Answering the “Conceptual Questions”

So Grant Wiggins blogged here about conceptual understanding in math classes, including a challenge for teachers to ask students these 13 questions and report the results. His claim was that no student would get every question correct. Since I’m in bed recuperating, I figured I’d take these on, no Googling allowed, and see if at least one former teacher could get all 13 correct, without resorting to jargon. Here we go!

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

You can divide by zero. To quote Rachel McAnallen, “You can do anything you want — YOU’RE the one with the pencil!” The problem is, there’s no way (assuming standard rules for arithmetic) to divide by zero in a way that gives a unique and logical answer. Division is the inverse operation to multiplication. Dividing a number by 2 gives an answer that can be multiplied by 2 to get the original number. So dividing by zero should give us an answer that can be multiplied by zero to give the original number. If I’m trying to divide 10 by 0, I should get an answer that can then be multiplied by 0 to get 10. However, since anything by multiplied by 0 gives 0, there is no real- (or complex-) value that we can multiply by zero to get 10. So the solution is undefined, which means we don’t have a way to describe that value (though if we wanted to create a new number system, we could come up with one — but it’s unlikely our new system would be as useful).

A nice concrete analogy helps. Imagine a table with eight 12-pack cans of soda. If we pick up 0 of those 12-packs and put them in our fridge, how many cans of soda are in the fridge? This models 0 groups of 12 cans, or 0 x 12, just like putting all eight in the fridge models 8 x 12, or 96 cans. So multiplication’s good.

Now open one of the 12-packs. Split the cans up into four groups and put one of those groups in the fridge. That models 12 cans divided into 4 groups, which gives 3 cans per group, so 3 cans in the fridge. (notice how this also helps us model dividing by 3/4… for exploration later). But if you try to split the 12 cans up into 0 groups, it’s very unclear what you should be doing. You can’t define the number of cans in each “group” because any number of cans in 0 groups leads to 0 cans, not 12. And soda cans, as matter, can neither be created nor destroyed.

Except like this

Except like this


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

Okay, this one’s boring. The = sign means that the value of the expression on the left of the sign is equivalent to the value on the right hand sign. When we solve a complicated equation like 3x + 4 – 2x = 6x + 8, all we are doing is changing that original equation (which looks tricky) to a set of “simpler” equivalent statements, using the axioms of equality and various properties of arithmetic. So the LHS can be rewritten as 3x – 2x + 4 via the commutative property, which is equivalent to x + 4 via the distributive property, so the whole equation is equivalent to x + 4 = 6x + 8, and we can proceed, adding -x and -8 to each side (properties of additive equality) to obtain -4 = 6x, and finally divide each side by 6 (division property of equality) to get x = -4/6.

At each step, we replaced or previous statement with a new statement, but by following the rules of algebra, we were careful to make sure each new statement was equivalent to the previous statement.

3)   You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.

I’m gonna assume a less rigorous definition of “prove” here, but the simple answer is that multiplication and division are inverse operations, so dividing by a number is the same as multiplying by its inverse, which via the inverse property, can be found by “inverting” the number (changing 3 to 1/3, changing 3/4 to 4/3, etc.). That’s rather unsatisfying.

Think of the 12-pack of soda. Multiplying by 1/4 means taking one-fourth of the cans. That’s 3 cans. Dividing by 4 can mean two things: Either splitting the 12 cans into four groups and counting the number of cans in one group OR splitting the 12 cans into groups of four and counting the number of groups. You can see that these processes give equivalent answers. But how can we model 12 divided by something like 2/5? How can I split something up into 2/5 groups? Well, soda’s a poor metaphor here, but I’ll keep going with it, because I’m thirsty.

Let’s first divide 12 by 1/2. We can think of this as splitting the 12 cans into 1/2 groups and counting the number of cans in one group OR splitting the 12 cans into groups of 1/2 cans and counting the number of groups. The second seems more intuitive, because I can’t make sense of a 1/2 group easily, but I can make sense of a 1/2 soda can. So I pour each soda into two glasses and count the number of groups of 1/2 soda can (number of groups of 1 glass). There should be 24 glasses, or 24 groups of 1/2 soda cans, so 12 ÷ 1/2 = 24. 

