I’ve been thinking a lot about teaching math lately, as I prepare for my entrance into grad school. I’ve tutored a lot of calculus and a little bit of lower-level math, and in a year I’ll be teaching classes of my own. I’ve worked with a number of students, and had a few friends, who fall into the (all too common) afraid-of-math category, and on a broader level I worry about the state of mathematics education. So many students leave high school afraid of math, unable to do basic algebra or even simple addition without reaching for a calculator, and I feel that’s wrong. With the exception of those few people with severe learning disabilities of one kind or another, it seems to me that everyone should be able to learn math, at least as well as they learn reading and writing. I’d like to think everyone can get a kick out of discovering patterns, and solving tricky puzzles in an elegant way. Maybe that’s too idealistic, I don’t know. I do know that more experienced minds than me have studied these problems (John Allen Paulos comes to mind), but I want to do what I can to combat them as well. Part of this means being the best teacher I can, and helping instill a sense that math is interesting, and doable.

To that end I’d like to reminisce on some exceptional experiences, good and bad, that I’ve had learning and teaching math. I’ve been lucky enough to have some very good teachers throughout my education, and my parents took plenty of opportunities to introduce interesting concepts. I’ve also witnessed some singularly terrible math teachers. In the next few posts I’ll describe some of each.

**Logarithms on a Pizza Box:**

One of my favorite stories to tell is how my parents taught me logarithms on a pizza box. One night when I was in maybe sixth grade, we were eating pizza for dinner and I decided to share an interesting fact I had picked up somewhere: “Did you know that as the pizza cools, the temperature drops in half every ten minutes?” Leaving aside the fact that this is an oversimplification (it’s the difference between the temperature of the pizza and the air that decays in this way, and I’ve no idea if the 10 minute figure is correct), this little tidbit sparked a dinner-long discussion about logarithms and exponential decay. My parents got out a marker and drew axes for time and temperature on the pizza box, then plotted the temperature at ten-minute intervals. When the points were connected I could see that they didn’t form a straight line, but rather a curve that got closer and closer to 0 and never quite touched.

My parents then explained that when graphed on a special kind of paper, the curve *would* become a straight line. My dad got out a piece of log paper, with equally-spaced vertical lines and strangely spaced horizontal lines, where every group of ten was spread out at the bottom, but bunched up at the top of the interval. My parents explained that on a normal axis, each line represented one unit more than the one below. But on a logarithmic scale, each line (actually, each group of ten lines) represented *ten times* as much as the previous one. When we plotted our pizza temperature vs. time graph on the log paper, sure enough, we got a straight line! On the other hand, a graph that would have been a line on the original axes now became a curve that looked very similar to the original temperature graph.

My dad also pulled out some log-log paper, where both axes have a logarithmic scale. (I must be the only person of my generation who’s ever seen real log paper–who needs it when we can plot things so easily on computers these days?) I didn’t understand why you would need logs on both axes–it wouldn’t straighten out the temperature graph, and lines on normal axes would still be straight on these (my parents weren’t about to try to explain exponents and properties of logarithms in a mathematically rigorous way to me at age 11!) Nevertheless, I thought it was interesting, and the odd-looking distribution of the lines had a kind of nice flow to it.

Now I don’t expect every parent to teach their kids about advanced math concepts at the dinner table. Most adults have never learned even this much math, or learned it but forgot it as soon as the final exam was over. I was extremely lucky to have parents with both the capability and the interest in math to teach me these kinds of things just for fun. (It is fun–how wonderfully geeky is it to draw mathematical diagrams on a pizza box? I wish I’d saved it.) And I don’t expect every kid to be capable of understanding this while in elementary school; I was pretty exceptional. But mathematics doesn’t have to be hard, and it doesn’t have to be boring–I firmly believe every high school student is capable of understanding the concepts my parents taught me that night. More than that, I believe every student is capable of having fun playing with ideas, discovering patterns and trying to figure out why they work.

