How can I counter a student response saying "Why are we bothered to reinvent the wheel when proving mathematical identities?"

Question

I am teaching an undergraduate mathematics course to software engineering students. I often ask my students to prove some mathematical identities as their homework. The identities could be, for example: trigonometry identities, Laplace transform identities, Menelaus', Stewart's, Ceva's theorems, etc.

One special student often just copies other works obtained from the Internet without doing any effort to prove them by himself. When I asked him, he replied "Why are we bothered to reinvent the wheel?". I have not replied yet and I will do later.

Could you give some advice for countering his argument?

Answer 1

Someone said:^*

When you reinvent the wheel, you end up learning a great deal about why wheels are round.

And that is really the point. When you’re at university to get a degree in wheels, you should fully expect not just to be told that wheels exist but to be asked to think deeply about them so that you develop an understanding, at the deepest level, of why they work the way they do. That is literally the whole point of going to college.

^{*Source: something I read recently somewhere on the Stack Exchange network. The quote is from memory, I can’t find the exact source now unfortunately.}

Answer 2

The student seems to have the misconception that mathematics is about "facts". Early education stresses elementary facts a lot, so this is pretty natural.

But mathematics is about understanding relationships, not memorizing facts. If you don't know why something in mathematics is true, you really don't understand it. The proofs in mathematics are more important than the theorem statements, actually, as the latter just capture the essence in a simpler statement.

Until he understands that it will be difficult for him to advance as a mathematician, assuming that is a goal. If it isn't a goal, then he has little incentive to change.

Answer 3

I was that student. And the reason is mainly not because I was a bad student, or a bad engineer (a 2:1 degree and 25 years in industry should answer those points). It was because my lecturers were bad teachers. And yes, you too may be a bad teacher right now, without realising it. The difference is that you've seen this student's question as a challenge to improve your teaching, and that's something you should be proud of. What matters for all of us is how we learn to do better.

You're teaching engineers, who are without exception practical people. (Or the good ones anyway. I assume you want your course to produce good engineers.) Remember that they aren't at university to become mathematicians, they're there to become engineers. If you want them to learn a subject which seems of no real value to them, you need to convey the value in doing it. It seems you haven't managed to get this across.

To divert slightly, think of The Karate Kid. How does Mr Miyagi teach Daniel? He gives him apparently pointless exercises to do. Daniel puts up with this for a while, but eventually tells Miyagi to shove it. Miyagi then demonstrates that there was some value to the exercises after all. The thing is though, this isn't the only way to learn, nor actually the way which works for most people. Go into a regular dojo, and you get taught how to block and punch by, well, blocking and punching.

You've got the Miyagi method though. You've given them seemingly pointless exercises to get practise in basic techniques. Now your Daniel has called you on it though. So what's the application for that technique? If it's widely used, what's even one application for it?

For concrete personal examples, we spent ages at university learning about phase lead/lag networks, PID controllers, and so on. What we didn't do was get an actual motor and position sensor, and actually control something. The result for me was that it didn't really stick, and I've had to relearn it all at work when I've had projects which needed it. Practicality and relevance are the keys for engineering.

You mention Laplace. Have you had them do any DSP? It doesn't get more hands-on with Laplace than filters. And that step response? That's what happens when your wheel hits a bump, or someone pops the microphone. Do you get a (Butterworth) ringing or a (Bessel) smooth step? And which do you accept, if the depth of filtering is also important?

There are lots of ways you can go which makes your subject relevant. All you need to do is find them. This may require you to talk to the engineering staff to find how they use these techniques (per @KevinArlin's comments), to give you examples you can hang your lectures on. If it means students they'll be teaching or supervising later have a better grasp of these basic techniques which they'll need, I'd expect them to jump at the chance.

Answer 4

I'm a mathematics graduate currently working in computer science and data analysis. In my experience it is difficult (if not impossible) to memorize every single aspect of mathematics, especially for identities which could be combined with one another in various complicated ways to produce infinite results. Instead I find it useful to start with the most basic parts for which I have a deep understanding and put them together - in essence "reinventing the wheel" - until I have my desired result.