Now let’s look at 12 ÷ 2/5. Again, we can split 12 cans into 2/5 groups and count the number of cans (confusing!) OR we can split 12 cans into groups of size 2/5 cans, then count the number of groups (maybe confusing, but more doable). First we make our groups of size 2/5 by pouring each soda into 5 glasses, then combining two glasses together. For the first step, we wind up with 12 x 5 = 60 glasses. Once we combine, we are left with 30 glasses. This is 30 groups of a 2/5 soda can each, so 12 ÷ 2/5 = 30. Notice this could be modeled by multiplying by 5, then dividing by 2, which is 12 x 5/2. Neat!

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

1/60, .002, .00156, .0015, .001

Trivial check for place value understanding. Would be more interesting with 1/600 as the second option, which would make it the same order of magnitude as the others.
[.002, 1/600, .00156, .0015, .001]

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

Multiplying by whole numbers works as repeated addition, but how do you explain multiplying by 1/4? Multiplication is better understood as stretching, scaling, or intensifying. Keith Devlin gave a great example of a volume knob. Twisting the knob cannot be described as repeated addition, but is a clear example of a multiplicative relationship.

This often causes misconceptions when teacher talk about multiplying by 1 as well. “Now, multiplying 6 by 3 is adding 6 to itself 3 times,” says the teacher, and the poor student sees 6 x 1 on a later problem and writes “6 + 6” for 6 added to itself once. It’s much better to talk about something of length 6 stretched to 3 times its length, so that students more easily recognize 6 x 1 as an object “stretched” to its full length, 6 x 5/4 as something stretched to be 25% longer than its original length, and 6 x 2/3 as something shrunk to 2/3 of its original length.

Repeated addition can be modeled by multiplication, but multiplication is not confined to repeated addition, as examples using rational values quickly demonstrate.

6)   A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?

Did students learn how to round in a context? Or do they see 30.375 and round that to 30, every time? You need an extra table to seat the 3-person “overflow.” (My favorite student response to a question like this is — “You only need 30, because there’s no such thing as .375 of a person.” What’s that misconception?)

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.

SBGers should own this question. Here are my two reasons.
1) Mean is sensitive to outliers (mostly, zeroes).

Assume a classroom where every grade has equal weight. Here are two sets of grades:

Student A: 100, 100, 100,  0,  100
Student B:  81,  80,   80,  80,  80

Student B has a higher grade than Student A. Do you think Student B has more mastery of the material? Do you think Student B is better prepared for the next math course?

Kids fail classes all the time due to compliance rather than competence, because it’s easier to give a 0 than track the student down and find out what they really know.

2) Mean is NOT sensitive to time.

Another set of grades, in sequence:
Student A: 25, 25, 50, 100, 100
Student B: 100, 100, 50, 25, 25

Same questions: Who has better mastery? Who will be better prepared for the next course?

Using a simple mean has a disparate impact on students who start “lower” than other students. Even if they do everything right and learn the material as well as anyone else, those low early grades will drag them down. Student A and Student B have the same grade, yet Student A seems like a star student and a great improvement story, while Student B is in a total tailspin. Thanks, mean!

8)   Construct a mathematical equation that describes the mathematical relationship between feet and yards. HINT: all you need as parts of the equation are F, Y, =, and 3.

F = 3Y, but dear lord this is SO HARD if you try to jump straight to the equation without thinking. Yet another nod to the CME project for having a process that starts with numerical “guesses” that are then refined and generalized.

9)   As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.). The order of operations is a convention. X(A + B) = XA + XB is the distributive property. It is a law. What is the difference between a convention and a law, then? Give another example of each.

Conventions are how we doing things that are largely irrelevant to the core of our mathematical system. Laws are important basic properties of our number system which have far-reaching ramifications if changed.

