When Algebra Was Fun

I have heard the claim that most mathematicians either liked algebra or liked geometry in their early training. Part of my personal mathematical mythology is that I didn't much care for mathematics until I got to geometry.  (Partly, or perhaps largely, because of a really awesome geometry teacher.)  To this day, I have to admit that I mostly find (high school) algebra somewhat tedious, although obviously useful and necessary.  (What is sometimes called abstract algebra, starting with groups, rings, and fields, is obviously a totally different creature.)  I can appreciate some of the patterns and such in high school algebra, although of course at this point finding this type of algebra interesting is a little like finding the alphabet interesting: I'm much more interested in using it as a tool to do other things with.  But when I look back on my algebra classes in junior high and high school, I found those classes sort of boring, although not as bad as the arithmetic classes which preceded them.  I guess I just didn't find algebra that interesting.

Or did I?

I was recently talking about an "enrichment" program I participated in for one year in fifth grade, in which a small group of students from across the school district were gathered and bussed off to a special class one day a week.  On program days, we got to do all kinds of great things, like reading and discussing cool books, engaging in research projects, doing experiments, and working on a computer. It was actually pretty awesome. There were two real problems with the program, 'though.  The biggest problem was that the fantastic educational opportunities we got in this alternative class were really what everybody probably should have been doing all the time, instead of a special one-day-a-week pull-out activity for whosoever was judged to be the "best and brightest."  The second problem was that the program was a set of additional pull-out activities, because the students in the program had to make up all the work we missed in our regular classes.  (This, by the way, is why I only participated in fifth grade: I didn't do so well with keeping up with the other stuff, which was frankly mind-numbingly boring.)  So in the enrichment program, we'd research and report on ancient Egyptian burial practices, then come back and have to read a passage out of the social studies text book to fill in the blanks on a mimeographed worksheet.  Or we'd collect cell samples from our mouths and examine them under microscopes to learn about cells, and then come back to have to copy a diagram of the human nervous system out of the textbook.  (Interestingly enough, I remember that picture because I remember coming to the conclusion that we must be less sensitive in our forearms than our upper arms, because the diagram clearly showed more nerves in the upper arms.  This was not a misconception that I ever got to discuss in class.)  Or we'd go learn how to solve problems using algebra, only to come back to "Do the following 25 fraction addition problems."

Wait, what was the last one?  I'd forgotten about that! We actually learned some algebra in the program.  I don't remember all the details, but I think we had a worksheet, and I remember the idea of introducing a variable for an unknown quantity, setting up an equation to represent a problem, and how you could go about finding out what the x (or whatever) represented.  The problems were puzzles, and they were wonderful.  Some were quite difficult; I'm not sure we solved all of the problems.  I remember being fascinated by the very idea of working in some sense "backwards" to figure out an unknown quantity.  It was an exciting adventure for us to figure out, a marvelous mystery.  We were figuring stuff out, guided (loosely) by the teacher, who introduced just enough hints for us to make it through each new challenge. Each new idea and discovery was shared and traded with great relish.

I remember wanting to learn more about algebra and thinking it was wonderful.  Until of course I had some problems with finishing up the necessary arithmetic by hand, which led to various adults tut-tutting to me about how I obviously should have been doing more arithmetic drills. That was pretty much the end of my interest in algebra  since it was clear to me that expressing interest in algebra would lead to being punished with more arithmetic drills first.  So instead of picking up some of the arithmetic incidentally as I studied more interesting stuff, I just ground my way through the required math classes as best I could, hoping they would be over with soon, and forgot about algebra.

I finally took a regular algebra class in the eighth grade, but I'm not quite sure if I remembered how much I had liked it once.  But my eighth grade algebra class was a bit of a nightmare, taught by a man who was best known for yelling at the students and picking his nose.  (I suspect the latter would have been more tolerated and ignored were it not for the former.)  It was definitely not an adventure, and the problems were definitely not puzzles.  There were just a bunch of rules, and an algorithm of some sort for solving every sort of problem.  Every day was a new type of problem, and mostly an expectation to memorize an algorithm for solving it.  There was no "figuring" anything out, and the techniques were no longer mysteries to be discovered, but miseries to be endured.  Hell, I barely passed that class.

But as I think back on it, I realize that my interest was not completely crushed, even if I didn't realize it at the time.  I remember at one point during a summer vacation suddenly thinking about graphs, and wondering what feature in an equation made a graph "straight" versus "wavy."  I actually developed a hypothesis (by experimenting) that equations in x and y which didn't have any powers except for "1" were the only straight lines, and other powers gave bent curves.  (I have no idea whether I had already been told this before or not, but if so it hadn't stuck until I noticed it myself.)  And it's also clear that I must have had some interest left in math, because seriously, what high school student spends part of his summer vacation plotting multiple graphs by hand to test out a hypothesis about which graphs will be shaped which way?

So my personal mythology is wrong.  I did once love algebra  almost if not as much as I later loved geometry.  And I wonder:  What if my early interest in algebra had been allowed and encouraged, even if I was yet unsteady at arithmetic?  What if my first formal algebra teacher had been the same teacher who later taught my geometry class in high school, who encouraged my exploration and experimentation?  In retrospect, what I relished so much about the geometry class was that the problems were once again puzzles: No algorithms, no sequence of steps to memorize, just a statement starting "Prove that...," and it was up to us to figure out some way of getting from Point A to Point B.

