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 studies 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.