Research mathematician and educator Edward Burger describes himself as "an expert in failure." Moreover, he'd like for more math students to feel the same way -- and to discover how learning from mistakes can unlock their creative abilities.
Why is it that only one or two percent of the population actually ends up learning calculus?
Because when we teach mathematics to students, we teach in a vacuum. They see it as unimportant, uninteresting and uninspiring. Therefore, they treat it as valueless. These word problems — "the train leaves Chicago at 12:00 and another one leaves Los Angeles at noon, and blah blah blah" — no one cares.
How can we possibly take a young mind, that has seen all sorts of interesting things in the world, watching MTV and all sorts of neat movies and reading Harry Potter and having a great time in life, how can we expect anything else except for them to say, "Mathematics is dull, uninspiring, completely devoid of real application in my life and valueless and, therefore, why bother to take calculus?"
What’s the alternative?
That’s a great question. I don’t think we should be telling our students anything that they wouldn’t be naturally ask themselves. One good approach is posing a sequence of intriguing questions to students that they want to know the answers to.
You don’t say, "This is what calculus is." You say, "Here’s a question: suppose that I get on a bicycle and I ride for 30 miles and I do the whole 30 mile ride in an hour and a half. But at the 20 mile mark, there was a road sign that says, SPEED LIMIT 20 MPH. Did I break the law?"
Did I break the law? Well, what does that mean? What I am asking is, how fast am I going at the very instant I crossed that road sign?! That’s what calculus answers. It allows us to discover instantaneous velocity. Velocity at any one moment in time.
Now you could pose a very intriguing question which people would be interested in thinking about and then say, "Gee, that is an interesting question, how can we find out? We can find average velocity, change in distance over change in time." But how do you find instantaneous velocity, when there is no change in time? The denominator would be zero; you can’t have zero as the denominator.
Welcome to calculus. It relates to the real world in every single way.
It seems like you are advocating a non-theoretical approach to the subject.
Mathematics is a very practical subject. Even if you are not going to be a scientist or engineer, mathematics has so much to offer. It can touch and enrich your life and allow you to look and yourself and the world in a clear, more focused and more meaningful, deeper way. It can help people look at their lives, and the problems within their lives, both professionally and personally, and resolve those problems in a more powerful and more effective way.
Mathematics is basically an approach to solving difficult problems. The techniques and strategies that we can learn in mathematics are techniques and strategies we can use to solve any difficult problem.
I’ll give you some examples from "The Heart of Mathematics," the textbook I have co-authored. These are our favorite top ten mathematical ways of thinking.
1. Just do it.
Sitting around and talking about it is one thing, but getting your hands dirty, reaching in and trying something is the way to proceed.
2. Make mistakes and fail but never give up.
We tend to believe that making mistakes defines failure and weakness, and that being wrong and failing are bad. But any great innovation or great new idea, whether it’s a medical breakthrough, a sociological discovery or a scientific realization, occurred through a sequence or succession of failed attempts.
The mathematician is an expert in failure. I am a research mathematician; I do research in number theory. I fail 99% of the time, but the 1% of the time that I have success is what leads to a scholarly work. It’s through the succession of failures that I work things out. I gain insights.
The power of mathematics is to learn from failure. When we don’t teach the value and the power of failure to our children, their creativity and their imagination are, in some senses, stifled.
Every failed attempt is a new discovery. How can we possibly ask people in our society to go out and do great things, to make new discoveries, to be creative and artistic in all sorts of different realms, if we don’t teach them the value and the power of failure as a means to do that?
3. Keep an open mind.
We’re constantly biased by the initial reactions and ill-conceived notions that we carry with us. We have to keep an open mind to everything.
4. Explore the consequences of new ideas.
Once we develop a new idea, instead of moving on, explore its nuances and its unintended and hidden consequences.
5. Seek the essential.
We have so much information at our disposal, and most of it is not essential. When we are trying to resolve a problem it is extremely difficult, but extremely important, to distill it down to its essence and throw everything else away. That will allow us to see a clear solution.
6. Understand the issue.
So often a student will come into my office and say, "Professor Burger, I can’t do number 7." And I say, "Okay, what is number 7 asking?" The student will say, "No, no, I don’t know how to do it," and I will say, "I understand that, but what is the question asking?"
Most of the time, the student won’t understand the question. How could you possibly answer a question that you don’t understand? It seems so obvious, but you run into it all the time!
I’ll let you in on a secret: The moment we understand the question, we’ve already answered it. So understanding the issue is critical.
7. Understand simple things deeply.
Quite often people think that the world is complicated and chaotic. That is just wrong. The reality is that there are some very simple basic ideas, some simple basic and principles — and if we understand those simple things very, very deeply, we will have a better understanding of our world and of our nature.
It’s one thing to understand things peripherally or tangentially, but we need to demand more of ourselves and develop a rich, deeper understanding that enables us to see the nuances that exist in simple things.
8. Break a difficult problem into easier ones.
The idea of dividing and conquering is one of the most powerful techniques that can be used in anything we do. Mathematics demonstrates that beautifully in so many different areas.
9. Examine issues from several points of view.
The idea of not just looking at the thing with tunnel vision, but asking, "What if I look at it this or that way? Even though it seems so crazy, what if I do this?" Quite often, that’s where innovation lies, new ideas and things people never thought of. These come when someone looks at something in a different way.
10. Look for patterns and similarities.
The idea of looking for a pattern is such a potent way of discovering some underlying feature or some underlying fact that it’s amazing.
That’s quite a list. But beyond becoming a better problem-solver in life, what can you do professionally with a degree in mathematics?
There are a lot of things that you can do. Some of them include becoming a mathematician in industry — for example, at Lucent Technologies or the Jet Propulsion Lab, the National Security Agency, places like that.
Or to do anything else, Law School, for example. Having an undergraduate degree in mathematics is the best preparation for law school, for medical school. Why? Because in mathematics we train people how to think, and that is what is required in law and medicine. Architecture, statistics, banking, financial things, Wall Street — I can’t imagine what mathematics doesn’t train you for.
You’re quite welcome.
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