harder problems

Challenges like the AMC Competitions

No rush through foundational courses

No courses in high school beyond freshman calculus

What some colleges are doing now:

Demanding 4’s or 5’s for credit

Changing freshman calculus to a course that assumes students have already studied calculus

How students are reacting:

They are less likely to seek credit.

They are more likely to play it safe.

They re-take calculus in college…

…unless they opt out of college mathematics and “take the credit and run.”

Other secondary-to-college articulation problems: Technology.

Pedagogy.

Placement tests.

Math wars continued.

Graduate students.

Etc.

Teaching mathematics is so hard because:

Technology is out of control.

Technology has transformed our classrooms…

Graphing calculators Computers Math software

TI Navigator Smart Boards TI Smart View

We must honestly assess every advance in technology for its appropriate uses in the classroom.

As noted before, we must also determine what is meant by important mathematics.

2 4

2

b b ac

a

Important?

Expendable?

The Skandu 2020:

It has the potential to scan any “standard” algebra textbook problem directly into its memory for an analysis of key instructional words, solve it with CAS, and display all possible solutions.

It will do the same for “standard” geometry textbook proofs.

The Skandu 2020

(Not its real name)

HA HA! I’m only kidding.

At least for now.

If there is no Skandu 2020 in our classrooms in five years, I doubt it will be because the design is impossible.

It will be because teachers do not feel that it would improve the teaching and learning of important mathematics.

Obviously, the CAS conversations continue.

It just might be time for another change!

They are not just about technology, nor should they be. They are about the teaching and learning of mathematics.Stay tuned. Be informed. Join the conversation.

Now it may be time to change this…

Imagine…the SchoolPad 2012

Teaching mathematics is so hard because:

There is so much more to teaching mathematics than just teaching mathematics.

WHAT MATHEMATICS TEACHERS HAD TO DO FOR HIGH SCHOOL STUDENTS IN THE PAST

1. Prepare the college-bound for calculus.2. Prepare the non-college-bound for employment.3. Identify which students were which.

WHAT MATHEMATICS TEACHERS HAVE TO DO FOR HIGH SCHOOL STUDENTS TODAY

1. Prepare them all for calculus.2. Prepare them all for employment.3. Prepare them all for a life dominated by computer technology.4. Prepare them all to pass state-mandated competency tests.

WHAT HIGH SCHOOL MATH TEACHERS HAD TO KNOW IN ORDER TO TEACH PROBLEM SOLVING IN THE PAST

1. Algebra.2. Geometry.3. Probability.4. Trigonometry.

WHAT HIGH SCHOOL MATH TEACHERS HAVE TO KNOW IN ORDER TO TEACH PROBLEM SOLVING TODAY

1. Mathematics.2. Astronomy.3. Economics.4. Statistics.5. Physics6. Biology.7. Etc.

The former paradigm:

Learn the mathematics in a context-free setting, then apply it to a section of “word problems” at the end of the chapter.

I can do the math…I just can’t do

the stupid WORDPROBLEMS!

Graphing calculators have made word problems more accessible to students. The emphasis has shifted much more toward modeling.

An example of a problem that used to be hard for students but that now is easy:

Three families order lunch at a fast food restaurant. The Jacksons pay $19.40 for 5 hamburgers, 3 small fries, and 5 soft drinks. The Garcias pay $11.05 for 3 hamburgers, 2 small fries, and 2 soft drinks. The Lorenzos pay $21.25 for 6 hamburgers, 4 small fries, and 3 soft drinks. How much would a person pay at this restaurant for one burger, one small order of fries, and one soft drink?

5 3 5 19.40

3 2 2 11.05

6 4 3 21.25

h f d

h f d

h f d

After modeling the problem, there are two easy methods of solving it:

For teachers, these changes have not come easily.

We have made changes, hopefully for the better.

You might think we could pause, reflect, and enjoy what we have accomplished.

But that is not how technology works!

Here are a few changes we have yet to make…

We need to stop thinking of a student’s mathematics education as a linear progression of skills that must be mastered.

Arithmetic Fractions Factoring

Equations Inequalities Radicals

Geometry Trigonometry

FunctionsCalculus Statistics

Proofs

If students who have not mastered our traditional mathematics skills can solve problems with technology, should it be our role as mathematics teachers to prevent them, or even discourage them, from doing so?

Dr. Retro, I’ve got it!That does not count,

Miss Nouveau. Put that

thing away.

We ALL must teach fundamental mathemics skills to our students, who probably will not have mastered them.

Patiently. Casually. As a matter of course.

Mr. Oiler, if there are twice as many dogs as cats, doesn’t that mean

that 2d = c?

Mr. Jones, if that is all you learned last

year, you had better drop this course

before it drops you.

Good question, Mr. Jones. Let’s see

what would happen if there were 4 cats…

We must honestly confront the goals of our current mathematics curricula.

Just because it is good mathematics does not mean that we have to keep teaching it.

Nor is it necessary, advisable, or perhaps even possible to teach everything that is in your textbook.

We should treat every mathematics course as a history course – at least in part.

We will probably always teach some topics for their historical value.

MATHEMATICS

Culture

QuantitativeLiteracy

Technology

Mental Discipline

Research(College Prep)

But whatever you do, have fun!

Learning is enjoyable.

So is teaching.

Any other philosophy of teaching and learning is counterproductive.

Even in mathematics!

dkennedy@baylorschool.org