Terms, Notations, and (Mostly Needless) Confusions

I wanted to call this “Logician General’s Warning: Confusion about Terminology is Hazardous to Your Understanding”, but it takes too much space…


Needless Confusion Over Terminology and Notation

A friend of mine who has a degree in statistics commented a few years ago that he couldn’t understand why people were confused about the terms random variable, probabilistic variable, and stochastic variable; after all, they all mean the same thing. I instantly realized that I myself had been confused because I didn’t know that. Or, quite likely, I once knew but had totally forgotten! I’ve seen confusion — usually needless confusion — over terminology cause serious problems many times, both inside and outside the classroom.

And while I’m talking about probabilistic things, how about Bernoulli “processes”, Markov “chains”, and Hidden Markov “models”? In my experience, those are the usual terms for the three phenomena; but they’re all “processes”!

The same thing happens with notation. I was guest-teaching a lesson on Zeno’s paradox of Achilles and the Tortoise to a high-school math “exploration” class (see my post about it, https://whymystudentsdidnt.wordpress.com/2012/11/30/zenos-achilles-and-the-tortoise-paradox-and-geometric-series/). As an example of a convergent infinite series, I wrote on the board


A lot of students had trouble with the 1/(2^n) part until their regular teacher pointed out that it means the same thing as (1/2)^n — a more familiar notation to them. And I probably would have used the latter form, if it had even occurred to me it might make a difference 😦 .

Hard-to-Avoid Confusion Over Terminology and Notation

How many students confuse quadratic expressions, quadratic equations, and quadratic functions? Many of my own students certainly did, but at least the terms are as consistent as possible. I’d say the situation with the two common notations for derivatives — dy/dx and y’ — is somewhere between “Needless” and “Hard-to-Avoid”. There’s some justification for both notations, but I wonder if it’s worth it.

It’s vitally important that students understand and remember the terms and notation we throw at them. If they’re mechanically following rules but they confuse widgets and wodgets, they’re dead; even if they’re really going for understanding, confusion about terms and notation can waste a lot of their time, and ours.

Early Number Sense and Success in Math

For the last few months, I’ve been teaching Algebra II to a class of one, a high-school senior who’s struggled with math for years. After spending two or three hours a week together for 11 weeks, I’m happy to report that Ellie (not her real name) has unlearned a lot of wrong ideas and learned enough right ones to get an A- on her first test (factoring quadratics, solving quadratic equations by factoring, etc.). Ellie told me the last time she felt she understood math in school was 6th grade and she hasn’t gotten better than a C or D (she’s not sure which) on a test since then. So she was happy with the A-, too!

I’ve always wondered how strong the connection is between number sense and learning math. The other day I mentioned to Ellie the recent news item (widely reprinted on the Web, e.g., http://www.8newsnow.com/story/21786760/early-number-sense-plays-role-in-later-math-skills) about University of Missouri research showing kids with poor number sense in primary grades have trouble with math years later. She responded that she remembers struggling with concepts like adding 4 and 4 in 1st grade — unless she used her fingers so she could see what the numbers meant. Yep.

Why I’m Better Than High-School Students at High-School Math

TableOfSmallPowersOfSmallIntsSo why is the person whose mathematical ability I’m most familiar with (me) better at high-school-level math than nearly all high-school students I’ve encountered? Knowing the answer might help me help the other students! I think the rather short list below pretty much explains things. (Of course I’m better in a lot of cases because I already know exactly how to solve the problem, but of course there’s more to it.)

I’m certain I’m better at:

1. Understanding “word problems”, i.e., translating into mathematical form the kind of verbal descriptions of situations that commonly appear in textbooks and on standardized exams. However, it’s not clear how much this has to do with math skill as such, as opposed to reading skill; Jo Boaler has some interesting comments on this point in What’s Math Got to Do With It?.

