Finessing Grades

Put these numbers in order: , , and , assuming that .

Here’s the context: Suppose (hypothetically, you know) you’re grading exams on a percentage scale (93% and above is an A, 90-92% is an A-, etc) and after adding scores you realize that maybe because of an unusually difficult problem, the points don’t quite match your overall impression in the sense that the exams that indicate a deep understanding should be the ones that get an A. Now suppose that by adjusting the scores by 5 points they do seem to match up and this is simpler than going back and revising the partial credit schemes on the whole exam.

Shall we play a game?

No, not Global Thermonuclear War* — more along the lines of Chutes and Ladders. The Pittsburgh Post-Gazette reported today on a study by Geetha B. Ramani and Robert S. Siegler in which children who played a numerical board game for four 15-20 minute periods over two weeks showed improvement in certain math skills even two months later.

A nearly ubiquitous form of fallacious reasoning

How should one prove that two algebraic expressions are identically equal to one another?

Many people first learning to do proofs are tempted to start with the identity they seek to prove, then use various algebra (or trig or logarithmic or…) identities to manipulate both sides of the equation, seeking to reduce the equation to an easily-recognized identity.

But of course starting from and showing that this eventually implies that (or some other true equation) does not constitute a valid proof that .

One of my favorite (simple) examples:

Thus (the reasoning goes), , which is a fabulous result with many wonderful applications in the world of personal finance.

The problem is that each of the inferences being drawn is an if-then inference, for which knowing the consequent is true (9=9) in fact tells us nothing about the truth of the antecedent (-3=3). This form of argument can only be valid if every step is reversible — in effect if each step in the proof involves an if-and-only-if inference. But since most people learning proof are not particularly savvy at bearing this distinction in mind, good pedagogy suggests we completely avoid proving by reducing it to . (Instead, one works from one side the equation to the other, applying identities to show that .) [There are other ways to show that two expressions are equal, but this is the one most commonly relevant to elementary algebraic, trigonometric, and logarithmic identities.]

These issues hit home for me today when I read the March 2008 issue of Student Math Notes, published by the National Council of Teachers of Mathematics as an insert in the NCTM News Bulletin. This month’s issue focuses on the Pythagorean Theorem, and has a student worksheet exploring properties of right triangles.

Toward the end of the worksheet the students are told that if are whole numbers, then will be a Pythagorean triple. The worksheet’s proof of this fact is to start with

perform several algebraic operations on both sides of the equation, and reduce the given equation to the identity

“Since both sides are equal, it follows that and the three numbers form a Pythagorean Triple”, the authors conclude.

Technically, I guess their proof is okay, because in fact each step in the derivation (which I’ve omitted here) is in fact reversible. But the concluding sentence is dangerous and highly misleading, and I must say I am rather dismayed by it all.

A better proof would be to start with , note that that is equal to , which simplifies to , which by inspection is .

What we lose in simplicity we gain in validity.

Happy Birthday Dr. Seuss!

In honor of Theodor Seuss Geisel’s 104^th birthday today, I thought I’d do a post on (fiction) books that have some math in them. But when I started to think about what I’d write, I quickly became overwhelmed. Should I mention The Phantom Tollbooth by Norman Juster? What about The Curious Incident of the Dog in the Night-time by Mark Haddon, a novel written from the perspective of a mathematical 15-year old who is also autistic? And then there is The Number Devil: A Mathematical Adventure by Hans Magnus Enzensberger, which I haven’t actually read but which came highly recommended. I started collecting titles of books with a-little-to-a-lot of math in them about two years ago, and the list could be a web site in and of itself.

Multidigit multiplication: vertically and crosswise

I think I have a new favorite way to multiply numbers: the vertically and crosswise technique. I learned about it from George Gheverghese Joseph’s Crest of the Peacock [a book that is sadly out of print], and was recently reminded of just how cool it is.

Root extraction, part II: cube roots

As you might guess, this post builds on “Root extraction, part I“, which gave a way to visualize the traditional square root algorithm geometrically, an approach that has the advantage that each step appears natural and easily motivated.

Our goal herein is to do much the same for cube roots. The point is to find a geometric construction, ideally one well-suited to physical manipulatives, in which the steps in building the successive digits of the cube root of a number are transparent.  As with the post on square roots, I make no claims to originality in what follows.

Example:  Find . 

Root extraction, part I: square roots

I recently discussed the traditional algorithms for computing square and cube roots in my History of Math class.   Our reading, on mathematics in ancient China, gave both algorithms as a set of rules for manipulating number rods.  For me, it was fascinating to see past the text: the rules as given would transfer directly to an abacus/soroban calculation, and were essentially the same as the rules that prior generations of American schoolchildren would have been drilled on in school.

