Weekends mean getting out during "daylight" hours, so I get to see what this town looks like with some light on it. Looks... snowy.
Problem solved! Here's the process I've been on today:
Basic math skills (familiarity with numeric computation, +/-/x/÷ of integers and fractions) are not on par with grade level, nor at a comfort level that allows me to proceed further with a math curriculum. It's very difficult to teach the methods of solving a two-variable system of linear equations when cross-multiplying two simple fractions brings the conversation to a standstill. The solution to this, in my experience, is math drills. Spend a couple minutes at the start of every lesson just drilling math computation way below the learning threshold (if you're working algebra, drill basic integer multiplication; if you're learning how to combine fractions with different denominators, drill integer addition, etc.), so I set out today to produce some worksheets for speed drills at the start of lessons. There's a problem inherent in that, though, and it seems to be cultural. If I give everyone a copy of a worksheet, I will get back a class average of what the kid with the best grasp of numerical computation could do, multiplied by some ratio of how fast people can copy off of her spreading outward away from her desk. It's not a widespread acceptance of cheating, to my eye, it's more like a lack of cultural understanding of the idea of cheating. Somewhere along the way in my schooling experience (and most of the schooling I've run into in the lower 48) it is stipulated between educators and students that the point of school is to do your own work, copying is cheating and invalidates the point of the lesson, and while students still choose to copy off work due to laziness or whatever, it's done with an understanding that it is working at cross purposes with the educator. Here it doesn't seem like anyone's internalized that idea. So when I'm constantly asking people to put their eyes on their own paper for a quiz I'm not met with "oh you caught me cheating", I'm met with a frustrated lack of comprehension of why I'm not letting them work on it in a better way - by working with someone who's better at it. I'm clearly just being obtuse and difficult by not letting them work on it in the most efficient way.
Far be it for me to try and effect some massive cultural shift in these students, or single-handedly get major buy-in to the Western educational process. Instead, the solution seems to be (and this worked for the quizzes on Friday) to just produce many versions of any worksheet, so no person is sitting near someone else working on the same material. We've all seen this in schools where cheating is a problem, on major individual tests. The onus of work that puts on me, however, is huge. To do speed drills with 40 students a day means writing up 40 individual worksheets every single day. I'm willing, but would prefer to spend some of my out-of-school time working on the actual lesson plans and the material I'm *teaching*, not the material they're behind on that I'm remediating. So I sat down at the computer today, and wrote up a python script that generates, based on random numbers, any number of worksheets with simple +/-/x/÷ problems using positive or negative integers or fractions, based on user prompt. I fire this program up, tell it I want to work on adding fractions with different denominators ranging from 1 to 12, and I have 8 students in the class, and it spits out 8 pdfs of different worksheets using LaTeX (so they actually look like math worksheets, not some MS Word garbage with fractions formatted like an English major would format them) that I can print off and distribute. Tomorrow we'll see how passing your sheet to the left and having your neighbor check your answers with a calculator goes over... but for the moment I have the material to drill computation with. I'm feeling pretty proud of that. Now, of course, spending the day working this solution up has put me a day behind on lesson planning for the week, but tomorrow I think I'll have my hands full going over the quiz and giving them an opportunity to correct their work and turn it back in for partial credit - not a single student aced the quiz (I think I had an 80%?) which is my problem, not theirs, I overshot my estimation of comprehension based on the first week, but at least the first step in reaction to that is to remediate exactly what they missed on the quiz. So that'll take me through the day tomorrow, and tomorrow after school I figure out where I want to turn this ship this week.
The nautical metaphor feels really apt out here. I've been adjusting the throttle all week, and the quiz was a big throttle-adjustment signal. But now that I think I'm dialing in the speed, I've gotta get a clearer idea of the heading. Do we just keep plowing ahead towards that beacon of calculus like most high school math classes (if you look at any high school math curriculum, it's one big steamship burning for calculus, taking detours to the probability and statistics islands, with a quick stop off at analog clocks and measuring or whatever silly state standard got passed off on the math department), or do I loop back and do a thorough trawl through the poorly-charted waters of intro algebra? Is less progress learned more fully more useful to these students? The number of students I have even considering college as an outside possibility is incredibly low, so it's hard to see what the actual benefit to a calculus-leaning curriculum would be at all...
More to come soon.
Cameron, that solution is amazing! The great variety of your skills, and ability to see ways to use them to solve problems, will make you the one teacher who will succeed with these kids. It's a great example of why learning a little of everything is so valuable.
ReplyDeleteStupid iphone won't publish my comment.
