Tag Archives: CGI

Children’s experiences with partitioning [TDI 3]

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).

I did not have to work very hard to find additional examples to support my claim. Here is a tutorial on Sophia. Here is something from “Your destination for math education”. And here is a self-paced math tutorial at Syracuse.

I am not cherry picking straw men here.

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.)

This leads us to the work of the Cognitively Guided Instruction (or CGI) research project from the University of Wisconsin. This project studied the ideas about addition and subtraction situations, and strategies for working them out, that children have before formal instruction begins.

It turns out that they know a lot.

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.

You can see this in a conversation I had with my children over the weekend (written up in full on Talking Math with Your Kids).

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.

Photo Oct 12, 2 13 41 PM

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:

  • money (where the connection to fractions is weak, see also week 2’s discussion on Canvas),
  • 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.

Guess the temperature

Griffin and I play a little game called Guess the Temperature. It goes about how you would expect. We step outside on the way to his bus. I ask him to guess the temperature. If I don’t already know, I get to guess after he does. If I do already know, I don’t cheat; we just remark on how close his guess was.

In Minnesota, this means we get to study integers.

Me: Griff, guess the temperature.

Griffin (eight years old): Two below zero.

Me: It’s three degrees above.

G: So I was off.

Me: Not by much, though. How much were you off by?

G: [muttering to himself, then loudly] Five degrees!

Me: How did you know that?

G: It’s two degrees up to zero, then three more.

Let’s pause for a moment here. You know how I just won’t shut up about CGI (Cognitively Guided Instruction)? It’s because they’re right. Children know mathematics before it is formally taught.

Consider the grade 6 (for 11-year olds) Common Core Standard 6.NS.C.5


Griff pretty much has this nailed down and is making progress on grade 7. But no one has formally taught him how to subtract integers. He reasons his way through a problem by making sense of the relationships in the context. He can find 3-(-2) without knowing keep-change-change.

But it’s not just Griffin. CGI demonstrated that children—all children—develop mathematical models of their worlds that precede instruction, and that instruction sensitive to these mathematical models is better than instruction that ignores them.

Back to our conversation.

Me: So what if it 10 degrees out, and you guessed 3?

G: [quickly] I’d be seven off.

Me: Right. How do you know that?

G: Ten minus three is seven.

Me: Nice. Subtraction. Do you know that you can always express the difference between your guess and the actual temperature with subtraction?

So in that last example, you subtracted your guess from the actual temperature. You could do that with your real guess today.

So three minus negative 2 is five.

G: [silent]

By this time we were nearing the bus stop. I had offered this tidbit as an intellectual nugget to chew on, rather than a lesson I expected him to absorb. But that is what it means to have instruction be sensitive to children’s mathematical models.

Another cool thing…

In our discussion the other day of whether the difference between van Hiele levels 0 and 1 is the use of official math vocabulary terms, or whether there is something more there, a student asked this (I am paraphrasing):

Is this like the CGI research, where children know things about addition and subtraction before they are taught in school?

Here’s what I love about this question:

  1. It demonstrates the power of having these students for a second semester;
  2. It demonstrates that students will—over time—make these ideas their own; if we keep at it, and build cases for the importance of the ideas in our courses, students will play with these ideas, and they will look for opportunities to apply and connect them; and
  3. It made me stop and think. I’m still not 100% sure whether it’s like that or not. I have something new to think about.

Salami-slice math

I was making lunch for Tabitha (who had just turned 5) one day. It was one of her favorites-salami quesadilla (house specialty). She was impatient and hungry. She asked for some salami while she waited.

Me: How many slices do you want?

Tabitha: Four.

Me: I’m sorry to say, we only have three.

T: Oh. That’s one less than I wanted.

Me: That’s right. Do you want them?

T: Yes.

Me: (Giving Tabitha the remaining salami slices) So if three slices is one less than four, what would two less be?

T: Two.

Me: And what would three less be?

T: (Thinking) One.

Me: Nice. How did you know that?

T: I don’t know.

Me: What would four less than four be?

T: (Thinking) Zero!

Me: Nice.

T: Ask me another.

Me: What would five less than four be?

T: (Thinking) Zero?

Me: Interesting.

T: Do a different “less”. Like less than five or something.

