Monday, August 25, 2014


Here in New York, it's back-to-school starting next week. Of course, if you're in the teaching profession (or really just know about it), you've probably been doing academic work and preparing for the fall semester throughout the summer and year-round. I'm in the very fortunate position that I'll have a new permanent position at CUNY this fall, so here's how I've spent my August:

I've developed a new website for practicing basic numerical skills that are prerequisite for algebra and other classes like statistics, calculus, and computer programming: I've written a few times in the past about the need for certain skills to be automatic, skills that have been taught but not mastered by most students who arrive in a remedial community-college algebra course, and therefore causes continual disruption and frustration when we're trying to deal with actual algebra concepts (links one, two). Like, for the algebra course itself: times tables, negative numbers, and order-of-operations. Or for a basic statistics course: operations on decimals like rounding, comparing, and converting to percent.

So what you get at the new site for each of these skills is a brief, 5-question quiz for each of these skills. Here's how I designed them:
  • Timed, so that students get a very clear portrayal of what the expectation is for mastery of each of these skills (15, 30, or 60 seconds per quiz). For example: sequential adding and counting on fingers for multiplications will not suffice.
  • Multiple-choice, so the site is usable on a variety of devices, including touch-screen mobile devices. For example: you can stand on a bus and drill yourself on a smartphone by just tapping with your thumb.
  • Randomized, so once you take a quiz you can click "Try Again" and get a new one, and drill yourself multiple times in just a few minutes each day.
  • Javascript, so the quiz runs entirely on your own device once you download the page the first time. The site doesn't require any login, accounts, server submissions, recording of attempts, or any transmitted or collected information whatsoever after you initially view the page.

This is something that I've wanted for a few years now, that no existing website really implemented and consolidated the way I wanted as a reference for students. Thanks to the new full-time track I feel I'm in the position to warrant developing it myself and leveraging it for numerous classes of my own in the future.

Feel free to check it out, and offer any comments and observations. If you feel it might help your incoming students this fall semester, give them the link and maybe it will lift a whole lot of our boats all at the same time. Do you think that will be of assistance?

Monday, August 11, 2014

When Are Parentheses Required for Substitution?

In my remedial algebra classes, on introducing the substitution of numerical values for variables, I've always said that it's safest to perform this substitution inside parentheses, especially for negative numbers. Of course, we all intuit times when that's not strictly necessary. So in this lecture I usually get one of the brighter students asking, "Exactly when is it necessary?". I've found this to be a surprisingly difficult question to answer. After a rather embarrassingly long consideration, here's what I've come up with.

Parentheses are basically required in the following two situations:
  1. Separating juxtaposed signs and numbers, and
  2. Collecting expressions with one operation under a higher-order operation that is not also a grouping symbol.

For situation #1, I'm assuming that we're not ever inserting new operational symbols like × or in cases of juxtaposed multiplication -- just the substituted expression and possibly parentheses. Parentheses are probably only needed for factors after the first one (i.e., after the coefficient).

For situation #2, we're mostly talking about multiplication and exponents, with some lower-order operation in the expression being substituted. Contrast with fraction bars (for division) and radicals, which have grouping built into the symbol, and thus no general requirement for new parentheses.

Here are a few examples of each. For the following, let x = 1, y = –2, z = ab, and w = a + b. Examples of separating juxtaposed signs and numbers:

Examples of collecting expressions with one operation under a higher-order operation:

We can explain the first example immediately above in that the negative sign acts the same as multiplying by –1, and therefore must be collected under the exponentiation operation. However, this does get slightly complicated by the use of the minus sign for both unary negation (i.e., multiplying by –1), and binary subtraction, which have different placements in the order of operations. For example, the following may be taken as a slightly ambiguous case:

Here, in substituting any numerical values at all for x and y, parentheses will definitely be necessary. However, this particular instance doesn't have juxtaposed numerals -- the real reason may be taken to be that without the parentheses, this would read as subtraction (lower order than the initial juxtaposed multiplication).

A few notes on specific cases of substitution:
  • If substituting one variable for another, then parentheses are never needed (the order of operations is clearly identical before and after).
  • If substituting a whole number, then only the situation of juxtaposed numbers after the coefficient can apply. Obviously a whole number has no written sign, and includes no operations to interfere with higher-order interactions.
  • If substituting a negative number, then any of the situations are possible. It does have an attached sign, may need separation from an advance factor (as above), and operates similarly to a multiplication (and thus needing collection under an exponent). 

