Category: News Clips

Quantitative Literacy

From http://devlinsangle.blogspot.com/2016/03/the-math-myth-that-permeates-math-myth.html:

March 1 saw the publication of the book The Math Myth: And Other STEM Delusions, by Andrew Hacker. MAA members are likely to recognize the author’s name from an opinion piece he published in the New York Times in 2012, with the arresting headline “Is Algebra Necessary?”

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On page 48, Hacker presents a question he took from an MCAT paper. It provides some technical data and asks what happens to the ratio of two inverse-square law forces between charges of given masses when the distance between them is halved. The context Hacker provides for this question is that medical professionals needs to be able to read and understand the mathematics used in technical papers. His claim is that this requirement does not extend to the physics of electrical and gravitational forces. In that, he is surely correct… What this question is asking for is, Do you understand what a ratio is? Surely that is something that any medical professional who will have to read and understand journal articles would need to know. Hacker completely misses this simple observation, and presents the question as an example of baroque mathematical testing run amok.

On page 70, he presents a question from an admissions test for selective high schools. A player throws two dice and the same number comes up on both. The question asks the student to choose the probability that the two dice sum to 9 from the list 0, 1/6, 2/9, 1/2, 1/3. Hacker’s problem is that the student is supposed to answer this in 90 seconds. Now, I share Hacker’s disdain for time-limited questions, but in this case the answer can only be 0. It’s not a probability question at all, and no computation is required. It just requires you to recognize that you can never get a sum of 9 when two dice show the same number. As with the MCAT question, the question is simply asking, Do you understand numbers? In this case, do you recognize that the sum of two equal numbers can never be odd…

You get the pattern surely? Hacker’s problem is he is unable to see through the surface gloss of a problem and recognize that in many cases it is just asking the student if she or he has a very basic grasp of number, quantity, and relationships. Yet these are precisely the kinds of abilities he argues elsewhere in the book are crucial in today’s world. He is, I suspect, a victim of the very kind of math teaching he rightly decries—one that concentrates on learning rules and mastering formal manipulations, with little attention to understanding.

My favorite response came from a very perceptive high school students in the New York Times’ letters to the editor (http://www.nytimes.com/2016/02/19/opinion/maths-place-in-the-classroom.html?ref=topics&_r=1):

In “Who Needs Math? Not Everybody” (Education Life, Feb. 7), Andrew Hacker, who teaches quantitative reasoning at Queens College, says that since only 5 percent of people use algebra and/or geometry in their jobs, students don’t need to learn these subjects.

As a high school student, I strongly disagree.

The point of learning is to understand the world. If the only point of learning is job preparation, why should students learn history, or read Shakespeare?

And while your job may never require you to know the difference between a postulate and a theorem, it will almost certainly require other math-based skills, like how to prove something or how to understand a graph. If nothing else, people need math to understand finance, which is a part of everyone’s life.

I also disagree with the logic that if people are failing algebra, then they shouldn’t take algebra. If people approach life that way, they will get nowhere.

Algebra and geometry have a place in the classroom. If students are failing, then the way math is taught may need to change. But what is taught needs no alteration.

Which is crying shame, because Hacker does have good ideas about what a quantitative literacy course should look like (again, from Devlin):

The tragedy of The Math Myth is that Hacker is actually arguing for exactly the kind of life-relevant mathematics education that I and many of my colleagues have been arguing for all our careers. (Our late colleague Lynn Steen comes to mind.) Unfortunately, and I suspect because Hacker himself did not have the benefit of a good math education, his understanding of mathematics is so far off base, he does not recognize that the examples he holds up as illustrations of bad education only seem so to him, because he misunderstands them.

The real myth in The Math Myth is the portrayal of mathematics that forms the basis of his analysis. It’s the same myth you see propagated in Facebook posts from frustrated parents about Common Core math homework their children bring home from school.

How many ways can you arrange 128 tennis balls?

I found this bit of computational mathematics fascinating. From http://www.joh.cam.ac.uk/how-many-ways-can-you-arrange-128-tennis-balls-researchers-solve-apparently-impossible-problem:

Imagine that you have 128 soft spheres, a bit like tennis balls. You can pack them together in any number of ways. How many different arrangements are possible?

The answer, it turns out, is something like 10250 (1 followed by 250 zeros). The number, also referred to as ten unquadragintilliard, is so huge that it vastly exceeds the total number of particles in the universe.

Far more important than the solution, however, is the fact that the researchers were able to answer the question at all. The method that they came up with can help scientists to calculate something called configurational entropy – a term used to describe how structurally disordered the particles in a physical system are.

