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September 20, 2011

Music and Math Lessons Learned

Nick Diaz

The following are not my words, but they could well be. They belong to Gary Rupert, long-time acquaintance of mine and band director per excellence at a nearby public high school. Here’s what Mr. Rupert wrote on his blog, “Today’s Task.”


“They received the music six weeks ago and were asked to learn it. There are only 22 measures and of those, six of them are rests. So let’s see, that means they would need to learn 2.7 measures a week. And since there are seven days in a week, they would have to learn roughly .39 measures a day. Now, there are 24 hours in a day so that means they would have to learn .018 measures an hour and since an hour is made up of 60 minutes, that’s .0003 measures a minute. Who said math isn’t a valuable subject?


“I am not an excuse person. I would rather my students look me in the eye and say, “I didn’t do what was expected of me” than to hear a list of excuses designed to relieve them of their responsibility. The truth is, they chose not to learn the .0003 measures, electing instead to do something else. Oh, I know they are busy…we are all busy. Yet we waste enough time each day that they could have had the entire piece learned using that time alone. Interestingly, when I challenge them about not knowing their part, they will give me every reason except the real one….and they wonder why I get frustrated!”


Mr. Rupert’s sentence about math in his first paragraph is music to the ears of teachers like me – pun fully intended. His writings and musings about education can be found at


In similar fashion, I insist that my 8th-grade students learn the perfect squares and square roots, from 1^2 to 32^2. I quiz them every week on the perfect squares, the same way students in elementary schools are (or should be) quizzed on their multiplication facts.


Some of my eighth graders have been performing very well on these quizzes, while some, expectedly, come up with excuses for not performing. They spend more time and energy thinking of excuses for not knowing that the square root of 484 is 22 than in learning, once and for all, that 22^2 is 484.


I do demonstrate methods of memorizing these; one does not memorize in a vacuum. There are many “tricks” and mnemonic devices, all of which I teach.


* * * * * * * * *


As for the importance of square roots, it’s one of the building blocks of mathematics. Mathematics as a whole builds upon itself, and knowing square roots will allow a student to understand higher mathematics. Square roots allow students to solve second-degree equations (quadratic equations); it’s also needed in graphing, trigonometry and in physics. In applied physics, for example, people use square roots to find the “magnitude” of scalar/vector objects.


Most numbers we deal with are not perfect squares; the higher they go, the larger the gap between two perfect squares. Knowing the square root of perfect squares allows us to estimate more efficiently what the square root of non-perfect squares is.


For example, say we are trying to find the square root of 18. We know that ’18′ falls between the perfect squares ’16′ & ’25′…this means that our answer will be between 4 & 5 (since those are the square roots of the perfect squares). Now to figure out if it’s going to be in the lower half (4.5 and below) or upper half (4.5 and above), we can test 4.5 * 4.5 and see if it gives us an answer higher or lower than ’18′.


By now my students should know that 4.5 * 4.5 is 20.25. (If you want to learn a quick trick for squaring numbers that end in 5, such as 45, 75, and 105, e-mail me and I’ll lead you to the answer).


We see that 20.25 is higher than 18, so our answer must be between ’4′ and ’4.5′. So this method (for perfect squares), won’t get an exact answer…it will at least allow you to guesstimate.


Our students, however, would rather “do” than “estimate.” “Doing” is mechanical, involves no thinking, and it’s done by the hand instead of the head. “Estimating” requires bracketing, interpolating, extrapolating, and involving number sense – a product of correct analytical thinking.


“Doing” brings on all kinds of errors in computation. “Estimating”, because it’s based on higher-order thinking skills, is much more reliable, despite its built-in inaccuracies.


In other words, the mind is quicker and more reliable than pencil-and-paper arithmetic, or even, in most practical instances, the calculator.


That’s what our children need – numeracy, number sense, substituting quantitative reasoning for humdrum arithmetic. I am convinced, after many years in this profession, that knowing the perfect squares, perfect cubes, powers of two, and the two-digit prime numbers – all by heart – is essential to student success in higher mathematics. Number sense is what brings understanding in algebra II and higher mathematics – not the blind manipulation of digits and symbols.


“Don’t show me why it works, Mr. D., just tell me the formula so I can plug it in my TI-84.’


I enjoy hearing from high school and college students, former victims of mine, about how they know and recognize a Pythagorean triple when they see one, while many of their classmates are reaching for their calculators. Number sense, acquired in their early years, leads to efficiency in their math work.


Efficiency – the ability to perform the greatest amount of work, of the highest possible quality, in the least amount of time, and with the lowest degree of effort.


Efficiency – the quality that Gary Rupert insists his students develop, simply by memorizing a few measures, so that they can make music, rather than just play notes.


Efficiency – the quality I insist my students develop, simply by memorizing math facts beyond the elementary ones, so their thoughtless algorithms and mechanical calculators are replaced by insightful thinking.


Efficiency is learned by students who make the conscious decision to learn, and invest their time and energy learning musical measures or perfect squares, rather than invent excuses for not learning.


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