kitchen table math, the sequel: inflexible knowledge

Saturday, February 28, 2009

inflexible knowledge


I've told C. he has to start doing a page of problems in Greenhall's Acing the New SAT I Math, which I ordered after comparing Amazon reader reviews for the various prep books. 

Today both he and I were stumped by this problem:

8. S is the sum of 4 consecutive integers, the smallest of which is n. In terms of S, what is the sum of four consecutive integers of which the greatest is n?

I solved it via "mathematical reasoning," a phrase I'm putting in quotation marks because while I do possess sufficient mathematical reasoning abilities to solve this problem via number lines and "n + 1s," apparently I do not possess sufficient mathematical reasoning power to solve it via substitution. (Or, rather, to recognize the fact that I am looking at a simple substitution problem.)

This is maddening.

However, the book is obviously going to be good for me to work through. 

Fun, too.

Does it take 10 years for inflexible knowledge to start easing up a bit?

Do we know?


Anonymous said...

S = n + (n+1) + (n+2) + (n+3) = 4n + 6

T = (n-3) + (n-2) + (n-1) + n = 4n - 6

"in terms of S":

T = 4n - 6
= 4n + 6 - 12
= S - 12


bky said...

Catherine, if you want to know how to go from what you know to what rocky knows, i.e. how to simply and more or less directly solve the problem, the key word might be "translation" rather than "reasoning" or "substition". You read "S is the sum of 4 consecutive integers", and then write down what that means

"S is" <--> S =
"the sum of 4 consecutive inegers " etc --> S = n + ... and so on. Then do the semi-obvious thing, to collect the ns. You do not always know where you are going with this, but two numbers are mentioned, the one starting with n (S) and the one ending with n (T in rocky's notation), so you need to translate them into a useful form.

Then you have to figure out what T is "in terms of S" and there is no general method of problem solving; there is a fairly general method of reading problems, however: translate. Often they seem tractable after that whereas before they seem to require cleverness.

Allison said...

I agree with kby. I bet most of us who knew immediately how to solve the problem did it with inflexible knowledge of what to do when you read "In terms of S".

We read the question and knew the code. "In terms of Y, ..." means "substitute Ys in where possible."

What did you think "in terms of S" meant? When you read that, what did you write down?

Lots of math solving feels like "tricks" because the trick is seeing how to translate the problem rapidly into something you know how to solve.

Now is a good time to read Polya's little How to Solve It. It's short! it's dense but short! :)

VickyS said...

The "T" confuses me.

Here's how I do it:

S = n + (n+1) + (n+2) + (n+3) = 4n + 6

so n = (S-6)/4

Next, (n-3) + (n-2) + (n-1) + n = 4n - 6

Finally substitute (S-6)/4 for n:

= 4[(S-6)/4] - 6 = S - 6 - 6 = S - 12

I could *only* see it as a substitution problem. That may also indicate inflexibility!

Anonymous said...

VickyS, when you solve simultaneous equations, you can choose between substitution and elimination. There's your flexibility.


vlorbik said...

whatever makes you see it clearly
is the right way. not that comparing
techniques is at all a bad idea...
but just wading in and looking at
what's going on looks to me
a whole lot more promising
than trying to *classify* techniques
and start looking for *technical terms*.

"i did this; you did that; he or she did the other",
followed by stuff like "mine is the most elegant",
"yours is the easiest to follow", "theirs follows
the most obvious line of investigation",
or what have you... we're *mucking about*.
changing the subject to "how can we understand
our *understandings* of this problem" instead of
"...understand *this problem*", and then trying
to do it with the same kind of precision
that we do mathematics or empirical science
smacks of edschoolism. do the math!

2 cents on the exercise.
i'm supposed to be able to teach this stuff
so naturally i'll first try to do it in my head:
an opportunity to talk about it might bust out
far away from any blackboard. let's see.

like rocky sez, the sum-starting-at-n is
n + (n+1) + (n+2) + (n+3)
---and i know this pretty much without thinking.
so i can chunk that up into a single thought
and leave it on my mental desktop.
which i suppose is my working metaphor--
for the nonce, mind you!--for something like
the "flexibility" one seeks (this skill...
whatever it is i'm waving my hands at...
is at least *part* of what we mean by
"mathematical maturity" [yep: all the time
they've been going on about chunks
on mental desktops and it took vlorbik
to make this clear]).

okay. next i *compare*

the sum-starting-at-n,
n + (n+1) + (n+2) + (n+3),
the sum-ending-at-n,
n + (n-1) + (n-2) + (n-3).

i've laid it out this way because it's the way
i organize it mentally; i think i'm doing it
this way because it feels like it use the least
of my "system resources" this way:
the second "formula" *resembles the first*
so there's less burden on my memory
(i only have to remember one-up-one-down
or something like that).

rocky laid it out his way because
"natural number order" is rightly so-called
(or so i imagine... far be it from me
to put reasons in anybody else's phenomenology).

but the point here is that with mine,
the mental computation is easier
(or seems so to me):

"n+1" and "n-1" have a difference of *two*;
"n+2" and "n-2" have a difference of *four*;
"n+3" and "n-3" have a difference of *six*.

add 'em all up; the two sums in question
differ by 12; our "answer" is S-12.

