Is it even useful to ask this question because there can be so many ways to approach the same problem? How would you describe the process of learning how to solve word problems? Actually, I’m sure there are component skills, and I’m just as sure I have no idea how to describe them.
The reason I’m trying to understand this is that I secured an unenthusiastic agreement that our school would attempt to incorporate shorter-term benchmark objectives into my daughter’s annual IEP goals. One of the goals is word problems, and it seems that the benchmarks should be the component skills involved.
The IEP committee chair remarked at the meeting that the teachers are the “experts”, and therefore they should have a good handle on this. On the other hand, maybe they don’t.
A basic idea in learning theory is that complex skills can be analyzed to see what the component parts are, so that these components can be taught individually to develop component skills. In reading, for example, a student learns how to decode words (phonics) and that the page is organized from left to right and top to bottom. As these skills become automatic the teacher and student pay less attention to them and a shift is made to more complex reading tasks. Constructivism, Complete Math and Integrated Content all work against the idea that component skills in math can be identified and taught, and that these form building blocks for subsequent learning.
Mathematically Correct
I’m a little afraid of what they’ll come up with, so I want to be prepared.
One annual goal might be: “When presented with grade-level word problems dealing with a variety of curriculum concepts and skills, the student will read and solve the problems.” What would be the component skills (to be used as benchmark objectives) for this?
To use an example:
Tim earned $7 for each of the 4 days he raked leaves in his neighbor’s yard. He spent $10 to buy a new rake. How much did he have left? Explain.
In order to solve this problem, you need to know:
-- Reading comprehension – Seems obvious, but how exactly would you assess? By using standard reading comprehension tests?
-- How to apply an appropriate heuristic – ?? With reform math, anything goes. Including guess & check.
-- Which mathematical operation(s) to use – Seems easy to assess.
-- Plug correct numbers into the selected operations - ??
-- Computation, multiplication and addition – Easy to assess
I’m really completely lost in trying to understand this. Wouldn’t a teacher know the skills, the steps? Maybe not, considering the type of teaching commonly in use today.
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I would definitely not trust them on this one.
ReplyDeleteMy district's philosophy seems to be that you can't teach strategies for solving word problems. They're wrong.
Of course you can. I use the book "Become a Problem Solving Genius" which is for grades 4 to 12.
It's divided up by types of word problems.
I'm sure there are other good resources as well.
Yes indeed, folks, you can directly teach strategies for solving word problems. AND you can identify types or general categories of word problems.
Once those strategies are taught, the student can then solve more complex word problems that require more than one strategy.
This used to be common sense.
You can't win on this one.
ReplyDeleteThere's no reason the think their premise is correct. There's no reason to think that the skills needed to perform a complex task can be broken down into a list of independent components.
Emphasis on independent.
The skill components relevant to the task interact with each other.
But you aren't likely to be able to get the school dweebs to process that.
oh, I'm glad you reminded me about Problem Solving Genius -- which I own, but had forgotten.
ReplyDeleteC. was "taught" 4 classic algebra 1 word problems in the past 1 or 2 weeks:
* number problems
* distance problems (2 trains left the station)
* coin problems
* age problems
I'm going to look up the date of the first homework and calculate EXACTLY how much time has been spent covering these.
Needless to say, Mildred Johnson is now in play.
I secured an unenthusiastic agreement that our school would attempt to incorporate shorter-term benchmark objectives into my daughter’s annual IEP goals
ReplyDeletewoo hoo!
You know what's a good thing to look at?
ReplyDeleteThe mini-word problem posts back on ktm-1.
Hang on; let me see if I can scare those up.
Did she actually say, "Teachers are the experts?"
ReplyDeleteYes, I think that's the right track, rather than try to identify the component skills, approach it with increasing levels of complexity.
ReplyDeleteStart with the easy problems then work towards more complex. It probably doesn't even matter much how you define complexity in this situation.
The problem-solving strategies everyone lists all the time are actually not bad (no surprise, given that they're drawn from Polya's book)...it's just the packaging that makes them seem dippy.
ReplyDeleteOff the top of my head, the one that is CRITICALLY important, IMO, and that I would want on my daughter's IEP, is "solve a simpler problem."
