Teaching Three-Dimensional Geometry, I

I have recently had a rather unusual opportunity.  I’ve talked a bit over the last few months about my consulting work producing online videos for a flipped classroom; I’ve been working busily on the Geometry unit.

Now the last section of this unit is on three-dimensional geometry, and I’ve been given pretty free reign as to what to cover in this 20-lectures series of 5-7 minute videos.  And given my interest in polyhedra (which I could focus on exclusively with no shortage of things to discuss!), I felt I had a good start.

But the challenge was also to cover some traditional topics (cones, cylinders, spheres, etc.) — as well as more advanced topics — while not using mathematics beyond what I’ve used in the first several sections of the Geometry unit.

There is, of course, no “correct” answer to this problem.  But I thought I’d share how I’d approach this series of lectures, since geometry is such a passion of mine — and I know it is for many readers as well.  The process of reforming high school geometry courses is now well underway; I hope to contribute to this discussion with today’s post.

Where to start?  Cones and cylinders — a very traditional beginning.  But I thought I’d start with surface areas.  Now for cylinders, this is pretty straightforward.  It’s not much more difficult for cones, but the approach is less obvious than for cylinders.

Earlier in the unit, we derived the formula for the area of a sector of a circle, so finding the lateral surface area of a cone is a nice opportunity to revisit this topic.  And of course, finding the lateral surface area of a cylinder involves just finding the area of a rectangle.

Now what do both of these problems have in common?  Their solution implies that cones and cylinders are flat.  In other words, we reduce what is apparently a three-dimensional problem (the surface area of a three-dimensional object) to a two-dimensional problem.

This is in sharp contrast to finding the surface area of a sphere — you can’t flatten out a sphere.  In fact, the entire science of cartography has evolved specifically in response to this inability.

So this is a nice chance to introduce a little differential geometry!  And no, I don’t really intend to go into differential geometry in any detail — but why not take just a minute in a lecture involving spheres to comment on why the formulas for the surface areas of cones and cylinders are fairly easy to derive, and why — at this level — we’re just given the formula for the surface area of a sphere.

I try to mention such ideas as frequently as I can — pointing out contrasts and connections which go beyond the usual presentation.  Sure, it may be lost on many or most students, but it just may provide that small spark for another.

I think such comments also get at the idea that mathematics is not a series of problems with answers at the back of the book…on the face of it, there is no apparent reason for a student to think that finding the surface area of a cone would be simpler than finding the surface area of a sphere.  This discussion gets them thinking.

Next, I’m planning to discuss Archimedes’ inscription of a sphere in a cylinder (which involves the relative volumes).  This is a bit more straightforward, and it’s a nice way to bring in a little history.

I also plan to look at inscribing a sphere in a right circular cone whose slant height is the same as the diameter of the base, so that we can look at a two-dimensional cross-section to solve the problem.  In particular, this revisits the topic of incircles of triangles in a natural way — I find it more difficult to motivate why you’d want to find an incircle when looking at a strictly two-dimensional problem.

Now on to calculus!  Yes, calculus.  One great mystery for students is the presence of “1/3” in so many volume formulas.  There is always the glib response — the “3” is for “three” dimensions, like the “2” in “1/2 bh” is for “two” dimensions.

When deriving these formulas using integration, this is actually exactly a fairly solid explanation.  But for high school students who have yet to take calculus?

It is easy to approximate the volume of a right circular cone by stacking thin circular disks on top of each other.  If we let the disks get thinner and take more and more of them, we find the volume of the cone as limit of these approximations.  All you need is the sum

\displaystyle\sum_{k=1}^n k^2=\dfrac{n(n+1)(2n+1)}6.

I plan to prove that

\displaystyle\sum_{k=1}^nk=\dfrac{n(n+1)}2,

and then prove (or perhaps just suggest — I’m not sure yet) the formula for the sum of squares.

I think a fairly informal approach could be successful here.  But I do think such discussions are necessary — in calculus, I’ve routinely asked students why certain formulas they remember are true, and they struggle.  As a simple example, students can rarely tell me why the hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg.

When teachers just give students formulas and ask them to plug numbers in to get answers to oversimplified word problems, of course there is a sense of mystery/confusion — where did these formulas come from?  I’m hoping that this discussion suggests that there is a lot more to mathematics than just a bunch of formulas to memorize.

