Student self-teaching

In my high school physics class, we had an occasional event Mr. Rea called a "practicum". Here's how it worked: the class is given a problem, and can work as a team to solve it. The one I remember best was predicting how far a ball shot out of a spring-loaded cannon would travel, and putting a piece of carbon paper down to track where X marked the spot. The class gives their answer, and half the credit for the problem is based on whether or not they get the right answer. The other half of the credit for the problem is given when the teacher picks one student from the class (quasi-randomly, since I remember that if someone was goofing off they often got picked), and asks them to explain how to solve the problem.

The method forces the class to teach each other, and for the strongest students to lift up the weakest ones to a shared level of understanding. I think the social pressure of having the weight of everyone else's outcome on your shoulders also made it an effective technique.

I have long been considering how to increase student involvement in my class, to shake up the routine, and to get the wallflowers and malcontents engaged with what's going on. The recent success of my extra credit endeavor has given me an idea: a practicum with a prize of no assigned homework. Here's how it would work:

Mr. Lancaster and I pick out a problem or three that are representative of the evening's homework, and assign to the class as a whole. After, say, five or ten minutes, if the class can answer the question(s) correctly, they get to proceed to the "bonus round". Maybe this first question or questions will be weighted like a small quiz? In the bonus round, one student is selected to explain how to solve the question or questions, and if they can do so correctly, the class wins no homework for that night.

Optional additions would be to let the student in the hot seat have a "lifeline", to be able to ask someone else in the class what the next step is, or to make the homework assignment for the evening extra credit for anyone who still wished to complete it.

I'll have to run it by George, but I kind of like the way this idea sounds...

The sorry state of mental math

I don't know what to think of calculators. On the one hand, they are a great tool, and I sometimes wonder what else Isaac Newton might have had time to discover if I could go back in time and hand him a $5 scientific calculator. On the other hand, I despair of his ever having learned the facility with numbers he later showed if he had been exposed to it too early in his development.

The fact is, hardly any of my students can do basic arithmetic without the aid of the calculator. I mentioned in a previous post the blind-leading-the-blind aspect of class efforts to solve 22/4 by hand in 4th hour, and I am consistently floored by the reliance on calculators for everything from simplifying fractions to simple addition and subtraction. I am left wondering how much of the sad state of my students' math ability in general is due to a consistent reliance on this tool to perform simple tasks for them. How much practice in basic operations like adding and subtracting, multiplying and dividing, have my kids lost because of this crutch? How much more fluent in the language of numbers would they be if they had never been exposed to a calculator?

I remember learning the multiplication tables in 3rd grade, from 1x1 to 9x9. The whole class made rocket ships out of construction paper, and the teacher put up nine planets increasing in size, labeled 1-9, against a black background on the rear bulletin board. Everyone started at planet 1, and was able to advance through the planets only when they could do all the problems for that number for the teacher, Mr. Murphy. The best part was that, for each planet, we got a number of jolly ranchers equal to that planet's number (that's 45 for the whole circuit, if you're keeping count). We also did algebra in elementary school occasionally -- we just didn't know it, because we were told it was 3 + square = 4, or 9 - triangle = 7, and find what goes in the square or the triangle. We did the "Mad Minute" in grade school too, where we were given a page of simple math problems and were challenged to see how many we could solve in 60 seconds. It is staggering that my students are just now reaching this point -- what did they do in grade school?

There is an argument, a compelling one, that the ability to do mental math is overrated, unnecessary even, and that forcing children to do without calculators is a relic of a bygone era, like learning to do long division. I'm not terribly qualified to comment on this, but here's my take anyway: insofar as mental mathematics for basic operations nurtures a sort of mental flexibility, a comfortable ease working with numbers on the small scale, I think it is invaluable. How many times in our lives will we need to add 8 and 6 without a calculator at hand? Or how about getting a rough idea of whether the gallon of milk at 3.20 or the half-gallon at 1.89 is a better deal?

Long division is an algorithm. That's why no one remembers it when they reach adulthood -- it's because it's a precise series of steps that removes the necessity of thinking about the problem. This is similar in spirit to the manner in which the book teaches my students to add numbers: If the numbers have the same sign, then add their absolute values and attach the common sign. For opposite signs, subtract the smaller absolute value from the larger, and attach the sign of the number with the larger absolute value. Seriously? It's like the intuitive sense of relating numbers to zero and that subtracting more than you have leaves a debt has been willfully excised.

So, while it is troubling that I have a whole class of algebra students who can't divide 22 by 4, it is not entirely surprising.

The power of extra credit

I've been wrestling recently with how to keep better attention when I'm trying to show my kids something cool. For my most recent talk, I suggested to Mr. Lancaster that I be allowed to offer extra credit problems to the students, based on what I was talking about, in order to nurture those fledgling attention spans. We settled on three extra credit questions, interspersed throughout the talk, each worth the same as a homework assignment.

I am both impressed with how well it worked, and distressed by how challenging some of the problems were to these students. Since many of the kids in the class are new from last semester, I gave an updated version of my "Who I am" presentation, talking in general terms about my research, about rockets and space propulsion, plasmas, satellites, vacuum chambers, and moon rovers.

I like to try to get students involved when I give a talk, and since my talk was going to be about plasma rockets (Hall thrusters, for the advanced reader), we began with talking about what a plasma was. This worked pretty well, because almost everyone has heard of a plasma TV, and at least one student knew that stars were plasma too. (I thought that was pretty impressive!) So I told them about a bunch of different things that use plasma, like neon signs, plasma TVs, CFL bulbs and welder's arcs, and also about some in nature, like lightning, the sun or the aurora. This led to our first extra credit question just a few minutes later -- to recall four of the seven or eight plasmas we had discussed. I was pretty pleased with this as an easy introduction to the format for the questions, plus I was sure someone could get it right, and referring back to information like that helps reinforce it in students' minds. In both algebra classes, someone got it right.

