Notice & Wonder, Part 2

If you missed the previous discussion, on 10/15 Steve’s freshman algebra class came up with a whole mess of observations on this problem.  On 10/19 I returned to put the same problem up on the board (to much groaning and general consternation -- “We already DID this one, Mr. McDonald!”), followed by a Powerpoint slide of all the N&Ws that I scraped from their notebooks into the previous post.

Last post, I made a tongue-in-cheek allusion to saturating the number of synapse connections you can make in a given time in the last post. I think this is a legitimate concern, and maybe just a science-ey way of saying something that’s common sense – you don’t go from ABCs to astrophysics overnight.

<pseudoscience>
My basic understanding of learning is that you are forming new synapses between neurons (brain cells), and that by repeated exposure to a new concept or way of thinking that you use and strengthen these new connections.  Given that framework (and any biology or biomed types who want to clarify this, feel free), I suspect you pretty quickly reach a limit to how many new synapses you can form in a given time.  Put simply, try to string too many neurons together too fast, and the chain breaks.
</pseudoscience>

Now, keep in mind that setting up and solving a system of equations like

Q + D = 8
0.25*Q + 0.10*D = 1.25

is really a rather smartypants and esoteric way of solving this spare change problem, especially for kids who aren’t yet skilled at seeing patterns, doing systematic trial and error, or indeed very good at abstracting mathematical concepts in general.  Mosquito, meet cannonball.  Rather than waste oxygen trying to write that sort of thing on the board, my goal is to get to a point where we are comfortable pulling information out of a word problem, testing out possible solutions, and noticing patterns and shortcuts to the answer.

With that in mind, we spent this next class period talking about all the things we noticed and wondered about the Spare Change story.  I put up the version of the problem with the question at the end,

then went through every last one of the questions they wondered about and answered them all – as it happens, Chan and Prashant are the names of two buddies of mine, so they’re college students.  Knowing Prashant, Chan probably didn’t get paid back.  Chan is nice because he’s just that kind of guy.

We briefly… discussed would be a stretch, let’s say I coerced replies from my captive audience about what types of wonderings were most likely to help answer the problem – that would be things involving money, not so much the background details like what school they went to, or how old they were.  I wanted to plant the seed of distinguishing between math and non-math details of the problem.

Finally, I spent a few moments asking what sorts of answers we could expect: would –4 be a good answer?  No?  You can’t have a negative number of quarters?  Okay, how about 9?  Well, we only had 8 coins, so maybe 2 would be better since we don’t even have 9 whole coins, much less quarters.  But the question asks how much of the money was in quarters, so that means the answer is an amount of money, not just a number of quarters.  Then I started asking them if we could have an answer of $0.20, or $.45 or $.55.  Eventually, I led the horse/class to water/the realization that the answer had to be a multiple of $0.25, since that’s how much a quarter is worth.

As a closing thought, I put up this new problem (below), had a student read it aloud for the class, and had them put their new N&Ws in their notebook for next time.

Notice & Wonder

Here's an activity I did on 10/15 with my algebra class. Note that everyone in the class has a class notebook, so I had them draw a line down the middle and make two columns, writing "I notice..." at the top of one column and "I wonder..." at the top of the other. Then, I put this problem on the board, titled “Spare Change”.

Now, take a moment and ask yourself what I asked the students – what do you notice, and what do you wonder? Take note, as I told them, that there is no wrong answer here – there’s not even a question! We’re just after what you see, and what you think could use some clarification. Here’s what I, as a grad student in the engineering and physical science fields, would say:

I (the teacher) notice:

• Prashant needs $1.25
• Chan has only quarters and dimes
• Chan has 8 coins total
• This is exactly enough money for Prashant
• (Dimes are worth $0.10)
• (Quarters are worth $0.25)

I (the teacher) wonder:

• Is there enough information to find the number of quarters and dimes?
• Is there even a solution?
• Could there be more than one solution?
• Are there amounts of money where quarters and dimes couldn’t add up right?
• What if Chan had nickels too?

