“... the brain’s interconnectivity makes such an assumption unsound.”

## Friday, February 12, 2016

### Link: The Learning Styles Neuromyth

A nice article reminding us that the whole idea of teaching to different "learning styles" is entirely without any scientific evidence in its favor:

## Friday, February 5, 2016

### Link: Study Time Decline

An interesting article analyzing the history of reported study time decline for U.S. college students.

- Point 1: Study time dramatically decreased in the 1961-1981 era (from about 24 hrs/week to 16 hrs/week), but has been close to stable since that time.
- Point 2: In that same early period, it seems that faculty expectations on teaching vs. research flip-flopped in that same early time period (about 70% prioritized teaching over research around 1975, with the proportion quickly dropping to about 50/50 by the mid-80's).

## Monday, February 1, 2016

### When Dice Fail

Some of the more popular posts on my gaming blog have been about how to check for balanced dice, using Pearson's chi-square test (testing a balanced die, testing balanced dice, testing balanced dice power). One of the observations in the last blog was that "chi-square is a test of rather lower power" (quoting Richard Lowry of Vassar College); to the extent that I've never had any dice that I've checked actually

Until now. Here's the situation: A while back my partner Isabelle, preparing entertainment for a long trip, picked up a box of cheap dice at the dollar store around the corner from us. These dice are in the Asian-style arrangement, with the "1" and "4" pip sides colored red (I believe because the lucky color red is meant to offset those unlucky numbers):

A few weeks ago, it occurred to me that these dice are just the right size for an experiment I run early in the semester with my statistics students: namely, rolling a moderately large number of dice in handful batches and comparing convergence to the theoretically-predicted proportion of successes. In particular, the plan is customarily to roll 80 dice and see how many times we get a 5 or 6 (mentally, I'm thinking in my

So when we did that in class last week, it seemed like the number of 5's and 6's was significantly lower than predicted, to the extent that it actually threw the whole lesson under a shadow of suspicion and confusion. I decided that when I got a chance I'd better test these dice before using them in class again. Following the findings of the prior blog on the "low power" issue, I knew that I had to get on the order of about 500 individual die-rolls in order to get a halfway decent test; in this case with a boxful of 15 dice, it seemed was convenient to make 30 batched rolls for 15 × 30 = 450 total die rolls... although somewhere along the way I lost count of the batches and wound up actually making 480 die rolls. Here are the results of my hand-tally sheet:

As you can see at the bottom of that sheet, this box of dice actually does fail the chi-square test, as the \(SSE = 1112\) is in fact greater than the critical value of \(X \cdot E = 11.070 \cdot 80 = 885.6\).Or in other words, with a chi-square value of \(X^2 = SSE/E = 1112/80 = 13.9\) and degrees of freedom \(df = 5\), we get a P-value of \(P = 0.016\) for this hypothesis test of the dice being unbalanced; that is, if the dice really were balanced, there would be less than a 2% chance of getting an SSE value this high by natural variation alone.

In retrospect, it's easy to see what the manufacturing problem is here: note in the frequency table that it's specifically the "1"'s and the "4"'s, the specially red-colored faces, that are appearing in a preponderance of the rolls. In particular, the "1" face on each die is drilled like an enormous crater compared to the other pips; it's about 3 mm wide and about 2 mm deep (whereas other pips are only about 1 mm in both dimensions). So the "6" on the other side from the "1" would be top heavy, and tends to roll down to the bottom, leaving the "1" on top more than anything else. Also, the corners of the die are very rounded, making it easier for them to turn over freely or even get spinning by accident.

Perhaps if the experiment in class had been to count 4's, 5's, and 6's (that is: hits against light armor in my wargame), I never would have noticed the dice being unbalanced (because together those faces have about the same weight as the 1's, 2's, and 3's together)? On the one hand my inclination is to throw these dice out so they never get used again in our house by accident; but on the other hand maybe I should keep them around as the only example that the chi-square test has managed to succeed at

*fail*the test.Until now. Here's the situation: A while back my partner Isabelle, preparing entertainment for a long trip, picked up a box of cheap dice at the dollar store around the corner from us. These dice are in the Asian-style arrangement, with the "1" and "4" pip sides colored red (I believe because the lucky color red is meant to offset those unlucky numbers):

A few weeks ago, it occurred to me that these dice are just the right size for an experiment I run early in the semester with my statistics students: namely, rolling a moderately large number of dice in handful batches and comparing convergence to the theoretically-predicted proportion of successes. In particular, the plan is customarily to roll 80 dice and see how many times we get a 5 or 6 (mentally, I'm thinking in my