I think I may have come up with this strategy myself at some point in high school.
I've been using it ever since, and it is hard as HELL to teach it to your own kid. Major resistance.
Start with simpler problems and continue to increase complexity!!!
ReplyDeleteThat makes so much sense, and it sounds so simple to implement. (Famous last words?)
I’ll talk with the teacher about this. Thank you!
Your example is very provocative because there is quite a bit of implicit information. You can see this if you rewrite the second and third sentences as follows.
ReplyDeleteAfter he was finished, he spent $10 of the money he had just earned to buy a new rake. How much of the money he had earned did he have left after buying the rake?
(Actually I wish he'd bought something else like "a birthday present for his friend." His buying a new rake with his rake earnings makes it seem as though it is important to know exactly what he bought.)
What skill is it that helps us determine the most obvious interpretation? Was he paid each day or only at the end? If he was paid each day, he could have bought the new rake the second day, so he'd have $14 - $10 = $4 left at that point.
Did he have no money at all before he raked his neighbor's leaves?
I agree that reading comprehension is important but it's more subtle than that. If I were asked, "When did Tim buy the rake?", I think it would be reasonable to say that the problem doesn't provide that information.
Here are some possible component skills. (1) Read the entire problem aloud. (2) Restate the problem in one's own words. (3) Define specified terms. (4) State if there is any missing information.
*choosing a variable (or variables):
ReplyDeletethe "component skill"
that beginners most often
need to think more about
(and that they're least
willing to hear about ...
to the point of high weirdness).
a closely related "component skill":
*dimensional analysis
(i.e., making sure that
the "units match" all across
each equation.
thus, for "i have 100 coins,
all dimes and quarters;
their total value is $14.80",
the variables are
d = number of dimes &
q = number of quarters
(*not* d=dimes, etc ...
d is a number, not a pile of metal!)
one then has d + q = 100
(# of coins) + (# of coins) = (# of coins).
moreover, 10d is the *value*
of all the dimes (in cents;
one could of course also use
.10d [their value in dollars]);
similiarly 25q; one then has
10d + 25q = 1480:
value [in cents]) + value = value.
now it's not a word problem
anymore (just a mere system
of equations; students can "easily"
be taught by rote how to solve these).
whatever makes word problems hard
is somewhere in *there*. what makes
the whole thing so frustrating
(for me as an instuctor) is that
students want to assume that
distinctions like "d= # of dimes"
versus "d = dimes" are pedantry
and don't bear thinking about.
they know they're struggling;
they know i'm an experienced hand;
but they *take it for granted*
that this point that i *repeatedly stress*
has nothing to do with *their*
particular difficulties.
my theory is that, to understand
that, "yes indeed, this issue
*is* what's hanging me up"
amounts to admitting to oneself that
"i don't know what i'm talking about".
i mean this literally --
the difficulty is that,
for example, the distinction
between a *number* of coins
and the *cash value* of those coins
hasn't become clear enough;
sometimes we're talking about one
and sometimes the other; we can't
say *anything* clearly enough
to be of any use without being clear
at all times *which* one
we're taking about (in a given
equation, say).
when a student at the present level
of discussion sees "x", they *want*
to think "some gimmick that lets me
solve algebra problems in some
mysterious fashion" but *need* to
see "x" as (the name of)
*some particular number*
(of coins, or miles travelled,
or gallons of beer or what have you;
units matter [as i expect i've
stressed enough for *this* audience]).
i'm increasing convinced that
a lot of our subjects not only
don't understand that it's *possible*
(never mind *necessary*)
to achieve perfect clarity
with the use of mathematical symbolism,
they don't *want* to understand it
and will come up with all manner
of (unconscious) ways to *avoid* it.
VME
I wonder whether it would help to include the units explicitly in each equation in Vlorbik's problem. So, using the same definitions:
ReplyDelete.10 dollars/dime * d*dimes + .25 dollars/quarter * q*quarters = 14.80 dollars.
.10d + .25q = 14.80 (after simplifying)
and
d*dimes * 1*coin/dime + q*quarters * 1*coin/quarter = 100
d + q = 100 (after simplifying)
Dimensional analysis strikes again.
I figured out the exact time sequence.