In fact this spirit of investigation, of figuring things out, is at the heart of my favorite movement in mathematics education, known as Inquiry Based Learning, or IBL.  In IBL, students are set problems of some sort to solve, something to figure out.  The steps are small enough for the students to figure out on their own, and they are led along a path of discovery.  That's what happened back in the fifth-grade enrichment program:  We were introduced to the idea of using a variable, or of "doing the same thing to both sides of an equation", and asked to figure out how to solve the next problem using what we knew.  We figured the stuff out "on our own" (in actuality with plenty of guidance), and we were excited to be doing it.  When I got to the eighth grade class, what I got instead was "Day 23: How to Solve a Digit Problem.  Step 1: Let t be the tens digit in the unknown number...."

Now I have a bit of a dilemma:  I now remember what joy in algebra felt like, but can I bring that to my students?  In particular, I've recently been teaching a remedial algebra class.  It's required for many students who have poor math placement scores on entering the university, and it covers a great deal of material in fairly short order to make sure the students have all the needed algebraic skills for their next mathematics class.  Because of this, it is very algorithmic, using a very step-by-step, one-topic-at-a-time approach--the very approach I was bored to tears with. Can I bring any of the joy of algebra to my students?  I can imagine running an algebra class in the spirit of that first encounter I had, following an IBL approach, but I also think it would require more time than the one semester I would generally have.  (Now in high school, algebra is usually spread over two years, which I think would be ample time for a careful, and ultimately quite rigorous and thorough IBL algebra course.)

I'm sure that if our remedial students had a more inquiry oriented algebra class, they would be more likely to find some enjoyment in the mathematics (as I once did), and they would probably grasp some of the basics more fully.  I wonder what the longer term effects of such a remedial program would be.  Would the students with a stronger basic foundation in algebra and an interest in the material do fine in a later class without learning all the needed techniques, more or less filling-in material as they went?  Or would they end up struggling and failing to keep up because they did not know the assumed prerequisite?  Maybe I need to think about this question.

Ponderings on Geometry

At a conference this summer I saw an interesting invited talk about teaching geometry. The speaker discussed some of the historical challenges to Euclidean geometry, and discussed how these were eventually overcome. The problem for schools being that while mathematicians like Hilbert managed to fill in all the gaps left by Euclid, the resulting geometry was very complex. It took a great deal of work to get from the axioms up to the interesting Euclidean results. In fact it took too long for a single year high school course to even get to the beginning of Euclidean geometry. As a result, the speaker claimed, most high schools eliminated axiomatic and proof-based geometry from the curriculum, replacing it with practical, computational geometry only. (He went on to detail a new approach, which allows a more rigorous approach to Euclid's geometry in a high school without starting from scratch, and using a book which is designed to be used in an inquiry based class. It looks fascinating, and I'm hoping to teach a class based on this book soon, but that's a digression.)

What caught me off guard was his statement that rigorous axiomatic approaches to geometry using proofs had been all but eliminated from most public high schools due to difficulty dealing with the challenges to Euclid. I had to wonder, was this actually true? I certainly had a proof based and axiomatic geometry in high school, and that was... ok, a little over twenty years ago. (Geez I feel old now.) And while my high school was mostly fine, it wasn't really an elite school. (When I taught an analysis class, I actually used several problems phrased in the form: "My high school calculus teacher said ___. Prove rigorously that he was wrong.") So had schools actually eliminated proofs from geometry since I had taken it? Or had my school been an odd stand out for not eliminating it?

Tonight I was about to start a geometry unit with a class. I asked the students for a show of hands of how many people remembered doing ruler and compass constructions. Less than a third--and maybe less than a quarter--raised their hands. Now I know from experience that some of the students may very well have seen constructions and just have forgotten everything five minutes later. But still, it seems that a number actually didn't do constructions. And I'm inclined to think that if they didn't do any constructions, they probably didn't do any proofs, although I could be wrong on this. (And of course some may have done constructions but not proofs.) So maybe the summer speaker was right, and axiomatic geometry is mostly gone.

But I question his reasoning about why the proofs are gone. He believed it to be because it proved too difficult to fill the gaps found in Euclid. I can't imagine boards of education, state legislatures and such largely care--or even know--about that. My first suspicion was that someone, somewhere decided it was "too hard", or that students just couldn't do it, and replaced it with other things.

But why too hard? Abstract reasoning and logic is hard, but given time and effort, most people can make progress. Ah, but there's the problem: Time. To do a good job with a class about proofs, it takes time spent working on problems, trying ideas, failing, and trying again. Learning to reason is a long and difficult process. And curriculum tends to fill up with all sorts of nonsense as everyone and their dog proposes new lists of things that "everybody" ought to know. Most of those things are lists of facts and formulas. I can almost hear the litany start for a geometry class: "Students must be able to give the formula for the area of a circle, a semicircle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus; Students must be able to find the perimeter of a circle, a rectangle, a square, a triangle, a trapezoid, a parallelogram, a rhombus..." (And it's worth noting that the only really interesting things in the list I just gave are probably the area of the circle and the rectangle, and perimeter of a circle. The rest should be easily derivable from some good geometric thinking. But the students will instead be given a list of formulas to memorize.)

We add and add to curriculum, and nothing is ever taken out. We add numerical approximations and work with computers. We add calculators, then graphing calculators, then geometry software. We add three dimensional shapes, and trigonometric functions, and whatever else anyone can think of because ... well, because someone else happened to remember it and thought it would show how rigorous we were being if the list of things for students to know was really long. (It should be noted that I am not opposed to any of these topics in high school math classes. But we must recognize that we cannot do every possible topic all at once.)

But a long list of topics is not rigor. That's just memorizing a bunch of stuff. Take any of those topics and spend some time with the students doing a long and careful analysis of some challenging problems, and you'd have a recipe for a great math class. The central questions will always be the following: What do you think is true? How do you know that? Why? Is this like anything we have done before? Can we generalize this result?