Of course, I would just about have a heart attack if a student in freshman algebra, still struggling with the very idea of word problems, wrote all this, but then that’s the point isn’t it? I gave everyone about three or four minutes to write down their noticing and wondering (hereafter, N&W), then I tried (operative word here!) to have a discussion, asking everyone to volunteer their N&Ws and making a class list. This was a big long session of what I like to call crickets, because everyone was silent until I started conscripting volunteers by calling on them.

Still, since I only ended up with about three or four N&Ws each, I got sneaky and went through their notebooks after class. I think there’s a pun involving shy and spy floating around here, but I can’t quite find the words. Anyway, below is a transcription of everything the students wrote in their class writing notebooks:

I (the class) notice:

• Where it says any it means the same thing as nothing
• You do laundry with quarters
• $1.25 is what I pay for lunch
• That it’s lunch time
• They’re at lunch
• It’s talking about money
• It’s money for lunch
• It’s a money problem
• Prashant needs $1.25
• Prashant has no money
• Chan has 8 coins
• It doesn’t say how much money Chan has in coins
• Prashant has to ask for money
• Chan only has quarters and dimes
• Chan has exactly enough money
• $1.25 / 8 = $ 0.156

I (the class) wonder:

• Who’s Prashant?
• What grade are they in?
• Why are there coins on the paper?
• How much money does Prashant need for lunch?
• What school do they go to?
• Did Prashant ever pay Chan back?
• How many quarters and dimes  each did Chan have?
• What did Prashant buy?
• How much money in coins does Chan have?
• Is he going to have enough money?
• Who’s Chan?

There is some overlap between the two lists, since one person may well have noticed the answer to another one’s wondering, but to the extent that we can imagine a class consciousness, this is a decent snapshot of that state of mind.

Look at how many stray, random, irrelevant thoughts are in there! Two thoughts on this. First, I purposefully gave some really broad, vague instructions to make sure everyone wrote something, along the lines of “Write down everything you think is important” for the noticing, and “Write down anything you don’t understand or you’re curious about” for the wondering. I only obliquely mentioned that they might try to find math-related things to notice or wonder, and I definitely didn’t tell them to try to solve the problem.

My second thought is, well, this is what your brain looks like before you’ve learned to filter out the important information in a word problem. After 15-odd years of doing math of some sort or other, I the teacher and probably you the reader are a point where we instinctively filtered most of that irrelevant stuff out. Indeed, my wonderings were pretty meta- in nature, abstracting the problem to the point of wondering about uniqueness and degeneracy of solutions (bonus points if you know what I mean by degenerate here!).

Most of the key information of the problem has been observed, at least by the class group-mind, though only a few students had all those key pieces assembled on their own, and fewer still deduced the answer to the unstated question – 3 quarters and 5 dimes. Before we’re ready to solve, we need to learn to filter all that extra stuff out.

But there’s a limit to how many new connections between neuron synapses you’re going to make in any given class, or any given blog post, so until next time, ciao!

Postscript: This post is my first made using Windows Live Writer, a free blogging composition program that can upload to Blogger.  The interface is WAY nicer than Blogger’s default editor, and includes handy things like full-screen composition and preview windows, real-time updated wordcounts and  much easier link and media content insertion.

The first day, for the second time

Pardon the stolen title, for those who recognize it, but it's a fitting sentiment for my first visit to Ypsi High on 9/17. It's both exciting and a little frightening to walk in on the first day to a strange new class. Still, I've done it before, I know what to expect this time, and I expect to do a lot better job getting to know these kids and influencing their lives. I'll be in two classes twice a week this year, instead of four classes once a week last time around. Plus, if there's any justice in the world, I'll be able to stay with these kids for the full year, rather than having them switched at the semester. Put it all together, and I'll be spending 4x as much time with these students as I did with most of my kids last year.

First thing in the door in 2nd hour calc, K (a bubbly extrovert, I soon discover) says:

"Hi! Are you a student teacher? A pre-student teacher?"

I was surprised to be noticed and addressed before I began walking around and introducing myself -- K noticed me getting my things out behind the teacher's desk and preparing for class. Still, this is the usual question, the one that confused several students last year, so I wanted to clear it up straight off. I told K, "No, I'm a pre-physicist," which got some interesting responses from her table ("Whoa!", "Weird!", "No way!") and I left it at that until I could talk to the whole class. My usual introduction presentation had to wait until my next visit, so I did a quick 60-second schpiel (Name, I'm a grad student, what's a grad student, I'm here because… math is awesome, etc.) but otherwise spent the day as a wallflower.