Outside of translating words to math, I’m better at:

2. Factoring small integers, especially recognizing perfect squares.

3. Seeing simple algebraic relationships (e.g., in equations), simplifying expressions, etc. To a considerable extent, this is just a matter of being really comfortable with mathematical notation.

4. Intuition for what’s reasonable — intermediate values and results: among other things, estimating numerical values.

5. Avoiding specific common mistakes, e.g., “everything is commutative”, confusing what works for addition/subtraction vs. multiplication/division.

Is this a good list? How would you describe why you’re better than students?

Now, accepting my list as a starting point for discussion, what can be done to help students with these skills? Or are all of these skills truly important, and if not, what can be done to reduce their effect on students’ grades and mathematical confidence?

To keep this post short, I’ll just mention one item in my list that strikes me as both not very important and easily mitigated: #2. I discovered last year that many of the students in both of my Algebra II classes — both honors classes, in fact — were struggling on tests with what I thought were easy problems simply because they didn’t recognize small perfect squares as such. When I realized that, I started putting the above table of Powers of Small Integers on the whiteboard, and that seemed to do the trick.

Be realistic about your strengths and weaknesses and adapt! (WIL #7)

It’s important to be realistic about your strengths and weaknesses and adapt accordingly. An important special case: Grading homework and exams can easily overwhelm the teacher. My brief tenure as a regular, full-time classroom teacher was as a maternity-leave replacement, and by the time I started, I knew I was a slow grader. I had four (count ’em) preps, and getting four lessons ready every day took so much time that I quickly realized that I’d have very little time to grade homework. From the beginning I didn’t try to grade as much homework as the very experienced teacher I was filling in for did. But I couldn’t even do what I thought (and had told my students) I could. It wasn’t until I graded the final exam that I realized how slow I was! I can see several reasons for this: wanting to give students really useful feedback, wanting to grade as consistently as possible, my inexperience, etc. If I teach much more, I’m sure I’ll get more efficient, but not by so much that it won’t always be an issue. But — as Ofer Levy pointed out to me later — the teacher usually has quite a bit of flexibility in how much grading they have students generate. One reason is that there are often ways to substantially reduce the load of grading without harming learning, so the teacher can avoid being a victim of a system they created themselves. Some ways that seem appropriate for teaching math, along with many other subjects: assign group instead of individual work; let students correct each other’s work; go over homework and quizzes in class but don’t collect or grade them. I did some of this, but could have done much more… and if I had, not only would I have been better off, my students would have been too because I would have had more time for their more serious problems.

Some students’ problems are beyond what a classroom teacher can solve (WIL #6)

Some students’ problems are beyond anything a classroom teacher can solve. Here are two examples from “Brief Survey of Calculus”, a 100-level college course I once taught—but the same problems could occur at the secondary-school level.

(a) Problem with course content. “Colton” told me he always struggled with math, and he certainly did in my class. He spent as much time in my office as all the other students combined; that seemed to help some, but not that much. And he wasn’t at all stupid. Like so much mathematics, calculus requires a fair amount of algebra, and I already knew many of my students’ problems were more with algebra than with calculus. So I finally asked Colton a very basic question: How much is 3 × (5 × 4) ? I wasn’t too surprised that he tried to use the Distributive Law, as if the question was 3 × (5 + 4) ! Colton was a senior, and older than most—probably 23 or 24 years old, and the Distributive Law is generally taught by 6th or 7th grades (under the current Indiana standards, it’s taught in both). So he’d undoubtedly been confused about this for well over 10 years, and it takes a long time to unlearn something that deep-seated.

(b) Problem unrelated to course content. At the beginning of the semester, “Isolde” really seemed to want to do well, but she did several odd things that hurt her grades enormously. For example, she had a graphing calculator but not a non-graphing one. I didn’t allow graphing calculators on quizzes/exams, so she started the first quiz/exam without any calculator until I noticed and loaned her one. Exactly the same thing happened on every other quiz/exam! She kept missing classes and even exams, and she never came to my office hours; she blamed all of that on transportation problems. And she repeatedly forgot important things. Perhaps she had tremendous family responsibilities, or she might have had ADD, or both. But she never told me what was going on, and, sadly, I couldn’t find a way to help her.