Anyway, this remains one of my favorite experiences with learning mathematics. What about you? What are your some of your best and worst stories about learning (or teaching) math?

on August 20, 2008 at 9:31 pmEfriqueNo parental experiences with mathematics at all, unfortunately – when I was seven, my mother told me never to ask another question relating to mathematics again. (At that point, my homework had hit adding fractions, and that was the end of that…)

I

wishI had some stories like that, to be honest.Maybe with my own kids, I’ll have a few good stories soon.

on August 20, 2008 at 10:43 pmJohn ArmstrongMy parents knew better than to try to teach me any mathematics. They just started giving me their old undergraduate textbooks to read.

on August 20, 2008 at 11:50 pmmillerI vaguely recall being taught several things by my dad throughout the years. I think those included triangle numbers, imaginary numbers, and the basic concepts of calculus. During my high school days, I was a big puzzle enthusiast, and I picked up a lot of math on my own, out of necessity. But nothing I could make into an interesting anecdote.

on August 22, 2008 at 4:35 pmAlexandre BorovikMy most formative experience was reading during summer vacations, aged 11, “Three Days in Karlikania” by Vladimir Levshin. A few months later, at school, I discovered that the images from the book were telling me that most natural way to deal with word problems in arithmetic was to compose a system of linear equations and solve it by formal manipulations. I was not supposed to do that and on my first attempt to submit my solution for marking I got “failed” mark for two reasons:

(a) it was too early for me to attempt algebraic equations, they had been on the syllabus yet;

(b) It did not matter that I got a correct answer since I solved the problem incorrectly anyway: I denoted the unknown by letter “a” instead of “x”.

Therefore, I had to conceal for another year that I could actually use basic algebra. Interestingly, I had a clear understanding that the reason (b) was utter rubbish. I did not get, however, any moral trauma because my mother was a school teacher of great experience and warned me in advance that I had to treat all teachers as human beings and forgive them their weaknesses. The list of human weaknesses which I had to be prepared to encounter in my teachers was very explicitly formulated to me by my mother and included, but was not restricted to, drink problems, lack of education and basic stupidity.

Levshin’s was a wonderful book; I read it as a fairy tale, and it planted into my brain seeds of algebra. Of course, it worked because I had seen on several occasions how my father solved, for my big brother, maths problems by composing and solving systems of linear equations, and did that with obvious enjoyment. When reading Levshin’s fantasy, I knew that the book was explaining to me what my father did.

on August 22, 2008 at 4:46 pmAlexandre BorovikAnd a request: Susan, may I quote your story in the texts that I am currently writing?

on August 22, 2008 at 7:59 pmmusesusanYikes, that’s horrible. I was lucky enough never to have a teacher who treated problems as though there was only one right way to do them, but I’ve known such teachers (and my next post will be on one of them). There are times when it is appropriate for a teacher to insist on a particular method being used to solve a problem, but only when the teacher needs to ensure that the student learns that method. In your case it should have been clear that you had not only already learned the desired method but knew more advanced and elegant methods for solving the problem. When a student solves a problem in a new way that is still mathematically correct, he or she should be commended, not criticized!

Please feel free to quote any portion or all of the story, just as long as I get appropriate credit, linkage if it’s online, etc. And tell me when and where you do–I have enough of an ego that I’m delighted to be quoted!

on August 23, 2008 at 8:53 amjd2718Nice story. Your parents knew logs? I didn’t have adults around me who were likely to teach me much math

Once when I was about 7, a friend of my step-father’s taught me to write numbers in other bases, and how we had 1, 10, 100, 1000 no matter which base. It was on graph paper, which I think I held onto for a while.

I wish I could remember which bases he showed me. But it did stick – I never relearned that bit from scratch. Hm. In fact, I don’t know that I was ever taught anything about different bases. I played with them on my own.

Jonathan

on August 23, 2008 at 10:02 ammusesusanMy mom was a math major in college although she went into business, and my dad in an engineer and inventor. So they both knew a good amount of math, even if it was things they had mostly forgotten. It’s still a little startling for me to realize that I finally know more math than them.

on August 23, 2008 at 11:01 amJohn ArmstrongIt gets weirder the more they know. Then you realize that you know more in your specialty than they in theirs.