If we changed PEMDAS, all that would happen is that notation would be slightly different. For example, if I was going to model taking 5 cans of soda from one pack, and 3 cans of soda from another pack, then doubling the number of cans, I would use the expression 2(5 + 3). If we changed PEMDAS to PEASMD, I would model it with 2*5 + 3, as the parentheses would no longer be necessary. We’d write equations and expressions in different but equivalent ways if PEMDAS was changed, but it wouldn’t change the nature of our number system. However, if X(A + B) wasn’t XA + XB, we’d be in a whole different world. We’d be talking about some system that was NOT the real numbers. (Similar to how when we change the commutative property so that X*A  A*X, we talk about matrices or other objects, not numbers).

Another convention is “simplified form.” Why is 1/2 the preferred way to write 5/10? We could make a convention that all fractions be written with denominator 10. It’d be the same math, just different optics (for example, 2/3 would be written as 6.66…./10). Why do we rationalize radicals instead of leaving √(2)/2 as 1/√2? We could easily switch that convention, with no far-reaching ramifications.

But change a law like 0*a = 0, and we’ve shook mathematics to its core. [EDIT: yes, this isn’t a “law”, but rather of consequence of identity, inverse, & distributive properties, but I still think it gets the point across]

10)  Why were imaginary numbers invented? [EXTRA CREDIT for 12thgraders: Why was the calculus invented?]

Solving cubics with real solutions sometimes requires taking square roots of negative numbers. Later in the process, those square roots of negatives are re-squared, so you still wind up with real answers, so no harm no foul. It was only later that imaginary numbers started to be accepted as solutions themselves. After I post, I’m Googling this to brush up on my history.

11) What’s the difference between an “accurate” answer and “an appropriately precise” answer? (HINT: when is the answer on your calculator inappropriate?)

I assume this is about “exact” answers, like 2/3 or √2 as opposed to “approximate” answers like .667 or 1.414, the latter type of answer being inappropriate when the teacher says to give exact answers, and the former being inappropriate when the teacher says to give approximate answers (j/k, but not really.) The answer on your calculator is inappropriate when giving it reveals that you have no idea what you’re actually doing (“The intersection happens at x = 1E-19”, “-3 squared equals -9”); alternately, when you hit the wrong keys.

12)   “In geometry, we begin with undefined terms.” Here’s what’s odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos will ensue if there are no definitions or familiar images for the basic elements.

Undefined terms can’t be rigorously defined, but they still have properties. There’s no such thing as a point with a length, width, or radius, or a line with area. There are canonical ways of drawing these things that help illustrate these properties (even though the representation of a point does have a radius, and a line does have a height, the drawing suggest that they do not). However, though the drawing suggest these properties, the drawings do not define these properties. We could draw points as squares, or lines as caterpillars, and these representations might suggest the correct properties in similar ways, but we could also draw points as triceratops, and lines as tyrannosaurs without changing the underlying mathematics (Hilbert liked tables, chairs, beer mugs, but c’mon man! DINOSAURS!), but it would make the representations much less intuitive.

Back when I still taught, I made a prezi for this: you can find it here:

13) “In geometry we assume many axioms.” What’s the difference between valid and goofy axioms – in other words, what gives us the right to assume the axioms we do in Euclidean geometry?

Valid axioms lead to logical systems of geometry in which theorems can be proven, and ideally, in which the world can be modeled. For example, the various version of the parallel postulate all lead to internally consistent systems of geometry that model flat space, spherical space, and saddle-curved space. Whereas if we changed the “Between any two points, there exists exactly one line” to “Between any two points, there exists exactly 41 lines,” it quickly gets silly. How do we define those lines? What interesting theorems can be proven about this geometry? What things can we usefully model?

I do dislike the language “what gives us the right.” We’re here, we’ve got brains and pencils, we can assume whatever crazy axioms we want! Saying “To hell with it, what if THIS is true” is how we got non-Euclidean geometry in the first place.

You don’t need a “right” to do math, you just need motivation.




2 thoughts on “Answering the “Conceptual Questions”

  1. Pingback: In which I take the “Grant Wiggins Challenge” :: The Max Ray Blog

  2. Fabulous. What fun! Though I quibble a bit with your last one. It ignores the fact that axioms were thought to be self-evident but postulates were not – they had to be ‘assumed’ to prove what we later want to prove. At least that’s how I learned Euclid and understand Heath’s Commentary on the Elements.

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