Coffee, Love and Matrix algebra – a review

by

Gary Ernest Davis
Publisher:  Republic of Mathematics
Publication Date: 2014
Number of Pages:  389
Format:  Paperback
ISBN: 9780692262306
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David A. Huckaby is an associate professor of mathematics at Angelo State University.

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Coffee, Love, and Matrix Algebra is a delightful work of fiction that chronicles the events of roughly a year in the life of the mathematics department at a university in Rhode Island. Can there be entertainment value in a book whose principal characters are math professors? Believe it or not, this is a page turner: The reader becomes emotionally invested in the ten to fifteen central characters and humor abounds throughout.

Many of the familiar features of academia are present. Faculty spar with administrators and faculty spar with faculty. In this particular department, there are essentially two groups: an energetic society of junior faculty engaged in exciting research, mostly in applied mathematics and/or statistics, and a smaller collection of senior faculty who no longer are — or never were — doing research. Members of the latter group for the most part stand in the way of the various initiatives that the younger faculty propose and strive to realize. Which group will exert more influence?

The main character is the very self-absorbed Jeffrey Albacete, whose fame in mathematics circles is due entirely to his very popular textbook Matrix Algebra, now in its 9th edition. Jeffrey is content, knowing that he is regarded internationally as an expert in matrix algebra. In Jeffrey’s opinion, the International Linear Algebra Society puts too much emphasis on linear algebra. The breakaway International Matrix Algebra Society, of which Jeffrey is a founding member and past president, puts the emphasis in the right place — on matrix algebra — and holds his textbookMatrix Algebra in high esteem. This is academic math humor done right.

Jeffrey enjoys sitting in his office and looking over all of the editions of his influential textbook. He also enjoys drinking coffee. Part of his routine is to walk to the campus gym, spend a few very leisurely minutes on the exercise bike, and then head over to the Daily Grind, a campus coffee shop. While he waits in line to order coffee, he counts the number of customers ahead of him in the queue and estimates the number of bricks on the wall. He muses about how humans start with counting and then progress to advanced topics like matrix algebra, of which he is an acknowledged expert. (Readers who share Jeffrey’s counting compulsion might want to count how many blueberry muffins he consumes over the course of the story.)

Fortunately, most of the abundant humor in the book derives neither from mathematical compulsions nor from the perceived strangeness of mathematicians. Most of the characters would be at home in any intellectual line of work, and much of the book’s humor in founded in the characters’ humanity, especially in their relationships with each other. Numerous ironic comments and observations, ranging from explicit to subtle, focus on characters’ foibles and interactions; the best of these remarks are worthy of Austen or Trollope. So one need not be a math professor to enjoy the book, although it is undoubtedly helpful. Granted, plenty of the mirth derives from the absurdities of academia, and the story delights those of us on “the inside” because it paints such an accurate picture of our work lives. Thankfully, however, perhaps even this academic humor can find an audience beyond the ivory tower: When situations arise that might be opaque to “outsiders,” the author routinely devotes a paragraph to spelling things out.

The characters who receive the most favorable treatment in the book — including those outside the university — are young, bright, driven, and for the most part technologically proficient. Their home is scientific computation, broadly defined, as they work to both advance and disseminate knowledge. One character who does not dwell in this realm is Alex the dog, who like Snoopy doesn’t talk but has powers beyond those of the typical canine. Another non-human character is perhaps the author’s favorite: Wolfram Research’s Computable Document Format. It is lauded throughout, even to the point that on more than one occasion someone conjectures that MathWorks must be sweating bullets.

Most of the main characters have good intentions as they strive to excel at their jobs and positively influence those around them. The characters frequently offer each other (and us) a healthy dose of simple wisdom, such as to view the vagaries of life not as problems but as opportunities. Over the course of the story, the optimistic and energetic characters do exert an influence on the pessimistic and moribund, but exactly how I will leave to the reader to discover. This is an enjoyable read and highly recommended.

