Category Archives: Problems (math)

Wiggins question #7

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.

Wiggins question #5

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)

 

Wiggins question #4

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.

Wiggins questions #2

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.

Ginger ale (also abbreviated list of Standards for Mathematical Practice)

We have some of these mini cans of ginger ale in the house this week. I am not sure where they came from; only that my wife bought them. Normally we only have sparkling water around, not pop (nor soda, nor…)

ginger.ale.1

So I’m looking at the can instead of grading like I should be and I notice the “25% fewer calories than regular ginger ales” claim.

And I think what any skeptical consumer ought to think. Sure fewer calories in the mini can. Duh.

Then I see this:

ginger.ale.2They have controlled for the size of the can. Nice. This one has 60 calories per 7.5 fl. oz. Regular ginger ales have 90 calories per 7.5 fl. oz.

I am briefly satisfied. And impressed.

But wait! 60 is 25% less than 90? ARGH!

Two possible explanations:

  1. 25% means at least 25%, and Seagram’s chose this nice simple number over the more complicated 33\frac{1}{3}%.
  2. It really is exactly 25%. But we know that calorie counts are rounded to the nearest 10 calories.

This second explanation leads to a sort of lovely task. How can we characterize the set of possible calorie counts for 7.5 fl. oz. of Seagram’s and of regular ginger ale so that, (a) the counts round to 60 and 90, and (b) one number is exactly 25% less than the other?

Extra credit: Which standards for mathematical practice are you using as you solve?

Double extra credit: Which of my abbreviated list of standards for mathematical practice (see below) are you using as you solve? And which was I using as I gazed at my can of ginger ale?

Prof. Triangleman’s Abbreviated List of Standards for Mathematical Practice.

PTALSMP 1: Ask questions. Ask why. Ask how. Ask whether your answer is right. Ask whether it makes sense. Ask what assumptions you have made, and whether an alternate set of assumptions might be warranted. Ask what if. Ask what if not.

PTALSMP 2: Play. See what happens if you carry out the computation you have in mind, even if you are not sure it’s the right one. See what happens if you do it the other way around. Try to think like someone else would think. Tweak and see what happens.

PLALSMP 3: Argue. Say why you think you are right. Say why you might be wrong. Try to understand how someone else sees things, and say why you think their perspective may be valid. Do not accept what others say is so, but listen carefully to it so that you can decide whether it is.

See also my Desmos graph of this relationship.

Betty’s Pies

In Two Harbors, on Minnesota’s North Shore of Lake Superior is a popular stopping point for vacationers: Betty’s Pies.

Photos - 6719

This restaurant moves a tremendous amount of pie each year. How much? Well, I’m not really sure because there was a smudge on the fact sheet on my menu.

Photos - 6717

Middle of the image: “Our bakers make…” (Please ignore the bold, oversized italic typo in the last line)

So how about it? How many pies do you think Betty’s Pies makes each year? Let me know how you think about it in the comments, OK?

Then you can click here to see the rest of Betty’s sign. Is it a good representation of how she cuts her pies?

Finally, you can view an unsmudged version of the menu to check your answer. (No cheating!)