I spent that time comparing and contrasting Mr. MacGregor's (Steve's) class with Mr. Lancaster's (George's), where I spent last year. George started each day walking around the class and checking homework; Steve greets each student at the door with the warmup exercise for the day and, in the algebra classes, puts up a stopwatch website on the board with a few-minute timer. (Note: in later weeks, he didn't use the stopwatch anymore -- they'd been trained by then!).

I like the new approach. It largely short-circuits the chaos caused by the homework walk-around, especially at the beginning of the freshmen classes -- most of them didn't do it, need to give an excuse why they didn't do it, or else look for (or put on a show of looking for; see: excuses, theatrical execution of) their homework, all while their classmates chat, run around, otherwise behave like the 13-year olds they are, or perhaps frantically try to finish the work. In the calc classes, of course, it's pretty chill, this is just the freshmen we're talking about here.

Steve collects homework after he goes over it in class, which is right after the warmup, and then goes on to the day's lesson. This has the effect of letting all the students ask all the questions they want while their work is still in front of them, so they can correct it if they want to get full credit. The students who didn't do it have a chance to try to make an effort, if they care, and if they still don't care there's not much to be done about it anyway. Considering that the point of homework in my eyes is just to get practice doing the work, as opposed to quizzes and tests that actually evaluate your skill, this approach suits me just fine.

On another note, after a year of using them the teachers (well, at least Steve) seem much more comfortable with the smart boards (electronic whiteboards; basically a whiteboard-as-tablet-computer). In Steve's class the board is the forum for all of the class discussion; the lecture notes go on the smartboard, so do the warmup and homework reviews, and he saves all the notes to PDFs that go up on the class website. This is pretty awesome, in a 'wow-that-is-so-sensible-it-floors-me-that-someone-actually-does-it' sort of way. Plain chalkboards or 'dumb' whiteboards are just anachronisms these days.

As expected, volume and drama increased dramatically between the calc (2nd hour) and algebra (3rd hour) classes, though I want to note a different desk situation this year too. I've never really liked the gridded setup with students in rows and columns and each person as a little island in a sea of chairs and desks. Steve's class is set up in… I don't know, continents, to continue the analogy, but groups of 4 desks together all around the smartboard. I can't really say why for sure, but I feel like this is a more comfortable setup -- maybe because each student has a little group of peers or friends around them, or something. I remember liking it better when I was in school, anyway. What I observed was a lot more class participation than I would see last year, without a huge uptick in screwing around. Of course, this is also a different teacher with a different teaching style than last year, so we're comparing apples to bananas here at best. Not even oranges!

More next time on how the introductions went and the plan for my role in the class.

If I tell you one thing, kid, it's this: Know your audience!

Well, I've had good days, and I've had flops. This particular incident falls in the flop category, and the lesson is simple: know your audience.

I gave a talk on the Fibonacci sequence, which is the recursive sequence where every number is the sum of the previous two: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc. There were some cool problems you can look at with this sequence, including some combinatorics puzzles, the old rabbit breeding problem of the Fib-meister himself, and lastly some interesting examples of the sequence showing up in plant patterns in the natural world, called phyllotaxis.

So, to review, I gave a talk about an abstract sequence of numbers and how it connects to flowers. In retrospect, I should perhaps have seen this one coming. It was just about impossible to keep anyone's attention during the talk, possibly because the combinatorics puzzle I led off with was just too hard. Normally a solid opening can land you a good free five minutes of attention span -- instead, I gave them this puzzle:

There are five steps leading up from my apartment door to the main entrance to my apartment building. Since I hurt my knee (true story!), I can only take steps one at a time, leading with my good leg. If I'm in a hurry, I can take two steps at a time, still leading with my good leg. However, sometimes in the morning I don't think of it right away and I'll take one step at a time for the first couple steps, then start taking two at at time. Or, I'll get tired or my leg will hurt so I'll take two at once at first, and then just take one at a time the way to the top. How many different ways could I take one or two steps to the top of the stairs?

