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C. did not recognize this expression as a case of the distributive property:
2x(x+1) + 3(x+1)
Can't say I blame him.
update
from Barry:
This isn't real obvious when you first come across it. So do this. Let's let x + 1 = T.
Now substitute T in the expression and you get:
2x(T) + 3(T)
Can you factor out the T?
Yes. You get T(2x + 3)
Now substitute the x + 1 back in. You get:
(x+1)(2x+3)
I think it was Ron Aharoni who referred to seeing expressions such as (x+1) as single entities as "chunking". To help students do such "chunking" it helps to do what I did above so they can see that x + 1 represents a number, and as such it can be factored.
I'm amazed by how difficult it is to see that "expression X" is the same as "expression Y." This is an ongoing source of pain in my mental life these days (not to put too fine a point on it), because I came to feel, shortly after Animals in Translation was completed, that Temple's & my thesis concerning hyperspecificity in animals and autistic people is wrong in some important way -- either wrong or perhaps right for the wrong reasons.
We argued that autistic people, children, and animals are hyperspecific compared to typical adults. Autistic people, children, and animals are splitters; nonautistic adults are lumpers; etc. (I know I've said all this before, but feel I must repeat in case newcomers stop by.)
The classic hyperspecificity story re: autistic children is the little boy who was painstakingly taught to spread butter on bread and then had no clue how to spread peanut butter on bread.
Until I began to reteach myself math, this kind of thing seemed to me incontrovertible evidence of the otherness of the autistic brain. But now that I'm factoring trinomials I've discovered I have something in common with that little boy. That's probably why God or the universe decided I should take up math. I needed an object lesson.
Still, the observations Temple has spent a lifetime making of animals' (and autistic people's) hyperspecificity aren't wrong. Normal adult humans aren't hyperspecific in the same way animals and autistic people are hyperspecific.
Sometimes I wonder whether the issue is simply that non-autistic adults pass through the hyperspecific stage of knowledge more quickly or more frequently than autistic people do. When a "typical" adult (typical being the preferred term these days) encounters brand-new material he, too, is hyperspecific, as I am with math. Everyone starts out a splitter.
But I don't think that's quite it, either.
I'm getting the feeling that animals may not be hyperspecific across the board, but perhaps only in certain realms. Maybe animals are more hyperspecific than adult humans when it comes to sensory data? e.g.: To a horse a saddle feels completely different at a walk, a trot, and a canter -- so different that he will buck his rider off when he moves from a trot to a canter if he hasn't been carefully trained to tolerate the saddle at all 3 gates individually.
Better story: Temple's black hat horse.
This was a horse who was terrified of people wearing black hats. He wasn't terrified of people wearing white hats or red hats. Just black hats.
I'm thinking, this morning, that humans may be relatively oblivious to "sensory data," that we're lost in words -- so perhaps words are the place where you'll see us being hyperspecific ? (There's evidence that language masks sensory data, but I don't know it/remember it well enough to summarize.)
Temple complains about this all the time. She'll give a talk and her audience will take away a too-specific meaning from her words; then they'll go out and apply her advice all wrong & bollocks things up. "People get hung up on the specific words," she'll say. (I'll write down the next example of this that crops up - can't think of one offhand.) These conversations have gotten to be quite funny because, after years of reading countless articles on autistic people being literal-minded and "concrete," I am now spending my time listening to an autistic person complain that normal people are literal-minded.
Well, she's right. Looking at an expression like 2x(x+1) + 3(x+1), I'm like the horse with the saddle and so is my 13-year old son.
"x+1" next to 2x is completely different from "x+1" next to 3.
Different enough to make us start pitching our riders into the haystack.
percent troubles
Robert Slavin on transfer of knowledge
rightwingprof on what students don't know
Inflexible Knowledge: The First Step to Expertise
23 comments:
This isn't real obvious when you first come across it. So do this. Let's let x + 1 = T.
Now substitute T in the expression and you get:
2x(T) + 3(T)
Can you factor out the T?
Yes. You get T(2x + 3)
Now substitute the x + 1 back in. You get:
(x+1)(2x+3)
I think it was Ron Aharoni who referred to seeing expressions such as (x+1) as single entities as "chunking". To help students do such "chunking" it helps to do what I did above so they can see that x + 1 represents a number, and as such it can be factored.
What was C. supposed to do with that expression?
I remember wondering what I was supposed to do with an expression. Often, they just said simplify. I remember thinking that there was just one direction to go. I ran into this with my son who seemed to think that factoring was quite unnatural or backwards.
My reaction to your problem was to expand and combine. I could then find the factors, if needed, but I didn't think of that first.
I've just added an "update" for anyone who's interested.
Barry - I wonder whether it would be a good strategy to....tell kids to be looking for "the same thing" - and to substitute "T"??
I'm going to try it with C.
Just have a formal, conscious "rule" that you "look for same."
I ran into this with my son who seemed to think that factoring was quite unnatural or backwards.
I think that's very interesting....I think it might be another aspect of the "unnaturalness" of math....
