Airtight Logic

  1. Hey, that person is wrong.
  2. When a person is wrong and are informed of their error, they recant.
  3. Hey, that person didn’t recant!
  4. Therefore, that person is dishonest.

(From the internet.)


But They’ve Never Heard of Positive Numbers

Ancient Greek mathematicians had no notion of negative — or positive — numbers. Similarly, for most young children, numbers are not signed. Children learn to count with natural numbers. At some point, they learn about zero. Later, they encounter (regular) fractions, decimals, and percentages. Typically, a child’s formal introduction to the notion of sign comes after all this experience. Interestingly, we have found that many students in the elementary grades have some familiarity with negative numbers but have never heard of positive numbers. These children inhabit intermediate worlds that consist of regular numbers and negative numbers before they begin to (intermittently) inhabit worlds of positively and negatively signed numbers.

Regular Numbers and Mathematical Worlds

Too Much of A Good Thing

Greg Ashman:

If more guidance makes minimally guided approaches more effective then why not use a fully guided approach? Won’t that be still more effective? It is an argument that plays out again in the book and one that offers little comfort to proponents of open-ended problem solving in high school maths classes.

But, Jordan Ellenberg:

The difference between the two pictures is the difference between linearity and nonlinearity, one of the central distinctions in mathematics…Mitchell’s reasoning is an example of false linearity—he’s assuming, without coming right out and saying so, that the course of prosperity is described by the line segment in the first picture, in which case Sweden stripping down its social infrastructure means we should do the same.

But as long as you believe there’s such a thing as too much welfare state and such a thing as too little, you know the linear picture is wrong. Some principle more complicated than “More government bad, less government good” is in effect. The generals who consulted Abraham Wald faced the same kind of situation: too little armor meant planes got shot down, too much meant the planes couldn’t fly. It’s not a question of whether adding more armor is good or bad; it could be either, depending on how heavily armored the planes are to start with. If there’s an optimal answer, it’s somewhere in the middle, and deviating from it in either direction is bad news.

Also, John Sweller:

That is not to say that there are no disadvantages to the use of worked examples. A lack of training with genuine problem-solving tasks may have negative effects on learners’ motivation. A heavy use of worked examples can provide learners with stereotyped solution patterns that may inhibit the generation of new, creative solutions to problems.

Greg’s argument is, “If a bit is good, isn’t a lot better?” But this sort of falsely linear thinking isn’t compelling, no matter what you think about direct instruction.


How did 69 turn into 29?

Last year, while reading and writing about cognitive load theory, I came across something weird that I couldn’t explain. A paragraph from Greg Ashman’s latest reminds me of this puzzle. It’s really small and inconsequential, but it’s been bugging me. Maybe you can figure it out.

He writes:

One of my PhD supervisors did an experiment in the 1980s. Undergraduates were given as series of problems. Each problem involved a starting number and a goal number. The participants had to get from the first number to the second using only two moves which they could repeat: multiply by three or subtract 29. The problems were designed so that each one was solved by alternating the steps. Although the students could generally solve the problems, very few ever worked out the rule.

Great. Multiply by three, or subtract 29.

Except you go back to that paper, and it’s actually subtract 69.

Screenshot 2016-09-11 at 6.31.37 PM.png

Where did Greg get the “subtract 29” from? I don’t know, but it could be from this piece by Sweller in 2016.

Screenshot 2016-09-11 at 6.33.29 PM.png

Anyway, totally unimportant. Completely uninteresting. But. Did he forget? Was it a typo? Did he decide — as so many before — that 69 is a funny number to talk about in classes?

If you see me and I’m looking pensive, this is probably what I’m thinking about.

Who do I need to know first?

Most of the time I find it pretty useless to talk about kids with their old teachers before the school year starts. It’s not that the teachers don’t have important things to say. They do, I just have no idea who they’re talking about. Let’s talk in a week!

Yesterday, though, I had a really nice chat about my class roster with a colleague. She’s teaching kids I’ve taught; I’m teaching a lot of her kids from last year. At first we fumbled for something useful to say, and we landed on the usual: this kid’s great, this kid loves to gab, this kid can fall through the cracks, etc.

The breakthrough was when we narrowed things down. Who do I need to form a relationship with first? She had some ideas, and then I shared mine.