I thought this was a pretty darned cool puzzle, since the situation is basically a true story and I was in fact doing exactly this on many morning while my knee was still in really bad shape. The key is to think about how you take the last step to the top -- you either reach it taking a single step, or a double. That means that really the number of ways to climb five steps is really the sum of the different ways to climb three steps or four steps. If we think of a sequence of numbers, where the n-th number denotes how many unique ways there are to climb a set of n steps, we've just proven that the n-th number is the sum of numbers n-1 and n-2. Sound familiar? That's right, it's all tied into the Fibonacci sequence, and the solution to the puzzle above is F(6) = 8 unique ways. (Note: F(1) is the number of ways to climb zero steps -- you just stand there. Just one way to do it. That's why F(6) is the number of ways to climb 5 steps.)

So that went over like a lead balloon. I skipped over the rabbit breeding problem at this point, because that one is ever trickier, and instead I went straight to the plant stuff -- I figured pretty pictures with neat designs might re-capture the lost attention of my students. But, they were pictures of flowers, pineapples and pinecones. If they were pictures of rockets, maybe. Pictures from video games, sure thing. Plants no. So, despite an array of neat images showing patterns in how the Fibonacci sequence describes flower petals, pinecone ridges and many other things, I got no traction at all.

I did fare a little better with an activity where we drew out a simple model of a nautilus shell, which can be modeled by drawing a sequence of squares with sidelengths given by the Fibonacci numbers (see here). I thought this sort of break from math-type stuff and detouring into a more geometric application might appeal to some of my students who I've seen doodling occasionally. I actually think it's a shame that geometry and probability aren't more integrated into the algebra classes like they were in the AlGeoStat sequence of classes in my high school (Plymouth-Canton School District). Frankly, basic algebra is really boring until you learn enough to start looking at more interesting problems, and geometry and intro stats bring a little bit of spice into an otherwise bland offering.

Anyway, the reason the Fibonacci sequence actually spits out something that looks like a nautilus shell is because the limit of the ratio of successive Fibonacci numbers is a constant value (in fact it's the golden ratio phi = 1.61....), so you're really approximating a logarithmic spiral. The upshot of this is that the spiral is self-similar at every scale, so a growing baby nautilus will grow such that it always fits in its shell and doesn't need a new one. While we discussed that effect, going into the math reasoning behind it in class would have been straining credulity at best, and by this point I was belatedly remembering that very important lesson:

Know. Your. Audience!

Student self-teaching: results

The practicum: see a few posts back, but it's basically "group quiz with a twist", namely, after the class does the group quiz together, one person selected at random has to get up in front of the class and explain how to do the quiz. The prize is no homework for the whole class.

This went surprisingly well, but not exactly as planned (surprise!). By treating the first part of class like a real quiz, with everyone working on a what looked exactly like a quiz sitting right in front of them, I think we really got everyone's attention -- in fact, despite my explanation I'm sure some students took a while to realize it wasn't a real quiz. The promise of no homework helped as always, which I still find shocking considering how few students turn in the homework anyway.

Nevertheless, I think that ten minutes of confronting a problem you have no idea how to do while the class is silent all around you can have an illuminating effect on the limits of your knowledge. One benefit of a practice quiz during class time is that you experience the sinking feeling you get from taking an exam you weren't prepared for, but before you've actually done any damage to your grade. Thus, sinking feeling of despair transforms into motivation, hopefully. A glimpse into the fiery depths, if you will, but with a chance for salvation immediately afterward.

Anyway, the first not-as-planned part came during what was supposed to be the self-teaching section. As it turned out, it ended up being more of one-on-one teacher-teaching, with little helpful student-student interaction. We rolled with this, though, and just devoted more time to helping everyone get their questions answered. In retrospect, I think I should have more actively encouraged the students I could see getting it to help their classmates.

Once George and I explained to everyone how to do the problems (without actually giving answers), the class voted on the answers to each problem. There were some disagreements on some problems, but the majority of the class that voted was usually right, and both 2nd and 4th hour were able to democratically arrive at all 5 correct answers.

