Category Archives: Courses

Summary and wrap up [TDI 7]

It’s finals week here at the Triangleman Decimal Institute.

Let’s review for the final exam, shall we?

Is that decimal point in the right place? How can we know?

Is that decimal point in the right place? How can we know?

Week 1 (Sept. 30): Decimals before fractions?

In this week, we considered the question, Should we treat decimals more like whole numbers or more like fractions? Associated with that question, we strived to make explicit the ways in which decimals are like whole numbers and the ways in which they are like fractions. We covered a surprising amount of ground on these questions.

Week 2 (Oct. 7): Money and decimals.

Our consideration of money and decimals led us to think very, very hard about units. In the case of money, it seems that we reached consensus that the place value aspects of the American monetary notation system are subordinate in many people’s experience to the different units aspects. That is, we are more likely to think of $3.50 as three dollars and fifty cents—two different units—than as three and fifty one-hundredths dollars.

While both ideas are correct, this conceptual difference has fairly strong explanatory power. It helps us understand a lot of the errors we see in classrooms and in the larger world.

Week 3 (Oct. 14): Children’s experiences with partitioning.

In this week, we explored experiences students have with cutting things into pieces, which is the real-world knowledge children can bring to classrooms when they study fractions and decimals.

Week 4 (Oct. 21): Interlude on the slicing of pizzas.

Dozens of math teachers on the problem led to our documenting 6, 8 and 10 slice pizzas directly (where a slice is interpreted to be the result of complete and equal partitioning of a pizza). We have strong claims that other numbers of slices exist, but no photographic evidence.

Week 5 (Oct. 28): Grouping is different from partitioning.

We argued this week about whether moving to the right in the decimal place value system is really a simple extension of moving to the left. It seems assumptions are everything here. Maria and I brought differing assumptions to this question, and this led to some very interesting and spirited debate.

Week 6 (Nov. 4): Decimals and curriculum (Common Core).

In week 6, we thought about the relationship between the ideas we had been working on and standards documents.

This brings us to week 7.

Your final exam consists of one question and one task. The question is this:

How can you show the world what you have learned these last several weeks?

The task is this:

Do it.

And let me know what you do, OK? I need data for my funders.


Decimals and curriculum (Common Core) [TDI 6]

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.

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The Common Core State Standards build decimals from the intersection of fraction and place value knowledge. Fractions are studied at third grade and fourth grade before decimals are introduced in fourth:

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.

Grouping is different from partitioning [TDI 5]

After last week’s pizza-slicing interlude, we are back on task for the closing half of the Decimal Institute.

This week, I want to invite discussion of the question, How much are decimals like whole numbers?

In case you are this far into things and cannot guess my answer (and in case you haven’t read this week’s title!), I offer the following clue.

From a purely abstract and logical perspective, decimals are exactly like whole numbers. No matter what place you are considering, the place to the left is worth 10 times as much, and the place to the right is worth \frac{1}{10} as much.

But there are many important ideas that this logical analysis ignores. And people do not always find abstract logical arguments compelling. So we’ll dig deeper than that.

I have four major ideas for us to consider. You, class, will surely have more.

1. Grouping groups is different from grouping units. Thanheiser (2009) demonstrated that some preservice elementary teachers could work competently with two-digit numbers yet make important errors with three-digit numbers. These teachers could explain the grouping inherent in writing a number like 23, but did not extend this reasoning to numbers such as 235. If decimals are really just like whole numbers, we should expect that all whole numbers are the same for learners. Thanheiser has demonstrated that they are not.

2a. Grouping patterns and partitioning patterns are often mismatched. The metric system was established by the scientific community for ease of working with our base-10 numeration system. It was developed intentionally at a moment in time when correspondences between numeration and measurement were of increasing importance.

Other measurement systems probably reflect the informal and natural ways people have of working with measurement. The Imperial system, for instance, is probably based on how people naturally view quantities.

In that case, consider the inch. Inches are grouped in twelves. They are partitioned in twos and powers of two.

The teaspoon is grouped in threes (making tablespoons) and partitioned in twos and fours.

Cups? Those are partitioned in twos, threes and fours. But they are grouped only in twos.

Time and again, the size of the grouping is not related to the number of partitions. Perhaps this is because partitioning and grouping are not closely related processes in people’s minds.

2b. This is borne out in my own work with preservice teachers. Go read my post titled, Measurement explored for full details. My experience in having students develop length-measurement systems includes these observations:

  1. Students nearly always partition in 4ths, 8ths and 16ths.
  2. Students almost never partition into 10ths.
  3. Students may group in threes or sixes, but they never ever partition this way.
  4. Students rarely think to group the same way they partition. That is, if they made 8ths, they might very well group in sixes. The convenience that would be afforded by consistency does not tend to occur to them in advance.