SteveH said...

"Does it take 10 years for inflexible knowledge to start easing up a bit?"

It doesn't take 10 years, but it isn't necessarily inflexible knowledge. Some if it depends on whether you've see this type of problem before. When I read "4 consecutive integers", my mind immediately thought of that class of problems. If I hadn't seen a problem like this, then it's not clear to me that any other problem I've done would help. Maybe, or maybe not.

What is flexible are practice and confidence (!) that you can find equations and perhaps use substitution (or any other technique) to get there. I find that some textbooks expect you to "think" and come up with the final equation in your head or with "mathematical" reasoning. This is often true before books formally introduce solving systems of equations. They force kids to solve what are easy two equation and two unknown problems using mental substitution to eliminate one or more variables. Chicken and cow leg (or tricycle and bicycle) problems are common here. Math isn't about reasoning out problems in your head. It's about using math skills that allow you to do things that you can't possibly do in your head.

My position is that math should allow you to think and reason less. Don't try to "figure out" the problem. Just start assigning variables and writing down equations that you know are correct. These can be really dumb equations, but if you know they are correct, then all you have to do is get an equal number of equations and variables.

I would say that it's more about inflexible skills than inflexible knowledge. Unfortunately, educators like to say that skills are rote. Math skills are thinking tools. Let them do the work so that you don't have imagine the final answer in your head. That's their whole purpose. For some of the problems I work on, I just let the equations lead me to the solution. If I don't have enough variables or equations, then maybe I have to do a little bit more thinking, but I don't have to figure it out in my head beforehand.

Barry Garelick said...

I agree with Vlorbik and Steve. Once you know certain skills and forms you use those to solve problems and start using them before you even know how exactly it's going to work out (i.e., substitution or whatever). In a good math class, the teacher might work out a tough problem at the board asking for student participation and along the way show there are various ways to solve it. But the main thing is to get the central concepts straight; how do you represent S when n is the greatest and when n is the least? What you do from there is like splitting hairs over something like whether it's better to pull a bandaid off slowly or fast.

vlorbik said...

"don't try to figure out the problem"
is putting it awfully strongly, but i think
steve and i are somewhat on the same page
with this.

mental calculation skills generally come,
if at all, long *after* paper-and-pencil skills
and, indeed, are a pretty good indication
of the famous "mastery" everybody's always on about.

the phenomenon of using the symbols on the page
to "reason less" is real and important.
"trust the code", i like to say... because
evidently, neurotypicals prefer to trust
their vague understandings and need
frequent reassurances that math actually works.

but for heck sake.
"math isn't about reasoning
problems out in your head"?
--then i've wasted half my life.

Barry Garelick said...

Oops. I meant to say "how do you represent the sum when n is the greatest and when n is the least?"

I wish Blogger would allow editing of comments.

SteveH said...

"but for heck sake.
'math isn't about reasoning
problems out in your head'?
--then i've wasted half my life."

Perhaps you've never encountered problems that required more effort and skills than a textbook problem. I suspect you just couldn't resist a chance to spin my comments out of context.

vlorbik said...

b-but, steve! we're *in* context!
whatever it was i couldn't resist...
it wasn't quite that. it just seemed
like a pretty good *straightline*...

and a chance to remind you
that you might want to tone down
the rhetoric; you sure didn't offend me
but, heck, don't forget i'm usually
more or less *on your side*
as far as "what math matters most"...

"problems that required more effort...":
i wrote a dissertation & got it published.
it was pretty tough. you actually know this, right?

VickyS said...

Just start assigning variables and writing down equations that you know are correct.

Yes, that's what I do. It allows me to figure out the problem *while* I'm penciling. When my older son, now in Precalc, has trouble with a problem, he hands me the book and asks about it. Most of the time I cannot even begin to discuss it with him until he hands me paper and pencil, and I read the problem while writing. Sometimes I wander around, figuratively, letting my brain (through my pencil) explore several different avenues of analysis. This frustrates him--he thinks I should be able to talk about it before starting to work on it!

Must just be my learning style ;-)

Also--echoing Barry, Steve and Vlorbik if I understand them all correctly!--the kids need to be (willing and) able to pencil these things out even when they can solve them in their heads. It's the way you learn to set up and trust the equations in more complex problems when you can't see the answer.

This also incredibly frustrates my kids!

SteveH said...

"you might want to tone down the rhetoric.."