All of this of course misses the biggest elephant in the room as to why high school math has dropped most reasoning and replaced it with lists of tasks and formulas: It's very easy to write a statewide multiple-choice test which to see if students can choose the formula for the area of a circle. It's very difficult (or perhaps impossible) to write a state-wide multiple choice test which determines how skilled students are at reasoning and solving complex problems. But I'd much prefer a student who can work out how to find the area of a trapezoid based on what she already knows than one who has only memorized the formula. I regularly have students who have memorized a formula corresponding to a figure, but can't figure out if it's the formula for the area or the perimeter. (When I ask how to find the area of a circle, I can guarantee about half the class will respond "2 pi r.")

And I'd really love to have students who learned enough reasoning to write proofs in their geometry class, but apparently that's rare now. No wonder we have trouble teaching proofs later in college.

The semester so far

Today began the second week of a new semester.

I have two particularly small sections of one class, in part, I think, because I'm the only person teaching the second semester of the course who was not teaching the first semester in the fall. I love having small classes, so I'm not complaining. One of my students who didn't show up on the first day claimed she had switched to Professor D's section, but was still showing up in my roster. I double checked with Prof D, since today was the last day of add/drop. It turns out that Prof D had signed an override to allow her to enter his already overfull section. He now has 33 students. I went from 16 to 15. I'm laughing. I'm not sure Professor D was when he found out what had happened.

On the other hand, my Gen Ed class (aka, "So you think you can math?") is not small at all, and has stayed firmly at the enrollment limit of 40. Students have some incentive to pass the class this semester, since in the fall both the difficulty of the course and the prerequisites will increase. I'm being very straightforward (and maybe just a little easier than usual) to give them the best shot of getting through before the new course requirements kick in. We have just finished the second class (for a total of 2.5 hours of classroom time) doing nothing but unit conversions. Some students are completely lost.

I had a student show up in my office today wanting to do an independent study this semester. We discussed some possibilities, but I said we should talk to the chair to find out if it was possible such a study could be approved this late. The chair responded by putting her head in her hands and making a sound like a expiring mongoose. Since the student still needs to be in the chair's good graces, he wisely rescinded his request for an independent study.

But my joy this semester will be teaching the "Intro to Proofs" class for majors. I'm teaching the course using a technique commonly known as a modified "Moore method", which I've been interested in for some time. I went to a workshop for new practitioners two summers ago, and last semester I applied for and received a mentor so I could start trying it. It's the ultimate in student centered instruction, where most (or sometimes all) class time is spent with students presenting proofs to the class and the class dissecting the proofs as needed. I was prepared for all sorts of disasters to occur, but to my astonishment, my first two classes have gone extraordinarily well. My current major concern is to make sure all of the students stay involved in the class, rather than just a subset. But so far the discussions and presentations in class have been wonderful, and in fact beyond what I had hoped for. I still expect plenty of challenges ahead, but at least I feel like I've been well prepared for them, so I'm cautiously hopeful that we may have a really good semester.

Oh, and the unseasonably warm weather we have been enjoying is drawing to a close. Looks like we'll be down below freezing for a while now. Drat.

What students hear

I sometimes think about what students hear when we (their teachers) talk. I don't think undergraduate students (even good ones) manage to grasp the essentials of mathematics. I don't think they see what we are doing as logical reasoning based around some fundamental principles. They hear the specific words and explanations for a particular problem, but don't grasp that what we say comes from some coherent system. This is not necessarily because we don't say that this is so. It just seems that this concept doesn't completely register in the time we have them. It might later, or it might not.

When, for example, we want to find all solutions to x² = x, I might suggest dividing both sides by x, getting x = 1, which is one solution. I then note that we can only divide by x if we assume x is not zero, and in fact x = 0 is the other solution. We then have all solutions to the equation. To me, at this point in my life, I see division of both sides of an equation as a legal operation with very specific restrictions (namely, that we cannot divide by zero). I also recognize that since we can view both sides originally as multiplied by x, that x = 0 is certainly a solution. But I remember when things were not quite so coherent, and then the step back to note that we cannot divide by zero seemed like a trick just to justify a zero solution. It seemed that a lot of algebra (and some other mathematics) was made up of a bunch of special rules and exceptions, and it seemed like teachers had a never ending supply of these to pull out to justify whatever they said the answer was. It seemed a bit like playing pretend with a small child, where there is an exception to everything, to be made up on the spot: "Oh yeah? Well I had my invisible anti-force field magic belt on, so I could escape from your force field!"

How do people learn to see mathematics (or any field) as a unified whole? Perhaps part of it comes from time and experience. It may just take a certain amount of time working with the concepts before they become solidified and can be deftly manipulated. It may be similar to the feeling I can still remember in college, when I became sufficiently comfortable with algebraic manipulations that I could use them to do faster mental arithmetic, by disassembling and reassembling the numbers in convenient ways, making the numbers dance as needed. It's not that no one had ever suggested the idea before; it's popular to present these ideas to students, but until those concepts are internalized, it doesn't make mental arithmetic any easier. It just seems like a trick--and a somewhat painfully difficult one, at times.

Or maybe we need to give students more opportunity to see mathematics as a unified whole from the beginning. That would be one reason I've for years pushed for depth over breadth. Whenever a curriculum issue comes up, my first thought is usually, "What can we cut out?" Skimming quickly over dozens of topics and techniques encourages that sense of mathematics as a big collection of tricks, rather than something that has meaning and can be reasoned about. We need time to think about ideas, process them, wrestle with them, and make them our own. If students can understand a few concepts deeply, they'll have a better chance of figuring out something new on their own.

