Tag Archives: decimals

Reflections on teaching

I am working on a ton of interesting projects right now. Not least of these is my classroom teaching at the community college. My fingertips are sore from typing.

And yet there is always more to say. More to think about. More conversations to have. Here is a peek into one that is ongoing.

Malke Rosenfeld and I have been going back and forth about math, dance, Papert and learning for a few months now. I am learning a lot from the conversation. She asked some questions this morning.

Malke: A thought just entered my head — why are you offering TDI? Is it based on a question you are unsure of and want to see what others think? Or are you seeing a deficit in math teachers’ thinking that you want to shore up?

Me: When ranting on Twitter, I could see that some of my assumptions about baseline teacher knowledge about fraction/decimal relationships as they pertain to developing children’s thinking were unfounded. That is, I was assuming teachers knew a lot more than they seemed to. Which has implications for my Khan Academy critiques, and lots of other writing on my blog. Yet people were also curious. So I wanted to say more in a way that would draw from and build on a larger collective knowledge, so it’s not just my spouting off.

Malke: Is there a reason you offered it specifically as a course, and not a moderated discussion (which it sort of seems like right now)?

Me: When you view learning as a social process, you tend to think of courses AS moderated discussions. I mean this quite seriously. I know that it goes against the grain of online (and face-to-face) course design. But that’s not because I think of online instruction differently from others; it’s because I have a particular view of learning that runs much deeper than that. If I tell and quiz, you’re not learning very much. If I propose a set of ideas, listen to what you have to say, encourage you to interact with others and move the conversation in directions that seem useful based on those interactions, you’re probably going to learn a lot.

As long as I can keep you engaged in that process. Which is a different challenge online than in the classroom.

Malke: Is there a place you specifically want your students to get to by the end of the seven weeks?  Or are you just curious to see what develops?

Me:  I am curious to learn what I can about teaching at every opportunity. I want to produce “students” who can articulate important questions (see? learning as having new questions to ask?) about curricular approaches to decimals. Ideally, I would help them to develop a critical voice that speaks to/through them when they work with individual students, when they plan lessons and when they talk with their colleagues in a variety of settings. In short, I want to change the way teachers view the territory of decimals, fractions and children’s minds. Strange mix of lofty and specific there, huh?


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 \frac{1}{4}=0.25); 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?

I report my height as 6 feet 1 inch. I do not report it as 6\frac{1}{12} feet, although I know that I could. Likewise I don’t think of 1 hour and 5 minutes as 1\frac{5}{60} 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 1\frac{25}{100} 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!

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 \frac{1}{100} 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?

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

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

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

I get that there is no perfect lesson

I get that there is no perfect lesson. Really I do.

And I get that students leave my classes with wrong ideas. But the thing is, when I come across these wrong ideas, I try to do something about it.

A couple of tweets from the field last week (sender’s tweets are locked, sorry).

S came in today claiming to have used Khan academy last night to learn about decimal place value. Was adamant 0.63 > 0.7 cc @Trianglemancsd

Was also adamant that 0.4 < 0.40. Not feeling overly confident about Mr. Khan and his explaining abilities…

Now, people get fussy when I fault Khan Academy for bad decimal instruction.

If I don’t like the videos, I am told it’s not about the videos; it’s about the exercises.

If I don’t like the exercises, I am told there are new ones in the queue.

If I don’t like the trial versions of the new ones in the queue, I am told that the particular exercises don’t matter; it’s about the knowledge map.

When I say that the knowledge map is flawed, I am told that it doesn’t matter because students can move around in Khan Academy in any way that they like.

And then every day kids are going to Khan Academy for help with decimals. Some of these kids, such as the one in the tweets above, are going there independently. And some of them are going there because their entire state is piloting it as a primary instructional resource!

Whoa there! they say. Khan Academy isn’t meant to be a primary instructional resource. 

But then here is a video that Khan Academy produced…

At 20 seconds in, a student teacher in mathematics says this:

When I first [learned] about Khan Academy, it was mostly “my teacher said this, but I can’t remember what he said, so I’m going to go check it out on Khan Academy. So it was more of a personal resource.

That’s kind of where I was thinking it would be in my classroom down the line. “If you’re struggling with this, go check out Khan Academy.”

But now, after coming to this, it can be that first step. It can be the go-to. “Hey, go learn this. Go learn the foundations, and then we can take it to the next level in our classroom, and put in those hands-on activities.”

Just to be clear, Khan Academy produced this video. I am not misrepresenting KA here. They are proud to share that a math teacher at a training views Khan Academy as a good primary instructional resource.

Now, I have long been critical of textbooks that introduce decimals as though they were a logical extension of the whole number place value system (just ask my students!) I am no fan of what Hung-Hsi Wu calls Textbook School Mathematics.

But if you are going to get introduced by the publisher of The New York Times  [at about 3:00 in the linked video] as  “a true pioneer” who is “breaking down barriers” with “heart”, “guts” and “innovation”, I think you have a responsibility at least as great as that of the average textbook author. You have to strive to do better and you have to pay attention to what people already know.

If you are going to repeat that your mission is “a free world-class education for anyone anywhere,” you need to spend some time concentrating on the meaning of world-class, rather than imitating the bad textbooks that presently exist.

I have taught many crappy lessons, and I surely have many ahead of me. I do not fault Khan Academy for having a few crappy lessons.

But I seek feedback from my students on what they are learning.

I consult research on learning for the topics they are struggling with. I collaborate with colleagues near and far to improve my lessons.

I do not defend my crappy lessons by calling them unimportant. I own their crappiness.

And I strive to do better next time.

[Comments closed. Hit me on Twitter if you want to talk.]