Some other thoughts: perhaps you could give me some examples of specific teaching goals you have, and specific problems you often encounter?
Sure, I can see that would be helpful. Right now I have a bunch of SAT prep students, and I teach college kids calculus when there’s a demand, but for the sake of argument let’s consider Algebra II. One of the goals in Algebra II is to get the student comfortable with polynomials: factoring, multiplying and dividing them, and understanding the relationship between those processes and things like zeroes and asymptotes of functions. So maybe we should talk about factoring?
Nearly all my students get the hang of factoring polynomials once I can convince them to sit down and practice for a while (which presents its own set of difficulties), but I’m sure I’m not teaching it optimally. Problems I run into: confusion about which term in a quadratic comes from what (“It’s supposed to multiply to this and add to this, right? Or is it the other way around?”); neglecting to look for common factors first; confusion/frustration when the leading coefficient isn’t 1; not recognizing special cases like difference of squares (only sort of a special case), higher degree polynomials in quadratic form, or sum/difference of cubes; not knowing when to use factoring by grouping. I have my own ad hoc ways of dealing with these problems, but I have no reason to believe they’re the best possible.
Maybe this is still too broad, or I’m assuming too much familiarity with the subject matter? I’m just tossing it out.
ETA: If the lowest hanging fruit in improving your own skills is to “stop doing stupid shit”, then it follows that the lowest hanging fruit in improving your teaching is to figure out how to get your students to “stop doing stupid shit”. :P
I like this idea. I do pretty much re-teach how to use fractions (and to a lesser extent exponents) whenever they come up, but much as I would like not having to do that, I’m not sure the problem is easily solved. Kids don’t learn how to use fractions partly because they don’t believe they need to; they decide in elementary or middle school that “decimals are way better and you can use a calculator,” and once they’re in high school they find out about “Ans=>Frac” on their graphing calculators. In my experience they really, really resent being drilled on fractions, and forget what they’ve learned very quickly because they refuse to use it for anything else. Maybe I’m being too cynical, though.
I do think it would be very useful to put together a “quit making stupid mistakes” program—if I could get kids to stop making sign errors, or doing the wrong operation because they didn’t think about it for half a second, their test scores would probably soar—and indeed I’ve seen this happen with at least one student before, so I should probably try to implement it systematically.
Hmmm, there could be lots to reply to in that post, but I’ll try to keep it brief...
Can you give me a few specific examples of actual tasks that your students have problems with most commonly? Like, show me exactly what the students are presented with.
With that, I might be able to do a transformed task analysis, and develop an example cognitive routine.
Actually, factoring is used as one illustration of a cognitive routine in Theory of Instruction. I’ll scan that section when I get time.
Ok. Well, we were talking about factoring. Here’s a factoring problem I would not expect most of my students to get:
Edit: Sorry, I guess you wanted more than one example? Not sure whether these are supposed to all be examples of the same basic type of problem, or different, or what, but I added a couple more factoring problems.
The last one might throw me off if I didn’t remember off hand that a^3+b^3 has a factoring—is there a reason one should be especially likely to mistake the others?
These are not intended to be hard factoring problems for LessWrong, as if there were such a thing. They’re hard factoring problems for most high school algebra students because none of them have a leading coefficient of 1, and because on all but the first you have to remember to look for common factors before they even look like something you can cope with. The first is mainly hard because most kids, when they even remember how to deal with the 4 at all, will try to factor it as (2x+c)(2x+c) and then give up.
These are not intended to be hard factoring problems for LessWrong, as if there were such a thing.
Successfully factor inferential distance into how hard it will be to convince people in your life of the following:
Probability is subjective
Science and its method are a social model for humans to avoid believing in untrue things, not a logically correct way to discover what is most likely true.
Absence of evidence is evidence of absence (only applies for those who were or will be in college)
Maybe the “completely” is inappropriate here? In my defense, that’s what textbooks at this level usually say when they mean “factor as far as possible using real, integer coefficients”.