For the same reasons, I also feel drawn to inquiry based learning or Moore method teaching, where students are asked to figure things out with minimal guidance. We start with a few ideas--say, some definitions and a few axioms--and students are asked to build on that framework. Each new step has to be justified, and students need to figure out what works and what doesn't, without being told specifically what to do or what's right and wrong. They should rather be led by careful questioning to notice for themselves what works and what doesn't. If instead of just saying, "No, that's wrong," the teacher can simply present another problem and let the students figure out that their previous approach is flawed, they may internalize the structures much better. After all, structures you yourself have built are already essentially internalized. And it's hard to view a subject as filled with arbitrary tricks when you yourself have built up those "tricks" because they were needed.

In any case, what I would most like is for the students to come away with a sense of my subject (and others) as a unified whole, as something they can investigate and reason about, rather than just a collection of tricks and techniques. The techniques are useful, and the tricks are powerful, but without a strong foundation, it all falls apart.

Random thoughts

Random thoughts for today:

• I ended up with leftovers of everything but turkey yesterday, since I supplemented the contents of a frozen dinner for my Thanksgiving feast. So I may have been the only person in the world to go out and buy frozen turkey dinners today, so I can use up the rest of the stuff.
• From the "Things I wish my Students Knew" file: You cannot tell which of x or -x is negative without knowing something about x. (Of course, if x = 0, neither is. But I'm not looking to completely blow their minds.)
• Two more weeks of classes to go. The end is in sight. But the spring semester is too near. There is a monster at the end of the semester!
• The weather is finally turning cold. My fingers are getting cold while I'm at home. I've noticed I have more of a problem with my "mousing" hand while I'm at the computer, I think because it's the one I can never put in a pocket for a while.
• There are occasional television shows that I keep up with a little because I think I'm hopeful the show will get better. The premise is interesting, and there's cool stuff going on, but the writing is pretty mediocre. Often the writing never improves. I find myself wishing some particular episode had a better writer, because I often walk away disappointed.

This is so hard, you should get college credit for it! Wait...

Our university has a math sequence for elementary education majors. The courses mostly take a very deep and comprehensive look at the underlying mathematics involved in about K-6 education. Since we have a number of mathematics education professors in the department, the courses are extraordinarily well designed. It meets in a room with large hexagonal tables where students can sit together in groups of six, and we have two large cabinets filled with all kinds of wonderful manipulatives which are used in many class activities. (A manipulative is any sort of physical object which can be manipulated to learn math. We have various types of colored chips, geometric shapes, and other cool toys for demonstrating mathematical concepts.) Many of the activities are actually similar to and based on activities which could be used to introduce concepts to elementary students, although of course the college students are expected to go a little deeper and are asked to do some things that we don't ask elementary students to do. (For example, we have the students in the course perform various operations in bases other than ten to emphasize the basics of a place value system. No one teaches base four or base twelve to elementary students anymore.)

It's actually a really fun class, and full of all sorts of wonderful discoveries waiting to be made. I personally find myself fascinated by the fact that in many cases, the way we explain a concept to our students parallels the abstract definitions which can be used to define that concept in advanced mathematics. So whereas in class we may use groups of red and yellow counters to define the integers, a mathematician might start tossing around scary sounding phrases like "sets of ordered pairs" and "equivalence classes", but ultimately mean pretty much the same thing. I personally found the demonstrations hugely enlightening the first time I did the class. It provided me with very concrete way to think about and explain concepts such as why a negative times a negative is a positive and why dividing by a fraction is done by multiplying by the reciprocal. The idea of the class of course is to provide our future elementary educators with similar insights.

Unfortunately, the class is always a struggle to one degree or another. One particular point which the students never seem to get (no matter how often they are told) is that this is not a class in elementary school mathematics. We obviously expect them to have already learned how to do things like add and subtract integers and fractions, how to multiply and divide multi-digit numbers, and the like. After all, they were supposed to have mastered these topics in grade school. (Except of course we know many of them actually can't do these things reliably, so the course also helps back up these concepts. But I digress.)

As a result, the students sometimes ignore instructions on how to complete an activity. For example, they are supposed to learn how to represent integers with sets of colored counters and then use the counters to add and subtract integers. (This is actually a really cool activity; I'll have to write about it sometime.) But since they know what 7+(-4) is, and following the directions to make representations of the numbers using the colored chips seems complicated, they instead just write down "7+(-4) = 3" and explain to me that "the model was too hard, so we just did it." Since they feel the class is about (or should be about) learning to add, subtract, multiply, and divide just like they did in grade school, there is no need to learn anything else about these topics as long as they know what the right answer is. They sometimes fail to understand that the colored counter model they have been asked to use is, in essence, the content of this course: we want them to learn to use a physical model which represents basic operations on integers, and to use that model to derive various known properties of addition and subtraction with integers. This issue is usually an uphill fight all semester with the students.

But this semester I'm getting even another argument from some students in one class. With almost every activity we do and with almost every mathematical model we describe and learn to use, the students complain to me that "this is too hard for any little kid to understand!" Which is completely irrelevant, since I'm not asking any little kids to do this work, I'm asking my class full of college students to do this work. I've told them I don't address the issue of how to teach their future students, but rather just teach them mathematics. I leave it to other people to teach them how to teach math. This doesn't sway the students.

I tell my students, "I'm not asking your students to to this."

The students respond, "Yes you are!" against all evidence to the contrary.

My students somehow feel that any topic which they consider to hard for a third grader should be too hard to ask a college student to do either. I suppose they want a refresher of third grade math without any of the "hard stuff." Remarkably, I seem to have little success with convincing the students that they are not, in fact, third graders. Do they really think that in a college math class they should learn nothing more than what grade school students are expected to learn?