Years after taking a certain course, I have found myself working on projects that need certain identities and facts (I've needed a surprising amount of geometry and trigonometry for processing image data). There is no way I would remember them all, but I can still remember the basic parts and the process needed to put everything together. Sure I could look it up (as long as I remembered the name or enough details about the identity, which isn't trivial), but that is not always an option...

Even more important, there are times when I am working with very well known identities or processes, but I need to put two things together, add something new, or make a change to get a slightly different result. If I only ever used what could be easily looked up, I would be out of luck in this new situation. But hopefully after "reinventing the wheel" many times over, I will have developed the problem solving skills necessary to tackle the problem.

Do that enough times, and complex things like "the wheel" become the basic parts for which you have a deep understanding, and you can use it to build greater and grander things, a "car" for instance.

Answer 5

You can tell the student the following:

The goal of homework is not to prove new things that the instructor doesn't know, but rather to give the students knowledge and experience in using the tools given to them. This is called "learning" - probably your course/institute has some goal the students "learn" something. You could refer him to that.

It's the same when a teacher teaches: The teacher does not reinwent a wheel but rather tells/shows known facts. Yet schools, universities are important.

This is also not related to mathematics or university. Most likely, the students had to learn a foreign language (or their own language) at school. Then they had to apply it (by speaking or by doing grammar exercises). Here they also do not invent new things (as one could say a novel does), but deepen their knowledge.

Answer 6

Counter by

1. Having the student consider why a musician or athlete practices routinely, and
2. offering an unsolved problem to prove as an alternative.

Together these points should prove compelling, assuming the student's objections were truly sincere.

Answer 7

Feynman cut to the heart of this issue in The Feynman Tips on Physics:

I have a few moments left, so I’d like to make a little speech about the relation of the mathematics to the physics—which, in fact, was well illustrated by this little example. It will not do to memorize the formulas, and to say to yourself, “I know all the formulas; all I gotta do is figure out how to put ’em in the problem!”

Now, you may succeed with this for a while, and the more you work on memorizing the formulas, the longer you’ll go on with this method—but it doesn’t work in the end.

You might say, “I’m not gonna believe him, because I’ve always been successful: that’s the way I’ve always done it; I’m always gonna do it that way.”

You are not always going to do it that way: you’re going to flunk— not this year, not next year, but eventually, when you get your job, or something—you’re going to lose along the line somewhere, because physics is an enormously extended thing: there are millions of formulas! It’s impossible to remember all the formulas—it’s impossible!

And the great thing that you’re ignoring, the powerful machine that you’re not using, is this: suppose Figure 1-19 is a map of all the physics formulas, all the relations in physics. (It should have more than two dimensions, but let’s suppose it’s like that.)

Now, suppose that something happened to your mind, that somehow all the material in some region was erased, and there was a little spot of missing goo in there. The relations of nature are so nice that it is possible, by logic, to “triangulate” from what is known to what’s in the hole. (See Fig. 1-20.)

And you can re-create the things that you’ve forgotten perpetually —if you don’t forget too much, and if you know enough. In other words, there comes a time—which you haven’t quite got to, yet—where you’ll know so many things that as you forget them, you can reconstruct them from the pieces that you can still remember. It is therefore of first-rate importance that you know how to “triangulate”—that is, to know how to figure something out from what you already know. It is absolutely necessary. You might say, “Ah, I don’t care; I’m a good memorizer! I know how to really memorize! In fact, I took a course in memory!”

That still doesn’t work! Because the real utility of physicists—both to discover new laws of nature, and to develop new things in industry, and so on—is not to talk about what’s already known, but to do something new— and so they triangulate out from the known things: they make a “triangulation” that no one has ever made before. (See Fig. 1-21.)

In order to learn how to do that, you’ve got to forget the memorizing of formulas, and to try to learn to understand the interrelationships of nature. That’s very much more difficult at the beginning, but it’s the only successful way.