Individual gain and engagement with mathematical understanding

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My colleague Mercedes McGowen and I examined a measure of individual student gain by pre-service elementary teachers, related to Richard Hake’s use of mean gain in the study of reform classes in undergraduate physics.

The gain statistic assesses the amount individual students increase their test scores from initial-test to final-test, as a proportion of the possible increase for each student.

We examined the written work in mathematics classes of pre-service elementary teachers with very high gain and those with very low gain and showed that these groups exhibit distinct psychological attitudes and dispositions to learning mathematics.

We showed a statistically significant, small, increase in average gain when course goals focus on patterns, connections, and meaning making in mathematics.

A common belief is that students with low initial-test scores will have higher gains, and students with high initial-test scores will have lower gains. We showed that this is not correct for a cohort of pre-service elementary teachers.

Apparent puzzles with goodness-of-fit tests

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Given two data sets it is a not unreasonable question to ask if they have similar distributions.

For example, if we produce one data set of 500 numbers between 0 and 1, chosen uniformly and randomly (at least as randomly as we can using a pseudo-random number generator), and another data set of 500 numbers distributed normally, with mean 0 and variance 1, then eyeballing their histograms tells us , even if we did not know, that they are not similarly distributed:

A more considered way of approaching the question of whether two data sets are similarly distributed is to utilize one or more goodness of fit tests. There are several of these in common use, including:

Mathematica® incorporates these, and other goodness-of-fit tests in the function DistributionFitTest[]

These goodness-of-fit tests basically perform a hypothesis test with the null hypothesis $H_0$ being that the  data sets are identically  distributed,  and an alternative hypothesis $H_a$ that they are not.

The goodness-of-fit tests return a p-value, and a small p-value indicates it is unlikely the data sets are similarly distributed.

So, if we carry out the Pearson Chi-Square test on the uniform and normal data sets, as above, we get the exceptionally small p-value $5.32085\times10^{-85}$ indicating, very strongly, that the two data sets are not similarly distributed.

The p-value from a Pearson Chi-Square test is a random variable: if we carry out the test on two other data sets, one from a uniform distribution, the other from a normal distribution, we will get a somewhat different p-value. How different? The plot below shows the variation in p-values when we simulated choosing 500 uniformly distributed numbers and 500 normally distributed numbers 1,000 times:

We see that despite some variation the values are all very small, as we would expect.

Now let’s see what happens if we choose  500 points from a uniform distribution, and 500 more points from the same uniform distribution. We expect the Pearson Chi-Square test to return a reasonably high p-value, indicating that we cannot reject the idea that the data come from the same distribution.  We did this once and got a satisfying 0.741581 as the p-value.

But what if we repeat this experiment 1,000 times. How will the p-values vary?

The plot below shows the result of 1,000 simulations of choosing two data sets of 500 points, each from the same uniformly distribution:

These p-values seem reasonably uniformly spread between 0 and 1. Are they? The Cramér-von Mises goodness-of-fit test indicates that we cannot reject the hypothesis that these p-values are uniformly distributed in the interval [0,1].

We set the confidence level for the Pearson Chi-Square test at 0.01, so we could expect that 1 time in 100 the Pearson Chi-Square test will indicate that the two data sets are not from the same distribution, even though they are. In 1,000 trials we could expect about 10 such instances, and that is more or less what we find.

The uniform distribution of the p-values is, at first glance, quite surprising, but since the p-values themselves are random values we expect that they will indicate something other than what we know to be the case every so often, dependent on the confidence level we set beforehand. For example, with the confidence level set at 0.05, we see that about 5% of the time the Pearson Chi-Square test indicates that the two data sets are not from the same distribution even though they are. :

Notes:

1. We randomly reset the seed for the pseudo-random number generator in Mathematica® at each of the 1,000 simulations.
2. The result of uniformly distributed p-values for data sets from the same distribution  is not peculiar to the Pearson Chi-Square test.
3. The uniform distribution of p-values under the null hypothesis is proved here.

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