My class roster has three names marked. I will get to know them first.

Teaching for Learning Styles is Inconsistent with Teaching for a Growth Mindset

A few years ago, some math teachers were discussing a book on twitter. The book had made the case that the existence of “learning styles” for kids is a myth. To some of the teachers in this discussion, this was very surprising.

Shouldn’t it be? We’ve all seen kids that seem stuck on an activity…until we present the material in some new way. Note-taking leads to learning for some kids, but not others. Other kids seem to lose track of an explanation halfway through, but thrive when given a chance to read it instead. And we’ve also all taught kids who seem to think through movement — these are kids who seem to be intellectually confined when physically constrained.

This is a tricky question, and I’m familiar with the anti-learning styles studies.

I don’t think the big issue with learning styles is that there’s no evidence for it, though. On its own, that’s only a bit troubling to me. Instead, I think there’s a risk that our learning styles work will go against our efforts to promote a growth mindset about intelligence.

The growth mindset literature encourages us to help kids see intelligence as plastic; their smarts can grow with effort. We would never tell kids that they just aren’t smart enough to understand a verbal explanation. How much different is it to tell them they’re a “visual learner,” and therefore less likely to understand that same verbal explanation?

This all comes from the best of intentions, of course. We want to make sure that different kids get their different needs met. But we have to be very careful not to do this in a way that encourages kids to identify with what they’re naturally better or worse at. We need to give that individual help in a way that sends the message that through hard work and the aid of teachers, learning will happen.

As I was discussing this on twitter, another educator mentioned that we also risk lowering expectations for students, either implicitly or explicitly, when we start designing tasks that seem to avoid areas where they’re perceived to be weak. That’s rough for a kid, and probably not what’s best for the class.

That’s my case against learning styles.

Chatting with @KentHaines @AranGlancy about Contexts for Subtracting a Negative

I love Kent’s work with integers. But I can’t stop thinking that this lesson uses contexts in a slightly different way than contexts are usually used to support the learning of arithmetic.

In my 4th Grade classes I teach kids how to multiply larger numbers. For this learning, I often use an area context to help. How does it help? Essentially, it’s a scaffold. For example, it’s often easier for kids to find the area of a 12 x 34 field than it is for them to calculate 12 x 34. This is, I think, because of two things many kids know about fields. First, that if you split up the field into a bunch of parts, the field is just made up of those parts. Second, when you split up the field on a page it’s easier to keep track of how some dimensions change while others stay the same. These are two benefits that come from treating 12 x 34 as an area problem.

But it does us no good to just ask kids to solve an easier problem for the sake of ease. We use this easier problem to help kids solve a more difficult problem, 12 x 34, without the context. What makes the area problem such a helpful context is that area is a perfect model for multiplication. So the area context is very close to the arithmetic problem.

What is it, then, that my kids can bring with them from the area problem to the multiplication problem? Strategies, or put another way, actions. There are things they know how to do with area problems, and extraordinarily similar actions will apply to the multiplication context. The strategies are the same in the two problems. You develop a strategy in an easier context, and then bring it over to the tougher context. Roughly, I think this is how it works.

How about subtracting a negative? The context that Kent has us using involves balloons and sandbags. We start with 10 balloons and 5 sandbags on-board our hot air balloon. What’s our height? Then we take away 3 sandbags. What’s our new height?

Here are my questions:

  • How similar is this context to the integer context?
  • What is the strategy/actions that kids are learning how to do in this context?

And I don’t know the answer to these questions. That’s why I’m stuck in trying to understand how the integer game works.

While thinking about this, I’ve been tempted to say that the strategy is “treating subtraction of a negative as adding a positive.” The issue is that it’s unclear to me whether this is a strategy that kids are really using in the hot-air balloon context. It’s certainly now how I think about all of these problems. Take this one: “You start at a height of 10, and you throw away 5 sandbags. What’s your new height?”

Here’s how I think about this problem:

  • Throwing away 5 sandbags? That means that my equilibrium is off by 5 things.
  • 5 things that make me go up.
  • So I’ll go up 5.
  • That puts me at 15.

It’s unclear to me what this habit of mental calculations looks like in the realm of negative integers. I’m wondering if this way of thinking doesn’t carry over to negative arithmetic particularly well.

