Posts
9-27-07
TIME: 1/2 hour
8:30 pm start
9:00 pm finish
HOME SCENE: both dogs have been skunked; I need to mix up skunk solution and bathe them but can’t because I must help C. with test; Andrew is tantruming upstairs; Ed has evening meeting; etc.
C. is still stacking equations vertically in order to solve; says it’s easier; says the other 8th graders do it this way, too
I tell him grownups don’t do it this way; he insists he must do it this way; yes-no, yes-no, yes-no
I attempt to demonstrate to him, a la Mathematics 6 by Nurk and Telgmaa, that when you “move” a term from one side to the other you switch signs, but he is resistant - will leave this 'til weekend & then insist
life would be far easier around here if his (male) teachers would encourage him to adopt and practice the classic, efficient mode of solving a simple linear equation, as opposed to having me do it
“Your father doesn’t stack equations vertically” etc.
[thought balloon: not fun, taxes too high....]
still mixes up and omits negative signs constantly; handwriting wasn’t taught to proficiency in IUFSD, & continues to be illegible & an obstacle to doing math problems and calculations correctly
has trouble distributing a negative [e.g. -5(x+2) ]
C. is confused by this problem, but solves it correctly:
P. 182: 35
3 - 5/6y = 2 + 16y
Needs substantial help with this one:
P. 187: 15
Solve 7(x+2) + 4(2x-3) = 47 for x.
Doesn't realize that “solve for x,” in this context, simply means “solve the equation.” Has only seen the phrase “solve for x” in the context of literal equations; his knowlege is inflexible and does not transfer.
I will need to monitor this, as the school will not.
I point out that there is only one variable in this equation; therefore it is solveable; he can find a numeric value for x. He does not know that if you have 2 variables you need 2 equations; if you have 3 variables you need 3 equations, etc. Has no sense why some equations have one number as a solution, while others have “no solution” or a “variable” as a solution.
Ed says C. doesn’t really know what a variable is (based on last night’s work).
C. can now set up and solve a simple distance problem in one variable (below). He has learned this in approximately one week’s time.
tonight’s problem:
A cargo plane left an airport at noon and flew toward New York at the average rate of 400 miles per hour. At 2 P.M. a passenger plane left the same airport for New York and flew the same route as the cargo plane at the average rate of 560 miles per hour. How many miles did the passenger plane fly before it overtook the cargo plane?
cargo plane
rate 400 mph
time x
distance 400x
passenger plane
rate 560 mph
time x-2
distance 560(x-2)
Chris sets up chart correctly, then writes equation correctly:
400x = 560(x-2)
He solves the equation correctly, coming up with:
x = 7
400x = 2800 miles
However, he does not then realize that 560(x-2) must equal 2800, too, nor does he realize that solving 400x and 560(x-2) and making sure the values are equal is the way to check his answer. Lacks conceptual understanding of what the equations he has set up actually mean in terms of the story problem.
Also, he doesn’t seem to have heard about the concept of checking one’s answer.
brainstorm
homework log #2
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21 comments:
I feel your pain about getting answers checked. The real problem is when it does NOT check. What now? It's hard for my son to find his own mistakes. Sometimes starting over fresh works, but sometimes he'll make the same mistake over again.
I'd try to avoid the whole concept of "moving terms". What he is doing in that case is adding or subtracting the same thing on both sides of the equation. Without variables:
1=1
Add 1 to each side of the equation:
1+1 = 1+1
Done like this, it's obvious that the equation remains true.
The object of this process in algebra is to cancel out terms to simplify the equation. A slightly more complex equation:
3-1 = 2
Add 1 to each side of the equation:
3-1+1 = 2+1
Simplify the equation:
3 + (-1+1) = 3
3+0 = 3
Done this way, it's again obvious that the equation is true. It's also obvious that you aren't "moving" a term, you are cancelling a term in one place, which has an effect in another place.
Now, replace 3 with X:
X-1 = 2
X-1+1 = 2+1
X = 3
One of the reasons to make this very explicit is to notice when you do something that might cause an undefined result (the most common example is dividing by 0).
Once this is well understood, then you can start in with the "moving a term" language as a shorthand.
