# Category Archives: Problems (teaching)

## Question 7

### Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.

All measures of center hide variation.

This is what makes them useful, and it is what makes them problematic.

Using the mean makes zeroes a problem in grading. Wildly divergent values (such as a zero in a gradebook) will greatly affect the mean. It is hard to argue that 2 A’s and a zero is the same as consistent D work. Yet this is how the mean plays out.

But going too far down this road will only lead to critiques of the whole system of grading students at all. I find that system to be indefensible and counterproductive. I have made my peace with it, and I try to do as little harm as possible with the responsibility I have to assign grades in my work.

All of which is to say, it is not using the mean that leads to a poor measure of achievement. It is mistaking quantitative measures for accurate ones that leads to a poor measure of achievement.

## Question 5

### “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.

Now this is when things get sticky.

It is a strong and presumptuous claim to say what an idea is.

In recent years, I have come to an understanding of why repeated addition is not the strongest foundation on which to build the idea of multiplication. But that is a far cry from making claims about what it is.

You see, any rich enough mathematical idea has multiple meanings. What is subtraction? Is it the inverse of addition? Is it the distance between two points on a number line? Is it takeaway? Subtraction is all of these, sort of.

And what is a fraction? The only definition of a fraction that is good enough to withstand the test of time and mathematical scrutiny is that a fraction is an equivalence class resulting from the equivalence relation,

$\frac{a}{b}=\frac{c}{d}$ if and only if $a\cdot d=b\cdot c$

Is that what a fraction is?

No. But I am off task.

I suspect that my answer may vary from some others out there. (Although perhaps it will not.)

Repeated addition is shaky ground for establishing multiplication because it doesn’t capture the unique structure that multiplication represents.

There is additive structure, and there is multiplicative structure. Additive structure is about comparisons and changes involving the same units. Apples plus apples gives apples. Miles plus miles give miles.

Multiplicative structure is about comparisons and changes involving different units. Hours times miles per hour gives miles. Three different units; one of them a unit rate. Always.

These are related structures but they are different.

Multiplicative structure is captured better by this idea: $A\times B$ means A groups of B (I am pretty sure I first ran across this particular characterization in Sybilla Beckmann’s textbook for math courses for elementary teachers). A, in this interpretation, is expressed in one unit. B is the unit rate (things per group). The product is expressed in a third unit.

This difference shows up in the following conversation between a mom and her daughter as they count the number of things in this array of meatballs.

Image from The New York Times.

Maya counted the top and bottom, 4 + 4 = 8. Then she counted L and R. 3 + 3 = 6. 8 + 6 = 14. 14 + 2 in the middle = 16. When I asked her why, she said, “Because you double count the corners when you count an array.”

She asked me to count so she could show me how. I counted 4 across the top and 3 down the side. “See, Mommy! You’re counting the corner one twice.”

Why do we count the corner one twice in this scenario? This seems to violate a fundamental principle of counting—one-to-one correspondence. One number word for each object, and one object for each number word.

The answer is that mom really did not count the corner meatball twice. The first time, she counted the meatball to establish that each row has 4 meatballs. The second time, she counted the rows. There are 3 rows, so there are 3 groups of 4 meatballs.

Much, much more on arrays in many places in my writing. Especially these:

Beyond the textbook wrap up (or What does this have to do with mathematics?)

Twister (on sister site, Talking Math with Your Kids)

## Question 4

### Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002

One sixtieth is the biggest of these, as the others are between one and two one-thousandths (or between one one-thousandth and one five-hundredth).

Then, in descending order, we have .002, .00156, .0015 and .001.

It is common that students will order the decimals this way:

.00156, .0015, .002, .001

I cannot predict where a person who does this will place one sixtieth.

Nonetheless, when we read these decimals as point zero zero one five six, we encourage students to ignore place value, and we encourage the misapplication of whole-number rules to the right of the decimal point.

We really do need to use place value language for decimals in classrooms. One-hundred-fifty-six one-hundred-thousandths.