There was also a lot of shyness in the second half of the practicum, where one person picked at random was supposed to come up and do the quiz on the overhead now that we'd gone over everything together and voted on the answers. I had a student pick a name from a hat at random in each class, and in one class that random person flat out refused to participate. It wasn't a belligerent thing, either, just an eyes-downcast, mumbling, embarrassed cop-out from a student who usually won't shut up! This was a good reminder of an important point. These kids try to act all tough and invincible and too-cool-for-anything, and it's all just as much of an act now as it was when I was their age -- with all that hard exterior they are just marshmallows inside!

I ended up taking volunteers in that class, one person for each of the five questions. Several students were really itching to get up there and show that they knew how to do the problem, and it's hard to refuse that kind of enthusiasm.

So, all in all it wasn't a disaster in terms of people bouncing off the walls or hanging from the ceiling. But did it work? Was it worth doing this versus the usual pre-quiz review of similar problems on the board?

By the numbers, correcting for students who missed the Thursday practicum and students who did not take the quiz at all, class averages were up slightly (about 0.5-1 points out of 15 total) compared to the most recent quiz. Now, is that meaningful? Perhaps the subject matter was just easier that week than it had been the week before. This would be a good problem for the stats classes in 1st and 3rd hour, but I think the answer would come down to insufficient data to draw a statistically significant conclusion.

Anecdotally, yeah, it was seemed like it was working. George and I talk a lot about what I should try to bring to the classroom, whether I'm presenting, just sitting in and helping out, or planning an activity and teaching like this. We've come to a conclusion that one of the most important things I can do is shake up the routine that the students are used to. If they're used to coming in and tuning out because they know what's coming and they don't like it, then anything I can do to rattle their cages and get them engaged in something new is worth it. And, judging by the increased attention span during the practicum relative to their usual quiz review, I'd say that's about right.

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.

The best way to make a difference?

Between exams, confusion over the start of the semester and, finally, an ill-timed "wind chill" day, I've only been in a few times this semester. Still, I've reached a dilemma -- how can I best use my time at Ypsi High to positively influence my students? Do I attempt to inspire or enrich as many students as possible by dividing my attentions over the whole class, or do I focus on bringing a small, struggling subset up to par? How can I balance these priorities?

I initially gave presentations on technical topics, trying to bridge the gap between contrived textbook examples and the students' or my own experiences. This can lead to some nice eureka moments ("You mean somebody actually uses this stuff?", or, "Wow, that's cool", sneakily making math cool by association), at least in the classes of older students, but it sometimes spreads thin on a class of 25 freshmen, over half of whom are failing basic algebra. It begs the question -- is my time better spent coming up with entertaining uses of math designed to appeal to a broad audience, but having little measurable effect, or else in concentrating on those few struggling students who would most benefit from extra attention?

This dilemma became acute in the leadup to semester exams. After observing so many students struggling with solving basic algebraic equations for lines, slopes and intercepts, presenting some bit of engineering entertainment seemed... irrelevant. Instead, with George's help I picked out a few borderline students in each class, kids on the knife-edge of passage or failure, that I hoped could make the leap with some extra time in a small group. We looked for a profile of students struggling to pass but still caring enough to put in an effort -- the kind of student who would benefit from a couple of days of tutoring, one-on-one or in a small group -- and I cleared my schedule to come in the Thursday and Friday before exams. We ended up with four students, three in a group and one by himself spread over two hours, and without the distractions of twenty other kids I walked them through as much material as we could manage, guided only by the areas that they found most confusing. While the group of three only included one student who fit both criteria -- one girl was doing well enough to pass (though struggling with recent material), and another just didn't want to be left out of going with her friends -- I figured I could handle three as well as one.

Despite the weather holiday that Friday depriving me of half my teaching time, one of my borderline students passed (I have yet to hear the result from the other). A small victory, to be sure, but a victory nevertheless. Still, while the imminent exam made this victory clear, it's less clear how effective my prior efforts were. Will someone who enjoyed one of my science talks and passed last semester discover a subject they might once have thought dull and dry? Where is the balance between lifting up the struggling students and helping the gifted ones soar?

For now, I'll enjoy my small victory. But next semester we return to square one, teaching those students who failed Algebra I and must repeat the course. I hope a few of them will soar, too.