Example (b) is clearly related to my What I Learned item #1, “students really are responsible for their own learning”; but example (a) is something else.

Triangle Types, Quadrilateral Quotas, & Hexagon Hierarchy

Christopher Danielson’s great post on The Hierarchy of Hexagons begins with the observation that “School geometry seems to me one of the most lifeless topics in all of mathematics. And the worst of all? The hierarchy of quadrilaterals.” Then he describes his attempt at “breathing some life into this dead horse” by having students classify hexagons instead. Seems to me he was wildly successful: I never heard of future elementary teachers proving anything, but his class “proved that a Bob cannot be equilateral.” !! But what is a Bob, you say? Read his post — no, wait: read the rest of this post, then read his.

The chart above simply shows the relationship among all types of triangles in a Venn-diagram-like way that I think is exceptionally clear. I’ve never seen it done like this before, but if someone else has, please tell me.

All I’ve done with this chart myself is to show it to my geometry classes and ask them to draw examples of some of the types. But there other ways to use this idea, and one that seems much better to me now is: ask students to come up with their own graphical ways of showing the relationships among all types of triangle; then show some or all of them (and perhaps mine) to the class, and lead a “compare and contrast” discussion. Much better because much more likely to result in students really understanding! Comments, anyone?

For that matter, a classification like this begs the question of what types of figures there are of four or more sides, and of course the number of possibilities goes up rapidly with the number of sides. But how many are there with four sides, and how rapidly does the number go up? Well, what if a shape — say, a hexagon with five interior right angles — doesn’t have a name? (Actually, according to Christopher Danielson’s students, that’s a “Bob”.) Is a pentagon with two long sides and three short ones where the long sides are adjacent different from one where they’re not? What if the two long sides are parallel? What if the angles are all equal, or is that even possible? Etc. etc.
So a discussion of classifying triangles could lead to a discussion of partitions, of classification in general, of hierarchies, of (abstract) trees, and, no doubt, other topics.

In classroom management, the more concrete, the better (WIL #5)

For my first semester as a student teacher, I taught 6th grade math. Many of my students had a hard time controlling themselves, and one of my standard tactics was to make a list of offenders I would make stay after class. A very common problem, and a common way to handle it. But it was obvious that the clearer the connection between the behavior and the consequences, the more effective a deterrent it was. So my partner and I started putting a timer on the smartboard to keep track of how much class time the offending students had wasted, and announcing that they’d be staying after class for the same length of time.


To make the connection even more concrete, I looked for a simple, easy-to-use program that would count down whatever length of time it had gotten up to, and preferably one that could run in a Web browser with no installation. But I couldn‘t find one: every program I ran across treated going up and down as unrelated functions, typically calling them “stopwatch” and “timer”.

Well, I know how to program a computer: I spent years working as a software engineer. So I finally (long after my semester in middle school) wrote it myself. Following a suggestion by my friend and colleague Jeremy Sebens, you can even set a time of day — the time the class ends is the obvious candidate — at which it’ll start counting down automatically!


“Don’s Up/Down Timer” is available at


There are also instructions for using (and, if you want, customizing) it there, but it requires no installation and it’s very easy to use.

Another classroom application might be the opposite of the above scenario: you want to reward students for doing something they don’t like — and the longer they do it, the longer they get to do something they do want to do.

I think Don’s Up/Down Timer would be most helpful in middle school, but it could be useful in any situation where you want to measure an unknown interval of time, then set a timer to go off when the same interval elapses again. Prof. Kathy Marrs of IUPUI tells me she’s used it with her college-freshman biology class — I’m not sure for what. Certainly there’s nothing specific to math about it.

(4 Feb. 2013: revised to add the “rewarding” application and update the URL.)