Math Conference

Initial Thoughts on the CUNY 2012 Mathematics Conference

About two weeks ago, I had the good fortune of attending the CUNY 2012 Mathematics Conference on Effective Instructional Strategies. To a large degree, this is a product of CUNY's Improving Undergraduate Mathematics Learning (IML) program, which in 2009 vetted and funded 10 research programs at various of the school's colleges. Many of the final papers were presented at this conference, and I'll plan to spend several posts presenting some of my thoughts on them -- you can see the final reports here.

As noted in many places, several of the reports, and on this blog previously, at least half our (my) job is teaching non-credit remedial Arithmetic and Algebra classes for students who can't initially pass a placement test for those subjects, and so many of the research projects were looking for ways to improve that teaching. (Again: Nationwide, about 60% of CC students need such remediation, and only about 30% complete it after some number of years.) Interestingly, within the hour of my sitting down to write this, a top AP headline crossed the news wire on exactly this issue: "Experts: Remedial College Classes Need Fixing".

The strategies looked at in the different CUNY research projects were fairly wide-ranging, and usually required total overhauls of the classes in some way. In general, they seemed to usually be one of: (1) group work/project-based exploratory learning, (2) online/software-based homework and exercise applications, and (3) "inverted" classrooms where video lectures were watched before class, and discussion and exercise drills run in-class.

Compare to my previous blog post. On the one hand, the American Educator article by Clark, et. al., on "Fully Guided Instruction" had me thinking that the group-work-exploration craze was petering out, but apparently that's not quite the case. Consider also, say, David Klein's anti-reformist essay "A Brief History of American K-12 Mathematics Education in the 20th Century" (section "Public Resistance to the NCTM Standards"):

To understand the public backlash against the NCTM math programs of the 1990s, one needs to understand some of the mathematical shortcomings of these programs... Student discovery group work was the preferred mode of learning, sometimes exclusively, and the guidelines for discovery projects were at best inefficient and often aimless... Arithmetic and algebra were radically de-emphasized. Mathematical definitions and proofs for the higher grades were generally deficient, missing entirely, or even incorrect. Some of the elementary school programs did not even provide books for students, as they might interfere with student discovery. 

I would say that each of these identifiers from the 1990's programs appeared in at least one of the research programs from the IML.

At a certain point, one of the speakers referred to Clark's research that expert learners do well with discovery-based methods, and novices do better with fully-guided instruction -- that being the same Clark who inspired my previous blog post (see more here). I was nodding along with this line of reasoning as very important, and then the speaker concluded with the line, "And since most of our students are aged 20-25, they count as experts, and need self-directed methods", at which point I almost fell out of my chair. (If our students can't pass a basic arithmetic/algebra test, then I don't see how it's valid to conclude that they're experts.)

More thoughts later.

Fully Guided Instruction

Highly Recommended Reading by Clark, Kirschner, and Sweller in American Educator Magazine: "Putting Students on the Path to Learning: The Case for Fully Guided Instruction".

I suppose that the "math wars" of the 1990's aren't entirely over yet. For example, for several years I've gotten this one magazine every other month called the NEA Higher Education Advocate (which in this household is referred to as "the crappy teaching magazine"). Every edition has a central keynote article under the heading of "Thriving in Academe", which almost uniformly features a call to "reform" type instructional techniques such as group-work projects, exploratory/discovery-based learning, and the like. When it comes in every other month, I tend to say, "Ah, the Pravda has arrived."

Among the funny recurring jokes of the feature is that it always has a "Issues to Consider" section (FAQ, basically), which frequently fields the question of "Won't this take more time and effort/ Not allow us to cover as many topics?" And the answer is usually some flavor of "Oh, yes!" For example, from the Dec-2008 article on "CRISP" (there's usually someone peddling a new acronym/system in every issue):

Won’t being C.R.I.S.P. cause me to sacrifice coverage? 

Of course. The sciences are especially concerned with complete nomenclature. They are worried that if a Biology 101 student doesn’t learn every bone, muscle, and organ in the body, the student won’t be prepared for Biology 102, not to mention advanced study in related fields such as nursing and exercise and sports science. However, since studies indicate that students will “forget” (they actually put the information in their short-term memories only) 75 to 90 percent of the material in three months anyway, shouldn’t you worry more that students develop skills and fundamental concepts? If students truly comprehend, for instance, how the bones work in general, shouldn’t they be able to figure out how a specific bone functions or know where to look it up? 