There are two important strategies here. The first is turning Tabitha’s request for salami into a situation that involves numbers. She asked for salami to eat while waited for the main part of her lunch. I could have just given her a few pieces of salami, but that’s a lost opportunity. It’s an easy move to ask her how many slices she wants, and then to compare that to the number of slices we actually have.

The second important strategy is the What if? questioning style here. This is where the abstraction happens. When I ask her What would two less be? I am offering here the opportunity to think in terms of salami slices, or to just think about numbers. She has already imagined the slices so they are available as a tool. Or she can count numbers in her head.

Note Tabitha’s comfort with zero as a number. When I ask her what four less than four is, she comfortably answers zero. Notably, she has to think about it. She hasn’t developed the rule that any number minus itself is zero, although this conversation could certainly go that direction by asking about three less than three salami slices and five less than five salami slices, etc.

Tabitha is five years old. There is plenty of time for her to learn about negative numbers. I did her no harm asking What is five less than four? I didn’t expect her to be able to say negative one, but I was curious about what she would say. Remember that being interested in children’s ideas is a key to talking math with them.

It matters, though, that she didn’t just reverse the numbers and say “One”. The numbers 5 and 4 have very specific meanings for her, and she uses those meanings the best she can.

Kindergarten addition strategies

A few months back, Tabitha (5) wanted me to ask her some math class questions. This led to some disagreement between the two of us on what constituted a math class question.

On our way back from our fall camping trip last weekend, Tabitha announced that she knows what 8+8 is.

Me: Ok. What is 8+8?

Tabitha: [longish thinking pause, eyes gazing up and to the right] 16!

Me: Wow. I could really see you thinking that through. How did you know that?

Tabitha: I counted.

Me: OK. Good. But how did you keep track of how much to count?

Tabitha: huh?

Me: How did you know when to stop counting?

Tabitha: When I got to 8. That was 16.

Me: Right. But how did you know when you had counted eight times?

Tabitha: I just counted to 8. That was 16.

Me: Oh! I know! Was it like this? Eight…nine (that’s one), ten (that’s two)…like that until you got to sixteen (that’s eight)?

Tabitha: Yes.

Me: How did you know to do that? Did you learn to do that in Kindergarten? Or was it your own idea?

Tabitha: [long pause] OK. I admit it; it was my idea.

I have written before about how sticking with this stuff pays dividends eventually. Tabitha announced that she knows what 8+8 is, but what she really meant was that she knows how to find 8+8. Tabitha knows that talking math is not about telling answers, but about actively thinking things through. That doesn’t come for free in classrooms, and it doesn’t come for free at home. It is a result of lots of prep work and lots of encouragement and lots of conversations.

Returning to the theme of the knowledge required for me to have this conversation with Tabitha, though, I needed to know something about possible strategies, and I needed to have thought some of those strategies through for myself-by trying them out.

If the sum in question had been 8+2, I wouldn’t have questioned her claim that “I counted”. You can subitize two, so you can know that you have counted twice without having a separate system for keeping track.

But counting 8 times without having a way to ensure that you have done so? This seems pretty much impossible. As an adult, I know to stop at 16, but that’s only because I already know the sum. As I count 9, 10, 11, 12, I am conscious that the only way I know I have counted four times is because I know that 8+4 is 12. I’m not able to keep track of the number of counts past three.

I had watched her fingers; no motion there.

So there had to be something else going on. I was only able to press because I knew what she knew (how to count) and what she didn’t know (addition facts) and some possible ways that this knowledge could be used to find the sum of 8 and 8.

The more I talk with these kids, the more impressed I am with the work of CGI. Much of what I know in this area is the result of studying their work.

UPDATE: The original CGI link in that last paragraph went to a tag search of CGI on my blog. That was confusing. It has been replaced by a particular post. Here’s the original link. And here’s a link to CGI on Wikipedia.

Further adventures in Kindergarten fractions

I took the kids camping this past weekend. Fall along the Mississippi River, mid-70-degree days and 50-degree nights. Pretty much perfect. Having read information about our state park together earlier in the day—including the park’s acreage—Tabitha posed a question.

Tabitha (five and a half): How many acres or miles is our campsite?

Me: It’s only a small fraction of a square mile, but it’s about \frac{1}{8} of an acre.

Tabitha: What’s a fraction?

Me: It’s like when you cut something up. It’s a number bigger than zero, but less than one.