Our brains are put together to see "wholes." (The Betty Edwards books on drawing are brilliant at explaining this, btw. Best I've seen, probably.)
When you factor, you are forcing yourself to "see" or find component parts.
That's really not what we're built to do - not visually, anyway.
The instructions in this case were clear because these expressions are in the lessons on factoring trinomials.
He can factor a trinomial, including a trinomial with a coefficient before the squared term, but then he didn't recognize this expression, on its own, as what he comes up with in the middle of factoring a trinomial.
Barry - I wonder whether it would be a good strategy to....tell kids to be looking for "the same thing" - and to substitute "T"??
It might be good as a scaffolding technique. When I first came across factoring problems such as the one talked about here, I had trouble seeing it at first. But the algebra book we were using (similar to Dolciani's in organization and drills) had provided enough exposure to expressions such as (x+1) that had to be operated with as a single entity (chunking) that it didn't take much work to make the leap.
With today's typical algebra books in which the math is a sidebar to photos of kids in roller coasters and grannies riding motorcycles, this point is no doubt lost in the noise.
You've raised a sprawly subject, as my daughter used to say
First, somewhat off topic:
saddle feels completely different at a walk, a trot, and a canter depends upon the horse and the construction of the saddle. I am refraining here from a rant about saddle fit, professional horsepersons who should know better and don't, etc. etc. etc. Feel free to email me about saddle fit, "rein lameness" and the like. It is very common in horses on the show circuit, for example. And let us talk about shoeing-caused lameness--not the farriers' malpractice, but what the client demans. STOP! Rant off.
What I really want to talk about is developmentally appropriate education. I know that sounds like some deep-constructivist bushwa, but...
What if the standard school curriculum regularly presents concepts (my classic example is long division) before the average child (AC) is developmentally ready to master it? I mean, suppose at age 8, AC can master it in say, 100 hours of teaching and practice (and that age 8 and 100 hours is a figure I grabbed out of thin air)...but if the curriculum is re-jigged to present it X months later, when the AC is ready, it takes only 10 hours of instruction and practice to master.
The issue of transitioning from print to cursive writing is another iconic issue. Why teach cursive at all? Books aren't printed in cursive. What does handwriting have to do with reading? Why teach cursive in third grade?
Back to 2x(x+1) + 3(x+1).
Cognitive asynchrony
That too was totally opaque to me at 16 -- but when I went back, at 26, preparing for the GMATs-- it was actually, well, not fun, but satisfying to solve. I concluded that my very verbal brain had finally grown some ability to enjoy fooling around with abstractions.
"What if the standard school curriculum regularly presents concepts (my classic example is long division) before the average child (AC) is developmentally ready to master it?"
Long division is usually taught in fourth grade. Schools didn't pull this grade out of thin air. (Many reform math curricula skip it completely, but that's a separate issue.) If you think that it would be better presented in a later grade, then you can try to make a case for that, but topics can't be evaluated out of context. You have to look at what topics come before and after. The problem I have with developmentally appropriate is that I have never seen it used to raise expectations. I have never seen anyone quote any study about the number of hours it takes to learn a subject for a particular grade. It seems to me that it would depend less on age than on the quality of the curriculum or prior preparation.
With full-inclusion, the idea is that kids can move along at their own developmentally appropriate pace. So now, developmentally appropriate is not an average statistic, but an individual one. Without statistics, how does one tell if an individual child is ready for a new topic? The only way you can make this decision is by giving it a try. (or see how they handled the prerequisites) But then, how do you tell the difference between not ready yet, laziness, bad teaching, or a bad curriculum? I don't trust schools to make that decision correctly.
"The issue of transitioning from print to cursive writing is another iconic issue. Why teach cursive at all? Books aren't printed in cursive. What does handwriting have to do with reading? Why teach cursive in third grade?"
You're mixing up two separate issues here: why and when.
"Back to 2x(x+1) + 3(x+1)."
"That too was totally opaque to me at 16 ..."
But it's not opaque for algebra. Some might not "see" that the (x+1) can be factored out directly, but they better be able to expand and then factor it like this:
2X^2 +2x + 3x + 3
2x^2 + 5x + 3
(2x+3)(x+1)
This is standard algebra and I expect schools to prepare kids for this material by 8th or 9th grade - when the kids are 13 or 14. This is developmentally appropriate. If you struggled with this as a junior in high school, then the reason for the opaqueness would most likely be with your school's curriculum, not your brain. So many times in my life I have come across topics that seem completely opaque to me. I have figured out, however, that the problem (usually) isn't me. It's just that I haven't found the right textbook or instruction.
I don't think that developmentally appropriate is a "sprawly subject". Schools should dive right in and get to work instead of dancing (coloring) all around a subject in the hopes that kids will learn by osmosis. If kids have trouble learning, schools should figure it out and deal with it directly.
The problem is that many educators think that hard work is developmentally inappropriate. That's number 2 on my list of three things I want for my son; to know the value of hard work.
"The problem is that many educators think that hard work is developmentally inappropriate."