Much, much more about this and related ideas in the Triangleman Decimal Institute posts from last fall. Short version: learning decimals is WAY more complicated than most people have any reason to imagine.

## Question 2

### “Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

This question is a strange one. It really isn’t how I would define problem solving, and I certainly wouldn’t include equality as a major component underlying problem solving.

Nonetheless…

I suppose he is getting at the idea that expressing equations in equivalent forms sometimes reveals different details of a problem.

For instance, I have created a new measure for cylinders: the circumradial measure. You add the radius and height. Then multiply this sum by the circumference.

$C_M = (r+h) \cdot (2\pi r)$

In exploring this measure, one might end up restating this formula in equivalent terms, as:

$C_M = 2\pi r^2+2\pi rh$

This is more recognizable as a formula for surface area of a cylinder. The form of the equation affects how we think about the relationship it expresses.

What does the equal sign mean?

This is an important question. There is lots of research about it (CGI folks have worked on it, for instance). Three quick points:

1. The equal sign means that the two things on either side have the same value as each other.
2. We often teach in ways that lead students to think that the equal sign means and now write the answer.
3. You can’t really understand much about algebra with the conception that (2) fosters. You need (1).

Finally, there are deep ideas underlying the equal sign. Equivalence is the mathematical way of talking about sameness. Stating the meaning of sameness precisely in mathematics turns out to be tricky and interesting work, and is a foundation of modern algebra.

## Wiggins questions #1

Math folks online have been all atwitter (heh) about a recent post by Grant Wiggins on conceptual understanding in math. Within that post (which I have not read in its entirety for reasons to be explained later), he proposed a series of questions that we should offer students as a way of opening our minds to what conceptual understanding means in mathematics.

Max Ray expressed a wish for some math ed bloggers to answer these questions in writing. I am obliging. One question at a time. One per week. I have not read the post so as not to bias myself.

I reserve the the right to critique the questions along the way.

## Question 1

### “You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)

Fact families.

Division is defined in relation to multiplication. For every one multiplication fact, there are two division facts.

3•2=6 is matched with 6÷2=3 and 6÷3=2.

Zero is a special case. 0•2=0, 0•5=0, 0•a=0 for all possible values of a.

This is no problem for multiplication. But it is a problem for division.

0÷0=2 would be a fact from the fact families. 0÷0=5 is another one. 0÷0=a for all possible values of a.

That is, 0÷0 can equal anything. And if it equals anything, it actually equals nothing. So 0÷0 is undefined.

More generally, though, 2÷0, 5÷0 and a÷0 for all possible values of a are problematic. Let’s say we decide that a÷0=12 (and let’s say that isn’t 0, since we took care of that case already). Then the fact family tells us that 0•12=a. But 0 multiplied by anything is 0. So 0 can’t be 12. But it can’t be anything else either. So a÷0 is undefined.

Conclusion: We cannot divide by zero for two reasons.

1. Division is defined in relation to multiplication, and
2. Zero has a special role in multiplication: 0•a=0 for all values of a.

We can use intuitive notions to establish that division by zero is a strange beast, but we can’t really firm up why without these more formal mathematical ideas.

## Brief thoughts on being ready for calculus

A smart friend (whose permission I have not asked) read an article of mine that will be published in Mathematics Teaching in the Middle School sometime soon. The article is based on my NCTM talk last spring, titled “They’ll Need It for Calculus”.

For clarification: are you arguing that the sorts of problems that you point to will help students better understand calculus, or that these sorts of problems will help students do better in their calculus classes?

I was pretty sure that you were making the first argument, but not the second.

My reply, which I stand by, is this:

That these two things are different from each other is a pretty damning critique of the whole affair, is it not?

You know what will help them do well in their calculus classes? Memorizing about 20 of these:

## The Fundamental Theorem of Calculus

I have taught Griffin (9 years old) the Fundamental Theorem of Calculus.

That is…

$\frac{d}{dx} \int_{a}^{x} f(t)dt=f(x)$

Details and discussion coming soon.

In the meantime, see Kristin’s related post.

Full disclosure. Griffin was paid a sum of \$0.25 for his performance.