I’ve spent a lot of my time in the past few days working around the house, constructing bookshelves (from store-bought Ikea-like kits), tearing down some interior walls in our basement, and whatnot. I found myself thinking about the physics courses I’ve taught in the past, and how some key points in the courses were perfectly illustrated by what I was experiencing with my own hands.

For example, some of the finishing nails used to secure baseboards are extremely thin, and to remove them is just like pulling weeds, which I discussed in a previous post. If you try to jerk them out, they just snap. One must pull gradually if one wishes to pull the nail out whole.

Large spiral nails come out with a creaky sound, and require a lot of effort. As a result they are a little warm upon removal … the work-energy theorem in action. (Sure, some of the work done in pulling the nail out is converted to the creaky sound and to the kinetic energy of the nail being pulled, but a lot of the work goes to thermal energy.)

Breaking the dry wall with my hands is just like pulling weeds, too. Pulling too sharply leads to a handful of dry wall breaking off, but pulling more gradually is more likely to break off a larger chunk, which is then easier to transport out to the garage. (By the way, dry wall is pretty hard, which makes this video clip of an NFL player pretty amazing, if it is legitimate. I played it frame by frame, and it looks real.)

One of the shelves has facings on the front edge of each shelf and the uprights, which I had to slide onto the rough edges. They were a tight fit, and I found that if I got the facing going, then I could slide it on relatively easily, but if I stopped it would be a real pain to get it started again. Yes, the coefficient of static friction is greater than the coefficient of kinetic friction, as anyone who has ever dragged a refrigerator across a floor knows.

My point is that what we learn in elementary physics classes is somewhat abstract, yet it is so apparent in our world. That is, it is *taught* abstractly, not that it is intrinsically abstract. And it appears to be abstract and somewhat meaningless to students, because they either have no experience of the concepts or they do not connect the concepts to their experiences.

For example, a fellow was about to back out of the driveway this afternoon, when I reminded him that his trunk lid was still up, blocking his rear window. He told me to “watch this,” then started to back out and then abruptly stopped, at which point the lid slammed shut. Newton’s first law in action. (And yes, I watched carefully to make sure that nobody was nearby as he was backing out.)

It is said that children who spend a lot of time moving about on jungle gyms (“monkey bars”) have an easier time visualizing in three dimensions when they grow into university students. More generally, students who have a rich experience of working with their hands have an easier time understanding basic physics, particularly if their teachers help them connect the concepts with their experiences.

It’s true in mathematics as well. One of the worst aspects of mathematics education is the definition-theorem-proof-example style that I suffered with so much as a student, and which is still very common, particularly in upper-level undergraduate and graduate courses. Much better for the student is lots of examples first, especially if they include concrete calculations, so that they have a repertoire of experience to draw upon. This sets the stage for abstraction, which is thereby much more meaningful.

This is how it has worked historically, time after time. Only after intensive work with concrete problems have mathematicians been able to invent the right definitions that allowed them to state and prove interesting theorems. This is highly creative work, and we do our students a terrible disservice to hide this natural sequence from them in the interest of “efficiency.” It might allow instructors to cover a lot of material efficiently, but it certainly does not lead to deep understanding very efficiently. On the contrary, the poor students are left to figure things out on their own, and good luck to them. Most students have no idea that they ought to seek out lots of concrete experience before tackling the lecture material, or they just don’t have the time because they can’t keep up with the pace.

There are a number of very nice books that are leading the way towards a much more humane way of teaching, but we still have a long way to go to bring the pedagogy of our upper-level courses in line with a natural way of learning. Elementary and high-school teachers are doing a much better job of this nowadays then when I was a student, and more power to them.

very well said

Thanks!

You’ve got a very nice site!

What if you’re studying pure mathematics and there are no real life applications to what you are learning? (e.g. As of right now I can’t think of any application of higher dimensional geometry other than the fact that it is very interesting to learn)

For physical concepts, “experience before instruction.” For abstract mathematical concepts, “examples before theorems.” I agree that for abstract concepts it’s not necessary (and often not possible) to make useful connections to real-world applications.