As usual, I realize I have much more to say on this topic than I had originally supposed…I’ve only discussed up to the fifth lecture so far!  Since I have not had extensive experience teaching more traditional topics at the high school, it has been an interesting challenge to tackle the usual geometry topics in a way that grabs students’ attention.  It’s a challenge I enjoy, and of course I’ll have much more to say about it next week….

 

Guest Blogger: Scott Kim, IV

Well, this is the last installment of Scott Kim’s blog post on transforming mathematics education!  These are all important issues, and when you think about them all at once, they seem insurmountable.  It takes each of us working one at a time in our local communities, as well as groups of us working together in broader communities, to effect a change.  What is crucial is that we not only discuss these issues, but we do something about them.  Those of us who participated in the discussion a month ago at the Bay Area Mathematical Artists Seminar are definitely interested in both discussing and doing.

Scott suggests we need to move past our differences and find constructive ways to act.  No, this isn’t easy.  But we need to do this to solve any problem, not just those surrounding mathematics education.  It’s time for some of us to start working on these issues, and many others of us to continue working.  We can’t just sit and watch, passively, any more.  It’s time to act.  What are you waiting for?

Level 4. Resistance from SOCIETY (quarreling crew)

Sailing is a team sport. You can’t get where you want to go without a cooperative crew. Similarly, math education reform is a social issue. You can’t change how math is taught unless parents, teachers, administrators and policy makers are on board. Most adults cling to the way they were taught as if it were the only way to teach math, largely out of ignorance — they simply aren’t aware of other approaches.

Here are three ways society needs to change the way it thinks about math and math education in order for change to happen.

4a. Attitude. The United States has an attitude problem when it comes to math teachers. First, we underpay and under-respect teachers. And the situation is only getting worse as math graduates flock to lucrative high-tech jobs instead of the teaching profession. The book The Smartest Kids in the World and How They Got That Way describes how FInland turned their educational system around — they decided to pay teachers well, set high qualification standards, and give teachers considerable autonomy to teach however they think is best, with the remarkable result that student respect for teachers is extremely high.

Second, it is socially acceptable, even a badge of honor, to say that you were never good at math. You would never say the same thing about reading. Many people do not in fact read books, but no one would publicly brag that they were never good at reading. Our society supports the idea that parents should read to their kids at night, but perpetuates the idea that being no good at math is just fine.

Solution: respect teachers by paying them well, and value math literacy as much as we value reading literacy.

4b. Vision. The national conversation about math education in the United States is locked in a debate about whether we should teach the basics, or the concepts. As a result we see over the decades that the pendulum swings back and forth between No Child Left Behind and standardized testing on one extreme, and New Math and Common Core Math on the other extreme. As long as the pendulum keeps swinging, we will never settle on stable solution. The resolution, of course, is that we need both. In practice, schools that overemphasize rote math find that they must supplement with conceptual exercises, and schools that overemphasize conceptual understanding find that they must supplement with mechanical drill. We need both rote skills and conceptual understanding, just as kids learning to read need both the mechanical skills of grammar and vocabulary, and the conceptual skills of comprehension and argument construction.

Solution: We need a vision of math education that seamlessly integrates mechanical skills and conceptual understanding, in a way that works within the practical realities of teacher abilities and schoolday schedules. To form a vision, don’t just ask people what they want. A vision should go further than conventional wisdom. As Henry Ford is reported to have said (but probably didn’t), “If I had asked people what they wanted, they would have said faster horses.” Or as Steve Jobs did say, “It’s really hard to design products by focus groups. A lot of times, people don’t know what they want until you show it to them.”

4c. The will to act. As a child I grumbled about the educational system I found myself in. As a young adult I started attending math education conferences (regional meetings of the National Council of Teachers of Mathematics), and was astonished to find that all the thousands of teachers at the conference knew perfectly well what math education should look like — full of joyful constructive activities that challenged kids to play with ideas and think deeply. Yet they went back to their schools and largely continued business as usual. They knew what to do, but were unwilling or unable to act, except at a very small scale.

Solution: Yes, a journey of a thousand miles starts with a single step. And change is slow. But if we’re to get where we want to go, we need to think bigger. Assume that big long lasting change is possible, and in the long term, inevitable. As Margaret Mead said, “Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it’s the only thing that ever has.” I’m starting my small group. Others I know are starting theirs. What about you?