So in contrast, the American Educator magazine is what I call the "good teaching magazine" and seems to have much higher-quality, more in-depth, and more interesting articles in each issue (it's published on a quarterly basis). The current "Fully Guided Instruction" article by Clark, et. al. was a bracing breath of fresh air, representing the opposite point of view, that attempting to have students "discover" principles on their own is a weaker technique than instructors cutting to the chase and simply telling them how things work in a straightforward manner (and modeling proper usage, and then overseeing practice). It cites seemingly strong research that the two techniques can be appropriate for different groups of students: in particular, strong students (with pre-existing deep background knowledge) work well with discovery-based learning, whereas weak students (those with deficiencies) do better with explicit direction. In fact:

Worse, a number of experiments found that less-skilled students who chose or were assigned to less-guided instruction received significantly lower scores on posttests than on pretest measures. For these relatively weak students, the failure to provide strong instructional support produced a measurable loss of learning.

So this seems particularly relevant to my work, over half of which is teaching remedial arithmetic and algebra at a large, urban community college (the stats for us, and nationwide, being about 60% of students taking remedial math, and only about 30% successfully completing it after 3 years).

And the other fascinating thing in the article was the description of a decades-long history of similar "discovery-based" reform efforts since at least the 1950's, each of which have been given a new name and similarly came up empty with research-based results for it. Highly recommended reading.

Necessary but Not Sufficient

Here's a random math/logic-book exercise I'd like to see:

The picture above shows a door with triple locks. Answer the following questions:

(a) Do you need the key to the top lock to enter this door? Yes.
(b) Will you be able to enter this door if you have only the key to the top lock? No.
(c) What's the technical term for this relationship? Necessary but not sufficient.


[Photo by LiGhtSynC under CC2.]

"Something Highly Unlikely"

On the Need to Establish Hypotheses Prior to Testing; Or, The Fact that It Is Overwhelmingly Likely that Something Highly Unlikely Will Happen in Any Experiment.

Lately I've gotten in the habit of doing several card-drawing demonstrations in my statistics classes (as concrete examples of sampling, estimating a population mean, hypothesis testing, interpretations, etc.) Here's one for a test question that I sometimes ask: "Is it acceptable to decide what type of test to conduct [left, right, or two-tailed] by examining the sample data?" Say that I bring in a deck of playing cards, shuffle, and deal out 6 cards. For example, when I just did this at my desk I got this:

Now, what follows in this paragraph would be an example faulty reasoning -- Note that I just got duplicate 4's in this draw, and of course, the probability of that happening is highly unlikely in a standard deck (specifically, a 6% chance to get two or more 4's *). Therefore, one might conclude that I doctored that deck with extra 4's.

What's wrong with that reasoning? Well, we didn't establish the hypotheses prior to testing, so it's unfair and biased to use this as data in support of that hypothesis. Or perhaps it's better to look at it this way: It's overwhelmingly likely that something highly unlikely will happen in any such experiment (if you look at the data post-facto and labor to draw out some weird numerology-like pattern). Specifically, the chance of getting some duplicated card value from a standard deck in this case (not necessarily 4's) is actually 65%. **

So let's try this again: It is fair to use the first draw as suggestive of a new hypothesis. Let's hypothesize: "He doctored this deck with extra 4's". So if we shuffle and draw 6 cards again, then we should expect to see one or more 4's. And when I ran this experiment just now the result was:

Which rather obviously destroys the hypothesis; in this case I didn't get any 4's at all. (I did get duplicate 8's, but again, normal probability says that you'll usually get duplicate somethings from a standard deck when drawing 6 cards, so it's not really surprising or interesting at all.) To be doing interesting science, you have to establish coherent hypotheses in advance, and be able to predict and replicate your results.

---

Calculation Footnotes:
* Drawing 6 cards: Chance to get zero 4's is: 48P6/52P6 = 0.603. Chance to get an initial 4 and then all non-4's is: 4×48P5/52P6 = 0.056; so chance to get a single 4 in some order is 6×0.056 = 0.336. Sum of these is 0.603+0.336 = 0.939. Therefore, the chance to get two or more 4's is P(not zero or one 4)  = 1 - 0.939 = 0.061 ~ 6%.

** Drawing 6 cards: Chance to get no duplicates is 52/52×48/51×44/50×40/49×36/48×32/47 = 0.345. Therefore, the chance to get at least one duplicate is P(not zero duplicates) = 1 - 0.345 = 0.655 ~ 65%. (And as a sanity check, the preceding should be approximately 1/13 this, i.e., 65%/13 ~ 5% which does check out.)

