Math

Meeting Mathematicians

Introduction

In 2002, Paul Lockhart wrote his classic critique of American K-12 mathematics education.

It is certainly true that certain math topics are important for quality of life. People are happier and more capable as citizens if they understand ratios, percents, measurement unit conversions, health formulas, kitchen math, basic probability and geometry, interest, investing, retirement planning, and enough about statistics to notice when someone is lying with statistics.

I have already written that textbook. It is a free resource that can help you if your math education neglected any of those topics.

As Paul Lockhart points out, these topics are learned by practicing rules and routines discovered by other people. Students are taught the answers to situations; they do not explore the situations from scratch. There is no open-ended exploration, which means there is nothing like what professional mathematicians actually do.

The next challenge is to put together the complimentary curriculum. Math students deserve a class that asks them to behave as mathematicians really behave: to invent or find situations and discover answers through creativity and open-ended exploration.

Such a math class requires two kinds of structure.

First, as a high school or college class, it needs assessment. How are students graded when they are discoverers and inventors? Since traditional math tests only measure various kinds of following directions, other types of assessment are needed.

Second, it must have a Big Plot that provides a scaffolding for students to lean on. Most Americans have never been shown how to behave like a mathematician and would not know where to start.

This web page is my "thinking aloud" about such a class. It is my first draft at the other math class we all should have had in high school.

Assessment

Attendance is worth 15% of the overall grade. I would like it higher, but at the community college where I teach this is limited.

Each student, each week, must present one of his or her mathematical discoveries before the class. Together these are worth 50% of the overall grade (so 5% each for a ten-week term.) A student who does not present during a certain week, for any reason, can "make up" that assignment during a future week by sharing the missed presentation and an extra presentation. Presentations will be graded using the following five-point rubric:

1. Is the math accurate?
2. Is the presentation clear?
3. Is the presentation organized and coherent?
4. Is the presentation interesting?
5. Do most students learn something they did not already know?

A term paper is worth the remaining 35% of the overall grade. This paper must answer specific questions. Which means I need to think of some.

The Historical Context

The most authentic way to learn how to behave like a mathematician is to learn about the types of thinking done by historical mathematicians. The curriculum will examine famous mathematicians from many cultures and use the issues they dealt to generate ideas for exploration and investigation.

Ahmes (Egypt in about 1650 B.C.E.) - Arithmetic with Clay and Forks

Multiplying two values by hand can be a lot of work. Ahmes had a shortcut. He would designate one of the values as the multiplier, then create a "doubling table" for the other value (the multiplicand). This was quickly done using a fork with as many tines as the multiplicand. Adding the appropriate rows from the doubling table yielded the product.

How could have Ahmes done division? Exponents? Formulas?

Ahmes (Egypt in about 1650 B.C.E.) - The Nicest Fractions

The ancient Egyptians only used fractions with a numerator of 1. So Ahmes could write down lengths of 1.5 cubits (1 + 1/2) or 1.125 cubits (1 + 1/8).

Now, Ahmes would have also worked with 5/8 by writing it as 1/2 + 1/8. But he probably would not have used 13/10,000 = 1/1,000 + 1/5,000 + 1/10,000 because it was a somewhat complicated and very small value, and measuring distances with ropes and wheels is simply not that accurate.

How would Ahmes have done fraction arithmetic? How much fraction math could Ahmes have done without going way beyond what was practically needed for construction, irrigation, and taxation?

Pythagoras (Greece in about 500 B.C.E.) - Math Facts with Pebbles

Pythagoras used pebbles to prove facts about numbers. For example, he could place pebbles into grids to show that the square of an even number was even, whereas the square of an odd number was odd.

What else could have Pythagoras have discovered about even and odd? How about if he used a different color pebbles for negative numbers? What if he was willing to break a pebble into equal pieces to work with fractions?

Aristotle (Greece in about 330 B.C.E.) - Geometry with Diagrams

Earlier Greek mathematicians had pioneered geometical diagram-thinking, but Aristotle wrote the book on the topic.

He mostly used two types of diagrams. The first variety consist of multiple copies of the same shape.

(Insert diagram showing area of a trapezoid.)

The second variety consist of one shape inside another shape, usually with an extra a line or two. Here is a complicated example of a question answered with the second kind of diagram.

This diagram includes a right isosceles triangle. Its hypotenuse is incommensurable with its other sides. This can be shown with a proof by contradiction.

Assume the diagonal (purple) and other sides (blue) are both whole-number multiples of a largest common factor. At least one of the multiples must be odd, or that "largest" common factor would not be the largest. As measured with the multiples, the purple and blue lines have whole-number lengths.

So the area of the brown square (the square of the hypotenuse) has whole-number area, and its area is twice the area of the blue square, so the area of the brown square must be even. If a square's area is even the length of its sides must also be even, which shows that the area of the brown square must not only be even but also be a multiple of four. But then the area of the blue square, being half, is also even. That would mean the blue line also has even length, which contradicts our assumption.

Note that if the blue square has sides of length 1, the proof shows that there is no fraction equal to the purple diagonal with length square root of two. So with a simple diagram Aristotle proved that the square root of two is irrational.

Using either kind of diagram, the favorite game among Aristotle and his peers was to draw the diagram and then explore which questions could be answered about the diagram's lengths, angles, and areas. Draw your own diagrams or use the examples below. What questions do the diagrams lead to?

How about diagrams drawn on a sphere instead of a flat paper? For example, a triangle made by connecting the north pole with two places on the earth's equator is a triangle with three right angles!

More to come...