Note that Wikipedia articles do show use of juxtaposed signs, e.g., 7 + –5 = 2, and discusses possibly superscripting the unary negation in elementary contexts and the computer language APL (link one, two), something that I've also seen on some calculators, in which cases parentheses would not be necessary. However, that's not something I've ever seen in textbooks (either college-level or otherwise), so I take that as nonstandard and not qualifying as well-written algebra. 

What do you think? Have I missed any important cases or examples?

Monday, July 21, 2014

Automatic Negatives

In my developmental (remedial) courses, I have been thinking more lately about where and how to communicate which skills need to be automatic -- that is, instantaneous and always correct. As I wrote earlier (link), these are skills which we take for granted in higher level courses, but they frequently get lost in the underbrush of all the other math topics, and students expect that struggling with them for several minutes is acceptable and normal behavior. A list of these remedial automatic skills that students often lack includes:
  • Times tables
  • Signed operations (add, subtract, multiply, divide)
  • Rounding whole numbers & decimals
  • Comparing decimals
  • Converting between decimal & percent
  • Etc.
These are the kinds of things that seem "obvious" to properly-educated people, but if no one ever communicated them to a student in that way, they are opaque. For example, in each of the last few  semesters in my sophomore statistics class, I've had "A" students, otherwise doing very well, but who were completely mystified at how we were converting decimal probabilities to percent on the fly. Notice that these are all one-step, immediate mental tasks: I wanted to include order-of-operations knowledge here (it's so critically important), but the truth is those tasks are inherently multi-step problems, so they don't really belong on the list above.

But let me focus more on the issue of negative numbers (signed operations). I find these to be the greatest stumbling block in students getting through the bottleneck remedial algebra course -- I can pull up any test, including finals, and see that usually at least half of the errors are simply signed-number mistakes. A student can know everything in the algebra course, but if they routinely trip over negatives even after I've begged them to practice it for a whole semester, then they have practically no chance of passing the final.

In June, I had the opportunity to teach an immersive one-week workshop for students who narrowly missed passing our department's prealgebra final (basic arithmetic with different types of numbers: integers, fractions, decimals, percent). This was a great experience, the students were hard-working and highly appreciative, and it gave me a chance to further focus on this issue. I was trying to do frequent one-minute speed drills on things like negative operations, and some students were having what seemed like an inordinately difficult time with them -- particularly the subtractions. So that night I sat down at the bus stop and tried to think through really carefully what we really do in practice as proficient math people.

Here's the thing: Not all negative operations are single-step. In particular, consider subtracting a negative number, written inside parentheses. I find that a lot of students are taught this bumfungled "keep change change" methodology: they will transform expressions as follows:
  • 3−(−9) = 3+(+9)
  • 5+(−8) = 5−(+8)
  • 1−(+7) = 1+(−7)
  • 4+(+6) = 4−(−6)
Now, all of these are true statements. But only some of them are helpful. It's not like the students' prior instructors were lying to them, except in regards to when this fact is useful in simplifying an expression with signs (namely, the 1st and 3rd cases above). Let's look at how a math professional would really do it. These are two-step problems; really, we would follow the order-of-operations and get rid of the parentheses in what's really a multiplying step, then combine like terms in an addition step.
  • 3−(−9) = 3+9 = 12
  • 5+(−8) = 5−8 = −3
Etc. So the lesson is that if an instructor shows students how to mangle signs in & around parentheses, they are really missing the point; when simplifying (evaluating), we will remove the parentheses entirely in the multiply step, and then always perform add/subtracts without any parentheses in the picture.

So in the current discussion, this informs us as to what we should be drilling students for "automaticity" in terms of negative number operations: namely, combining terms with no parentheses has to be the automatic one-step skill. If you want to explain this as effectively adding terms that's fine; but don't fail to clearly communicate that this is expected to be instantaneous and immediate, in one mental step, in practice.