Tabitha: Huh? That doesn’t make any sense!

Me: Well, let’s say you, Griffy and I had three s’mores, and we wanted to share them equally. We would each get one, right?

Tabitha: Yeah.

Me: But what if we only had 2?

Tabitha: Well, then you’d have to cut them in half.

Me: Right. So \frac{1}{2} is a fraction

Tabitha: Oh.


Tabitha: But what’s the number? You said a fraction was a number bigger than zero, but less than 1.

Me: One-half is more than zero, but less than one.

Tabitha: Half of what?

Me: Half of anything is more than nothing, but less than the whole thing.

Tabitha: But what’s the number? A half isn’t a number!

I have been thinking about the moment when there is a choice to talk math with my kids. I have been trying to understand what I need to know in order to recognize that a choice exists and in order to pursue a mathematical conversation.

Fractions are tough because there really is a lot of specialized knowledge about how people learn them. I have been reading and teaching from the book Extending Children’s Mathematics over the last year or so. The authors make the argument that fair sharing is the best entry point for children’s sense-making about fractions. Not part-whole. Not number line. Fair sharing.

Notice how this plays out in my conversation with Tabitha. I start with part-whole, move to (arguably) number line and she protests that these ideas make no sense.

But as soon as I go with fair sharing, she’s on it. She gets that things sometimes need to be cut up in order to be shared equally.

She also understands—and this is crucial—that halves are meaningless without a referent whole. “Half of what?” is a brilliant and essential question.

So what did I need to know in order to pursue this conversation? I needed to know that there are multiple ways of thinking about fractions, and that fair sharing is going to be helpful for a young child to think about. And that part-whole and number line are going to be dead ends.

Tabitha learns from the conversation that fractions have to do with fair sharing. She doesn’t understand one-eighth—the fraction that initiated the conversation. She doesn’t understand anything more about the size of our campsite, nor about acres, miles or even square miles.

She learns that fractions have to do with sharing. That’s a pretty good Kindergarten-level idea, right there.

Pushing back on some pushback

New comment in one of my Tabitha posts, from Steve Prosser (who, not coincidentally, has an app he’d like to sell you),

It is important starting around entry to first grade, IMHO, for children to: a) have memorized the patterns of basic arithmetic equations, and to b) understand how more complex problems can break down into simpler step-by-step arithmetic. Doing this in a way that promotes right-minded learning (pattern recognition) is vital. I’ve done my best attempting this for my daughter with my app – mathflashapp – and her performance through the third grade suggest this is the right track.

I have lots to say here and struggled with whether to say it in the comments or on the main part of the blog. So forgive me if this gets too detailed.

The post in question was really about the relationship between research findings and a child’s development. I wasn’t expressing a belief or an opinion in that post, nor a philosophy. I was using the example to make the research come alive.

This points to a more general principle here on the blog. I am interested in examples (whether my own or others’) when they either:

(1) Present a puzzling case that needs explaining, or

(2) Illustrate research findings,

and when they

(3) Are offered with enough detail that others may propose alternate interpretations or hypotheses.

In Boolean algebra, I strive for (puzzling OR illustrative) AND detailed.

Claims of the sort, “It worked for me and I’m OK” or “I did this with my class; they seem to be unharmed” are not particularly helpful to the cause of each of us learning something.

I have no beef with fact memorization, nor with apps that help students to memorize these facts. But we have good research evidence that far more time is spent on low level rule recitation, practice and review in American mathematics classrooms than in other countries with more successful mathematics education programs.

One of the agendas of this blog (there are many) is to explore what other possibilities are within our grasp at a variety of levels. So I’m never going to devote much space to techniques for memorizing arithmetic facts. There’s no new ground for me to cover there. Yes, an app makes it more efficient and gives instantaneous feedback. It’s a marginally better training device than flashcards would be. I have no problem with that.

But I don’t think we learn much from it as a field.

On the other hand, I know for sure that few people outside of the hard-core elementary math education circles know anything substantive about CGI. We have a lot to learn from that project. Examples can help bring the findings to life, and can help people understand the importance of these findings.

Selling a group of math teachers on the proposition that it would be nice if more students knew their addition and multiplication facts? That is not a particularly difficult challenge.

Helping them (and me) to understand the thinking that’s going on in kids’ minds as they learn new stuff? That’s a life’s work.

See also my post on setting norms.