Why learn Mathematics?

I found a link to a blog posting on Wolf’s blog about the question: What is the point of math? It is quite an interesting read for me, but at some places very annoying.

Why do we learn the quadratic formula (also called the midnight formula in Bavaria because you are supposed be able to repeat it when someone awakens you ad midnight). After all, you may never have use this formula in your professional career. In fact, this is true even if you are mathematician working outside research. So what is the point of this formula then?

\(x_{1,2} = \dfrac{p}{2} \pm \sqrt{\dfrac{p^2}{4} – q}\)

You will be surprised to hear my answer as a math professor. Indeed, I agree, it is a useless fact to learn. You can absolutely do without that formula. If you asked me I’d cancel it from the syllabus altogether.

First of all, you can solve quadratic equations without the formula quite easily. The technique involved in this is called quadratic expansion. It is using algebraic tools to transform the equation into an equivalent equation which solves the problem. The underlying theory is the theory of fields. The technique can be applied in the field of complex numbers or in finite fields. So why learn a formula if you can derive it with just a few steps? Moreover, learning the algebra behind the calculations is far more useful than to remember the result.

Then, if you want to solve equations, why exactly quadratic equations? Just because you can solve an equation easily does not mean it is important. Admittedly, I can think of some applications besides elementary geometry. One is acceleration in physics or the ballistic curve in vacuum. But they are far to sparse to support a „midnight formula“.

Moreover, if you want to solve a formula, then start a computer. Install a software like my EMT, and it will print the answer in numerical and algebraic form (with the help of Maxima) in no time, even if the formula looks a bit more complicated than usually.

>&solve(a+x-b^2+a=(a^2+b^2)/x,x) | expand

                     4              2      2     2
               sqrt(b  + (4 - 4 a) b  + 8 a )   b
        [x = - ------------------------------ + -- - a, 
                             2                  2
                                     4              2      2     2
                               sqrt(b  + (4 - 4 a) b  + 8 a )   b
                           x = ------------------------------ + -- - a]
                                             2                  2

Learning how to compute this by hand and practicing is a waste of time. We have computers to do this.

If I say such things some people understand that math should be abandoned altogether because it can be done by computers. This would be a complete misunderstanding. Besides the fact that you still must be able to do some simple computations in your head or with a pencil, there is more to math than computing.

So what is math all about? If you want to learn about this, but are not mathematically inclined enough to look at some research paper, visit one of the pages of the modern branch of school teachers that try to envision new ways to get students to be interested in math. Dan Meyer’s blog is an example. They start with real world activities. Their path is to have the students ask the questions by themselves if possible. Their conclusion is that math can be made to appear interesting or even necessary to answer the questions.

As a math professor I wished the students would understand the principles that prove the equivalences in the quadratic expansion and the logic behind the square root function, and not just memorize the quadratic formula. It is logic that students fight with the most. And that deficiency affects all branches of mathematics, and it also affects thinking in real life.

So please, teachers, give up on memorizing formulas or practicing boring examples. Have the courage to change the school book approach to something closer to real life and the things that really matter in math nowadays. I know that some school syllabuses have already changed. But the switch is not yet in the brains of most teachers. And it has not yet reached the public opinion about our subject.

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