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 *a *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 *a÷*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.

- Division is defined in relation to multiplication, and
- 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.

One of my favorite teaching stories to tell is centered on this fact. In an early discussion with a Precalc class on rational functions (I know, I know why rational functions?!?!?) students were debating whether it was 0 in the numerator or 0 in the denominator that caused problems. A student named Henry confidently declared the denominator was the problem because you cannot divide by zero. I replied in the affirmative and asked him to remind his classmates why they cannot divide by 0. His response? “My teacher told me so.” A pretty good response, all in all. We spent the next ten minutes working on a better one and our conversation was essentially yours above.

Zero can make everything Uncountable, break the limit…

Hi,

your treatment of this question is largely formal, as you freely admit at the end. This seems to me to be at odds with the introductory idea that the issue in question is conceptual understanding.

I thinking you have proven why (formally) but I’m not sure that you have explained why (conceptually).

For me, the intuitive notions that you seem to dismiss are exactly what needs to be explored in order to ensure a sound conceptual understanding.

I generally give students a number of options about how to think about this. If all else fails, I recommend James Tanton’s video: https://www.youtube.com/watch?v=x9K0FKsDxGg

My favorite intuitive explanation was one I developed “before I knew better.” Take as an sample dividend 6. Consider 6/6 as asking, How many times can I subtract six from six until I have less than 6 and greater than or equal to 0 left? The answer is clearly 1. Now about 6/3? The answer is 2. And 6/2? That would be 3.

6/1 = 6. Now we ‘raise the price of poker’ a bit. What is 6/(1/2)? Once we see that this must be 12, we’re pretty much home free. As we make our divisor a fraction closer and closer to 0, the quotient gets larger and larger without bound. Many students, including adults, see that we can make the quotient arbitrarily large by making the divisor arbitrarily small but positive. I’ve had a lot of people tell me this was more impactful than the formal argument based on multiplication and the idea of inverse operations. However, the first time I mentioned it when I was doing graduate work in math education at U of Michigan, my explanation (to an elementary school teacher) was overheard by the late Joe Payne, an elementary math education professor. He frowned at me and said, “That’s not really a good explanation.” I was a bit too green to want to argue with him about it, but over time, I’ve come to the conclusion that it’s a very good explanation, simply not a formal one. And I believe we need as many ways of helping students (and ourselves) come to grips with these “obvious” but not at all obvious ideas.

It’s a perfectly fine explanation. I think that your comment that it came to you “before you knew better”, and that it was a math ed professor that disapproved, speaks volumes.

I’d also like to take this opportunity to thank you for your thoughtful insights into the common core.

Interesting point,

Neil Brown. Your objection to the formality seems to suggest an opposition betweenconceptualandformal. I am not so sure this opposition is real.The explanation of

MPGworks with division as defined in relation to subtraction. For him, division isrepeated subtraction, and the consequences of that definition suggest that there is no finite number that can be the quotient when dividing by zero.My explanation works with division as defined in relation to multiplication. Besides the particular choice of operation we have made for grounding the thinking about division by zero, are they so different? That choice is debatable (and it sounds as though

MPG‘s math ed prof was pretty judgmental about this choice), but the spirit is quite similar.As I mentioned, I have not yet read Wiggins’ post. But if his definition of conceptual understanding proscribes all formalities, I will have a bone to pick when I finally get to it.

I talk with my future elementary teachers a lot about the differences between

conceptual talkandprocedural talk. The latter focuses onhow toandwhat to do next. The former focuses on relationships among ideas. Sometimes these ideas are purely intuitive notions from real world experience. Sometimes these ideas come from prior work in mathematics. Butconceptual talk, for me, is about ideas, and it is about relationships among these ideas.I would be genuinely curious how others view this relationship.

