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.
The Decimal Institute is winding down. This week, I have a short post outlining the relationship between our discussion these past weeks and the Common Core State Standards (with links). Then next week we will wrap up with a summary of what I have learned and an invitation to participants to share their own learning.
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
One of the issues we have been wrestling with in the Institute has been how much decimals are like whole numbers and how much they are like fractions. In light of this conversation, I found the following statements about comparisons interesting.
CCSS.Math.Content.1.NBT.B.3 Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
CCSS.Math.Content.2.NBT.A.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
CCSS.Math.Content.4.NBT.A.2 Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
These all refer to comparisons of whole numbers—at grades 1, 2 and 4. Comparisons of decimals appear at grades 4 and 5. For example:
CCSS.Math.Content.4.NF.C.7 Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. [emphasis added]
The phrase, Recognize that comparisons are valid only when the two decimals refer to the same whole, struck me as odd. If I am comparing 0.21 to 0.5, I need to make the whole clear, but if I compare 21 to 5, I do not?
This seems to be an overcommitment to decimals being like fractions rather than like whole numbers. Or not enough of a commitment to the ambiguity of whole numbers.
In any case, the treatment of decimals in the Common Core State Standards is probably one of the major challenges for U.S. elementary teachers, who may be accustomed to curriculum materials that emphasize the place value similarities of decimals to whole numbers rather than the partitioning similarities to fractions.
I will provide some examples of pre-Common Core U.S. curriculum in the Canvas discussion to support this claim. Join us over there, won’t you?
Non-U.S. teachers, please share with us your observations about how these standards relate to curricular progressions you are using. An international perspective will be quite useful to all of us.
And please start thinking about what you can do in the coming weeks to share/demonstrate/document/extend your learning from our time together. Consider it your tuition to the Institute.
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?
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.
I am working on a ton of interesting projects right now. Not least of these is my classroom teaching at the community college. My fingertips are sore from typing.
And yet there is always more to say. More to think about. More conversations to have. Here is a peek into one that is ongoing.
Malke Rosenfeld and I have been going back and forth about math, dance, Papert and learning for a few months now. I am learning a lot from the conversation. She asked some questions this morning.
Malke: A thought just entered my head — why are you offering TDI? Is it based on a question you are unsure of and want to see what others think? Or are you seeing a deficit in math teachers’ thinking that you want to shore up?
Me: When ranting on Twitter, I could see that some of my assumptions about baseline teacher knowledge about fraction/decimal relationships as they pertain to developing children’s thinking were unfounded. That is, I was assuming teachers knew a lot more than they seemed to. Which has implications for my Khan Academy critiques, and lots of other writing on my blog. Yet people were also curious. So I wanted to say more in a way that would draw from and build on a larger collective knowledge, so it’s not just my spouting off.
Malke: Is there a reason you offered it specifically as a course, and not a moderated discussion (which it sort of seems like right now)?
Me: When you view learning as a social process, you tend to think of courses AS moderated discussions. I mean this quite seriously. I know that it goes against the grain of online (and face-to-face) course design. But that’s not because I think of online instruction differently from others; it’s because I have a particular view of learning that runs much deeper than that. If I tell and quiz, you’re not learning very much. If I propose a set of ideas, listen to what you have to say, encourage you to interact with others and move the conversation in directions that seem useful based on those interactions, you’re probably going to learn a lot.
As long as I can keep you engaged in that process. Which is a different challenge online than in the classroom.
Malke: Is there a place you specifically want your students to get to by the end of the seven weeks? Or are you just curious to see what develops?
Me: I am curious to learn what I can about teaching at every opportunity. I want to produce “students” who can articulate important questions (see? learning as having new questions to ask?) about curricular approaches to decimals. Ideally, I would help them to develop a critical voice that speaks to/through them when they work with individual students, when they plan lessons and when they talk with their colleagues in a variety of settings. In short, I want to change the way teachers view the territory of decimals, fractions and children’s minds. Strange mix of lofty and specific there, huh?
In recent weeks, I have written several times about decimals and their treatment in curriculum. In discussions surrounding that writing, it has become clear to me that everyone involved in children’s learning of decimals can both learn and contribute to the learning of others.
Which is why I am excited to announce…
The Triangleman Decimal Institute
For seven weeks, starting Monday, September 30, I will invite all interested parties to an online conversation about decimals and learning decimals.
Each Monday, I’ll have a new post here to launch and focus our discussions. Comments will be closed in order to move the discussions to more productive venues (see below).
You may participate in any way that you like, including the following:
Self study. Read at your leisure. Discuss with yourself, your colleagues, your spouse and/or your Australian Labradoodle.
Twitter. I invite you to use the #decimalchat hashtag to respond, argue, offer evidence and discuss.
Canvas. It is no secret that I love this LMS. I have established a course in Canvas. The course is public, free and you may self-enroll. We will mainly use the discussion forums there, which function MUCH better than WordPress comments for our purposes. I will establish a new discussion forum there for each week’s post, but students (i.e. you) can also create discussions.
You may come and go as you please.
My promise to you is to keep myself on the schedule in the syllabus below and to engage to the extent possible in the discussions on Twitter and Canvas.