*Book of War*game, how many times can we score hits against opponents in medium armor -- but I don't say that in class).So when we did that in class last week, it seemed like the number of 5's and 6's was significantly lower than predicted, to the extent that it actually threw the whole lesson under a shadow of suspicion and confusion. I decided that when I got a chance I'd better test these dice before using them in class again. Following the findings of the prior blog on the "low power" issue, I knew that I had to get on the order of about 500 individual die-rolls in order to get a halfway decent test; in this case with a boxful of 15 dice, it seemed was convenient to make 30 batched rolls for 15 × 30 = 450 total die rolls... although somewhere along the way I lost count of the batches and wound up actually making 480 die rolls. Here are the results of my hand-tally sheet:

As you can see at the bottom of that sheet, this box of dice actually does fail the chi-square test, as the \(SSE = 1112\) is in fact greater than the critical value of \(X \cdot E = 11.070 \cdot 80 = 885.6\).Or in other words, with a chi-square value of \(X^2 = SSE/E = 1112/80 = 13.9\) and degrees of freedom \(df = 5\), we get a P-value of \(P = 0.016\) for this hypothesis test of the dice being unbalanced; that is, if the dice really were balanced, there would be less than a 2% chance of getting an SSE value this high by natural variation alone.

In retrospect, it's easy to see what the manufacturing problem is here: note in the frequency table that it's specifically the "1"'s and the "4"'s, the specially red-colored faces, that are appearing in a preponderance of the rolls. In particular, the "1" face on each die is drilled like an enormous crater compared to the other pips; it's about 3 mm wide and about 2 mm deep (whereas other pips are only about 1 mm in both dimensions). So the "6" on the other side from the "1" would be top heavy, and tends to roll down to the bottom, leaving the "1" on top more than anything else. Also, the corners of the die are very rounded, making it easier for them to turn over freely or even get spinning by accident.

Perhaps if the experiment in class had been to count 4's, 5's, and 6's (that is: hits against light armor in my wargame), I never would have noticed the dice being unbalanced (because together those faces have about the same weight as the 1's, 2's, and 3's together)? On the one hand my inclination is to throw these dice out so they never get used again in our house by accident; but on the other hand maybe I should keep them around as the only example that the chi-square test has managed to succeed at

*rejecting*to date.## Saturday, January 30, 2016

### Link: Tricky Rational Exponents

Consider the following apparent paradox:

\(-1 = (-1)^1 = (-1)^\frac{2}{2} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2} = (1)^\frac{1}{2} = \sqrt{1} = 1\)

Of the seven equalities in this statement,

\(-1 = (-1)^1 = (-1)^\frac{2}{2} = (-1)^{2 \cdot \frac{1}{2}} = ((-1)^2)^\frac{1}{2} = (1)^\frac{1}{2} = \sqrt{1} = 1\)

Of the seven equalities in this statement,

*exactly which of them are false*? Give a specific number between (1) and (7). Join in the discussion where I posted this at StackExachange, if you like:## Thursday, January 28, 2016

### Seat Belt Enforcement

Yesterday in the Washington Post, libertarian police-abuse crusader Radley Balko wrote an opinion piece arguing against mandatory seat-belt laws. He opens:

Here's the problem with that math: the Florida study would in fact be evidence that blacks are failing to wear seat belts at about twice the rate of whites. According to those numbers, the rate of blacks not wearing seat belts would be: 100% - 85.8% = 14.2%, while the rate of non-compliance for whites would be 100% - 91.5% = 8.5%. And as a ratio, 14.2%/8.5% = 1.67, or pretty close to 2 (double) if you round to the nearest multiple.

Now, I actually think that Radley Balko has done some of the very best, most dedicated work drawing our attention to the problem of hyper-militarized police tactics in recent years (and decades); see his book

But sometimes Balko gets his arguments scrambled up, and this is one of those cases. His claim that there's only one "possible explanation for the disparity" fails on the grounds that these numbers are evidence that, in general, there's actually no disparity in enforcement at all; enforcement tracks the non-compliance ratio very closely. He can do better than to hang his hat in this case on a fundamental misunderstanding of the numbers involved.

The ACLU of Florida just released a report showing that in 2014, black motorists in the state were pulled over for seat belt violations at about twice the rate of white motorists... Differences in seat belt use don’t explain the disparity. Blacks in Florida are only slightly less likely to wear seat belts. The ACLU points to a 2014 study by the Florida Department of Transportation that found that 85.8 percent of blacks were observed to be wearing seat belts vs. 91.5 percent of whites. The only possible explanation for the disparity that doesn’t involve racial bias might be that it’s easier to spot seat-belt violations in urban areas than in more rural parts of the state... even if it did explain part or all of the disparity, it still means that blacks in Florida are disproportionately targeted.