It wasn’t until the end of college that I was able to put down my own thoughts, coherently, about division by zero: http://www.eecs.tufts.edu/~mgolds07/div_by_zero.pdf This was after being exposed to both many formal systems (linear algebra, functional programming) and having plenty of time to “mull over it” to generate conceptual depth. Above, your students have recognized that 0 is “special”, but they haven’t given it a name (“the multiplicative singularity”). It doesn’t have to be a name, certainly not _that_ name, but to make the next step they have to see how it is “special” in addition as well, but in a different way. The terms support the paper’s ultimate conclusion, by identifying and reapplying a causal mechanism. That said, I am a proponent of not introducing names until the ideas are established, which I call “labeling concepts” in contrast to “defining terms”. Learning when to name something, that allows a distinction to be drawn, is a large part of learning how to learn.

Great question, love the ideas. I have one that works for my students.

As a middle school Special Education Teacher, I teach whole-number division as breaking up a certain number of objects into a specified number of groups. Ex: Give a student 6 pencils, tell them to give the same number of pencils to 2 people (have to be fair, each has the same number). Thus, I have taught 6 divided by 2. Dividing by one, you give all the pencils to one person. (I avoid teaching division of fractions using this model). However, dividing something into zero means that you have a certain number of objects that don’t exist in any grouping. Nobody has these objects, they don’t exist anywhere. I make a show of doing a bad magic trick and making the pencils disappear (throwing them to the floor). They point out the pencils are not gone. “Exactly. This isn’t mathemagic. Pencils can’t just disappear. They have to belong somewhere, in some grouping. That’s why you can’t divide by zero.”

Not sure of the formal correctness of the explanation, but it has been successful with my students more often than not.

Pingback: Wiggins questions #3 | Overthinking my teaching

I believe any explanation of why you cannot divide by zero should be developmentally appropriate for the child who needs to know. We can look at lots of different ways of explaining this that are conceptual but at various levels of formality, from what I explain below to looking at the behavior of the function 1/x as x goes to 0. The level of formality should appropriately match the child.

With our pre-service elementary teachers, we take division by zero from a beginning conceptual level of what it means to divide (partitive/sharing and quotative/measuring, using the latter terms). I believe a good strategy is to lead students down a conceptually solid path (sharing15 cookies among 5 groups or friends or measuring by groups or stacks of 5) where the answer to the division question (how many cookies does each friend get or how many stacks of 5 cookies can I make) is reasonable. Then we keep the context the same, but we change the divisor (sharing 15 cookies among 0 friends or making stacks of 0 or empty bags, if you don’t like stacks of 0). Keep in mind that IF division by zero was an okay thing to do, we would be able to get an answer to the questions we ask next. Then ask the division question (how many cookies does each of the 0 friends get or how many groups/stacks/bags of 0 can I make). These questions either don’t make sense or don’t have a (unique) answer or both.

I made a video with xtranormal (so yes, it sounds like two robots talking) that goes through the whole explanation with the pre-service teachers as the intended audience at https://www.youtube.com/watch?v=eEnGL-vojVk

I appreciate the comment by Neil Brown and the related example by SpTeacherJames. I think Neils’ comment isn’t necessarily a contrast between formal and conceptual. It could be interpreted based on the “math maturity” of the learners involved. An adult may see the “formal” development as satisfying. But if the adult is a teacher (or preservice teacher) of elementary or middle school, I don’t think the more formal is sufficient. For a child to understand, the development needs to be concrete (less abstract symbol stuff) in the beginning. So I think we could (or should) develop the idea out of the (already established) measurement and sharing approach to division.

So, for example (just looking at the easier context of measurement that Adam shared above) the child will easily see that even though the question, of how many stacks of zero can be made, makes sense, an attempt at a satisfying answer is elusive (can make stacks of zero forever). And I would be happy if a 3rd or 4th grade student recognized that when we get engaged in the situation and solution involving division by zero, some significant mathematical discomfort is generated. A similar discomfort (having to ask a question that is meaningless – if I have no friends, why would I be asking how many each of these non-existant people get) happens with sharing division. Iwould encourge all to watch the “extranormal” videl Adam shared above.