Answer 8

For the last student who asked me this, I pulled out Gradshteyn and Ryzhik (my copy from 1992). My first line (in joke voice) was "Alright, get to memorizing."

Then I flipped to page xxiii and pointed at the text leading up to "We then kept only the simplest formula." Then jumped to page xxiv to the line "Thus, before looking up an integral in the tables, the user should simplify as much as possible the arguments ... in the integrand."

The student responded that there exists software containing the entire book. I replied with my experience : "There are only a few pieces of software whose marketing departments make this claim. To date, none of them have been correct, as I have found fairly simple integrals that each version of each piece of software I have used fails to integrate."

If you can't bring your work in to a form that can be found in references, you are crippled compared to someone who can. This may involve algebra, trigonometric identities, Fourier, Laplace or other integral transforms, the Cauchy integral formula, et c. If you don't understand those tools and how they work, you will not understand when you should or should not be using them. This knowledge comes only from practice.

Answer 9

My answer would go along the line:

It's a bad idea to blindly believe everything you read. We all know that there's a lot of nonsense out there in the internet. So you better be able to judge yourself, and this exercise is part of teaching you how to evaluate whether some mathematical claim is true or not.

Maybe it would raise motivation to have the students decide whether some claim is true or false, and to present sound reasoning (proof) for their answer.

Answer 10

Doing math is different from 'knowing math'. With knowing math I mean memorizing formulas and concepts. We all probably have experienced this in an exam that contained math: you think you know some formula but when you're in the exam you notice you don't actually know how to implement it in practice. Proving identities is one of the best ways of getting some hands on practive with the material. Even though it might not be the most direct way to practice with it. One benefit you get by proving those identities is that you better know when you are allowed to apply these identities because now you know exactly what conditions are required to make the proof work.

A second argument is that proving an identity is an effective way to remember said identity. Personally proving a formula about 3 times over an extended time is enough to memorize it for me.

I'm still a student so I don't know how important it is to be able to make proofs in your career and this will probably vary a lot depending on your department. (it is very important in the maths/physics department though)

Answer 11

Let me formulate a contra-answer to the one from Graham:

I am also a mathematician, and I have worked as a software engineer for several firms. One of them had created their own development environment, which generated a whole bunch of binaries and one textfile. All those automatically generated things together were the actual product.

After six years, I was assigned as a functional tester, and one of the things I did, was parsing, reading, grepping, ... the automatically generated textfile, and my colleagues were completely astonished of the amount of bugs I found while doing that.
But most of all, I was astonished by the fact that litteraly no-one (we were several hundreds of colleagues) had ever thaught of reading that file as a human being: people were so focused on the fact that that file was auto-generated that they did not even think of doing that. My insight went as a shock through the company!

Therefore I'd like to encourage you continuing what you are doing: try to increase the insight of your students. They may resent you for it, they might even claim hating you for it. But later (I was about 35 years old), there might come a moment where that insight comes in very handy and for the rest of their lives, they'll love your for it!

Answer 12

I agree with your student.

I have a PhD in physics and used math as a toolbox. A wonderful, useful, shiny toolbox. The kind of toolbox where you know that in order to find the roots of 3x^2-9x+2=0 you would find delta etc. I do not care how it was found, because this is a tool (I am not interested how a screwdriver is built either).

This is really, honestly not to diminish the value of your field. Someone had to invent these tools in the first place.

My son is going through 2nd degree polynomials as we speak. I had to suffer through the whole proof part and he knows that at the end of the day, it is three formulas he will blindly use.

One thing you could consider doing - and this is a very useful exercise: to set up a problem where blindly using the formula will make them fail, because the formula has an introduction part ("If x belongs to ..., then (formula)"). This will probably not make them study the proof (who knows) but will teach them a valuable lessons on limitations.

And someday, as they will be engineers, they may remember that story with limitations when building the bridge you will walk on. This happened to me (not with a bridge, fortunately) and helped me to always remember that models have contraints.

What I am trying to say is that the proof may not be the most valuable thing to teach them.