But do you want to know what the worst part is? Most of the activities actually aren't beyond the grasp of moderately intelligent third graders. I consider it the dirty little secret of the course. Granted, it would take more time, but grade school students could certainly be taught rules for representing integers with colored chips. With practice, they could learn techniques for adding and subtracting with the colored counters and even explain how it works. Eventually they would find patterns in what happened when you add and subtract integers. The same is true for almost every other topic we discuss, from the most basic (addition of whole numbers), to the most advanced (division with fractions, perhaps). You couldn't do all of K-6 in a semester obviously, and children may not make as many connections as a college student ought to be able to, but they could do almost every activity we do in the college course, and learn a lot.

I don't even bother to argue the point with my students 'though, because whether grade school students could do what we do or not is entirely beside the point. My class isn't filled with grade school students. It's supposedly filled with college students. College students who want to be elementary teachers. The same teachers that will lay the next generations mathematical foundations. Which will, in another ten to fifteen years or so, become our next generation of college students sitting in my college classes. And that thought usually fills me with the urge to go lie down for a while.

Projects

I gave my numerical analysis students this week off to work on their projects (which they start presenting next week). I told them I would be available during class time for them to ask questions. The difference between doing this with upper level and lower level students is that the upper level students will actually do it. (I was actually a little surprised at how many people I saw.)

I'm actually really pleased with my numerical students right now. Most of them have been working hard on interesting projects. Most of my work in meeting with the students up to this point has actually been in getting students to scale down their proposals to a manageable size. One group of students was originally starting with an ambitious project of figuring out how to guide a robot through an obstacle course using GPS guidance. I initially got that scaled down to just working with the GPS, then got them to massively reduce the number of factors they include in their GPS model, and finally we have settled on doing linear least squares fitting, which is simpler than the non-linear least squares fitting that GPS requires. I'm relieved, and so are they. I knew at the beginning of the semester that this was simply too big, and I think it's finally something they can finish in a reasonable time. (Actually, they are almost finished now.)

Almost all of the projects are progressing nicely and look really interesting. I have student showing how calculators evaluate functions like sines and cosines, another student solving linear systems using iterative techniques, one studying efficient matrix multiplication for graphics applications, a pair of students working on Bezier curves and their use in graphics, a group of three presenting on a numerical simulation of the Tacoma Narrows bridge collapse, one doing arbitrary precision arithmetic, and one doing on the fly polynomial interpolation to compensate for lag in networked computer games. All of the presentations look good. I just need to make sure they don't run too long.

Sometimes I really like teaching.

Precalculus Despair

So this semester I'm teaching precalculus. Supposedly the students are already reasonably proficient in algebra.

Good parts of the semester include the fact that I found a book that I like. I really, really like it. It's beautifully focused. It feels like every time I start a new exercise, I think, "Yes--this is exactly what I wish my calculus students understood." And it's beautifully structured, spiraling through topics, adding layers of subtlety with each turn. I'm very happy about having such a good book in part because pretty much every semester I have been here so far, I've used books other people picked out or which were "typical" for a course at the university, and I've pretty much universally despised those books. I've stopped trusting anyone's recommendations.

I also like the way I have structured the semester, with lots of "mini" tests, which are cumulative, rather than two or three "big" tests that students cram for. It keeps the students up with the material, and it also keeps me apprised of where my students are. But this means I really do know how my students are doing, and I'm not feeling as happy about this at the moment. After the second quiz, I know there are many things that many of them cannot do. Many fairly simple things that many of them cannot do. Including things we have done repeatedly since the second day of class.

Part of the problem comes from previous deficits. Many of the students have trouble solving simple equations. Several need to be reminded repeatedly that there are real numbers between 2 and 3. Some are not sure what you might get if you were to square the square root of 5, or that -3 < -2, or whether a squared real number might be negative, or whether one might be allowed to take the square root of zero. (It's zero, by the way.) When students are struggling with these issues, it makes it difficult for them to learn about the domain and range of a function, and what the rate of change of a function on an interval might be, and how to sketch a piecewise defined function. How did these students end up in precalculus? Are they really expected to be able to complete calculus next semester?

So I have many failing students now. And tomorrow I must chide them to get the help they need if they wish to pass. There is still time, but the time to catch up is running out rapidly. And for so many of them, there is so much to catch up on.

Perspective

I had just finished a Calculus II class on antiderivatives using the natural logarithm, and as I was erasing the board (filled with indefinite integrals, u-substitutions, and things like "ln|sec(t)|"), one of my precalculus students came into class to ask me something. I shifted gears to answer his question, and realized that while I considered most of the stuff on the board pretty easy, to a precalculus student, it must look incredibly complicated, and perhaps like sheer gibberish. And of course it seems transparent to me; I've been doing calculus since 1988, so it's been over 20 years now. (Amazingly, things like this have stopped making me feel old.)

It's much like the conversation I had with another colleague once: We were talking about low-level, introductory courses at the university (Big State Tech U), and meant any of the various calculus sequences.* But we observed that for the general population, "calculus" is used as a metaphor for anything unbelievably advanced and difficult. (Sort of like "brain surgery" and "rocket science", although if you are a brain surgeons or rocket scientist, you mastered calculus long ago.) Most mathematicians (and a number of other scientists) see calculus at the starting point for our fields, while most of the general population sees it as the pinnacle of learning.

*Of course, that conversation was at Big State Tech U. Now I do teach at a school that teaches a wide variety of courses lower than calculus, including lots of algebra, a general education math course, courses for elementary teachers, and even remedial courses. Not that I don't still consider calculus the first real college level math class.