ReplyDelete9-21-2007: first integer word problem comes home
9-21-2007 - 9-27-207: all four classic word problems are taught (integer, age, coin, distance)
9-28-2007 test
Teacher ended up testing just two of the problem types, I gather because he could tell the kids' knowledge was shaky.
Chris got 60% right.
Off to the races.
I've asked for a copy of the syllabus or scope and sequence or whatever so I can resume preteaching the lessons.
btw, I should add that this is the curriculum set by the math department ---- set way back when, I believe, when the former chair was still her3e
ReplyDeleteAlgebra in 8th grade has always been taught as a "killer course."
This is the wash-out course. (This language isn't used to describe it, but the philosophy and practice are in place.)
OK, here are the stats:
* Chris got 15 out of 25 correct on 9-28-07 test
* 9 kids of 26 got scores below 20 correct
* class average was 86%
I think these figures show the problem
We've got a third of the class hosing the test; there were 1 or 2 kids who got scores lower than C's 15.
But the class still has an average of 86%.
We have a very large number of kids who can learn whatever you throw at them -- LARGE. (Also, I'm pretty sure that SOME of the best kids in the class aren't being tutored, though I also know that two of the best kids had very, very extensive tutoring.)
The course is "naturalized" and justified by the existence of these talented kids.
They are the ones who are "able" to learn algebra in 8th grade, not my child.
In fact, those kids should be learning geometry or algebra 2 at this point.
My kid doesn't belong in the same class with those kids, but he does belong in beginning algebra.
Or he would if he'd had decent curriculum and pedagogy over the past two years.
Ed came up from the basement tonight and said, "He doesn't know what a fraction is."
ReplyDeleteI don't want to hear it.
>>My kid doesn't belong in the same class with those kids, but he does belong in beginning algebra.
ReplyDelete>>Or he would if he'd had decent curriculum and pedagogy over the past two years.
Do you find any value in the class ? Or in classes labeled 'advanced' or 'accelerated'? I ask because I found that in my child's 'advanced' section of 6th, there was no problem solving or mathematical thinking of any kind going on - just rote memorization of definitions and a few algorithms. Even solving an equation for one variable (5x-3=17, solve for x) was dumbed down with an instruction to 'use the law of opposites', where I was looking for the student to use and know the properties. The only difference between advanced and reg. was that a few more topics were covered, which was no value as they were also rote memorized. No text, so the students are totally dependent on quality of instructor or tutor. I wonder if this philosophy is all over the Hudson Valley.
I pulled my son out, and put him in reg. ed. 7th so he could do a rigorous pre-algebra course at home in his preferred learning style (definitely not auditory) without having to keep up with the high memorization demand at school (advanced 7th does 7th plus pre-algebra this year). I am debating the worth of testing into Alg. I in the fall, as it will be only the second cohort after the switch from Math A. To me, there is no point in an accelerated or advanced class if depth or rigor in the subject is ignored. Might as well be in reg. ed. and learn more at home without the time pressure.
Speaking of pre-algebra, last night I learned my son was confused about the math lesson in class yesterday. By his explanation, it sounded like algebra, but I was unsure. He is in fourth grade and so far all he has done in math is place value...a few problems in expanded and standard notation and rounding. So, I emailed his teacher and this is the reponse I received:
ReplyDelete"That part of the math lesson yesterday was, I won't say purposefully confusing, but perhaps purposely thought-provoking. I was introducing the teeny tiniest snippet of an algebra concept: kids always think of 17-8 equalling 9, but hardly ever think of 17-8=1+8 or
17-9=3x3. That was the focus of what I thought would be a fun/puzzling lesson, and if anyone didn't "get it" right away, that's okay. The fact that it was in crossword puzzle form, with some clues and answers written vertically, served to further stump some of the kids. C. wasn't wandering off during this, but also wasn't happy, I believe, thinking outside of his comfort zone. He wasn't alone!! That's perfectly okay."
This was not thought provoking for my son, it was in fact very confusing. Keep in mind my district is using TERC math investigations. Perhaps this is one of the lessons.
My son's response to the lesson was that it was "brutal" and he didn't know why he had to know this stuff anyway.
Nice and very informative. I am studying this problem and I must say that your article has helped me a lot.
ReplyDelete