The comments on that post are thought-provoking and we should feel free to pick up threads of those comments in this week’s discussion.

3. Place value understanding does not seem to cross the decimal point easily. I do alternate place value work with my preservice teachers. Bear with me on this if you’re not familiar. In a base-5 system, we count 1, 2, 3, 4, 10. We make groups of this many: ***** instead of this many: **********; the latter is what underlies our usual base-10 system.

This means we write 10_{five} for our usual five and 100_{five} for our usual twenty-five. After mastering grouping with fives instead of tens, we move to partitioning. If decimals are just like whole numbers, this should present no difficulty.

But it presents tremendous difficulty. Even my strongest students have a common struggle, which is this: They view the whole and the part of a decimal number separately and treat them equivalently.

Here is what this means. Consider the base-10 number 20.20. This is “twenty and twenty hundredths”. My students tend to correctly interpret whole number part of this. Twenty is four groups of five so they write 40_{five}. But then they do the same thing with the decimal part, writing .40_{five}, so that 20.20_{ten}=40.40_{five}.

But this is not right. The decimal part represents 20 hundredths. But if we have changed bases, then the values of the decimal places change too. The first place is fifths; the second is twenty-fifths; and so on.

Through the use of grids and activities paralleling those from the Rational Number Project (Cramer, et al., 2009), they come to understand that 20.20_{ten}=40.1_{five}

The underlying difficulty seems to be that…

4. The unit changes when we add digits to the right of the decimal point. When you read whole numbers aloud, the unit is always the same—one. Thirty-two means thirty-two ones. 562 means 562 ones. Yes, the 6 has a value, and this value changes depending on its place. But no matter the number of digits, the number counts ones.

This is not true with decimals. 0.32 means thirty-two hundredths. 0.562 means 562 thousandths. Thousandths are different units from hundredths. The unit changes to the right of the decimal point in way that it does not for whole numbers.

To summarize, our question this week is: How much are decimals like whole numbers? My answer is that they are not very much alike at all. I outlined four reasons: (1) Even whole number place value is more challenging than logic suggests, (2) Our experiences with grouping and with partitioning tend not to parallel each other, (3) We tend to think of whole number parts and decimal parts as separate things, and (4) The units we count are different to the right of the decimal point, depending on how many digits there are.

How say you, class?


Cramer, K.A., Monson, D.S., Wyberg, T., Leavitt, S. & Whitney, S.B. (2009). Models for initial decimal ideas. Teaching children mathematics, 16, 2, 106—117.

Thanheiser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers, Journal for research in mathematics education, 40, (3), 251–281.

On the slicing of pizzas [TDI 4]

This week we get to the origins of the Decimal Institute.

One Thursday evening, I was sitting around thinking about the Khan Academy knowledge map, trying to put my finger on the exact argument that I wanted to make about why the decimals-before-fractions thing was so deeply disturbing to me. I was trying to formulate an argument that math teachers would find convincing.

I settled on this.

Twitter settled on money (see week 2) and pizza. I contended (and still do) that a 10-slice pizza is a rare beast. I may have overstated their rareness in my sign off that night:

Yes, I admit that this was an overstatement. But pizza slicing is just the thing we need to lighten the mood this week, so let’s investigate together the various ways pizzas are in fact cut.

For example, here are instructions for a 10-slice pizza. (Or are they? In any case, tip o’ the chef’s cap to Kate Nowak for the find)

And here is the closest thing I could find to light-saber pizza cutting (with thanks to Chris Robinson).

And here is a machine whose sole purpose is precision cutting of pizzas into 7 slices (props to Malke Rosenfeld for the find).

To finish my story, the Decimal Institute was born the following morning. My Twitter conversation made clear that not all math teachers were buying my argument made 140 characters at a time. So I offered to talk about these ideas in a (MUCH) longer format.

This week, let’s slow things down and have a bit of fun.

Our challenge as a group is to find the complete set of numbers into which pizzas are (or have been) equally partitioned.

For example, I have provided evidence in this post that 8 (the laser cutter) and 7 are in this set. I have not provided evidence that 10 is (did you watch that video carefully?).

Our standards for evidence are high. Photographs, videos and original documents are acceptable. Clip art for middle school textbooks are not.

We will collect and discuss on Canvas. I will curate and share what we find here on the blog in a week or so.



Leslie Billings asks:

Does it matter what shape the pizza is before cutting? ie Circular vs Rectangular? Or some other shape?

I feel comfortable leaving these issues to the community of interested parties.

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.