I agree with Christopher that the formal understanding can very much be a conceptual one. A formal understanding is probably gained through more effort over a longer time than an informal one, but that doesn’t make it “less real” of an understanding — just a different one.

IMO, the subtraction argument by MPG is harder to understand conceptually than the formal one. Specifically, in the sentence, ” How many times can I subtract six from six until I have less than 6 and greater than or equal to 0 left? “, I see no clear

reasonwhy we need “less than 6 and greater than or equal to 0 left”. To an elementary school student, surely we should simply reach 0.Perhaps the qualifying language is there to ensure (“formally”?) that the idea still works when dividing real numbers rather than natural numbers. “Less than 6” ensures the idea works when dividing by negative integers But the definition also says 6/pi is 2.

I cannot make an intuitive connection between dividing real numbers and subtracting until that complicated condition has been reached. To me, it is conceptually much simpler to view division as opposed to multiplication (having a conceptual understanding of multiplication of real numbers is itself a beast, but I’ll leave it in a black box) The beauty of Christopher’s argument is that the logic, the heart of the understanding it contains, works with a huge range of preconceived notions of the context. (But not all — what is division in a non-commutative context, like matrix multiplication?) True conceptual understanding includes an awareness of the context in which that understanding is reasonable. This is a huge reason why multiple understandings of the same idea are important.

The subtraction argument or grouping argument are superior for young kids, but they should be simple: subtract until you get 0. When you realize this doesn’t work when you throw in negative numbers, fractions, or other real numbers, is it better to make the subtraction argument more cumbersome or to toss it and look for something entirely different? I like the latter.

Lovely ongoing discussion here. I will ask only this of

Barry: Should we really be calling matrix multiplicationmultiplication? I get that it is an extension of meaning in that it is consistent with integer multiplication when we consider 1×1 matrices. But the extension is alsoinconsistentwith important aspects of integer multiplication. That inverses are rare rather than plentiful is among these inconsistencies. (That we can only multiply certain families of matrices together is another.)Follow-up question: Is it reasonable to expect an introductory notion of a thing to be good enough to encompass

everythingthis thing will be called upon to represent later on?If we agree on No to that follow up, then it is a very tricky art indeed to determine how much one wants the introductory notion to accomplish. Which gets me back to

is.Fun stuff here.

Just read your Wiggins #5 answer, so I now understand your penultimate comment. (Very nice answer, by the way). I hadn’t even considered that explaining “the introductory notion” was a goal in this experiment, and upon rereading Mr. Wiggins’ post, I still don’t see it.

I also see that I might need to disagree with his operational definition of conceptual understanding. Your response to #5 is excellent, IMO, but I don’t see any of the five criteria he gives for a conceptual explanation as encoding the essence of what you wrote. At first glance, this seems to me related to the nature of the question: why is multiplication not just repeated addition? The related question you pose, “what is multiplication”, seems much more setup for an answer meeting his criteria, although as you say, there is no objective answer to this question.

It is not clear to me that there are any rules to the game we are playing here,

Barry. I am making up my own as I go.Because I work with elementary teachers, I am very much concerned with introductory notions. You, by contrast, may be very much concerned with extensions of meaning. The ways in which these interact is really interesting territory for me.

It will also be interesting for me to go back and read Wiggins’ 5 criteria. I have not read them yet—on purpose—and am writing up what I consider to be conceptual explanations of these ideas. Embedded in that may very well be some important differences between what Wiggins considers to be conceptual work and what I do.

My favorite explanation of why we cannot divide by zero:

Six divided by two is asking us to divide six into two equal parts. The answer is three, because two threes (multiplying) equal six.

Six divided by zero is asking us to divide six into no (equal) parts. If we have six of something, how can we still have six while having “no” parts… = nothing? This is an inherent contradiction: we cannot have six of something while simultaneously having “no” parts to it.