Sure, I can see that would be helpful. Right now I have a bunch of SAT prep students, and I teach college kids calculus when there’s a demand, but for the sake of argument let’s consider Algebra II. One of the goals in Algebra II is to get the student comfortable with polynomials: factoring, multiplying and dividing them, and understanding the relationship between those processes and things like zeroes and asymptotes of functions. So maybe we should talk about factoring?
Nearly all my students get the hang of factoring polynomials once I can convince them to sit down and practice for a while (which presents its own set of difficulties), but I’m sure I’m not teaching it optimally. Problems I run into: confusion about which term in a quadratic comes from what (“It’s supposed to multiply to this and add to this, right? Or is it the other way around?”); neglecting to look for common factors first; confusion/frustration when the leading coefficient isn’t 1; not recognizing special cases like difference of squares (only sort of a special case), higher degree polynomials in quadratic form, or sum/difference of cubes; not knowing when to use factoring by grouping. I have my own ad hoc ways of dealing with these problems, but I have no reason to believe they’re the best possible.
Maybe this is still too broad, or I’m assuming too much familiarity with the subject matter? I’m just tossing it out.
I like this idea. I do pretty much re-teach how to use fractions (and to a lesser extent exponents) whenever they come up, but much as I would like not having to do that, I’m not sure the problem is easily solved. Kids don’t learn how to use fractions partly because they don’t believe they need to; they decide in elementary or middle school that “decimals are way better and you can use a calculator,” and once they’re in high school they find out about “Ans=>Frac” on their graphing calculators. In my experience they really, really resent being drilled on fractions, and forget what they’ve learned very quickly because they refuse to use it for anything else. Maybe I’m being too cynical, though.
I do think it would be very useful to put together a “quit making stupid mistakes” program—if I could get kids to stop making sign errors, or doing the wrong operation because they didn’t think about it for half a second, their test scores would probably soar—and indeed I’ve seen this happen with at least one student before, so I should probably try to implement it systematically.
Hmmm, there could be lots to reply to in that post, but I’ll try to keep it brief...
Can you give me a few specific examples of actual tasks that your students have problems with most commonly? Like, show me exactly what the students are presented with.
With that, I might be able to do a transformed task analysis, and develop an example cognitive routine.
Actually, factoring is used as one illustration of a cognitive routine in Theory of Instruction. I’ll scan that section when I get time.
Ok. Well, we were talking about factoring. Here’s a factoring problem I would not expect most of my students to get:
Edit: Sorry, I guess you wanted more than one example? Not sure whether these are supposed to all be examples of the same basic type of problem, or different, or what, but I added a couple more factoring problems.
Factor completely.
4x^2+11x-3
3x^3-13x^2-10x
3x^5-3x
2z^3+16
blinks
The last one might throw me off if I didn’t remember off hand that a^3+b^3 has a factoring—is there a reason one should be especially likely to mistake the others?
These are not intended to be hard factoring problems for LessWrong, as if there were such a thing. They’re hard factoring problems for most high school algebra students because none of them have a leading coefficient of 1, and because on all but the first you have to remember to look for common factors before they even look like something you can cope with. The first is mainly hard because most kids, when they even remember how to deal with the 4 at all, will try to factor it as (2x+c)(2x+c) and then give up.
Successfully factor inferential distance into how hard it will be to convince people in your life of the following:
Probability is subjective
Science and its method are a social model for humans to avoid believing in untrue things, not a logically correct way to discover what is most likely true.
Absence of evidence is evidence of absence (only applies for those who were or will be in college)
Good luck!
Its roots will be multiples of the three complex cube roots of unity, so it can’t be factored in the reals.
Maybe the “completely” is inappropriate here? In my defense, that’s what textbooks at this level usually say when they mean “factor as far as possible using real, integer coefficients”.
All I was looking for was 2(z+2)(z^2-2z+4).