  • 2−6 ← Automatic drill ok
  • −7+1 ← Automatic drill ok
  •  6−7+2 ← Automatic drill ok
  • 5−(−2) ← Not automatic drill ok (2-step problem)
I feel like this was an important lesson I got to learn from this summer's immersion workshop. A speed drill can include automatically multiplying or dividing integers, or combining terms with no parentheses -- but add/subtracts with parentheses don't belong in the same category, because you really do need to apply two separate simplifying steps for them. And perhaps most important of all: clearly communicate that the one plan that always succeeds is the standard order-of-operations, not a score of different random manipulations to memorize for different situations.

Monday, July 7, 2014

Multiple Choice Expectations

A while back I considered the chance to pass a standard multiple-choice final exam (link), granted a certain basis of actual knowledge of the material. Today let's look at it from the other perspective, i.e., what the expected score is for different levels of actual knowledge:

As you can see, if we pass a student with a 60% score on such a multiple-choice test, then the most likely bet (point estimate) for their true level of knowledge is around 11 or 12 of the questions, that is, less than half of the actual content for the course.

Saturday, June 28, 2014

Happy Tau Day!

Happy Tau Day! Celebrating τ = 2π = 6.28..., which simplifies all the math formulas that include 2π to make one full turn around a circle. At our house, we'll be having tacos, and tequila, and tarts. Check out the Tau Day manifesto, especially the excellent video by Vi Hart, linked at the bottom of the page --

(Coincidentally, it's also the Perfect Day, because 6 and 28 are the only "perfect numbers" that appear on a calendar --

Edit: A full course of full circles for Tau Day!

Wednesday, June 4, 2014

Reformatted Writing

A short observation that I sometimes use to my advantage: It helps me greatly if I write something in one format, and then look at it in some reformatted view, before finally distributing it publicly. This tends to dredge up a number of subtle errors or weaknesses which I otherwise can't see. In some sense it gives me "fresh eyes" to really read the draft from a new perspective (as opposed to one's memory blocking reception of what's actually on the page). In this sense we might say that "what you see is what you get" (WYSIWYG) is actually a hindrance instead of a help. A few cases:
  • Blogger. Editing blog posts here, one uses a rich-text editor that is different from the final HTML markup on the site. One clicks "preview" to actually see it reformatted as it will appear on the Blogger site. Something that happens occasionally is that I might have a duplicated "the the" (or something) across a line break, such that I don't initially see it; when the paragraphs and lines get moved around in Preview, I can catch this much more easily. More generally, I can pick up on weaknesses in sentence structure and places I can clarify much more easily in the second view perspective.

  • Java Code. Just recently I've committed to formatting all my personal coding projects in the javadoc format (special comments that can be parsed out automatically to external documentation). This required a change in style that I thought would be irritating, but was much less painful than I expected, and more than compensated for by the benefits. Again, if I write my code once and then generate the documentation and look at that, I'm finding there's a whole lot of places I can improve on variable names, method names, comments on use, etc. Looking at it from the perspective of a true "outsider", with only the cues someone would start with to theoretically get into my code, gives my end product much greater depth and robustness.
So in summary: Write once, view in a totally different format, and then edit. Results are improved to a surprising degree.

Monday, May 5, 2014

Basic Logic Errors

I constantly wish that students were taught rudimentary logic at an early age (links: one, two, three). Just musing about that today, here are three common stumbling blocks I see in different classes due to not being able to read logical statements properly:
  1. "If" Statement. In a basic algebra class, we have the rule "If the base is negative, then even powers are positive, but odd powers are negative". Immediately after that, I'll always have some students incorrectly evaluate something like: −5² = 25 (or worse, 2³ = −8) . Note that the base of the exponent is not negative, but some students overlook the check required for the "if" qualifier.

  2. "Or" Statement. In an elementary statistics class, we have the rule "To estimate a population mean, we must have either a normal population or a large sample size." Then when I ask the class "Do we need a normal population?", the entire class will always incorrectly respond with "Yes!" the first time. Of course that's not true; they're overlooking that only one case of the "or" needs to be satisfied (most commonly by just a large sample size). It takes several sessions of quizzing them on that before they are sensitive to the question being asked.

  3. "And" Statement. In practically any class, we might have the policy, "To pass this class you need at least a 60% weighted average, and a 60% score on the final exam." This constantly causes confusion and aggravation. Testy "So, the final exam doesn't count?", or "So, only the final exam counts?" are questions that I routinely have to address. Obviously, students are unclear on the fact that each of two requirements must be satisfied for an "and" statement like that.