I moved on to talking about rockets, and about how the space shuttle uses 6 gigawatts of power when it lifts off. This was a pretty good opportunity to explain about words like giga, mega, kilo, and also milli, micro and nano. I tried to get the class to think of words that use these prefixes, and they didn't do too bad. With the prompt of computers, 2nd hour came up with giga, mega and kilobytes, and of course the iPod nano got a mention. But for our second extra credit problem, asking how many 60W light bulbs would be equal to the space shuttle's 6 GW, there was a bit more of a struggle. Both classes eventually got it (mainly by guessing, I'm afraid... the answer is 100 million), but I think I could have done a better job talking about the different scales. Those numbers are awfully big to think about without a really firm grasp of the way you jump by a thousand between each scale, and I don't think I laid that foundation well enough to make them comfortable with all the zeroes in 100,000,000.

What I like best about presenting is that, if I can get the students' interest, some will ask questions that let me branch out and address their interest individually. For example, when I asked everyone to tell me what they thought of space, they came up with a lot of great descriptions (black, cold, way up there, empty), but one person also said, "no gravity." I let that slide for a minute, but later when a person asked how fast rockets went to get into space, I elaborated about escape velocities, and explained how something going fast enough would fall at the same rate as the curve of the earth fell away, so it never hits the ground. So, being in orbit is like being in free fall, hence the apparent lack of gravity. It's a good sign that the kids felt comfortable asking questions, and when I can answer one it helps cement their attention.

In fact, Mr. Lancaster and the student teacher Mr. T both commented on how today, after I gave my talk, was the best behaved 2nd hour has been all semester. So, I either bored them to sleep, or else I gave them something to think about that made class seem shorter and more relevant. Let's hope for option (b).

I also told the students about my lab, to get them interested in the field trip there at the end of the semester, and I explained how the lunar rover was tested in our vacuum chamber back in the '60s. For the last extra credit question, I looked up how far the rover had travelled on Apollo 17, and asked the students to figure out its average speed when it went 22 miles in 4 hours. 2nd hour got this one right away, but 4th hour never did get it at all, even after (literally) about 20 guesses. It's 5 and a half, and it's a little distressing that 22/4 was that difficult for them.

Of course, they didn't have their calculators.... but that's for another post.

I thought this went pretty well, using a combination of extra credit problems and frequent opportunities for involvement ("What do you think of when you think of a rocket", "Can anyone think of a word with giga or mega or kilo in it", "Tell me what you know about space", etc.). I was also able to make a pretty good connection with the current class material, which has been heavy on word problems and distance = rate * time type stuff lately. All in all, one of my more successful attempts.

Left with the Chaff

I find myself with a backlog of would-be posts here, so let's get started. Things got shaken up quite a bit at the semester break, with all the freshmen who passed their first semester of algebra departing for greener pastures, and those who failed sticking around for round 2 with Mr. Lancaster.

I'm a tad skeptical of the wisdom of taking all the students who failed this class the first time around and hoping for better results by trying the same thing over again en masse. Who was it that said that the definition of insanity is doing the same thing over again and expecting different results? Ah, right, Einstein. The same bright chap who assured us that, whatever our "difficulties in mathematics... mine are still greater," and noted that "it is a miracle that curiosity survives formal education."

What kills me is that, given the right environment, I feel like there isn't a kid anywhere I couldn't help have fun with math. Math is a game, one that challenges you to shift the pieces around like those blacksmith's puzzles to get the piece you want to come free. But even at this point, these kids are so convinced that math is boring and stupid (in some cases, substitute school for math) that they don't even want to play. And I don't blame them -- I'm bored, going over textbook problems by rote, doing the same thing over and over again: come in, everybody sit down, turn in homework, go over homework, get lectured at about more problems in preparation for more homework, get assigned said homework, leave.

But part of the environment problem is getting a critical mass of kids interested, willing to play along with what you're saying. When I give a presentation, whether on the space shuttle, or Fibonacci numbers, or how car engines work, I can keep a class' attention by sheer charisma if the number of really disinterested or unhappy students is small enough. From that perspective, I can't begrudge all the teachers whose lives are easier this semester because their failing students have left the class, freeing them to devote their attention to better students. From my perspective, on the other hand, I'm now working with all the kids they left behind, and sheer force of charisma often isn't enough. It's tough to blow a kid's hair back when they're wishing they weren't even there in the first place.

It's tough to come up with material to enrich the class beyond the book, too. I only do it about once a week, and I spend probably a good deal more time than the nominal 6 hours including prep time that we've enlisted for. The book, for all its flaws, is a recipe to follow when you don't have the luxury of that time, when you've got to be prepared to come in every day of the week. But man, the book is boring. That's the problem with all the frustration of doing things the same way -- it takes resources to change, whether time or money, that you usually don't have. What if the kids who had failed were in classes of 6, instead of 26? Suddenly the dynamic switches from a crowded lecture you can hide in to a conversation where involvement is difficult to avoid. Unfortunately, you also need 3 more teachers for that scenario.

Complicating matters is that I don't even really know most of these kids yet. I've got most of their names, sure, but the bonds of familiarity and affection that I made with all the students who were struggling but not failing last semester aren't there for all these kids. I'm still a stranger, and I've got to lay all that groundwork again to become someone they think has something worthwhile to tell them.