Tag Archives: place value

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.

Decimals before fractions? [TDI 1]

The Khan Academy knowledge map got me thinking about this recently, but the basic question at the heart of this Institute has been on my mind for a very long time.

Does it make sense to study decimals before fractions?

Decimals lie beneath addition and subtraction in the hierarchy. Fractions are not even in this part of the map; they are far off to the lower left.

The Khan Academy knowledge map. Decimals lie beneath addition and subtraction in the hierarchy. Fractions are not in this part of the map; they are far off to the lower left.

We do not have to answer that question right away. Indeed I do not think that there is a simple answer. I will argue in the coming weeks that the preponderance of theoretical and empirical evidence points to no.

You are not obligated to agree with me.

As I worked on formulating an argument the other night, I tried to make my question more concrete. Here is what I came up with (via Twitter):

Now, Twitter is a medium that makes nuance difficult.

So let’s strive to find nuance, subtlety and complexity in this conversation.

That last question is an important one for me. Traditionally, U.S. curriculum has had students working with decimals before they work seriously with fractions. Khan Academy isn’t going against the curricular flow in this area. What this means is that one-tenth is the first fraction students study. Is this justified?

The arguments in favor of studying decimals before fractions include these:

Place value. Decimals are the logical extension of the whole-number place value system. Just as you go from 1 to 10 to 100 by moving one place to the left, you also go from 100 to 10 to 1 by moving one place to the right. When you move left, the value of the place is multiplied by a factor of 10; when you move right, the value of the place is divided by a factor of 10. Decimals just continue that process.

Money. Children come to school with experiences involving money. They know what one dollar is; they know that 10 dimes make up a dollar; they have seen $1.25 and can talk about what that means. As a result, decimals are part of children’s everyday experience in a way that (say) sevenths are not.

Measurement. Metric measurements (and many but not all Imperial measurements) are expressed in units and tenths of units. Children are familiar with the meaning of “12.2 fluid ounces” or “3.2 meters”. So it makes sense to operate on tenths and hundredths even before formalizing the underlying mathematics of fractions.

How say you? Are these powerful arguments for you? Have I missed any arguments in favor of studying decimals before fractions? Do you have evidence to bring to bear on the question of whether it makes sense to study decimals first? Can you provide curricular examples to support (or refute) my claim that U.S. curriculum typically presents decimals before fractions? Can you provide an international perspective for us?

Follow this link to the announcement of this course for more information.

Instructions for joining the course:

This course has enabled open enrollment. Students can self-enroll in the course once you share with them this URL:  https://canvas.instructure.com/enroll/MY4YM3. Alternatively, they can sign up at https://canvas.instructure.com/register  and use the following join code: MY4YM3

The Triangleman Decimal Institute [TDI]

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:

  1. Self study. Read at your leisure. Discuss with yourself, your colleagues, your spouse and/or your Australian Labradoodle.
  2. Twitter. I invite you to use the #decimalchat hashtag to respond, argue, offer evidence and discuss.
  3. 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.


Come join us for some or all of the following.

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

Week 2 (Oct. 7): Money and decimals.

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

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

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

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

Week 7 (Nov. 14): Summary and wrap up.

There will be no grades, tests or tuition. Just the love of knowledge and the collective passion of teachers wanting to do their best.

See you in class on Monday!

Note from Canvas:

This course has enabled open enrollment. Students can self-enroll in the course once you share with them this URL:  https://canvas.instructure.com/enroll/MY4YM3. Alternatively, they can sign up at https://canvas.instructure.com/register  and use the following join code: MY4YM3

Number and numeration gone wrong

This came from a workbook bought by the kids’ grandparents.

Can someone please explain the purpose of the jars of bugs here?