Money and decimals [TDI 2]

Week 2 of the Decimal Institute begins with a claim that many experienced teachers will find obvious.

Namely: Decimals are difficult.

When students struggle with difficult things, it is the teacher’s instinct—indeed the teacher’s job—to help.

When students struggle with decimals, we frequently refer to money. The idea is this: Kids understand money. They are familiar with the notation of money which is based on decimal notation. They are familiar with the language of money: quarters, nickels, pennies, et cetera (bear with us, you folks from non-dollar nations, and do your best to follow the argument; we’ll wait for you if you need to Google something—I would need to do the same for shillings.) Students can bring their knowledge of money to bear on understanding decimals.

While I have no doubt that money has been helpful for students to get correct answers to particular problems, nor even that money can be the basis for students to build particular ideas about decimals (e.g. that ); I do have some critical questions about whether money is a strong foundation for building generalized decimal concepts.

Among these questions are the following.

1. If money is such a strong basis for decimal concepts, why do we so often see decimal errors with money?

The Gallery of Misplaced Decimals
(You may click to enlarge each one if you like)

2. Is it possible, as Max Ray suggests below, that the conception people tend to carry in their minds is of dollars and cents as separate units, as they do feet and inches?

@Trianglemancsd though I am curious what students use to make sense of cents prior to fractions. A dollars unit and a pennies unit, right?

— Max Ray (@maxmathforum) September 27, 2013

I report my height as 6 feet 1 inch. I do not report it as feet, although I know that I could. Likewise I don’t think of 1 hour and 5 minutes as hours, although I know this to be correct.

Is it possible that many people think of $1.25 as 1 dollar and 25 cents, rather than as dollars?

Maybe students are thinking of dollars and cents as different units that have a nice conversion rate, rather than of dollars as the natural unit and cents as a partitioning of that unit.

Follow-up questions: (a) Might Max’s insight help to explain the errors in the gallery of misplaced decimals? (b) What are the implications of this for using money to teach decimals?

3. Related to the foregoing: even when students do think of dollars and cents as more than just related units, is it possible that students are thinking of cents as the natural unit, and that dollars are built out of them? This would contrast with viewing dollars as the natural unit from which cents are partitioned.

I asked this question on the blog back in January, and readers answered it differently from the class of future elementary teachers I posed it to at the same time. What can we learn from that difference?

Is it just a coincidence that this table includes no fractions?

From Wentworth’s Mental Arithmetic (circa 1895). Thanks to Monica Cataldo for the amazing find!

4. Even if we do think of 1 cent as of a dollar, does money support the repeated repartitioning that is essential to decimals? E.g. Find a number between 0.04 and 0.05. Does thinking about money support a student in getting to 0.041?

5. Finally, ask 100 sixth-graders how much money $0.1 is. I bet at least 30 of them say “1 cent”. Again, money seems to support particular decimal special cases, but does money help students generalize beyond those special cases to the important and challenging ideas underlying decimals?

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Comments closed here. Let’s talk in the course and on Twitter under #decimalchat.

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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

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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?

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):

Someone needs to explain something to me…+

— Christopher (@Trianglemancsd) September 27, 2013

+ if decimals depend only on whole number place value, addition and subtraction, then what exactly is a “tenth”? http://t.co/Ght3VE2FpH

— Christopher (@Trianglemancsd) September 27, 2013

You can talk all you want about “tens” and “tenths” but if we don’t have a common language of fractions, you’re talking nonsense.

— Christopher (@Trianglemancsd) September 27, 2013

And if you want to claim that a “tenth” is the thing you need ten of to make a whole, then I will need to know what a “hundredth” is.

— Christopher (@Trianglemancsd) September 27, 2013

But seriously, what is your explanation for why one tenth is the best fraction to study first?

— Christopher (@Trianglemancsd) September 27, 2013

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?

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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