An Interview with Randall Munroe

FiveThirtyEight.com interviewed Randall Munroe, the author of the wildly popular xkcd webcomic. I recommend the whole interview, but I thought that the follow few paragraphs were exceptionally insightful.

One thing that bothers me is large numbers presented without context. We’re always seeing things like, “This canal project will require 1.15 million tons of concrete.” It’s presented as if it should mean something to us, as if numbers are inherently informative. So we feel like if we don’t understand it, it’s our fault.

But I have only a vague idea of what one ton of concrete looks like. I have no idea what to think of a million tons. Is that a lot? It’s clearly supposed to sound like a lot, because it has the word “million” in it. But on the other hand, “The Adventures of Pluto Nash” made $7 million at the box office, and it was one of the biggest flops in movie history.

It can be more useful to look for context. Is concrete a surprisingly large share of the project’s budget? Is the project going to consume more concrete than the rest of the state combined? Will this project use up a large share of the world’s concrete? Or is this just easy, space-filling trivia? A good rule of thumb might be, “If I added a zero to this number, would the sentence containing it mean something different to me?” If the answer is “no,” maybe the number has no business being in the sentence in the first place.

Sphere Packing Solved in Higher Dimensions

I enjoyed reading this bit of mathematical news: https://www.quantamagazine.org/20160330-sphere-packing-solved-in-higher-dimensions/

The opening paragraphs:

In a pair of papers posted online this month, a Ukrainian mathematician has solved two high-dimensional versions of the centuries-old “sphere packing” problem. In dimensions eight and 24 (the latter dimension in collaboration with other researchers), she has proved that two highly symmetrical arrangements pack spheres together in the densest possible way.

Mathematicians have been studying sphere packings since at least 1611, when Johannes Kepler conjectured that the densest way to pack together equal-sized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores. Despite the problem’s seeming simplicity, it was not settled until 1998, when Thomas Hales, now of the University of Pittsburgh, finally proved Kepler’s conjecture in 250 pages of mathematical arguments combined with mammoth computer calculations.

Do’s, Don’ts for Parents to Help Teens Build Math Interest and Success

I really enjoyed reading this article: http://www.usnews.com/education/blogs/high-school-notes/2016/04/11/dos-donts-for-parents-to-help-teens-build-math-interest-and-success

A summary:

  • Don’t project negative feeling toward math onto teens
  • Do talk to teens and teachers about what’s being taught in math class
  • Don’t be too quick to hire a tutor for struggling students
  • Do support students with the right tools

I recommend the whole article and the references therein.

Teens do better in science when they know Einstein and Curie also struggled

From http://qz.com/622749/teens-do-better-in-science-when-they-know-einstein-and-curie-also-struggled/:

The study, published in the Journal of Educational Psychology, divided 402 ninth- and 10th-graders from four New York City public schools in Harlem and the Bronx into three groups. One group read an 800-word excerpt from a scientific textbook on the accomplishments of Albert Einstein, Marie Curie, and Michael Faraday (an English scientist who made discoveries about electromagnetism).

Another group learned about the scientists’ personal struggles, such as the fact that Einstein had to flee Nazi Germany to avoid persecution, or Marie Curie had to study in secret because women were discouraged from academic pursuits at the time. The third group learned about the scientists’ intellectual struggles and how they confronted them.

After six weeks, the two groups who learned about how the scientists struggled significantly improved their science grades and increased their motivation to study science. The lowest performing students showed the greatest gains.

Meanwhile, the students who learned only about the scientists’ achievements performed worse. They believed the scientists were innately gifted—unlike themselves.

Statisticians Found One Thing They Can Agree On: It’s Time To Stop Misusing P-Values

From the excellent article http://fivethirtyeight.com/features/statisticians-found-one-thing-they-can-agree-on-its-time-to-stop-misusing-p-values/

A common misconception among nonstatisticians is that p-values can tell you the probability that a result occurred by chance. This interpretation is dead wrong, but you see it again and again and again and again. The p-value only tells you something about the probability of seeing your results given a particular hypothetical explanation — it cannot tell you the probability that the results are true or whether they’re due to random chance…

Nor can a p-value tell you the size of an effect, the strength of the evidence or the importance of a result. Yet despite all these limitations, p-values are often used as a way to separate true findings from spurious ones, and that creates perverse incentives…

If there’s one takeaway from the ASA statement, it’s that p-values are not badges of truth and p < 0.05 is not a line that separates real results from false ones. They’re simply one piece of a puzzle that should be considered in the context of other evidence.

The article above links to the statement by the American Statistical Association as well as various commentaries by statisticians about the proper use of p-values.