# Category Archives: Problems (teaching)

I find the tone of this article a bit over the top: It Ain’t No Repeated Addition. In it, Keith Devlin (as is SOP for mathematicians) takes a too-strong epistemological stance–multiplication is not repeated addition.

I am much more interested in the nuanced space between provocative stances. For instance, I am much more interested in a question such as, What is gained and lost in defining multiplication in relation to addition versus some other approach?

Exploring this question allows all knowledgeable parties access to the conversation, and it helps us listen to each other. Telling others that they are wrong tends to shut down the conversation, to discourage listening and make people defensive. Through that lens, I can can read Devlin’s piece in a productive way.

In that spirit, I have engaged with our good friend Michael Pershan on the topic of exponents (By the way; go read this piece—it is excellent.). In particular, I have attempted to ask the analogous question about exponentiation as an operation.

In particular, he has been exploring the conditions under which students confound exponentiation with multiplication. As seen in the very common algebra mistake, $100^{0.5}=50$.

I have suggested that perhaps the trouble lies in defining exponentiation as repeated multiplication. After a bit of brainstorming, I came up with an alternate definition: doubling (and tripling, etc.)

What if we think of the powers of 2 not as repeated multiplication, but as number of doublings?

This sounds like a trivial difference, and perhaps it will prove to be. But I think it is more than that.

For instance, repeated multiplication makes me think of $2^{5}$ on its own. But number of doublings suggests to me a starting value (which could be anything) and then we double that value some number of times.

Repeated multiplication doesn’t make clear what to do about $2^{1}$, nor $2^{0}$. What does it mean to multiply a single 2? Or no 2′s at all?

Number of doublings makes this more clear. $2^{1}$ means double once, while $2^{0}$ means do not double your original value.

Rational exponents? Start with mixed numbers and you should be in good shape. One and a half doublings is more than twice what we started with, but less than four times.

What would it be like to start instruction in exponents from the number of doublings perspective instead of from the repeated multiplication perspective?

No better playground for hypothesis testing than a truly blank slate.

Little man knows squat about exponents.

Griffin (eight years old): Ten.

Me: Then double it again.

G: 20. Then 40…80…160…2…no…320…640…1280…

Me: Wow.

G: Then two-thousand…five-hundred-sixty.

Me: Holy cow. I did not know you knew that many doublings!

G: Yeah. That’s all I can do, though. I can’t think of what comes next.

Me: Right. Next would be 5120. But that doesn’t matter. We started with 5. Then you doubled one time to get 10. You doubled two times to get 20. You doubled three times to get 40.

G: Yeah.

Me: What if you doubled one and a half times? What do you think that would be?

G: 15

Me: So if you double 5 one and a half times, you would expect it to be 15?

G: Yeah. Is that right? What would it be?

Me: Wait. I want to know what you think here. I will answer all of your questions after you answer a few more of mine. Why do you say 15?

G: Well, doubling once is 10, then half off the next one would be 5 less, so 15.

G: Yeah.

Me: What if you had zero doublings?

G: Well…it could be 5. Or maybe 0.

Me: What is the thinking behind 5?

G: You don’t double it at all, so it’s just the same.

Me: And what is the thinking behind 0?

G: Adding and timesing with zero…it’s usually zero. So I think it might be that. But it could be 5. What is it?

Me: I promise I’ll answer all your questions in a minute. One more…What if you doubled half a time?

G: Well…I don’t know….Seven and a half, maybe. I don’t know. I like whole numbers better.

The doublings approach led this third grader to:

1. Linear interpolation for rational exponents, rather than triggering a multiplication schema, and
2. The possibility that $2^{0}=1$ (albeit with a low degree of certainty).

These both seem like improvements over the intuitions Michael demonstrates in his piece—intuitions which certainly mesh with the misconceptions with which I am familiar in my middle school and college teaching.

## Problem solving

Too long for a tweet; not really worthy of a blog post. Here goes anyway…

I just walked past a computer science classroom, where the instructor was drawing an elaborate set of diagrams—presumably representing subroutines and whatnot.

This is the sort of thing I mean.

Students were dutifully taking notes.

I thought quickly to myself, How effective can this possibly be? Don’t you have to learn programming by solving problems?

And then I thought to myself, a bit more slowly, And how is mathematics any different?

