Problem A: 22 cookies. Each kid gets 10 cookies. How many kids can get a full share? How many are left over?
Problem B: 22 cookies. There are 10 kids. How many cookies does each kid get? How many are left over?
These are not copied verbatim from Tabitha’s third-grade homework this week, but the numbers and context are the same. (Forgive me; I didn’t think about the potential for large-group discussion until the homework went back to school.)
The point is this: One of these problems was very easy for Tabitha, and the other was very challenging.
I have written about the two major types of division problems before, and they are relevant here.
Problem A was a snap for Tabitha. She skip counts well, and she is a whiz with place value. How many 10s in 268? Why 26 of course! This is the sort of thing I’m talking about.
So Problem A above is a piece of cake for her. This problem—for Tabitha—is very clearly asking How many tens are in 22? For her, this isn’t really even a question worth asking. Each kid gets one ten. There are two tens. QED.
Problem B doesn’t submit to this strategy in an obvious way. It requires her to keep track of 22 things as they get shared among 10 kids. One for you, one for you, one for you, etc. That’s taxing work, and so it’s a much harder problem for her.
When we discussed this problem together the other night, I made the argument that you use up 10 cookies each time you give everybody one cookie. I wanted to help her see how her strategy from Problem A would be useful in Problem B, while respecting that—for her—the sameness of these two problems is not at all obvious.
What’s the moral of the story? Let me know your thoughts in the comments.
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?
This is a classic example demonstrating the danger of applying procedures without thinking. The quotient can be expressed either as 31, remainder 3; or as . Neither of these answers the question, though. According to unspoken principles of table renting, we will probably need 32 tables.
Of course, I can imagine a student thinking like a caterer and building any of the following arguments:
We need 31 tables (or fewer) because 5% of people on a typical guest list do not show up.
We need 31 tables because if everyone comes, several will be young children who will sit in their parents’ laps.
We need 31 tables—if everyone shows up, we can just stick an extra chair at each of three tables.
We need at least 35 tables: No one wants to sit on the side where they can’t see the band playing at the front of the room, so we need to allow for fewer than 8 people at each table.
Et cetera.
I would argue that we need to teach in ways that do two things:
Allow/force students to interpret their computational results in light of the context (there is a CCSS Mathematical Practice standard about this), and
Focus students’ attention on the role the computation plays in answering this kind of question. Why are we dividing? and What does the quotient mean? are the kinds of questions I have in mind here.
You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.
Oh dear. If anyone on the Internet has had more to say about dividing fractions than I have, I am unaware of who that is. (And, for the record, I would like to buy that person an adult beverage!)
Unlike the division by zero stuff from question 1, this question is better tackled with informal notions than with formalities. The formalities leave one feeling cold and empty, for they don’t answer the conceptual why. The formalities will invoke the associative property of multiplication, the definition of reciprocal, inverse and the multiplicative identity, et cetera.
The conceptual why—for many of us—lies in thinking about fractions as operators, and in thinking about a particular meaning of division.
1. A meaning of division
There are two meanings for division: partitive (or sharing) and quotative (or measuring). The partitive meaning is the most common one we think of when we do whole number division. I have 12 cookies to share equally among 3 people. How many cookies does each person get? We know the number of groups (3 in this example) and we need to find the size of each group.
I can mow 4 lawns with of a tank of gas in my lawnmower is a partitive division problem because I know what of a tank can do, and I want to find what a whole tank can do. So performing the division will answer the question.
2. Fractions as operators
When I multiply by a fraction, I am making things larger (if the fraction is greater than 1), or smaller (if the fraction is less than 1, but still positive).
Scaling from (say) 5 to 4 requires multiplying 5 by . Scaling from 4 to 5 requires multiplying by . This relationship always holds—reverse the order of scaling and you need to multiply by the reciprocal.
putting it all together
Back to the lawnmower. There is some number of lawns I can mow with a full tank of gas in my lawnmower. Whatever that number is, it was scaled by to get 4 lawns. Now we need to scale back to that number (whatever it is) in order to know the number of lawns I can mow with a full tank.
So I need to scale 4 up by .
Now we have two solutions to the same problem. The first solution involved division. The second solution involved multiplication. They are both correct so they must have the same value. Therefore,
There was nothing special about the numbers chosen here, so the same argument applies to all positive values.
We have to be careful about zero. Negative numbers behave the same way as positive numbers in this case, since the associative and commutative properties of multiplication will let us isolate any values of and treat everything else as a positive number.
Please note that you do not need to invert and multiply to solve fraction division problems. You can use common denominators, then divide just the resulting numerators. You can use common numerators, then use the reciprocal of the resulting denominators. Or you can just divide across as you do when you multiply fractions. The origins of the strong preference for invert-and-multiply are unclear.
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.
If you watch this video, you will see a pretty standard U.S. treatment of introductory fraction material.
PLEASE understand that this is not about Sal Khan or Khan Academy. What you see in that video is what happens in many, many elementary classrooms across the U.S. on any given day. It is what is written into our textbooks (pre-Common Core, of course—we’ll get to Common Core in week 6).