The real question is how they deal with it.

on August 23, 2008 at 3:47 pmKathy (Susan's Mom)We deal with it by living vicariously through her.

on August 24, 2008 at 2:31 pmthoughtcounts AMy parents didn’t know enough math to teach me very much, but they were always very encouraging and got me books and things that helped me learn on my own. My favorite story is me going up to my mom randomly one day asking whether 1/2 + 1/3 was 5/6. (I hadn’t been taught anything about common denominators.) An enthusiastic positive response to things like this does a lot to encourage young children.

Alexandre’s story sounds very familiar. In fourth grade I had a teacher who put the problem “.5, .10, .15, _?__” on a test, asking for the next number in the pattern. I assumed it was a typo and went up to ask if the first number should be .05. I was told no, it was right as it was, and incredulously asked how I could possibly not see a pattern. My mom took the test to the principal and got me switched to another class. There are a ton of incompetent math teachers out there. You really need to fight against bad ones, and schools need to have higher standards (and offer higher pay in subjects where there are shortages).

on August 25, 2008 at 5:00 amBlack Mask of Al-Jabr « Mathematics under the Microscope[…] August 25, 2008 by Alexandre Borovik This is my expanded and corrected comment to Susan Beckhardt’s post Musings on Math Education, part 1: Logarithms on a Pizza Box. […]

on August 26, 2008 at 5:20 amMartinBI have used log paper. Although computers do make things easier.

I really wish that my maths teachers had been more interesting at school.

on September 1, 2008 at 1:27 amStarting From An Early Age « The Twenty Eighth Line[…] From An Early Age There has been much chatter recently about early life experiencs of mathematics and the effect it has on subsequent […]

on September 26, 2008 at 11:48 pmMusings on Math Education, part 2: “Math is not about being creative!” « Intrinsically Knotted[…] 26, 2008 by musesusan Part 1 was one of my best experiences with learning math; this is one of my […]

on October 8, 2008 at 8:22 pmGoogleMasterJust found your blog and this fascinating first post of a series.

My parents (math teacher, chemical engineer) had lots of interesting books around when I was young. There was the algebra text that my mom had used to teach high school students, as well as many other math/science/engineering books.

One day my mom came home from a church rummage sale with an 1890-something correspondence school textbook on mathematics. I eagerly dug into the section on calculating square, cube, and higher roots with pencil and paper. The book works through examples of 2nd, 3rd, 4th, and 5th roots to show the pattern of solution and leaves you with the tantalizing notion that “higher roots can be solved in a similar fashion”.

Not only have I used log paper but I have also worked with Smith charts — and multiple colored pencils.

on October 8, 2008 at 10:58 pmmusesusanThe old textbooks are great. There’s a lot of techniques that we just don’t learn in school these days because we’ve become so used to calculators. (I just tutored a guy today who reached for his calculator to figure out 13 x 100.)

Once I borrowed my mom’s old calculus textbook, and I noticed in one of the appendices a in introduction to algebra and group theory. I was amazed at the idea that you could define 0 so generally and then

proveall the familiar properties from those few axioms.Smith charts look pretty interesting! I’ve never seen them before, but I can see how they’d be useful for working in the complex plane.

on October 20, 2008 at 9:22 pmCrux AustralisI bought a refrigerator from a department store a few years back. The deposit was 10%, and the store clerk had to use a calculator to figure it out. Three times. I just about had a fit.

on October 21, 2008 at 12:28 ammusesusanOoh, that’s scary. Although I had a calculus student pull out her calculator for 4×3 the other day.

Actually, that reminds me of a time a while back when I was buying something for exactly $100 and the cashier was supposed to give me 20% off. Unfortunately, the cash register only had a 10% off button, so he pushed that button twice and saw the new total of $81 instead of $80. He commented that for some reason, this always happened, and there must be something wrong with the register, but I assured him that no, the register was just taking 10% off the current total each time. He was completely unable to fathom why the second 10% off should decrease the total by only $9.