I know; this is what I tried to show him.
Absolutely refused even to CONSIDER it.
However, I will print this out, and possibly tell him a GUY WHO KNOWS A LOT OF MATH WROTE IT UP FOR HIM, and see if I can make some headway this weekend.
That's actually what he's doing -- I can't find an illustration for what I mean by "solving vertically."
These days the kids are taught to solve a linear equation by treating it like a vertical addition or subtraction problem.
Maybe I can do it?
x + 5 = 6
x + 5 = 6
-5 = -5
__________
x = 1
I like this way of teaching it, but the kids (or at least Chris) have never made the transition to doing this horizontally.
He absolutely doesn't see that the verbal description "move the term" or "isolate the variable" etc. IS WHAT HE IS DOING NOW.
Nor can he see that when he does this calculation vertically he ends up with a term whose sign has been changed.
This is BIG-TIME inflexible knowledge.
Sometimes starting over fresh works, but sometimes he'll make the same mistake over again.
I have gotten caught in near-infinite same-mistake-all-over-again loops myself.
Unreal.
The vertical format works fine for addition and subtraction; not so much for multiplication and division, especially with complicated terms. I think it does work fairly well to show the actual explicit steps, though.
"Moving a term" also works best with addition and subtraction, but it doesn't generalize the concept as well as the vertical format. (It's a shortcut that improves speed, but like many shortcuts I suspect that it could get in the way of generalizing.)
With any technique, until the concept is internalized, I'd like to see the explicit step of the unsimplified equation. It makes it a bit more clear exactly what you are doing and why. When you get to problems that require that you "divide by x^2-7x+6/5", this explicit step is really important.
Needed substantial help with this one:
Solve 7(x+2) + 4(2x-3) = 47 for x.
Didn’t realize that “solve for x,” in this context, simply means “solve the equation.” He’s only seen “solve for x” in the context of literal equations; his knowledge is inflexible and doesn’t transfer.
Hyperspecific, as am I. Or, maybe "hyperliteral" is a better description for me.
The phrase, "solve for x" makes me want to immediately write "x =," and then I'm completely lost. (And, by the way, for a very literal person who knows how to solve the equation, the phrase "solve for x" would introduce an OBVIOUSNESS that would make said very literal person believe that the answer must NOT be obvious, thereby completely short-circuiting said very literal person's thinking.)
(When people give me directions and say, "Take the first left," my brain says I TAKE THE FIRST LEFT, whether that left be in a driveway, a cowpath, or driving the wrong way down a one-way street. I've learned ways to compensate for this, but it's always [and will likely always be] a matter of compensation.)
Another thing that's really hyperspecific that really throws a guy off--the numbers that butt right up next to the parentheses (to indicate multiplication). You look at that "7(x + 2)" and THAT suddenly becomes the problem.
Okay, I'm just thinking out loud here. But, if you give C. this problem . . .
7 x [] + 4 x [] = 47
. . . and say, "The number in each box has to be the same number," how long do you think it would take him to solve that problem?
I'm going to guess that it would be less than 3 minutes.
If the problem were framed that way for me, the distractions would be removed, and I could solve it easily.
There are a lot of distractions in that problem. That's the thing.
The scary thing is that as far as I can tell, the middle and high schools never push them to go beyond this "vertical" method of algebra. My first year college students still do things this way, and don't see how it slows them down and makes some kinds of science problems more difficult than they need to be. Many of them can't follow any other method; it is too fast for them.
Mr. Person,
literalness is an interesting thing. The other day after giving my son a definition he became stumped on a problem that required it. When I told him that it was simply a matter of applying the definition he told me, "You said that A=B, but you didn't say that B=A."
With a hs sophomore, I'm only just now seeing the problems of too little distributed practice and too much graphing calculator use in Algebra II.
Teachers said by Algebra Ii and Geometry, they should be able to use calculators because they already know all that basic stuff. Not True!!
They used calculators for that basic stuff, now they can't balance an equation.
Due to his need to construct a model home out of paper, drop breakable stuff using cotton and string and popcycle sticks, and build a rubber-band powered car, he didn't have enough time to consistently work matrix multiplication to the point of mastery.
