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?

References

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.

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