Amen to that statement. I am sorry I think the term "developmentally inappropriate" is a sham or at least it is at my sons school. Why push kids or even teach them for that matter if you can always fall back on "they weren't ready for the material" excuse?
You know what my first grader is doing in math? He is cutting shapes out and gluing them on paper (thanks to TERC). Learning addition and subtraction in first grade is deemed developmentally inappropriate, but gluing shapes on a piece of paper is appropriate? Maybe for preschool, but not first grade. Thank goodness he is learning real math at home and he enjoys it.
So many times in my life I have come across topics that seem completely opaque to me. I have figured out, however, that the problem (usually) isn't me. It's just that I haven't found the right textbook or instruction.
Thanks Steve! That is so true we should make a poster of it.
It is very embarrassing now but when I first took calculus I didn't understand why I should care about the "area under a curve." I didn't have any trouble dealing with algebraic formulas like d = rt but I had never thought of distance as being the area of a triangle with time as its base and rate as its height.
In other words, I apparently thought calculus was explicitly used to determine actual areas (in square inches or whatever). Duh!
I apparently thought calculus was explicitly used to determine actual areas (in square inches or whatever). Duh!
Glad you brought this up. I was fortunate to have an extremely good physics teacher in high school, who showed how the area of a triangle represented distance traveled by an accelerating object and in fact used it to derive the formula of distance for a uniformly accelerating object. When I got to calculus in college the use of integration was obvious.
I think this is related...
In the past couple of years, I have heard several math teachers state that teaching the acronym FOIL (First, Outside, Inside, Last) is a bad way to teach multiplication of binomials. They say that it is more appropriate--and more general--to stress the multiplication of binomials as an application of the distributive property. FOIL won't work to multiply a binomial with a trinomial, but the distributive property will. I see the point, but I think that teaching FOIL is okay, too, if it works.
Anyway, a methodical, step-by-step explanation of the use of the distributive property to multiply binomials would cause students to recognize arrangements like 2x(x + 1) + 3(x + 1) as an intermediate step in multiplying two binomials, I think. Therefore, they might also recognize it as an intermediate step in the opposite direction of factoring.
Dan K.
"I think the term "developmentally inappropriate" is a sham or at least it is at my sons school."
Has anyone ever heard of a case where developmentally appropriate is used to set higher expectations? You could use this argument for GATE kids, but schools like to hold these kids back and use them as pedagogical role models. That, or they claim that these kids aren't smarter, they just learn differently. In this sense, developmentally appropriate is a sham.
["I think the term "developmentally inappropriate" is a sham or at least it is at my sons school."]
As SteveH wrote so eloquently, all too often the phrase translates into low expectations.
The phrase we need to succinctly describe educationist charlatanry is educationally inappropriate. Next time an educationist mumbles something about best practices and the like, tell him that what he means by "best practices" is educationally inappropriate. Cite the army of illiterati and innumerati produced by by "best practices."
But the algebra book we were using (similar to Dolciani's in organization and drills) had provided enough exposure to expressions such as (x+1) that had to be operated with as a single entity (chunking) that it didn't take much work to make the leap.
With today's typical algebra books in which the math is a sidebar to photos of kids in roller coasters and grannies riding motorcycles, this point is no doubt lost in the noise.
Good point.
I think Saxon does a terrific job "pre-chunking" these elements.
In fact, that might be a good way to describe the approach he takes throughout.
It's obvious that he teaches component parts before wholes, but he also teaches the chunked component parts.
Liz!
Horse rants are good!
Actually animal rants in general are good.
If you're still around....would a properly fitted saddle not require training at each gait?
Is that right?
You've raised a sprawly subject, as my daughter used to say
sprawly for sure
Anyway, a methodical, step-by-step explanation of the use of the distributive property to multiply binomials would cause students to recognize arrangements like 2x(x + 1) + 3(x + 1) as an intermediate step in multiplying two binomials, I think. Therefore, they might also recognize it as an intermediate step in the opposite direction of factoring.
boy, I'm going to have trouble responding (just don't know math well enough yet to be able to write fluently about it).
I've come to feel, strongly, that the more you can associate the properties with what the student is doing, the better. I very much appreciated the Russian Math problems where you had to write which property you were using on each line.
otoh, lately I've realized that FOIL is terrifically helpful for....and now I've forgotten what for, exactly.
I say this because I didn't really learn FOIL. I'm not sure Saxon teaches it at all (he may & I've forgotten) and I didn't learn it as a kid that I recall.
Saxon teaches you to multiply polynomials by lining them up vertically.
Recently it's been useful to me to use FOIL and look at it because.....I think I had this perception when I was trying to grasp why a trinomial in the ax^2+bx+c form factors the way it does.
I think.
So many times in my life I have come across topics that seem completely opaque to me. I have figured out, however, that the problem (usually) isn't me. It's just that I haven't found the right textbook or instruction.
It is AMAZING the way one text will leave you mystified and another will seem all clear.
I spent ages trying to figure out why in the hell one would invert and multiply.
Then one day I saw a fraction division problem written as a complex fraction and the scales fell from my eyes.
educationally inappropriate
I like that
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