My students (mostly high school math teachers) found the book’s explanation of the method obscure;  the key is to view the process geometrically, rather than as a mechanical set of rules for manipulating digits.

I make no claim of originality in what follows; I offer it here in part because I can’t find any lucent discussions along these lines on the web. 

How to write paragraphs and proofs

Remember when you first learned how to write paragraphs?  For me I think it was seventh grade, and the approach we were given was to craft five sentence long paragraphs:  the first sentence set out the topic (or thesis); the next three sentences gave further detail supporting the thesis; and the last sentence summarized the content of the paragraph. 

By the end of seventh grade, we were starting to write essays.  An essay consisted of five paragraphs:  the introduction, three supporting paragraphs, and the conclusion.  Each of those paragraphs were themselves five sentences long, following the same structure.

Egads, a fractal! 

Writing Review Sheets

I was re-reading the post Writing Tests recently, and thinking about how I prepare for exams. In particular, I was thinking about how I prepare students for exams, and one way that has worked well for me is giving an Outline of Topics as a review sheet.

Math Helps College Students

According to the August 4, 2007 issue of Science News (Vol. 172, No. 5, p.78), the more math and science classes students take in high school, the better they perform overall in college (emphasis mine):

Apparently, high school math is the key to good grades in college science classes.

A survey of more than 8,000 students from 74 colleges found that each additional year of high school math correlated with a 1-to-2-point advantage, on a 100-point scale, in college chemistry, physics, and biology grades. For example, 2 additional years of high school math typically corresponded to a 3-point improvement in college biology—the difference between, say, a B+ and an A-.

Kids who took more high school classes in chemistry, physics, or biology gained a similar edge when they took a class within the same discipline at the college level. However, no significant benefit crossed a line between science disciplines. Only math seemed to boost grades in other subjects. The study appears in the July 27 Science.

Coauthor Philip Sadler of Harvard University says that students who take advanced high school math classes are better able to handle the more basic math required in college science classes.

The results are “not surprising,” says James Milgram, a Stanford University mathematician and a member of a presidential panel advising the U.S. Department of Education. He points out that decadal surveys by the department have shown that as more students have taken advanced high school math classes, their chances of graduating from college have improved. “There is overwhelming evidence that the single most important factor that correlates with success in college is what is done in high school math,” says Milgram.

(Hat tip to Brian Witz.)

EdHeads at the STEM Blog

There was a great resource posted yesterday at the the STEM Blog (Discussions in Science, Technology, Engineering, & Mathematics Education): Are you an Edhead? This post directed people to the site Edheads.org, which features activities on investigating a car crash, performing surgeries, predicting the weather, and more. What impressed me when I visited the site was the level of detail in the activities: the car crash investigation walks visitors through the process of collecting data and applying it to mathematical equations, while the hip surgery gave more surgical steps than I would have expected, with optional photos as a bonus. Be warned: even in cartoon form, if you are squeamish about surgery then this isn’t the activity for you, although our seven-year-old declared it AWESOME! and I found it to be pretty cool myself. All of the activities come with teacher’s guides and grade level suggestions.

Thanks to the STEM Blog (a brand new blog — even newer than us!) for bringing this site to our attention.

Grocery Store Polygons: Cookie Cake Tops

The post two days ago (Junk Food Geometry) focused on edible polygons, but perhaps my favorite examples of grocery store polygons are inedible: the cookie cake tops at Wegmans. The aspect that stood out initially to me is that they are non-standard polygons. The medium sized one, shown to the left, is a heptagon! This cookie top and a pillbox we once found are the only two real-life examples I’ve seen of regular heptagons.  Edited to add :  of course, within days I found heptagons in a Harry Potter game and in coins.

Junk Food Geometry

I was about to post about the interesting shapes in the cookie cake tops at our grocery store (which I’ll do shortly), and I found myself getting distracted thinking about all of the interesting shapes in the snack aisle itself.

“Double Down” Dumbs It Down

Double Down is a quiz show for New York State high school students that airs on PBS (WCNY to be specific). I happened to catch a rerun last night, and one of the categories was “Math”. Here are a couple of the questions:

1. A polygon with 5 sides is called what?
2. A polygon with 8 sides is called what?

These are high school students, remember. My 20-month old daughter knows what an octagon is. Can we give these kids some credit?!

Counting on your Fingers

Three Finger Tricks for Multiplying also made me think of another finger trick: using both hands to count to 99. It’s not as clever as multiplication, but it’s one I use regularly for keeping track of numbers and for simple addition or subtraction.