## Questioning a premise: Money and decimals

We are working on composed units and place value in my math course for future elementary and special ed teachers. We were discussing examples of composed units that are composed of things exactly like one another. A pack of gum is an example of this, while a car is not. A car is composed of a whole bunch of parts, but those parts are not all alike.

A student asked whether quarters being composed to make a dollar count. I said yes and then took a poll. I now share that poll with you, and invite discussion.

### The poll

Both of the following are correct. However, one of them probably matches more closely how you think about this. So vote for the description that feels closest to how you think:

1. A dollar is the original unit, and it is partitioned into smaller units called cents. In this case a cent is a partitioned unit.
2. A cent is the original unit, and one hundred of them are composed into a larger unit called a dollar. In this case, a dollar is a composed unit.

### The Results

I won’t tell you the results of my poll yet. I will tell that I had an instinct about it, that my students favored one of these outcomes by a ratio of $\frac{2}{3}$ to $\frac{1}{3}$, and that my instinct about which would be more popular was correct.

So how about it? What are your thoughts? And what are the implications for using money to teach decimals?

## Decimal place value v. whole number place value

As the rest of my household heads back to school, I am getting down to the business of planning for my new semester. One instructional problem I am trying to solve is that of pushing my future elementary teachers to understand decimals more deeply. Whole number place value I have nailed down. But decimal place value is another story. The rules are so deeply embedded in them that it is very difficult to get them to question these rules, or to seek a better understanding of them.

I have collected substantial data over recent semesters (not research quality data, but consistent formative and summative assessment results) to demonstrate that my students operate on decimals using a combination of known rules and whole-number place value principles. They can switch back and forth between these fluidly and generate right answers to nearly any decimal task.
Which leaves me seeking questions that they cannot answer without digging more deeply.
I have a candidate question in mind: Why can we put zeroes after a decimal number without changing the value, but not after a whole number without changing its value?

It occurred to me last evening that I knew what my answer to this was, but that I lacked a wide repertoire of correct explanations. So I asked Twitter. Lots of good stuff followed. Read the full conversation on Storify.

Study Confirms Correlation Between Student Grades and Ratings on Instructor Evaluations

To those who are super concerned with grade inflation, this seems to be the smoking gun. Low grades lead to bad evaluations from students, and good evaluations are the result of high grades. There is an upward incentive pressure on grades that is degrading the quality of higher education. QED.

Only, that’s not at all what’s going on in the study.

The study is described thusly:

To test her hypothesis that poor teaching evaluations are a form of revenge, Vaillancourt designed three related studies, each involving between 150 and 176 mostly first-year university students.

Students were given up to 20 minutes to write a short essay on an assigned topic, and were told their work would be graded by professors or teaching assistants. The essays were then randomly assigned a high or low grade and, in some cases, comments such as “No suggestions, great essay!” or “This is one of the worst essays I have ever read!”

Students were then asked to provide feedback to their evaluator about his or her marking ability, fairness, helpfulness and general competence.

In the first study, participants given poor marks and negative comments were almost 10 times more likely to rate their evaluator harshly than those given high marks and positive comments.

It seems reasonable to conclude that anonymous scoring with zero useful, actionable feedback generates the results described. But I would also submit that anonymous scoring with zero useful, actionable feedback is crummy teaching.

And students in the study argued this quite strongly. Consider the epigraph from the published research piece:

My evaluator must not mark hastily and should re-read key points to offer better feedback. It is actually the poorest job in terms of evaluation that I’ve ever seen. It is easy for the evaluator to talk crap like “This is the worst essay I have ever read,” but when offering feedback about how it may be improved say nothing except “needs clarification.” How is this supposed to help me improve? This evaluator is a complete dumb ass, excuse my French.

This is far from a call for a better grade. It’s a call for meaningful feedback, and in fact supports a hypothesis that student feedback has a kernel of truth to which we ought to pay attention.

Indeed, Ken Bain and his colleagues argue that:

1. when we ask students to rate the quality of their learning in a course, they are actually decent judges of this (notice that this is not among the things asked in the study in question), and
2. the best college teachers sought to understand and meaningfully interpret feedback from students, using it to reflect on and improve their practice.

To restate, in Bain’s study, instructor quality was negatively correlated with instructor dismissiveness of feedback from students. Kernel of truth, people.