Contest Updates

This year I managed to organize our first ever Putnam team, and even got students together for practices. Unfortunately, the students dropped like flies. Two dropped after the first practice, and two more after the second. I ended with enough students to make a team, so we were going to be officially represented--but then one more student dropped at the last minute.

I enjoyed Putnam practice sessions, which I ran a bit like a Moore method class (with me in it). I didn't look up solutions to the problems; we just tackled what we could and tried to come up with ideas. It was interesting and moderately productive. (These are hard problems.) Unfortunately, two very strong math majors chose to participate because they are involved in too many other things. In the end, our Putnam team did not score any points, but our school does appear on the list of "Schools which took the Putnam exam", which I still think is cool. I also feel pretty good that I solved a few of the problems on the exam this year myself. (Did I mention these are hard problems?) Perhaps we will do better next year.

I also got together our department's "College Bowl" team, with significant help from another faculty member. We needed four players plus an alternate, and unfortunately we only had five people come to try out, but we did have some fairly good people. I had some schemes for this year, too. It basically tests how much useless trivia students have memorized, so I obtained for our team used copies of a great popular book on memory systems (to stuff pointless facts into their heads with) and a copy each of An Incomplete Education (full of pointless College-Bowl-style trivia to start stuffing). In the end, we did very respectably in our school; we almost made it to the finals.

I also had fun this year "consulting" with the programming contest team. The programming contests frequently involve some interesting mathematics, so I've been coming to practice sessions and helping them figure out how to tackle the mathy problems. The problems are often pretty cool. Between that and the Putnam, I enjoyed doing some occasional math last year. I also like working with the programming team because it lets the students see professors "bridging the gap" between math and computer science. For some reason, we seem to have a division between the math and computer science majors in the department, which is pretty weird, because there isn't much gap I've noticed among the professors.

So what new contests will the "I-don't-like-competition" guy find himself involved in? Funny you should ask; I actually thought about trying to get together a team for a mathematical modeling contest in the spring. Of course as with the Putnam, what I'm really interested in is getting together students to do some math, not really to compete. I didn't do it this year, but I'm looking into more information about mathematical modeling for next year.

Regional meeting

I went to a regional mathematics meeting last weekend in Pittsburgh, which was fun. One of my colleagues got a van and brought a group of students down to the meeting. One even gave a student talk that evening. Unfortunately, the student government ran out of funds and couldn't pay for the students to stay overnight, but they still got to see some cool stuff. Including a trip with us for Indian food. I think that may have been a rather big leap for them; I've never seen people so happy to see rice and bread. I even got them to try gulab jamun, which is actually the most fantastic dessert ever created, but is described as cheese balls in honey, so I'm not surprised they were a little hesitant. (I told them afterwards, "But now you can go tell all your friends you ate cheese balls in honey.") I did hear that our lunch necessitated a side trip to Wendy's later when the group split up for a while.

I did stay the night, and I actually gave a short presentation on a little problem I started thinking about last summer. So that makes two presentations at meetings this year. I'm beginning to suspect that getting accepted to talk is not so difficult. However, I still have to do the publishing part, and that takes a little more work.

Fun story from the conference: On the first day, the organizers had a survey in which everyone put stickers on a poster to vote for various activities at regional meetings. When we finished, we wrote our name on the back of the sticker sheet and dropped it in a basket for a drawing the next day. When it came time for the drawing, someone stood up and told us, "The janitor threw out the basket this morning..."

Spending money?

We have somehow ended up with a largish budget for our library funds this year. I have guesses as to why this is so, but no one really knows what is up or has been able to answer why the budget is larger than usual. However, I strongly believe that the excess is temporary and will be gone next year, so purchasing more journals or database access (which we probably need) is not a good choice, as that would be a recurring cost.

I'm on the committee which makes requests to the library for new acquisitions. Despite the likelihood of losing some of the funding next year, I actually have requested some database access anyway, but mostly I'm requesting books. The collection for mathematics is actually not too bad; previous committee members have made some excellent selections, and while the collection is not huge by any measure, it is of good quality. (Most the the "classic" books that I thought should be in the library turned out to already be in the library when I first looked into this last year, which pleasantly surprised me.) However this leaves me with a difficulty: How do I spend the rest of this excess money? I have not gotten many responses from the department from my e-mail asking for requests, so I'm largely on my own.

If there were a large backlog of standard works that every library should have that we were missing, this would be easy to solve. If I could safely request more periodicals and databases, again I would have no problem. If lots of people in the department responded to my request for suggestions, again it would be easy. Instead I end up poring through catalogs and lists of reviews and trying to determine which are the good books that are worth getting. In some cases, I'm trying to make judgments (usually from reviews) about whether a book in a field I know almost nothing about is any good. Usually this is not too bad, since in any year a few hours of perusal can generate a list of a dozen or so books that are highly recommended. But when the budget is larger, this takes much longer to do.

Not using all of the funds would be a bad idea. It might result in the regular budget getting cut. (If the larger budget actually continues next year, I'll be ecstatic; we'll assume it's permanent and can expand our journal selection accordingly.)

All of this leaves me with the peculiar sensation of trying to meet a deadline for spending large amounts of money, which sounds easier than it is. (OK, I guess if I just wanted to spend the money and didn't care what it was on, I could make a list really quickly, but it seems pointless to me to get a bad book just because the budget it large. Instead, I have to make a few hundred really good selections.)

I never want to look at another book review again.

The Perversity of Self-Referential Teaching

When I taught my students version one of the Fundamental Theorem of Calculus (the one which tells you how to take derivatives of functions involving integrals), I said:

Isn't this easy? Don't you wish the whole test was going to be this? Well, I will go ahead and tell you that when I put one of these problems on the exam, half of you will forget and miss it.