To be clear, we introduce fractions with a part-whole model. A circle (or rectangle) represents the whole. We cut the circle (or rectangle) into some number of equal-sized pieces—that number is the denominator. We shade some of those pieces—that number is the numerator. That I am pointing this out surely makes some Decimal Institute attendees uncomfortable because how could it be any different? I’ll get to that in a moment. Stick with me here as I build a case pertaining to decimals.
If you believe that defining an abstract mathematical object and then operating on that object is the most powerful way to teach mathematics, then there is no logical objection to starting fraction instruction with decimals.
After all, children know something about our base-10 place value system by the time they get to third grade. They know something about the decimal point notation by then, too, as the result of money and (sometimes) measurement. (Oh, and calculators—don’t forget the calculators.)
So why not put all of that together and have tenths—the very heart of the territory to the right of the decimal point—be the first fractions they study? If you believe that children learn mathematics as a logical system that is little influenced by their everyday experience then there is no reason not to.
From a logical perspective, halves and tenths are the same sorts of objects. Tenths come along with a handy notation and so—from a logical perspective—are simpler than halves.
Indeed, it is much much easier to train children to get correct answers to decimal addition problems than it is to train them to get correct answers to fraction addition problems—even when the fraction addition problems have common denominators. (Sorry, no research link on this. Ask your nearest upper elementary or middle school teacher whether I am talking nonsense here.)
But we cannot fool ourselves into believing that ease of obtaining correct results has any correlation with grasping underlying concepts. Children can be trained to give correct answers without having any idea what the symbols they are operating on represent.
Take the video linked here, for example. (In it, I do a Khan Academy exercise using a purposely flawed way of thinking and score approximately 90%—I get an A without showing that I know anything useful.)
Of particular importance is their finding that when teachers know how students are likely to think about addition and subtraction problems, and when teachers know the strategies students are likely to use, these teachers are more effective at teaching addition and subtraction.
In short, CGI demonstrated—for addition and subtraction—that better understanding the cognitive structure of addition and subtraction makes you a more effective teacher.
In the years since that first set of results, the team has extended their results to initial fraction ideas. In the book Extending Children’s Mathematics, they argue that the cognitive way into fractions with children is fair sharing.
That is, the ideas that children bring to school prior to formal instruction having to do with fractions are those that come from sharing things. Sharing cookies, cupcakes, couches and pears; children have cut or broken these things in half, considered whether the resulting pieces are equal in size, and decided whether the sharing is fair many times before they study fractions in school.
When you do start with fair sharing, children’s ideas about how to do this follow a predictable path. Halving and halving again are common early ideas even when sharing among three or five people. Similarly, children share incompletely early on. When they need to share one cookie among 3 people, they will suggest cutting into 4 pieces and saving the fourth for later.
This more recent CGI research demonstrates that paying careful simultaneous attention to (1) the number of things being shared, and (2) the number of people doing the sharing is a late-developing and sophisticated skill that comes as an end product of instruction.
In that conversation, we had 2 pears to share among 3 of us (real pears, not textbook pears). Griffin (9 years old) suggested cutting them into thirds, but then got distracted by the campfire before correctly naming the amount we would each get. Tabitha (6 years old) worked with me to half and half again. Only once we had a single remaining piece right there in front of us did she suggest cutting that piece into 3 pieces.
The concrete conversation created a need for thirds. But thirds only occurred to her once that need existed. As long as we had whole pears or halves of pears, we could keep cutting in half.
Here was the end result of that sharing.
Now back to decimals.
The CGI fraction work constitutes persuasive evidence that not all fractions are cognitively equivalent. While starting the study of fractions with tenths makes sense from a logical perspective, CGI demonstrates that children do not learn from logical first principles.
They learn by considering their experience.
Children have lots of experience with halves. We might expect thirds to be just as obvious to children as halves are, but it isn’t true.
So let’s take seriously the idea that experience in the world has an effect on how children learn. And let’s accept that this fact should have an effect on curriculum design.
Then if you still want to teach decimals before fractions, you would have a responsibility to demonstrate that children have anywhere near the real-world experience with tenths that they do with halves and thirds.
When we discussed on Twitter recently children’s real-world experience with tenths, we came up with:
pizzas (about which I am skeptical, see next week’s interlude),
metric measurements, and
not much else.
In comparison to the tremendous amount of work children have done with halves and halves of halves (and halves of those), how can tenths be the first fraction they study in school?
Summary
To summarize, I am arguing:
That part-whole fraction work makes logical sense to experienced fraction learners,
That children do not learn fractions by logical progression from definitions, but by connecting to their experiences with situations in which fractions arise in their everyday lives,
That we have research evidence for this latter claim,
That the truth of this claim should have implications for how we teach decimals to children, since their experiences with tenths are much less robust than their experiences with simpler fractions, and that chief among these implications is…
That we ought to reserve serious decimal work until kids have developed the major fraction ideas about partitioning, repartitioning and naming the units that result.
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. Neither of these answers the question, though. According to unspoken principles of table renting, we will probably need 32 tables.
of a tank of gas in my lawnmower is a partitive division problem because I know what 
will answer the question.
. Scaling from 4 to 5 requires multiplying by 
. This relationship always holds—reverse the order of scaling and you need to multiply by the reciprocal.
.
and treat everything else as a positive number.
Overthinking my teaching 