At a time when calls for a kind of academic disarmament have begun echoing through affluent communities around the nation, a faction of students are moving in exactly the opposite direction...
"The youngest ones, very naturally, their minds see math differently [said Inessa Rifkin, co-founder of Russian School of Mathematics]... It is common that they can ask simple questions and then, in the next minute, a very complicated one. But if the teacher doesn’t know enough mathematics, she will answer the simple question and shut down the other, more difficult one. We want children to ask difficult questions, to engage so it is not boring, to be able to do algebra at an early age, sure, but also to see it for what it is: a tool for critical thinking. If their teachers can’t help them do this, well... It is a betrayal.”
And while the proportion of American students scoring at advanced levels in math is rising, those gains are almost entirely limited to the children of the highly educated, and largely exclude the children of the poor. By the end of high school, the percentage of low-income advanced-math learners rounds to zero...
The No Child Left Behind Act... demanded that states turn their attention to getting struggling learners to perform adequately...The cumulative effect of these actions, perversely, has been to push accelerated learning outside public schools—to privatize it, focusing it even more tightly on children whose parents have the money and wherewithal to take advantage. In no subject is that clearer today than in math.
2016-02-26
Link: Math Circles at the Atlantic
An article this month at the Atlantic on the explosive rise of extracurricular (and expensive) advanced-math circles and competitions, to make up for the perceived deficiencies in math education in schools. Some telling quotes:
2016-02-19
Link: Common Core Battles
A nice overview of the history of the battles around Common Core. Starts with a surprising anecdote about Bill Gates getting the brush-off when he personally met with Charles Koch to discuss the issue. Re: George W. Bush's "No Child Left Behind", an aggravatingly familiar development:
To bring themselves closer to 100%, many states simply lowered the score needed to pass their tests. The result: In 2007, Mississippi judged 90% of its fourth graders “proficient” on the state’s reading test, yet only 19% measured up on a standardized national exam given every two years. In Georgia, 82% of eighth-graders met the state’s minimums in math, while just 25% passed the national test. A yawning “honesty gap,” as it came to be known, prevailed in most states.
2016-02-15
Hembree on Math Anxiety
Reviewing a 1990 paper by Ray Hembree on math anxiety; a meta-study of approximately 150 papers, with a combined total of about 25,000 subjects. (Note the high sample size makes almost all findings significant at the p < 0.01 level). Math anxiety is known to be negatively correlated with performance in math (tests, etc.), and more common among women than men.
Math anxiety is somewhat correlated with a constellation of other general anxieties (r² = 0.12 to 0.27). Work to enhance math competence did not reduce anxiety.
Whole-group interventions are not effective (curricular changes, classroom pedagogy structure, in-class psychological treatments). The only thing that is effective is out-of-classroom, one-on-one treatments (behavioral systematic desensitization; cognitive restructuring); these have a marked effect at both lowering anxiety and boosting actual math-test performance.
In short: Addressing math anxiety is largely out of the hands of the classroom teacher. Unless the student has access, or the institution provides access, to one-on-one behavioral desensitization therapy, no group-level interventions are found to be effective.
Also recall that elementary education majors have the highest math anxiety, and the lowest math performance, of all U.S. college majors. (It seems possible that some entrants choose elementary education as a career path precisely because they are bad at math and see that as one of their limited options; I know I've had at least one such student say something to that effect to me.) This clearly dovetails with Sian Beilock's 2009 finding that math-anxious female elementary teachers model math-anxiety particularly to their female students, who imitate the same and wind up with worse math performance and attitudes by the end of the year (link). And this general trend of weak education majors has been the case in the U.S. for at least a century now (link).
So we might hypothesize: A feedback loop exists between poor early math education, heightened math anxiety among female students, and those same students returning to early childhood education as a career.
See below for Hembree's table of math anxiety by class and major (p. 41); note that elementary education majors, and those taking the standard "math for elementary teachers" (frequently the only math class such teachers take), are significantly worse off than anyone else:
Math anxiety is somewhat correlated with a constellation of other general anxieties (r² = 0.12 to 0.27). Work to enhance math competence did not reduce anxiety.
Whole-group interventions are not effective (curricular changes, classroom pedagogy structure, in-class psychological treatments). The only thing that is effective is out-of-classroom, one-on-one treatments (behavioral systematic desensitization; cognitive restructuring); these have a marked effect at both lowering anxiety and boosting actual math-test performance.
In short: Addressing math anxiety is largely out of the hands of the classroom teacher. Unless the student has access, or the institution provides access, to one-on-one behavioral desensitization therapy, no group-level interventions are found to be effective.
Also recall that elementary education majors have the highest math anxiety, and the lowest math performance, of all U.S. college majors. (It seems possible that some entrants choose elementary education as a career path precisely because they are bad at math and see that as one of their limited options; I know I've had at least one such student say something to that effect to me.) This clearly dovetails with Sian Beilock's 2009 finding that math-anxious female elementary teachers model math-anxiety particularly to their female students, who imitate the same and wind up with worse math performance and attitudes by the end of the year (link). And this general trend of weak education majors has been the case in the U.S. for at least a century now (link).
So we might hypothesize: A feedback loop exists between poor early math education, heightened math anxiety among female students, and those same students returning to early childhood education as a career.
See below for Hembree's table of math anxiety by class and major (p. 41); note that elementary education majors, and those taking the standard "math for elementary teachers" (frequently the only math class such teachers take), are significantly worse off than anyone else:
Hembree, Ray. "The nature, effects, and relief of mathematics anxiety." Journal for research in mathematics education (1990): 33-46. (Link)
2016-02-12
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:
“... the brain’s interconnectivity makes such an assumption unsound.”
2016-02-05
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).
2016-02-01
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 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.
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.
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