Visualizing the Power Function

After some recent discussion on the power function, it occurred to me that if I tried to visualize the function b^a, I really didn't know what it would look like. Here it is below:



As shown here, the base b is along the x-axis, and the power a is on the y-axis. The range displayed is between +/-2 on both axes, with the origin (0, 0) in the center. Positive values are shown in red, negative values in blue; intensity is scaled to the highest value in the top-right corner (i.e., 2^2 = 4). Black pixels represent either very small values (on the right half, for b>0) or else undefined values (on the left half, for b<0).

So a few things are apparent. In the 1st quadrant, going towards the top-right, you get larger positive values (when b>0 and a>0); near the y-axis in that quadrant you get diminishing values, namely 0 when b=0. But in the 4th quadrant the situation is reversed: you get diminishing values towards the bottom-right and arbitrarily large value near the y-axis (hence the intense bright region on the bottom, with vanishing b and negative a, generating values much larger than what you get in the top-right). On the left-hand side (b<0), the graph is mostly black, with only narrow bands of value where the power a is an integer (alternating red and blue, as the powers alternate positive or negative values).

One discovery regarding that left side: I didn't realize how contentious it was to possibly define rational exponents for a negative base! Apparently some textbooks go either way with that. For example, the textbooks at my school permit it, but they have to institute a clunky "assume root exists and exponent on b^(m/n) is reduced to lowest terms" definition, so as to avoid a contradiction like -2 = (-8)^(1/3) = (-8)^(2/6) = ((-8)^2)^(1/6) = (64)^(1/6) = +2. On the other hand, you have academic papers such as from Tirosh/Even, "To Define or Not to Define: The Case of (-8)^1/3" (Educational Studies in Mathematics, Vol. 33 No. 3, Sep. 1997) which point out this problem and others, and recommend leaving them undefined. I think I'm personally convinced by that. Hopefully we all agree that irrational exponents to negative bases are undefined, so the left-hand side of the graph above really does need to be black almost everywhere.

And then of course you've got the case of 0^0, which I'm likewise convinced (again contrary to the books at my school) should be defined to be 1. On the one hand, the horizontal axis a=0 will definitely have a value of 1 for all b with that possible single exception. While on the other hand, the vertical axis has a value of 0 coming down from the top, but you're going to have a discontinuity at the origin no matter what; either a value of 1 when a=0, or, failing that, an undefined value as soon as you take an arbitrarily small step below the origin (since given negative power -a, 0^(-a) = 1/(0^a) = 1/0 which is undefined). So I figure you might as well define 0^0 = 1 and the loss in continuity on the vertical axis is immeasurably small.

In summary: Certainly not a picture I could have intuited, and one with unexpectedly complicated structure and more regions of controversy regarding definitions than I expected (granted such a fundamental function as b^a).

Possibly there's some additional use in seeing a spreadsheet of numerical values from the same region, below:


Download if you want:
- Java source code (JAVA) to generate the image
- Open Document Spreadsheet (ODS) for the numbers
- Wikipedia 3D graph of similar region (positive base only)

Say What?

In a story on the giant Mega Millions lottery this weekend:

Accountant Ray Lousteau, who bought 55 Mega Millions tickets Friday in New Orleans, knows buying that many tickets doesn't mathematically increase his odds, and that his $55 could have gone elsewhere. He spent it anyway.

"Mathematically, it doesn't make a difference, and intellectually we know that. But for some reason buying more tickets makes you feel more lucky," Lousteau said. "Even people who know better are apt to feel that way."

Um... having more tickets in a lottery doesn't increase your chance of winning? How the hell does that work? And how did this get by both an accountant and the journalist writing the story?

Google Divide by Zero

You know that Google automatically acts as a calculator, right? Type in any kind of math expression, and it automatically simplifies it in response -- including unit conversions of all sorts (very useful for that latter part, in my experience).

But here's something I discovered the other day: The calculator won't respond at all to any kind of division by zero. It won't say there's an error; it won't say it's undefined or not-a-number (NAN); it just won't trigger the calculator facility at all. It goes straight to a regular web search like it wasn't math at all. (I realized this after my first basic math class; carefully defined division and considered divide-by-zero, compared to a calculator error response, and then I asserted the same would happen in Google. Turns out that's not quite correct.)

This is true even if you try to hide the division-by-zero in some kind of very complicated expression (that's otherwise obviously math). Consider these:

• 6/2
• 15/(7-4)
• 1/(4-9/3*3+6)
  

And then contrast with the following:

• 6/0
• 15/(7-7)
• 1/(4-9/3*3+5)

I'm not sure if this is an oversight, or some tremendously subtle winking in-joke by our friends from Menlo Park. (Like: The calculator has to get triggered, do quite a bit of work before determing there's a divide-by-zero, and then decide to run away and hide itself from appearing.) Can you make Google Calculator admit to a divide-by-zero in any way?