Am I just using hyperbole for effect?

Perhaps a little, but I am very serious about my underlying point; that schools expect kids to use math as some sort of magic thinking process, not as a set of skills or tools. You don't have to mentally figure out problems and THEN use math skills to solve them. Let the skills lead you to the understanding.

You have 15 cycles. Some are bicycles and some are tricycles. There are 40 wheels in all. How many bicycles and tricycles do you have?

Kids have to solve things like this before they get to algebra, as if "thinking" and guess and check define math. What you really need to do is learn how to select variables, come up with legal equations, and turn the crank.

Too many kids don't even know where to start. At best, they are told do do things like draw a picture or work backwards. They aren't offered systematic practice on defining variables and equations. Perhaps educators think that understanding has to come first. They don't think you can't just blindly start writing equations.

For many problems, however, understanding comes from being able to write down something solid like an equation, even if you're not sure where its going to lead. Actually, it does lead to a solution because all you need are an equal number of variables and equations. This helps students break a problem apart. If they use one number for one equation, then maybe the other number will lead them to a separate equation. Too many kids are afraid of many variables and equations. They feel like they have to think their way to one equation in one unknown.

Barry Garelick said...

the kids need to be (willing and) able to pencil these things out even when they can solve them in their heads.

Right! Too many times, kids figure that if they can do it in their heads they don't need to write it down. Nor solve any equations. I saw this in Vern Williams' class once; a student got the right answer by just guessing appropriately. Vern asked him for the equation. The student said he didn't need it because "it was obvious". Vern responded: "What happens when it isn't obvious?" He then told the students that equations are your insurance policy to make sure you are getting the right answer.

Rudbeckia Hirta said...

Steve, that bicycle problem is the CLASSIC example of a problem that doesn't really need alegbra to solve. It's used to teach if-then thinking when you're making a hypothesis.

You start by pretending that all of your stuff is the one with fewer wheels/legs/heads/whatever. So in this case you pretend that all 15 are bicycles. If that were the case, you'd have 30 wheels. But you don't -- you have 10 extra wheels, so you know that you have 10 tricycles.

The whole point of this type of problem is to work on deductive reasoning.

SteveH said...

"...doesn't really need alegbra to solve."

Lots of problems don't need algebra to solve, and one could justify all of discovery learning by claiming it develops deductive reasoning. But then again, is there no development of deductive reasoning (figuring things out) when you approach math from a skills-first perspective? Unfortunately, many educators see deductive reasoning as an end goal, rather than a useful by-product of skills and practice.

The problem with deductive reasoning in the bicycle problem is that whatever kids might have learned does not necessarily translate to other problems and some are left with the belief that there is no concrete path to solving problems or mastering math. And before long, they will have problems that defy deductive reasoning. Math skills can take over when your brain leaves you high and dry.

Barry Garelick said...

And before long, they will have problems that defy deductive reasoning. Math skills can take over when your brain leaves you high and dry.

Again, Vern Williams: "What are you going to do when it isn't obvius?"

vlorbik said...

if the point of a certain type of exercise
is to work on deductive reasoning, good.
but then the point of working on *that*,
for students, is that "common sense"
and "algebra" can be made to reach
the *same* results. this is part of
how one *learns* to "trust the code"...

"when i'm i gonna use this?"
is trying to tell us something.
students tend to believe that learning
to use the symbolism makes no sense
*because* they have *nothing to say*
with it. and while *we* know
that you jollywell have to walk
before you can run and that
"stuff to say", somewhat counter-intuitively,
generally seems to *follow*
"the means to say it" in this arena*.

getting students just to pick up the pencil
is hard if they feel like you're gonna ask 'em
to write anything down that hasn't been
dictated letter-for-letter in their ears
(and sometimes even then).

you've got to *start* with stuff you already
understand, just to be sure the machine works.
but if one has convinced oneself somehow,
and this appears to be *very* common,
that, for instance, *variables* will *never*
mean anything at the level of "common sense",
this crucial phase of early algebra development
will *never* cease to be an issue...

also students tend to believe that we *should*
be telling them *just* "how do i calculate this",
and they believe this because we quite often *do*.
when what we should *really* be telling them
is "how do i write this up so it can be understood?"

this bit where you go on thinking
about a given problem long *after*
you've gotten the solution is pretty close
to the *heart of the matter* here...

vlorbik said...

the sentence at the end of my second paragraph
trails off without resolution. sometimes i agree
with barry that editing one's own comments
would be a darn good feature. other times
i enjoy knowing that my mad editor skills
are on display all over this blogosphere
and will be *until* editing becomes easy
(so it'll effectively become impossible to notice
how consistently i tended to get things right
in the oldschool hardway).

not having to pass this "word verification" twice.
now, *that* would be a nice feature.

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Art Reed