Part of the problem is that there aren’t a lot of “strategies” available for us as we think about 10 – (-5). There only seems to be one way: treat it like adding a positive. How do you create a context that makes this easier?

(Though I’m a fan of the “subtraction is the opposite of addition” approach. This seems like a strategy to me.)

But I do love Kent’s game. I bet it does help kids, and I would use it to introduce subtracting a negative.

I think this game functions differently than the area context, though I struggle to define exactly how. (Your thoughts?) I’ll take a shot at expressing one half-formed idea I had about how the game might work.

It could be that the game is particularly helpful as an introduction because it gives kids a story to tell about how come this all makes sense. I can think of areas of math where I’ve found stories like this helpful for remembering things that otherwise I can’t make meaning of. So maybe this game functions more like an analogy or heuristic explanation than a context. Putting that another way, maybe I’ve just been thinking too narrowly about how contexts can help learning.



Chatting with @jennalaib @kangoff @kvanduzer @heidifessenden about multiplication fluency

This is an area of my 3rd/4th Grade teaching that I want to do a much better job with this year.

Why does this stress me out?

When I first started teaching 3rd/4th, I was at a very fluffy moment in my teaching. I was just going to make sure that students conceptually understood multiplication, and fluency would just happen when it happened.

This worked for some kids, but I remember a number of kids that made very little progress in their multiplication fluency over the course of our time together. At the same time, I was seeing the benefits of the conceptual work because they could find a strategy for tackling bigger multiplication (like 31 x 12) but it would TAKE FOREVER. Because of the fluency.

Around this time I also got hired to do some research reading on multiplication fact fluency. I’m not going to do the citation thing, because no one cares, but that experience along with my teaching helped me clarify my views about how learning multiplication facts happens.

What’s the difference between conceptual understanding, fluency and having facts memorized?

There are lots of terms that we try to use. There are people who have “standard” definitions of all these things. Here is my current understanding of how to promote fluency, and how it relates to practice, timed stuff and conceptual understanding:

  • Consider a multiplication problem like 6 x 7. There are a lot of strategies that kids could use to figure out this answer. If a kid can find out the answer using any mathematically valid strategy, then I’m happy to say that this student has conceptual understanding of multiplication. By this I mean the multiplication operation, roughly “what multiplication means.”
  • To find 6 x 7 a kid might count by 7s, or by 6s. They might find 5 x 7 and then add a 7. They might find 3 x 7 and double that. And so on. All of these are informal strategies for finding 6 x 7. I have a lot of informal strategies that I use for mental multiplication in my own mathematical life. For example, 12 x 7 I can figure out by doubling 6 x 7. It’s not that I know 12 x 7, it’s that I can quickly figure it out using informal strategies. This is what I think a lot of us mean by “fluency” with multiplication — the ability to efficiently use informal strategies to derive an answer.
  • On the other hand, I basically never have to use informal strategies for single-digit multiplication because I’m the kind of kid that had no trouble committing these facts to memory in my elementary years. TERC Investigations calls this “just knowing” what 6 x 7 is, and this is what a lot of us call “having the facts memorized.”

This isn’t news to any of you, but I just wanted to clarify my terms before I used them a bunch.

But how do all of these things — conceptual understanding of multiplication, fluency with informal strategies, and having facts memorized — fit together in kids’ learning?

How do multiplication facts get memorized?

Let’s start with having facts memorized. From experience and my reading, I think the way you come to have facts memorized is by holding a fact in your head and trying to remember it. This is sort of like if I told you my middle name (I’m not gonna) and then it’s rattling around in your memory and a few minutes later you had to remember my name and you’re like….uh, oh yeah, it’s ______. Those moments, piled together, eventually commit my middle name to your long-term memory. It’s memorized.

I’m saying a lot of things and this isn’t really “writing” as much as barfing thoughts. But this is an important point to me: to have facts memorized, you need to try to remember them.

These moments where you’re trying to remember a fact that you’re holding in your head can occur either artificially or naturally. Artificially, through practice. But a lot of practice isn’t actually designed to create moments where you’ve got a fact in your head and you’re trying to remember it. Consider a test that just gives a child a multiplication problem to solve. A student could complete this task by just using some strategy to derive the fact, and they haven’t practiced remembering the fact at all.  Or consider those ubiquitous Mad Minute things. They can’t possibly help you practice remembering a multiplication that you don’t already have pretty available to you through fluent derivation with informal strategies.