Due to his need to construct a model home out of paper, drop breakable stuff using cotton and string and popcycle sticks, and build a rubber-band powered car, he didn't have enough time to consistently work matrix multiplication to the point of mastery.
Niiiiiiiice.
Great quote.
This curriculum sure does jump from topic to topic!
Just for fun I checked a 1945 "Intermediate Algebra" book I found at a book sale and discovered that "moving to the other side" has a name.
"Transposition is the process of omitting a term from one member [side] of an equation and writing the same term with the opposite sign in the other member. It is a short-cut method of applying the addition and subtraction axioms.
...
We should think of transposition, not as a process of algebra, but merely as a short way of applying the axioms of addition and subtraction. It is especialy useful when several terms are to be transposed."
"
I'd never seen the "stacked" method of applying these axioms and am not sure it is a good idea. However, I do understand that students have learned they are supposed to do it the way the teacher taught.
The vertical format works fine for addition and subtraction; not so much for multiplication and division, especially with complicated terms. I think it does work fairly well to show the actual explicit steps, though.
I agree!
I'd never seen this before, and I thought it was great when C. learned it in 6th grade.
But it takes up FAR too much time and paper; at some point he's going to have to switch to the horizontal mode.
Also, he really is failing to make the connection that the horizontal mode IS THE SAME THING.
I sympathize -- actually, I'll put that anecdote up front.
OK, must do another FIFTEEN MINUTES OF PAYING WORK.
Susan J
THANK YOU - GREAT FIND!!!
"Transposition is the process of omitting a term from one member [side] of an equation and writing the same term with the opposite sign in the other member. It is a short-cut method of applying the addition and subtraction axioms.
...
We should think of transposition, not as a process of algebra, but merely as a short way of applying the axioms of addition and subtraction. It is especialy useful when several terms are to be transposed."
"
I'm going to print that out and have C. read it.
btw, this is one of the things I keep harping on ---- math texts are EXTREMELY helpful when they make exactly this point.
As a person learning math, especially when I was still reteaching myself arithmetic, I REALLY needed to know what was a shortcut and what was "real."
It is HUGELY helpful to read this kind of statement.
What I liked about the "vertical" mode of solving an equation was that it made a connection between a simple addition problem written vertically and a simple linear equation problem, also written vertically.
Of course, if you're using Singapore Math you might not need to make that connection, since Singapore Math is careful to have kids do LOTS of "horizontal math" calculations in addition to "vertical" math calculations.
Also, and Susan J made this observation before, these kids have not been taught handwriting anywhere even close to proficiency.
It takes them forever to produce figures they can read, which means excess steps are VERY costly in terms of time and efficiency.
"Also, and Susan J made this observation before, these kids have not been taught handwriting anywhere even close to proficiency."
This came up at our parent-teacher conference this morning*. Our son's teacher was explaining that she knew that the class was behind where it should be in handwriting (backordered books), but that it was less important than other things. After all, she had made it through college without writing by hand at all.
I brought up the point that handwriting becomes critical for math fluency relatively quickly, which seemed to be a bit of a revelation for her. (I don't think it's intuitively obvious that this should be so, and I'm pretty sure nobody had ever explained it in those terms before.)
* If I have the time, I'll try to put up a real post about this conference and our son's school late tonight or tomorrow; it reinforced my opinion that we've chosen well.
PLEASE DO!
If I have the time, I'll try to put up a real post about this conference and our son's school late tonight or tomorrow; it reinforced my opinion that we've chosen well.
Doug--you make an excellent point with your comment about handwriting and its connection to arithmetic.
We worked and worked with this with M when she was in the primary grades. Kumon really gave us that opportunity in a way that wasn't being emphasized in elementary school. If you are going to get the correct answer when you are doing double digit multiplication, you had better line up the numbers correctly. Same way with long division.
Accuracy and neatness and legibility does matter. The point was that she had to learn to practice paying attention to detail and lining up the numbers correctly. I think these habits have helped her immensely with Algebra and now Geometry.
When we first started Kumon, my kids would do additional sheet just tracing numbers. It started out with larger numbers and then shrunk to smaller ones. I guess the good Kumon people figured that one out too!
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