Category Archives: Uncategorized

Webinar Thursday evening!

I was invited by Discovery recently to do a webinar in their Siemens STEM Academy Connect series.

I say yes to pretty much everything that doesn’t cost me money, so I said yes.

It takes place Thursday, Dec. 5 at 7:00 p.m. EST.

This will be version 3 of “Standards for Mathematical Practice: They’re Not Just for Students Anymore!” Version 1 I did for a Global Math Department meeting. Version 2 I did at the NCTM Regional in Louisville. Version 3 is substantially revised from those first two.

Among other things, we will discuss what this photograph has to do with mathematical practices.

There is a handout which I would encourage participants to spend time with beforehand.

It is free.

I have done my best to plan a maximally interactive hour, given the webinar format.

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.

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):

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?

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:

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.

Syllabus

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

Instructional creativity [#NYTEdTech]

As you may know, I spent the day at the New York Times Schools for Tomorrow Conference, in Times Square. Gorgeous building, very well organized event.

Let me tell you two stories from my day today.

First story

I sat out the panel session on the question, What’s the new era business model for higher education. But I listened in on most of it from an adjacent room. Tony Florence began one of his turns on the panel by asking us to, “Imagine a household that doesn’t have a DVR—unacceptable; that’s not TV right now.”

I do not need to imagine that house. I live in that house. Large numbers of students in America’s public schools live in that house. (And can we please follow up on the “house” assumption?)

But that was just a prelude to his next point, which had to do with education.

“Tivo for the classroom. It records the lecture, syncs up with content—PowerPoint or whatever—and is available for the student to pause, rewind and play as many times as necessary.”

Second story

Delaware governor Jack Martell was speaking about how data is being used in Delaware schools.

Evidently, data is collected about student performance (I do not have in my notes the details of how this occurs). Teachers have one hour each week that they are to meet in teams in their schools to pore over this data, to make sense of it and to inform instruction.

He offered the example of a group of teachers (at, say, fourth or fifth grade) who see in their data that their students are struggling with adding fractions*. But they notice (because data is shared within the district) that the students at the school down the street are doing better.

So the teachers in the first school go to the teachers in the second school and ask, “What worksheets are you using?”

Conclusion

For me, these two stories encapsulate the kind of instructional creativity that was on display today. That lectures, PowerPoints and worksheets are the media and formats we are working with is taken for granted. The question is how these will be delivered.

This is our revolution.