A lot of harm has been done to kids in the service of creating remembering practice for kids. So why not avoid them entirely?

Kids can get these opportunities to remember facts in more natural settings. Here is a way that this can happen:

  • Kids have a conceptual understanding of the multiplication operation.
  • You then teach kids a bunch of informal mental strategies for deriving multiplications.
  • Kids get fluent at these informal strategies.
  • As kids do other multiplication work, they frequently find themselves deriving multiplication through these informal strategies. Then, later in the problem, they have to remember what they derived. This creates remembering practice for kids.
  • And if they don’t remember it? They just rederive the multiplication using an informal strategy, and then they’re like “oh yeah!” Another remembering opportunity.

The above, I think, is how some writers, PD people and academics envision how kids could naturally come to have their multiplication facts memorized.

…but the natural approach wasn’t working for enough of my students. So I’ve adopted a more balanced approach that mixes the natural and artificial pathways towards getting remembering practice.

What do I think would be best for kids in my context?

In my 3rd and 4th Grade classes, I never want to ask kids to do something that isn’t meaningful to them. At the same time, I worry a lot about how numeracy impacts the math kids feel confident with in my high school classes. For this reason, I’m not satisfied with just fluency with informal strategies for my elementary students — I also want to help them come to have these multiplications memorized.

My ideal approach goes something like this:

  • Conceptual Understanding: Make sure kids have some strategy for solving a single-digit multiplication problem, either on paper or in their heads. I want them to have conceptual understanding of the multiplication operations.
  • Using Number Talks to Develop Informal Strategies: Figure out some small group of multiplication facts, and make sure that my kids become fluent in using informal strategies to derive these multiplications.
  • Create Practice Cards: I give students index cards and ask them to write that small group of multiplication facts on those cards. I ask them to mix those in with a bunch of cards they’ve mastered. (Flash card key #1: keep the success rate high.) Write informal strategy hints (e.g. “double 3 x 7”) at the bottom of the card, if they want. (Flash card key #2: it’s your deck, you do what makes sense.)
  • Practice with Cards: Practice your deck, alone or with a friend. Go through the whole deck at least twice. (Flash card key #3: try to keep it low stress).
  • Follow-Up With Quizzes to Assess: Figure out if we’re ready for a new informal strategy, a new set of facts, a new type of talk, a new set of cards to add to our decks, or whatever.

I was doing this in 4th Grade last year, but I was unsystematic and sloppy. I think I know how to do this so that kids don’t get stressed out and that it’s enjoyable and fun. Honest to god the kids loved the cards and often asked if we could work on them. The kids liked them so much that I started putting non-multiplication stuff into their decks. My sales pitch for this sort of practice — some things that we figure out are worth trying to remember — applied to lots of things. By the end of the year I was asking kids to put in a few division problems, some fraction addition and subtraction and multiplication into their decks.

What is wrong with this approach?

I am a sloppy, unsystematic teacher. Partly this is an artifact of my teaching context — I teach 4 grade levels in 4 different rooms, don’t have much wall space, don’t have a consistent stock of supplies in my rooms, etc. — but I know I can do better.

Part of how my sloppiness expressed itself was that practice with cards was whenever-I-remembered. I think it would be better if I scheduled this in to the week somehow.

The bigger issue was that I don’t have a good plan for how to break up the multiplication facts. I’m sure there are resources that I could use here, but what I need is something like Set 1: multiplication involving 2s and stuff you can figure out by using doubling strategies on those 2 facts. Set 2: multiplication by 10s(?) Set 3: Using halving and multiplication by 10s to figure out other stuff, etc. I want to make sure that the informal work with strategies is structured over the course of the school year to cover all the facts. I want the strategy work to precede the remembering practice work. Do you know of a resource for this? I’d be interested in it!

I think those are the major issues that I want to focus on this year.

What do you think?

This isn’t Michael’s Take On Multiplication. The above was “Michael’s response to a tweet.” This is a rough shot at describing how I’ve been handling this in my thinking and my classes lately.