For instance, there is a theorem in first-year calculus that states that a differentiable function is continuous. Before even stating the theorem, students should have lots of examples of continuous functions, discontinuous functions, differentiable functions, and functions that are not differentiable. Fortunately, most students do have lots of such examples, as they have been studying functions in high school for a number of years, but it’s good pedagogy to remind them of these examples before stating and proving the theorem. Drawing lots of pictures before, during, and after stating and proving the theorem is also helpful for students in creating a “sense” for what the theorem means, and to connect the symbolic proof with the geometry of the situation.

Going more abstract, I learned the axioms of a group at an insanely early age (I think it was Grade 4 or 5, back in the days of the ill-advised “New Math”). There was a mnemonic in the textbook that I remember to this day: clani — cl for closure property, a for associative, n for neutral element (what is nowadays called the identity element), and i for the existence of an inverse for every element. I had the properties memorized back then, but I don’t think I had any idea whatsoever about what a group is.

It would have been so much better to examine many concrete examples of groups before abstracting a definition. We could have looked at “clock numbers” (it’s 5 o’clock and 8 hours pass, which makes it 1 o’clock, so in some sense 5 + 8 = 1) and then played with clocks that have different numbers of hours (24-hour clock, 7-hour clock, etc.). We could have looked at groups of geometrical transformations (without calling them groups, just looking at them as examples), such as rotations and reflections of various geometric figures.

And then, finally, after looking at many examples, we could have said, “Hey. Notice that there is something that connects all of these examples … a kind of essence. In all the examples you have some kind of “objects” (or maybe it’s better to call them actions or transformations, or to use all of these words together), such as numbers, transformations, etc. And you also have a way of combining the objects (“adding” the numbers, combining the transformations by doing one, then following up by doing another) that results in an effect that could have been caused by just taking a single object from the collection (this is the closure property). And every object has a matching object that “undoes” the action of the first object (this is the existence of inverses). And so on.

And in this way, I think students would get a better sense for how mathematics is done … how the process of abstraction and generalization works. I believe this would help average students understand better and with less pain, and would accelerate the progress of top students.

One could argue that students should be doing this for themselves, and I certainly always advise students in advanced courses to do this. That is, when faced with bewildering abstraction, find in the textbook (or construct for yourself if there are none in the textbook) several examples of each new concept, definition, etc. Then study the examples carefully and intensively. Then it will be easier to go back to the abstractions (i.e., the theorems) and have a better chance of understanding them and their proofs.

However, I have found that most students don’t do this, both because they have never been trained to do this and because they have no time. And this is the real point of my post: We need to build time into courses for this kind of work, and also train students to do this for themselves.

As I said, top students probably do this for themselves already. But most students (including me when I was an undergraduate) can benefit a lot from systematic training along these lines.

Nicely put!

Although in my advanced linear algebra class, the professor, unfortunately, has decided to not follow the course outline and textbook for the first half of the course and doing a few topics of his own, leaving us with only a few, but interesting in-class examples to work with. It’s about halfway through the course and it looks like we’re moving back to the outline and textbook, having only covered around 2 of the 11 topics. I assume that we’ll be covering multiple topics in one lecture in order to finish.

For people in this situation, I would visit the professor’s office hours as usually the professors are quite friendly, interested, and very open on what they teach. I find that whenever I go, I always find new insight on and different ways of thinking about what we covered in class.

Your advice is very wise, Stochastic Seeker!

Good luck with the advanced linear algebra course! It’s always a challenge when the prof teaches concepts that are not in your textbook, because then you must search for material on your own if you wish an alternative treatment of the topics. Depending on what the topics are, if you desire some sort of additional reading, you might look at Finite Dimensional Vector Spaces by Paul Halmos, or Linear Algebra Done Right, by Sheldon Axler, or one of the linear algebra books by Gilbert Strang.

Hi Santo

Some basic maths websites for you to check out…

http://moebiusnoodles.posterous.com/#!/

http://www.naturalmath.com/

Hi hakea,

Thanks for the links! The second one is very interesting, but I was unable to access the first one. Do I have to join something or register first?