Answer 13

There are a couple of good reasons:

• firstly (at least when it comes to first-principles mathematical proofs), the skills involved in reinventing the wheel are very similar to the skills involved in inventing something new, so repeatedly reinventing the wheel is good practice for later inventing something new;
• secondly, undergraduate students are in training to join the elite of experts, who are likely to be the people called in to solve problems at speed, in disaster-recovery situations where infrastructure like the internet, electricity, running water, etc. has gone down, so they need to learn to solve problems without the internet available (this one is also the justification, in terms of authenticity, for using closed-book, timed exams in the summative assessment of undergraduates).

There's also an ultra-cynical reason: there are people who make a comfortable living out of reinventing the wheel (sometimes literally) and patenting it.

Answer 14

Maybe your student is seeing only the traditional aspect of the proof: a mean of verification, validation, conviction. If this is the case and the student believe the identities are true, there is no reason to bother to prove them. Then, you could try to explain and work with the other functions of the proof: explanation, systematisation, discovery, communication and intellectual challenge. See this paper by Michael de Villiers. Here are two quotes:

Who has not yet experienced frustration when confronted by students asking "why do we have to prove this?"

The question is, however, "what functions does proof have within mathematics itself which can potentially be utilized in the mathematics classroom to make proof a more meaningful activity?" The purpose of this section is to describe some important functions of proof, and briefly discuss some implications for the teaching of proof.

Answer 15

Because you need it to pass the exam.

Call my cynical, but that's the ultimate reason. Not wanting to reinvent the wheel is often very reasonable, e.g. if you were asked to prove Fermat's Last Theorem, you would be foolhardy to attempt to prove it yourself instead of searching up Andrew Wiles' proof.

The great benefit of attempting to prove the identities yourself is that you become familiar with the thought process, the basic equations, and so on. For example with trigonometry, one might see an expression involving sin^3(x) and think of converting that one of sin(3x). Without this insight, one might never make progress. Working on similar proofs before helps provide this signpost when it matters during the exam.

If the student is capable of learning the proofs enough that they can prove similar identities, under time pressure & in an exam setting, then there's nothing to worry about. Whether they get to this stage by working through someone else's proof, or by figuring it out themselves, isn't important.

Answer 16

I was that student

NOTE - Since I posted this there has been a similar answer by @Graham - However Graham's answer is so much better explained than mine that I have upvoted it and recommend it over mine.

Not as an undergraduate but before that in secondary education.

I resented doing chemistry and physics experiments because they were printed in books that we had to work through. As far as I was concerned, they weren't experiments because experiments are discovering something new, not repeating what had already been done. Instead I would do my own experiments at the back of the class.

I refused to learn a proof for Pythagoras' theorem that we were supposed to one year. I ended up having to prove it almost from scratch in an exam.

I rarely did maths homework because I had "understood" it when it was explained in the class and didn't feel the necessity to demonstrate the fact.

I did pass the exams in these subjects - barely.

By the time came that I was expected to apply for university I was so fed up with education that I wasn't interested. I ended up going to music college instead. It was only years later that I decided to pick up my studies again. I was working full-time and simultaneously did a part-time degree. By this time I was more mature. I worked hard, followed the rules and got a first and went on to do a Masters.

I really don't think anything could have changed my mind at the time. I was lazy and a rebel.

I suspect that this student is similar and has got by so far by surviving on natural ability without rigour. It may be that he will be more suited to researching one specific difficult problem and will make the effort at that time to put in all the necessary work. Think Andrew Wiles.

Suggestion

If like I was, this student is a lazy rebel (and somewhat immature), it will be difficult to motivate him to take the standard route. The best I can suggest is to find something that really motivates him and show how following the course will aid him in achieving that.

Answer 17

Since they are software engineering students, ask them if it is still relevant to understand different sorting algorithms. After all, there are libraries.

The answer, of course, is that one should be able to understand how one's tools work, at least when one has reason to look. I recall a bug where a programmer was unaware that quicksort was not stable.