In one of my jobs, I’m a paraprofessional in a high school science classroom. Last week in Biology we were in the unit on carbohydrates, lipids and proteins as polymers. Of course, the students have been struggling because we’re touching on biochemistry concepts, and they’ve not had chemistry.
So I asked the teacher, “If biology depends upon chemistry, and chemistry depends upon physics, then WHY don’t we start with physics, and then go to chemistry and then biology?”
(I mean like, duh!)
And he replied, “Because most of these students don’t have the math background for physics; they’re only in geometry.”
That’s Geometry as the class that for some reason is inserted between Algebra 1 and Algebra 2, before they get to Trigonometry. Actually, a person can get through basic physics without using calculus. But you do need algebra. And you especially need to be willing and able to tackle “story problems”.
I think this whole issue with “story problems” illustrates a very basic problem in a great deal of American maths instruction. Mention “story problems” and the students groan and whine, if not almost break out in a rash because “they’re confusing and hard and [we] don’t understand them”. Unfortunately, a lot of textbooks (and educators) focus too much on rote drills using the particular method for that lesson, for example, “how to use distribution” by multiplying 5(3×-2). The problem with such approaches is that it focuses the students upon recognising a particular “kind” of problem, and then using whatever method was taught with those kinds of problems in that section. The students often lose out on when to use particular methods, and why.
(The most daring and illuminating thing a mathematics teacher ever told me was in my college algebra class, when she said to us, “If you don’t like it, then change it!” Meaning, if the way the problem is set up is difficult to do, then you can manipulate it into an easier form. Naturally, that’s what a lot of algebra is really about — juggling the knowns to figure out the unknown — but it had never been explained as something where each of us were allowed to change things around to what we needed, rather than trying to guess what some anonymous person thought should be done to the problem.)
Another thing that students often miss when learning too much rote mathematics is determining how to identify and then create their own equations (& inequalities). You know, the “When am I ever gonna do THIS in real life!” whine. Well, that is to some degree a valid complaint; everyone wants to learn things that have some use in their own lives.
But in real life, no one is ever handed a textbook of problems to solve, much less one with the answers to the odd-numbered problems in the back of the book. This is my answer to those who complain (about any subject), “I don’t care how it’s done, just tell me the right answer.” Because the really important part of the instruction is not the answer, but knowing how to get the answer! (The teacher doesn’t really need dozens of students to tell them the answer; the teacher could calculate the answer, or look it up in the teacher’s textbook, assuming there’s not a typo. I’m always amazed at how many typos were in the answer keys!)
In real life, we don’t have ready-made equations, aside from various bookkeeping functions. Rather, we have an assortment of situations where we need to figure something out, usually as a part of a much larger project we’re working on. For example, this weekend I will have to calculate area of a yard, then do a concentration conversion, then calculate the amount of material to spread on my lawn. Of course, in real life I can’t go down to the hardware store with a number like “3.6666667.” In real life, I would end up buying a 5-pound or 2-kilogram bag of whatever.
That’s a story problem; I have to identify which variables I need, and which formulae to use, select conversion factors (if needed), and then create my own equations using whatever quantities I know, to figure out which quantities I will need.
Regardless of whether or not we use calculators, we still have to know how to set up the problems, and how to solve them. Calculators enable us to do the drudge-work more easily and quickly, so we don’t lose sight of what step we’re on, or what we’re trying to solve for. I have no quibble with using calculators for most kinds of mathematics or science instruction. Regardless of what kinds of tools you use, you still have to know when and how to use them. No matter what some students think, the calculator doesn’t tell you the answer to the problem; it only displays the result of what you punched into it.* GIGO, Caveat emptor, et cetera.
Back to teaching science. So in this simplistic sense, we can’t teach our science in a sensible physics-chemistry-biology order because our students can’t do the maths. At our local schools, the state has decided to start doing science assessments, which if I understand correctly, will have one particular grade getting a physical sciences assessment, and another grade getting a biological sciences assessment, but because the current biology students didn’t do the physical sciences assessment last year, they have to do it and the biological sciences assessment this year. (I don’t understand the rationale for that, either.) Which of course means that the science teachers are taking time from teaching the curriculum for doing assessment preparation.
As the saying goes, “No one ever grew by being measured.” Although I think most everyone agrees that we need good teachers, and that we need to hold schools accountable for providing the educations that our students need, I don’t think that the constant focus upon assessment tests is the way to achieve that. We end up placing too much focus on test-preparation instead of teaching, and reciting or identifying the “right answer” instead of knowing how or why it’s the right answer.
Bug Girl has a nice little piece, “We’re taking our math and going home in a huff” about the Bush administration not wanting to participate in the next global test of advanced math and science, quite possibly because US students don’t compare well to their peers in many other countries, and the results wouldn’t reflect well upon the 2002 No Child Left Behind Act.
When all is said and done, we still end up placing too much focus on getting the “right answer” on the NCLB stats, instead of knowing how or why it’s the right answer, or more importantly, if it answers the questions we’re asking, or if we’re even asking the right questions.
Such as the important question about the most sensible order to present different science subjects.
* No matter what some students think, the calculator doesn’t tell you the answer to the problem; it only displays the result of what you punched into it.
“Seven!” he crows, shoving the calculator in my face.
“And what does ‘seven’ mean?” I ask.
“That’s the answer — seven.”
“No, seven-what? What do you have seven of?”
Cross retort: “I dunno.”
“Well, what’s the problem asking?”
Student recites, “How many [blah-blah-blah]…”
“Okay, so what do you have seven of?”
I wait a moment, letting the mental cogs catch up.
“Oh,” he realises, “Seven pizzas.”
“Right.” I then model translating the numeric result into a verbal answer, “To feed everyone at the party you would need seven pizzas. This is what you use math for.”
“Hah! I would just buy a bunch of pizzas and eat them ALL MYSELF!” comes his grandiose reply, accompanied by exuberant arm-waving and squirming at the desk.
I sigh at this last response; this is a program for students with emotional and behavioural issues, after all…
(See my additional comment below.)