After earning an applied math degree and teaching math for years, Darren realized he doesn’t really understand math as well as he should.

In college, he “could calculate my butt off, but so often didn’t fully understand what I was doing.”

As an example, in differential equations I could calculate eigenvalues all day long, but to this day I don’t know what an eigenvalue is or what it does for me or why I need to calculate it. I’ve taught myself plenty –sometimes just days before I had to teach it to my students.

He told a student bound for Cal Poly “not to make the mistake I made; ask the questions, go for the deeper understanding.” And Darren decided to go for it himself.

I’ve never understood the Fundamental Theorem of Calculus. Why, exactly, are an integral and an antiderivative the same thing? I’ve followed the steps in my calculus books, and understood each step, but never really understood how they all fit together. So today I pulled a different calculus book out of my closet and I started studying. I found one that provided a very user-friendly explanation, which then allowed me to understand the very rigorous (read: dry and difficult) proof in a second text. It took a few minutes to replace decades of deficit.

In the fall, Darren will start a masters program in Teaching Math through the University of Idaho’s Engineering Outreach Program.

Most of us never have the courage to face our eigenvalues, much less blog about it.

“Facing our eigenvalues..” Love it.

Perhaps I have misplaced my faith in other members of my profession, but don’t all good teachers “face their eigenvalues” whenever they get a chance? I realize that in elementary and middle school it’s more about remembering content details than understanding sophisticated proofs, but many of the people I work with spend a lot of their own free time brushing up on dusty knowledge or learning something new. It’s just not interesting not to be growing your own knowledge while you teach.

Or maybe I’m hanging out with mutant teachers…

I think the key to your statement is “all

goodteachers.” I have seen far too many teachers who shrug off their deficiencies, especially in math. Many simply say “I just don’t get math,” and muddle through, not attempting to face their problem.In several cases, I have actually heard teachers laugh at things they don’t understand, as though they are proud to not be one of those ‘super geniuses’ who understands ‘complicated math concepts’. Mind you, the arcane concept I was speaking about at the time was the fact that being able to understand operations with exponents and variables requires understanding the multiplication represents repeated addition and exponents represent repeated multiplication.

Most of the teachers I have seen and worked with are constantly improving themselves. Far too often, however, have I encountered those who aren’t. Others have to try to pick up the slack, often years later.

It helps to read a good history of mathematics to see how these concepts arose and what physical or social situations inspired them. I was lucky as an undergraduate to have taken a course from Morris Kline, and I recommend his books. There are good modern histories; try T.W. Körner’s The Pleasures of Counting.

I second your recommendations. I also suggest:

Edna Kramer’s The Nature and Growth of Modern Mathematics

Lillan Lieber’s books on Mathematics

I wish I could remember the name now, but not too long ago I read of a physics professor who said that through 4 courses beginning in high school, he had never really understood Electricity and Magnetism. It didn’t click until he had gotten his PhD and was teaching it as a tenure track assistant professor.

I suspect there are lots of physics students who can “do the math” but don’t really understand why it works.

Which raises a difficult question for me. This guy was obviously very successful. Should physics teachers train people to do the math and figure that eventually there will be “enough” understanding to do engineering and design? But then what of the vast majority of people who don’t go on to do work in physics? Should we concentrate on understanding? And what if most students don’t want to understand? What if instead they want you to give them some recipes that they can apply mechanically, get a passing grade in the course, and then forget?

There are many acceptable scenarios in the “do versus understand” arena. Some people resist learning anything they can’t understand. To me, that is crippling because there are many things that we come to understand by doing them, repeatedly as the example above shows. Others are perfectly happy to do without understanding, and I would argue that there are certain things we must do whether we understand them or not. Like balancing a check book or checking the air pressure in our tires. It is true that understanding why something works (in physics, grammar, or any other complicated field of knowledge) enhances the ability to go further and to create new knowledge; but it certainly isn’t true that we have to understand everything we learn to use.

But how to pick which “do versus understand” scenario is appropriate?

Partly, I’m worried about something deeper. For most students in most courses, it’s do, don’t try to understand, then forget. Is this really the best use of their young years? Should we be encouraging them to live so much of their youth that way?

Yes, they will forget much…but most will remember one content area at least, and that is a great way to start a career.

Yes, they will forget much…but most will remember one content area at least…Someone who becomes a teacher will certainly remember that subject. Someone who becomes a political junkie will remember her American Government course. Someone who becomes an industrial chemist will remember Intro to Chem. I’m sure there are a few others.

Beyond that, I’m not at all sure. Everyone will remember smidgeons here and there but that’s not the same thing.

But the point is that at age 18, few know they’re going to be a chemist or teacher…so we’re giving them options for later on.

I think teachers are in somewhat of a unique situation here. An engineer who needs to calculate eigenvectors inherently knows why the eigenvectors are useful. A teacher is often in the position of teaching something they don’t use themselves, which is harder, I think.

Most of us eventually forget the math we don’t use, but if you’ve learned it once, it’s usually easy to go back and pick it up again. I was in this position recently (coincidently involving eigenvectors and some other linear algebra) and I found Khan Academy extremely helpful.

Joanne, thanks for the link! And everyone else, thanks for the suggestions and commentary.

Four out of three students don’t understand fractions.

There are three kinds of people: those who can’t count and those who can.

I don’t understand why my wife cries when she is happy…but, I know how to make her happy.

The list goes on…why bother with understanding, if, what you do, works?