Here's the problem with that math: the Florida study would in fact be evidence that blacks are failing to wear seat belts at about twice the rate of whites. According to those numbers, the rate of blacks not wearing seat belts would be: 100% - 85.8% = 14.2%, while the rate of non-compliance for whites would be 100% - 91.5% = 8.5%. And as a ratio, 14.2%/8.5% = 1.67, or pretty close to 2 (double) if you round to the nearest multiple.

Now, I actually think that Radley Balko has done some of the very best, most dedicated work drawing our attention to the problem of hyper-militarized police tactics in recent years (and decades); see his book

*Rise of the Warrior Cop*for more. And I'm pretty sensitive to issues of overly-punitive enforcement and fines that are repressively punishing to the poor; back in the 80's I used to routinely listen to Jerry Williams on WRKO radio in Boston, when he was crusading against the rise of seat-belt laws, and I found those arguments, at times, compelling.But sometimes Balko gets his arguments scrambled up, and this is one of those cases. His claim that there's only one "possible explanation for the disparity" fails on the grounds that these numbers are evidence that, in general, there's actually no disparity in enforcement at all; enforcement tracks the non-compliance ratio very closely. He can do better than to hang his hat in this case on a fundamental misunderstanding of the numbers involved.

## Monday, January 25, 2016

### Grading on a Continuum

Anecdote: I had a social-sciences teacher in high school who didn't understand that real numbers are a continuum.

On the first day of class, he tried to present how grades would be computed at the end of the course. So on the board he wrote something like: D 60-69, C 70-79, B 80-89, A 90-100% (relating final letter grade to weighted total in the course).

Then he looked at it and said, "Oh, wait, that's not right, what if a student gets 89.5%?". So he starting erasing and adjusting the cutoff scores so it looked something like: D 60-69.5, C 69.6-79.5, B 79.6-89.5, A 89.6-100%.

And of course then he went, "But, no, what if a student gets 89.59%?", and started erasing and adjusting

I think he took about 10 minutes or more of the first class iterating on this (before he gave up he'd gotten to maybe 4 places after the decimal point). I remember myself and a bunch of other students just looking back and forth at each other, slack-jawed from astonishment. It raises a couple of questions: Did he not know that

I always think about this on the first day in my statistics courses, when we are careful (following Weiss book notation) to define our grouped classes for continuous data using a special symbol "-<", meaning "up to but less than" (e.g., B 80 -< 90, A 90 -< 100, so that any score less than 90 would be unambiguously not in the A category, leaving no gaps). As I present this, my social-science teacher embarrassing himself is always at the back of my mind -- and I'd like to share it with my students as a case-study, but frankly the anecdote would take too much time and distract from the critically important first day of my own class.

But the initial reaction we got for that teacher was accurate; although he couldn't perceive it, he was about as dense as the real numbers all semester long.

On the first day of class, he tried to present how grades would be computed at the end of the course. So on the board he wrote something like: D 60-69, C 70-79, B 80-89, A 90-100% (relating final letter grade to weighted total in the course).

Then he looked at it and said, "Oh, wait, that's not right, what if a student gets 89.5%?". So he starting erasing and adjusting the cutoff scores so it looked something like: D 60-69.5, C 69.6-79.5, B 79.6-89.5, A 89.6-100%.

And of course then he went, "But, no, what if a student gets 89.59%?", and started erasing and adjusting

*again*to generate something like: D 60-69.55, C 69.56-79.55, B 79.56-89.55, A 89.56-100. And then of course noticed that there were still gaps between the intervals and went at it for a few more cycles.I think he took about 10 minutes or more of the first class iterating on this (before he gave up he'd gotten to maybe 4 places after the decimal point). I remember myself and a bunch of other students just looking back and forth at each other, slack-jawed from astonishment. It raises a couple of questions: Did he not know that

*real numbers are dense*? And had he never thought through his grading schema*until this very moment?*I always think about this on the first day in my statistics courses, when we are careful (following Weiss book notation) to define our grouped classes for continuous data using a special symbol "-<", meaning "up to but less than" (e.g., B 80 -< 90, A 90 -< 100, so that any score less than 90 would be unambiguously not in the A category, leaving no gaps). As I present this, my social-science teacher embarrassing himself is always at the back of my mind -- and I'd like to share it with my students as a case-study, but frankly the anecdote would take too much time and distract from the critically important first day of my own class.

But the initial reaction we got for that teacher was accurate; although he couldn't perceive it, he was about as dense as the real numbers all semester long.

## Monday, January 18, 2016

### Limitations

- Howard Eves,Whenever one learns a new mathematical operation, it is imperative also to learn the limitations under which the operation may be performed. Lack of this additional knowledge can lead to the employment of the new operation in a blindly formal manner in situations where the operation is not properly applicable, perhaps resulting in absurd and paradoxical conclusions. Instructors of mathematics see mistakes of this sort made by their students almost every day...

*Great Moments in Mathematics*, Lecture 32.

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