I say this every time I teach the section, in the hopes that it might make the material stick. This time I went one step further and added:

That will be true despite the fact that I have just told you this.

I thought possibly that this statement might make at least a few more students decide to remember it. (It seemed to make an impression at least; they did laugh.)

Well I gave the exam this morning, and it turns out that in fact more than half the class forgot it. I think somewhere around 80% forgot it. Maybe next time I will try: "Everyone gets this problem every time, so don't bother to study this at all."

Spring Semester

I have a schedule for Spring now. I don't have any more 8 am classes--Yea! I'm also scheduled to teach three sections of math for elementary ed students--also Yea! And I'm teaching Discrete Math, which is worth a cautious and qualified Yea (?)!

Discrete was my favorite class in college, and it was also the "sexy" class that convinced many people to be math majors. It is the course where students learn to write proofs, and the topics tend to include things like set theory, formal logic, and combinatorics. (Of course, I got to take it with a realio trulio set theorist, so how cool is that?) The only problem is that since then I've been at two other schools where Discrete Math seems to be a major problem (including this one), rather than a joy. I have no idea why this is, and I wonder if I'm up to doing the job my professor did with my class. I hope so. I think it's a very cool course, even though it's not in the area I ended up in.

I'm not sure why, but I've already had two students try to get me to sign an override form so they can take Discrete Math even though they haven't taken the prerequisite yet. Hint to any students thinking about this: The answer is no.

Applied Combinatorics

When I was in grad school, a friend and I noticed that one of our professors would appear multiple days a row in the same outfit. In one case, he showed up in the same outfit for a week, and only changed suddenly at the end of the week when he hosted a visitor to the school. (I hypothesized that a conversation in his household that morning probably started with "You are not wearing the same thing you've had on all week while your friend is visiting...")

This may have left me a little paranoid about what I wear to class. Sometimes (like this semester) I try to keep track of what I've worn to different classes over the course of the semester. This leaves me figuring out how to arrange my sartorial schedule to have the fewest number of repeats for each class. Fortunately, I have several classes this semester which only meet once a week, and that makes it possible to make it look like even I have a wide and varied wardrobe.

I was very relieved when the weather improved enough that I could wear some short sleeved shirts, because I was running out of long sleeved shirts for my twice a week class. Then it got cruddy again. We're down to the last week of classes, and I'm barely able to wear my warm weather clothes. At least I'm ready for my summer classes.

On the other hand, it occurs to me that this may not be necessary at all. I usually can't remember what anyone I know was wearing the previous day, so I'm not sure I'd notice if anyone was repeating. A professor in Psychology even assured me that people are very unlikely to remember. Still, I'll try not to wear the same thing all week. It'll only make future professors paranoid.

In which I play Real Professor (TM)

The last week or so has had me feeling like I'm a real professor, which I guess I am now. I'm just not used to thinking of myself that way.

I'm serving on the library committee for our department, and got to recommend books for the library to get. After encouragement from the department, I even requested some books that might be helpful for my own research interests. (This seems odd to me, since it's pretty much just an undergraduate institution, and as such, I may be the only person to ever look at one or two of these.)

Then I gave a talk for the department. Granted, it was just a rehash of my dissertation defense, but it was a talk. (Which meant I had to go remember what all that stuff in my dissertation meant. More or less. Thank goodness no one had any questions about asymptotic values.)

And finally on Friday, I met for the first time with a newly formed presidential commission on LGBT issues at the university. Me, on a commission. We should have a bat-signal style spotlight to summon us for emergency meetings.

All this is on top of all the usual stuff going on. I'm just now grading papers from students who think that paying off a loan early could save them over $100,000 off of the $982 finance charge. (They have a test coming up, too.) Earlier in the week I was trying to teach my elementary education students how to work in bases other than 10. (It's no wonder this tends to be known as a disaster for students; the elementary teachers may not understand it.) And my topics class is working on a take-home test which has generated plenty of consternation.

But hey, I feel like a professional sometimes now.

Academic Year Interstitial

Spring semester has started, much too soon... it always does.

Over the break, I put together my "notebook", explaining what I've been doing at the University for the last year, except in my case it was what I've been doing at the University for the past five months. I'm hoping that matters, because I have the feeling that most of the sections on scholarly development and university service essentially boil down to: "I got nothin'." At least I can say I won Time magazine's person of the year.

I feel the same way about the courses I'm teaching in the spring. I have three sections which meet once a week for two and a half hours, and I'm feeling slightly lost about what to do. One class now counts for a whole week. The classes will only meet 15 times! In one respect I like the longer sessions; it feels like I can do some things (like group activities) which take a little time to start--we don't have to break suddenly in the middle and assume the students will still remember what they are doing when we start the next class. But what do I do about tests? I can't really see successfully covering new material the same class as I give a test, but I don't see giving up three weeks of class to give three tests either. For the topics course, I'm thinking I should be able to get away with some take homes, but I'm not sure that will work in lower level courses. So I'm experimenting. One course I'm doing just quizzes, basically every week. Another I'm actually giving an hour test and then moving on to new material. We'll see what happens.

Then there's the textbook for one low level course I'm teaching. I was pressed for time last semester, and ended up selecting the textbook that most of the rest of the department uses for the same course. I've been reading it over the past week or so, and I hate it. It's annoyingly filled with bubble-gummy pop culture references to everything under the sun, from Jurassic Park to "Judge Judy", in some sort of attempt to be hip and exciting for what I'm sure the author thinks of as "those wacky college kids", or equivalently, "gen-y" or "gen-z" or maybe we've cycled back to "gen-a". There is also a formula, neatly boxed, for everything. In each section, we have a sequence of carefully labeled problems. Each problem has its own formula, and an example of plugging numbers into the formula. It's a frigging how-to manual on steroids. Nowhere does it ask students to actually think about anything; it asks them to memorize, plug numbers into formulas, and regurgitate regularly. (The author has the gall to then title the book Mathematical Thinking.) Oh, and then there's this other little kicker: sometimes it's just flat out wrong. I'll look for another book for the next time I teach this course, and if I don't find something better, I'll roll my own. The list of topics isn't that complicated.

