In precalculus we recently learned how to solve triangles. This involved such equations as sin 20 = 5/x, or tan 35 = x/7, or cos (theta) = 6/11. From past experience I know there are students that forget how to solve things. Once you throw "sin 20" or "tan (theta)" into the mix, all their algebra knowledge goes out the window. So I carefully walk through and remind them of how to isolate variables, and assure them that "sin 20" is JUST a number, and warn them that, "no, you are not multiplying 'sin' by '20', you must keep them as a unit". And then, WHOA, stop the presses, x is in the denominator?? Well, why don't I just take 'sin 20 = 5/x, and divide by 5 and POOF, an x is left in the numerator on the right hand side.
Anyhow, we walk through this, and most kids get it. But then I see one student struggling and struggling and getting more frustrated when I'm going through the different types of examples because he doesn't know why things change up and why I'm "solving differently" for the different situations. Well, he came in for tutoring this past morning, and by probing his mind, I see that he never internalized the algebra steps: what's being done to x? how do you undo it? do the same thing to both sides of the equation.
He would look at an equation like 3x - 7 = 5, and reason it out in his head: well, something minus 7 is 5, so that something must be 12. Then 3 times something is 12, so that something, or x, must be 4. Great reasoning, but then he never had to learn or practice: add 7 to both sides. divide both sides by 3. And now when he sees: sin x - 7 = 11/3 (or something like that), his old methods don't serve him well. I believe we got him to the point where now he can do it. I'm so glad he came in for tutoring, so I could take the time to probe deeper as to where his difficulties lie (lay/laid/???). I don't have that luxury in class with 33 other students of all levels waiting for "teaching".
Kids who learn to "reason out" one and two step equations, with integer solutions - they are a nightmare. I find that at the beginning of a (hard) algebra class, everything seems easy, they don't pay attention because they think they can already solve equations.
ReplyDeleteOnce in a while there is an answer like 11/7 that they don't get, but no big deal for them, since they get everything else.
And then the work gets hard, and they are lost, because, exactly as you say, their methods are not resilient.
We straighten most out, but it's so much harder than it has to be.
Now, go to another class, another school. How many algebra I courses avoid fractions almost entirely?
And that's how they get to trig like that.
At least, that's what I think.
Jonathan
I'm with you there. When I first came to this particular school, it was an eye-opener that I still needed to constantly reinforce fractions and bring them up and actually teach them to algebra 2 students and to precal students AND to calculus students. I've now made my peace with it (or I've been desensitized).
ReplyDeleteI guess it's how math is taught now, we work on a skill for a week or two and then move on and unless we're constantly aware of "spiraling back", the kids (being human), forget the old skill in place of the new.
Ms. Cookie
Here's the story of my appendectomy, which, believe it or not, is relevant.
ReplyDeleteWhen I was in my senior year of HS, I was out for 2 weeks recovering from an emergency appendectomy. It just so happened that during those 2 weeks our calculus class moved from differentiation to integration. My first day back to school, my calculus teacher was out, but there was a note on the board "reminding" us of the test scheduled for the very next day. So, I decided to catch myself up on 2 weeks worth of calculus in one night. ;-)
I don't remember if I didn't read as far as "integration by parts" or if I did, but hadn't really internalized it. I certainly hadn't practiced it. But we *had* been practicing differentiation for many months. So, I solved every problem on the test by inspection.
My teacher, shocked that I even came in intending to take the test, did not dock me for not showing my work, or for not using the method she had intended for us to use, but she did take me aside and quickly taught me integration by parts. ;-)
Of course, as the term went on, we also had problems that did not yield readily to inspection, which drove home the need for the method! It seems like that may be missing in some algebra classes. The time should come when you can't "eyeball it" and have to use the methods that have been taught. And that time should probably come sooner rather than later. (We had a similar issue when my youngest didn't want to learn borrowing, because he could use mental math much more easily. -- He borrows just fine now after his teacher decided to give him some "bigger" problems to work on.)