1. The engineering office in the village of Whitesboro has a map of the village that is laid out on a rectangular coordinate system. A traffic circle located on the map is represented by the equation . The village planning commission asks that the transformation be applied to produce a new traffic circle, where the center of the dilation is at the origin.
   -- New York State Regents Exam, Math B, 2008
    
2. In a pumpkin tossing contest in Morton, Illinois, a contestant won the catapult competition by using two telephone poles, huge rubber bands, and a power winch. Suppose the pumpkin was launched with an initial speed of 125 feet per second, at an angle of 45º, and from an initial height of 25 feet. How far did the pumpkin travel?
   -- Algebra 2, McDougal Littell

The Endless Argument

I was recently alerted by Scott McLeod to a blog post that:

"Stephen Andrilli, professor of math and computer science at La Salle University in Pennsylvania, suggests teaching "consumer math" to help students better manage credit cards, mortgages, and other important, real-world financial concerns."

As far as endlessly circular arguments go, incorporation of real life material into the curriculum is on the top of the list. Generally it goes like this:

A: Students are unmotivated! We need more real life math. Like what they’ll actually see in a career.

B: Students need the basics first. We need more rote memorization.

A: But they don’t care about rote memorization.

B: Enough of the cutesiness. Our students need to buckle down to be competitive with China and India. India teaches math traditionally and you don’t see them complaining.

A: I’m telling you, students don’t care about school any more. Our drop out rate is too high.

B: Too much tweaking the traditional curriculum and they don’t learn math at all!

B: If you try to get them to discover in the end they’ll learn only a fraction of what they need.

A: You need to get down here and see what it’s actually like!

C: Enough you three, otherwise I’ll have to move you to separate corners of the room again.

The problem with such arguments is they oversimplify. Both examples that start this article are have some element of the “real world” in them, yet almost universally students would rather solve problem b than problem a.

The mere presence of real world material does not guarantee an interesting question (for a student), nor does the absence guarantee a boring question.

A Movie Interlude

After renting videos from Netflix the website will ask you to rate them from 1 to 5 stars. Using your ratings the website will start predicting what other movies you might like. It will even try to guess what you’ll rate those movies as (so even though the average of all users may rate a movie as 2 stars, Netflix will be able to tell you if your personal rating will be 5 stars).

The Netflix Prize is a competition designed to improve the accuracy of the Netflix recommendation algorithm. Winning requires a 10% improvement in their original algorithm. (As of this writing the best improvement is 9.44%.)

One might think it would help to look at genres of movies, and maybe the sorts of categories the Internet Movie Database lists movies under (for High School Musical 3: Family Relationships, Interracial Relationship, Interfamilial Relationship, Sequel, Dancing, Graduation).

However, as one team attempting to win the Netflix Prize writes:

". . .my interpretation is that movie data is just too black and white. User tastes are infinite shades of grey (think floating point shades of grey). It's not true that someone likes all sci-fi movies. And no one can enjoy all the Tom Hanks movies equally. But the algorithms can figure out the subtle nuances that define user rating patterns. It can figure out that you really like sci-fi comedies that have a happy ending, but that you enjoy the sci-fi/horror genre, where one of the main characters dies, a bit less."

The categories as described by computer algorithms ended up being much more useful and powerful than any human categorization. In a sense, the reality is too complex to be described by words.

Rating Math Problems

Is it no wonder, then, that the single category of real-life-or-not is insufficient to describe the quality of a problem?

Movies ratings also makes a good analogy for a student’s mind when they come across a problem. Will they attempt it, or will they try to do something else? This is partly affected by the student’s personal “rating”.  While such ratings should not dictate what students are given – for example, many students consider easy problems better than hard ones, even though they should solve hard ones for their own good – it’s a useful factor to think of when assessing how much internal motivation a particular lesson plan has.

Questions to Ask

Is the math used how those in the related profession would actually use the math?

Anyway, I'm happy with this book, because it's the first example of applying arithmetic to science. I'm a bit unhappy when I read about the stars' temperatures, but I'm not very unhappy because it's more or less right -- it's just an example of error. Then comes the list of problems. It says, "John and his father go out to look at the stars. John sees two blue stars and a red star. His father sees a green star, a violet star, and two yellow stars. What is the total temperature of the stars seen by John and his father?" -- and I would explode in horror.
-- Judging Books by Their Covers by Richard Feynman 

Did the authors of the problem want to force some particular math operation (say, dilation of a graph) and just wrote an imaginary situation around it?

Context? Could you get more artificial? Bizarre? Village laid out in a rectangle, ok. But x-y equation for a street feature? Arbitrary transformation of a traffic circle?

-- Commentary from JD2718 on the Regents Exam problem

Is the situation described interesting in itself? Does it presume cultural-specific knowledge (say about ships on the ocean, in a textbook used in Kansas)? Does it give formulas the students could find themselves with experiment? Is there a way of solving the problem superior to what the question is fishing for?

In General

Note that for a problem to be interesting, it is not necessary for a question to relate to a calculation the students will have to do later in life. It is unlikely (from problem "a") that students will be setting up giant rubber bands with power winches any point in their lifetime. The situation itself is internally interesting enough to qualify for a high rating.

While matching with career interest can help boost a rating, that’s only one criteria out of many. It is hopeless anyway to make every student happy in that respect; a lesson about engineering might bore the budding fashion designers; a lesson revolving around fashion design will might prompt the engineers to take a nap. Teachers can only hope to rotate their repertoire enough that each student is happy (in a career-related respect) at least some of the time.

It also is possible to have an problem that’s interesting (for students) which does not relate to real life in any way. I will explore this idea in my next post.

Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.

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