Amusement from Students

Here I just collect a few of the random student incidents which have amused me over the last semester:

• Picture it: I'm in the middle of deriving the quadratic formula (which gives the solutions to a general quadratic equation), and the students are obviously having a lot of trouble with the abstraction, since this is supposed to cover all possible cases. (I also dropped a term or two and had to make corrections as we went, which didn't help.) In the midst of our derivation, a student asks: "Can we do another one after this one?"
• A student tells me he studied the wrong chapter for the exam. It was a chapter we hadn't studied yet. I asked, "Didn't this all seem pretty unfamiliar?" He said, "It always does..."
• Learned from tests: Three lines which meet at a point might be called (1) sedimentary, (2) a midpoint, (3) conclusive, or (4) parallel. (The last is my favorite answer.) A convent is made up of two angles with measures summing to 90 degrees. Many of my algebra students decided that the product of 6 and 9 is 15. One of my colleagues believes some of her students have trouble reading; I have no idea where she gets that idea.
• One student missed class and sent me a note with his homework from the day explaining he couldn't make it because he was in line for a Playstation 3. The note said he was going to sell the Playstation "for a large profit of which I could use for school related bills." He also let me know he was in several TV and newspaper interviews from that day, "if that helps validate my dilema" (sic). I considered taking the homework in exchange for a Playstation 3.
• Another colleague was despairing at the end of the semester because on the final he asked students to calculate how much total they would pay for a $200,000 home loan paid over 30 years at some standard interest rate. The answers ranged from $120,000 to eight million dollars. We concluded that we should quit teaching and go into the business of selling home mortgages to our students.

The probability is rapidly approaching zero

I've been teaching a class for students planning to be elementary and middle school teachers. Recently, we've been learning about probability. It's a very hands-on class; we meet in a room with nice big hexagonal tables instead of standard desks and do lots of activities. There are two big cabinets filled with wonderful manipulatives. (Note for non-math-education sorts: Manipulatives are physical objects used to teach mathematical concepts. They include items like plastic chips or other counters, different shaped tiles and blocks, and a variety of interesting and creative tools people have come up with. They're a lot of fun to play with.)

Naturally, I try to lead students to discover important results, then hilight and review the results. I'm however realizing that conducting experiments in probability with students who are not so good at following directions is often a recipe for disaster.

A typical day consists of me giving them an experiment to conduct and record results from. (Perhaps rolling dice or flipping coins.) I explain and briefly demonstrate what they are going to do, then set them to collect data. When they are done, I smile like the Buddha and ask them what they found.

They tell me.

Then I say, "You got what? How--wait, what exactly did you do? That can't be right..." Then we proceed to go over the procedure until I find the mistake.

Actually, after the first experiment, I had way to much faith in my students. I recorded their results on the board, looked them over, and declared that this was a very unusual outcome. Something you would only expect to see maybe one in a million times, but nonetheless possible. But after all, I explained, that doesn't mean it can never happen. That was when a student asked "Wasn't that how we were supposed to do it?" And that was when I discovered we had not had a one-in-a-million event, but actually what turned out to be closer to a nine-times-out-of-ten event^*: They had written down what they thought should happen instead of what actually happened. This tends to produce results that can politely be termed "wonky".

So new problem: What is the probability that my students will understand and follow all of my directions?
---
^* Granted, Terry Pratchett does point out that one-in-a-million chances do come up 9 times out of 10.

Chaos

The Chaos and Fractals course is on! Apparently, it was even pretty popular at the University level committee, and there is some interest in transforming this into some sort of permanent course offering. Still not much success in finding a book that I really like.

However, I did find the following quote, which I like a lot:

A dictionary definition of chaos is a "disordered state or collection; a confused mixture." This is an accurate description of dynamical systems theory today--or of any other lively field of research.

Morris Hirsch

I'm in the process of trying to collect some more quotes, plus cool pictures and so forth to try and advertise this course. I need to make sure I get sufficient enrollment. Of course, I've always liked the following by Terry Pratchett from the opening of Witches Abroad, but it's too long to fit on a flyer:

...the universe was full of ignorance all around and the scientist panned through it like a prospector crouched over a mountain stream, looking for the gold of knowledge among the gravel of unreason, the sand of uncertainty and the little whiskery eight-legged swimming things of superstition. Occasionally he would straighten up and say things like "Hurrah, I've discovered Boyle's Third Law." And everyone knew where they stood.

But the trouble was that ignorance became more interesting, especially big fascinating ignorance about huge and important things like matter and creation, and people stopped patiently building their little houses of rational sticks in the chaos of the universe and started getting interested in the chaos itself--partly because it was a lot easier to be an expert on chaos, but mostly because it made really good patterns that you could put on a t-shirt.

(So maybe I should call this 'Fermat's Review': "I have found a wonderful quote about this topic, but the flyer is too small to contain it.") If I recall correctly, I did put up this quote at my dissertation defense, 'though. When people ask what I study and I say "Julia Sets", I can always follow up with "the mathematics of T-shirts and coffee cups."

The Pratchett quote also seems to be about as close as we come to a criticism of chaos or fractals. I was hoping for a long list of quotes, at least one of which would have someone kvetching about it being some sort of abomination, but nothing so far. Any suggestions?