116_Why Teaching Math Is So Hard

This makes mathematics different from virtually every other subject in the traditional school curriculum... Teaching math is so hard because…Our professional leaders cannot agree on wh

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:

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?

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

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.

Teaching math is so hard because…

The people who must identify which

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

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

Teaching mathematics is so hard because:

“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!

• 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;

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:

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

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.

And speaking of logarithms…

Teaching mathematics is so hard because:

Assessment is out of control.

NYS Regents

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

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

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

x x

Teaching mathematics is so hard because:

Articulation is out of control

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.

“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.”

1986 51,273 exams

1993 101,945 exams

2003 212,794 exams

2008 276,004 exams



Unofficial 2009 point

What once was designed to

be a program for validating

college-level study of a few

What effect is this AP

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

8 th Grade… 7 th Grade… 6 th 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:

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

Teaching mathematics is so hard because:

Technology is out of control.

Technology has transformed our classrooms…

Graphing calculators Computers Math software

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

a

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

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.

They are not just about technology, nor

should they be They are about the teaching and learning of mathematics.

Now it may be time to change this…

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

I can do the math…

I just can’t do the stupid WORD

PROBLEMS!

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:

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.

Equations Inequalities Radicals

Geometry Trigonometry 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

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

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

Culture Technology

But whatever you do, have fun!

dkennedy@baylorschool.org