January/February 2007

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...On "Math... What Good Is It?" (October 2006)

Editor's Note: This article prompted more than 500 reader responses, most of them with answers to the math quiz that was given at the end of the article, and many of them including comments on the teaching of math. Here are just a few.

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Your evaluation reinforces my own opinion of math education in the United States. When I study (at Georgia Tech) with friends from the Northeast, I realize how much I missed in middle and high school. The most advanced track in my high school (one of the best in northwest Florida) got me to first-year calculus, but it skipped linear algebra (basic vectors and matrices) and so much more. Despite my dissatisfaction, I know I was relatively lucky to have access to three or four advanced placement classes, two dual enrollment classes, etc.

Not only is there a gap in math education between countries, but there exist large disparities among the states. We've got to somehow take action to remedy this.

— Ryan Westafer
IEEE Student Member

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I'm a Brazilian and moved to the United States a while ago. My oldest boy finished 5th grade in the Brazilian school system and until the 8th grade in the American system he was seeing topics already learned in Brazil! Isn't that amazing? I gave your problem to him (now he's in the 12th grade—GPA 4.7) and he didn't know how to solve it. . . . I hope he understands that what he learns in school is not necessarily enough.

I found your problem very interesting! First because it's simple to solve if you know the basics, and second because I never solved a problem like that! My solution is 12 and here is how I found it.

A number in a base b can be converted to base 10 by adding all digits multiplied by the base raised to nth power (where n is the digit position counting from zero and from the right to left.) So we have the following equations:

(24)b = 2b + 4
(554)b = 5 b^2 + 5b + 4
(2b +4)^2 = (5b^2 + 5b + 4) = >4b^2 + 16b + 16 = 5b^2 + 5b + 4 or b^2 – 11b – 12 = 0
Solving for b we find -1 and 12, and -1 is not a valid base.

Thanks for the challenge!

— Paulo S. Mendes
IEEE Member
Hopkinton, Mass.

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I read your article with great interest. After observing how math is taught in the schools my children attended, I can easily understand why high school students hate math. Teachers seldom make any attempt to explain the applications for the math they are teaching. If the teacher is competent to teach math, which is often questionable, they make few attempts to make it interesting.

One of my sons had good math skills, but he was burned out by the boring repetitious manner in which math was presented. When he was studying factoring of polynomials in algebra, I helped him study and determined that he had mastered this skill quite well. I could give him any problem in his textbook and he could reach the correct answer. But his teacher insisted on assigning dozens of the same type problems every evening. Even for someone skilled in factoring polynomials, this required at least one hour every night to complete the homework assignments. I talked to his teacher and asked if it might be possible for him to 'quiz out" of this section and move on to new material, but her attitude was that this was impossible. Apparently the teaching plan is to keep repeating the same material until nearly everyone learns it, but this is very frustrating for those who master the skill quickly.

At the college level, math courses for engineers can also be very frustrating. They are often taught by uninterested math-major graduate students who have absolutely no interest in applications, yet applications are what engineers do! I think engineers would benefit greatly if math courses were taught by engineers.

— James M. Cook
IEEE Member
Kansas City, Mo.

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Your article on the topic of math is useful and should be presented to as many teachers of science and math as possible. The problem is that we have dumbed-down too much of high school and now teach to the lowest common denominator; try to push them through and get past the FCAT. Maybe we need more engineers as visiting teachers to explain these things.

— Richard Lowrie
Life Senior Member

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Excellent article! When formerly an instructor at ITT Technical Institute, I taught remedial math. I had the same discussions with students, and fortunately, the ITT curriculum gets directly to applied mathematical situations fairly quickly. I did, however, try to make the connection between the abstract and the applied. As you say, some had the aptitude to make the connection; others did not.

For an old-time software geek, your problem was fairly easy to solve. The square of 24 in base 10 is 576. As all good hexadecimal geeks will tell you, the square of 24 in base 16 is 510. Thus the answer must lie between base 10 and 16. The odd bases are out, because the ones place will also be an odd number. 554 is closer to 576 than 510, so the obvious answer, without calculation, is base 12. Doing the math confirms that base 12 is the answer.

— Bob Gruszczynski
IEEE Member
Willis, Mich.

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The answer to your math puzzle, of course, is 12. Or maybe -1. Since -1 is hardly a base, the answer must be 12.

I don't have the answer to your bigger puzzle, how to teach math to kids, and it will be much harder to solve. Perhaps a part of the solution could be injecting more math into the popular video games. The math-oriented games and puzzles found on the net are not nearly as compelling to kids (or adults for that matter) as the shoot-em-up games (WarQuest, for example). Furthermore, it's harder to find competition when solving puzzles, than when shooting aliens, so the shoot-em-ups are more appealing to kids who naturally want to compete (in unorganized games). Yet the guys that write the games are really mathematically oriented. Perhaps the game writers could use some of their genius to include (disguised) math problems in their games. But what would motivate them to do so?

We certainly have a growing problem recruiting young scientists and engineers, and it won't go away by itself. Thanks for your article. More articles like this need to be published in non-engineering literature, too.

— Stan Fralick
IEEE Member

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Editor's Note: Bruce Blair, mathematician, engineer, and math teacher, easily solved the problem and added:

If one understands the principles and applies them, all manner of problems can be solved . . . that is the lesson that math should teach. Learning to appreciate the elegance and power of axioms, physical laws and relationships opens the door to creativity.

My students could see no value in geometric proofs . . . they were tooooo hard. That was the only argument they could make. 'Turning on the light" for them is a real challenge.

— Bruce Blair
IEEE Member

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Using a quadratic equation, the answer is b =12.

Regarding your article, I remember my high school math teacher telling us about the early days of his career—the 1940s. At that time, he was teaching math to kids in small ethnic communities in the far north of Russia—the areas where the tribal way of life was still predominant. He commented that even as the students became fairly proficient in arithmetic, they could do it as long as it was done in the context of familiar objects—people, deer, etc., and that they had great difficulties in transitioning from objects to abstract numbers. He also observed how difficult it was to introduce to them the notion of zero and negative numbers.

— Sergey Liberman
IEEE Member
Bedford, Mass.

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Editor's Note: James Morse easily solved the problem, adding:

While I agree that this type of problem may not have any practical application, I do believe that these types of problems are a good exercise to make one think about how to solve problems. When people ask me what I do, I tell them I solve problems . . . that's what engineering is all about.

— James Morse
IEEE Member
Poway, Calif.

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If you gave that question to a high school algebra class, I would hope that most students would happily type out an answer in less than a minute. Most wield CAS-enabled calculators, like the TI-89. Mine reports:

Solve (2*b + 4)^2 = 5*b^2 + 5*b + 4)
b = 12 or b = -1

I had the good fortune of working with number systems using arbitrary bases quite early in school — 2nd grade! It was an entirely fun concept. Math can easily be made the most fun subject in school. Just avoid spending a year on long division, and you'll be OK.

— James Eberle
IEEE Member
Burlingame, Calif.

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I found your article very interesting. I would add that I think they push the kids too hard in the lower grades. Mathematics requires maturity. Otherwise it can be a bore. Another observation is that the teachers are calling it mathematics while we called it arithmetic when I was in elementary school. Mathematics came when one got to algebra and geometry. It not only trivializes mathematics but it gives the children and some of their teachers the idea that majoring in mathematics means more years of boring arithmetic.

— Jim Cooley
IEEE Life Fellow

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