ReplyDeleteEach blog post I read I keep thinking: see how his theater background has prepared him for this! Long hours! No sleep! Scrapping plans! Problem solving!
But this one is straight up good teaching. Nicely done! Alaska is lucky to have you Cam. They WILL remember the maths!
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DeleteTheatre is the best possible teacher training.
DeleteGood immediate solution! My two cents:
ReplyDeleteThat beacon of calculus is, as you suggest, the wrong goal. If a student eventually gets calculus -- which they can do later in life, like in college if they go there -- they'll only do it with a good understanding of functions. So giving them a sloppy, incomplete understanding of functions does them no good. And if they don't go to college, they don't need calculus.
So what do they actually need beyond basic calculation? It seems to me that ONE point of algebra is to lead towards functions. And a deep point of functions is that in life, some mathy answers are not numbers but rather relationships. We don't HAVE to represent relationships symbolically, but there are genuine situations where the abstract symbolic representations we try to shove down the cherubs' throats are actually useful.
What else they really need is an interesting question. I think that the detour to stats is critical; one way to look at that is that in life, some mathy answers are not specific numbers, but rather fuzzy ranges. Students who have the mistaken impression that math is only good for precise calculation need to see how we can deal with uncertainty.
Whew. Anyhow, my only concern is that you not fall prey to "false prerequisites." I think you know this, that the computation drills are good only in small doses. You're right that they need them, but keep looking for ways to address the higher-order stuff -- stuff with meaning for them -- even when their skills are weak.
Keep up the good work!
Hmmm... maybe I was unclear in the metaphor - by beacon of calculus I of course didn't mean to imply that the goal was to reach calculus in high school. Perhaps I haven't made clear the distance that particular beacon is from our current location.
DeleteThis particular statement: "And if they don't go to college, they don't need calculus." I find to be incredibly dubious and in need of support, but that's another conversation.
I guess what I'm saying with the beacon of calculus is this (to use functions as a fantastic example): if I'm teaching functions, what am I teaching that they mean? All mathematics can be understood from either a linear algebra, geometry, or calculus perspective. What is a function? Is it the solution to a pair or group of equations? Is it a set of numbers on a graph represented by an equation that instructs you how to draw it? Is it a manipulable, differentiable, and integrable object that can be divided infinitessimally? More specifically, when teaching continuity of functions do we understand it primarily as a place where you pick up your pencil when graphing it? Do we understand it primarily as a place where the equation has no solution? Do we understand it primarily as a place where no matter how hard you zoom in on it it's the same function?
That's what I meant by "beacon of calculus". The lack of a coherent metaphor is perhaps my biggest pet peeve with shoddy mathematics instruction. As far as a metaphor as a guide and scaffold goes, it doesn't matter how close you get to the eventual destination (you don't have to ever reach a strong understanding of the hyperbolic plane to approach things from a geometric perspective, and you don't have to differentiate a single equation to have calculus as your guide), but it is, I think, vitally important that you pick *one* and stick with it as your primary form of introducing a new concept (growing extensively). This doesn't preclude filling in the other metaphors when growing expansively, and mapping the homomorphisms between them when growing intensively; but it means that the students know their first tool for reaching out to grab a new concept is always the same one for a course of instruction. The lack of that coherence in our primary and secondary instruction is, to my eye, the largest cause of students that don't want to reach out and grab new concepts - they don't have a go-to tool to take hold of a new concept.
"What else they really need is an interesting question. I think that the detour to stats is critical; one way to look at that is that in life, some mathy answers are not specific numbers, but rather fuzzy ranges." Sure... first I have to get them to believe that some of the answers in life are specific numbers. The reason why I'm hesitant to make the detour to stats yet is that the innate connection between mathematics and the real world is shaky at best here, and completely absent for many. I'm not exaggerating - the understanding that when the symbol shaped like an "8" and the symbol shaped like a "6" with a little x next to it are on the board, the teacher wants you to write the shapes "4" and "8" and you get rewarded is literally the extent of mapping math onto the world that has been done for many students here. So as far as "computation drills vs stuff with meaning" debate - I think you may be taking for granted an erroneous understanding of the degree to which the computation itself has meaning. It's a different world out here.
it is fascinating to read about your mathematics side Cam. It is not something that I have been that exposed within the context of our theatre work together. I admire your innovation and willingness to improve your way through these difficult waters to continue your metaphor. Keep up the good work as you continue to explore the skill level of your kids, the curriculum you feel you should be teaching and what kind of higher-order questions you need to be asking of them. What kind of guidance or expectations does the school itself have for these students and you as it relates to mathematics. Also have you done a village reading of A Noble Failure yet?
ReplyDeleteTake care and stay warm.
Don