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Ed and I are plotting strategy for the state math test & for the rest of the school year.
I need advice.
Sometime this weekend it came to us that we have a need for speed.
speed and accuracy: the KUMON motto
Christopher managed to move up from a D- on his next to last test (a highly inflated D- may I add) to a genuine C on the last test by dint of mighty reteaching and home-assigned distributed practice.
He needs to move to a B.
He also needs to return to a 4 on the state test this year.
Since we're not going to be able to achieve this through our preferred medium of sound curriculum and pedagogy, we're going to have to find a work-around, and that work-around is going to be speed.
He has to practice until he's fast. That way he can whiz through the problems he does know and have a shot at cracking some of the problems he doesn't know. (Meanwhile my friend Kris says she's begun to think like Ms. K; she's managing to figure out what will be on the test that the kids have never seen before. Kris took calculus in college, so she can do these things. Next test, she's going to brief me.)
Anyway, speed.
Here's the question.
These days kids are taught to "do the same thing to both sides" of an equation vertically instead of horizontally.
I can't find an image of it online, and I can't get the spacing to work, so you'll have to imagine it from this:
3x+2 = 5+2x
- 2x = - 2x
x + 2 = 5 + 0
Then you do another vertical subtraction, writing a below the 2 on the left side and the 5 on the right side.
I like this procedure because it makes obvious the fact that you're subtracting (or adding) when you "do the same thing to both sides."
But it takes forever, and it eats up space since Christopher's handwriting is still big and not neat. (Sure glad the schools don't bother teaching handwriting to mastery any more! Why would we waste time teaching legible handwriting when we've got keyboards??)
I'm going to teach him how to solve equations horizontally and have him practice until he can do it fast.
But how should I do this?
Russian Math shows students what happens when you add or subtract the same quantity from both sides: you change the sign.
From that point on the book has you simply move variables and constants from one side to the other, changing the sign as you move:
3x + 2 = 5 + 2x
2 = 5 + 2x - 3x
That's the fastest method, obviously.
Also the most space-saving when you have enormous handwriting.
But Ed says doing things that way gives you more likelihood of error, and I should have Christopher write out what he's doing:
3x + 2 = 5 + 2x
3x - 3x + 2 = 5 + 2x - 3x
etc.
Any thoughts?
We are also going to drill dimensional analysis until Christopher can do it in his sleep.
_________________
image source:
Linear Equations in One Variable
11 comments:
I'm happy with "move to the other side and change the sign." I think there's the least chance of mechanical error with this procedure.
You can explain why it works later.
3x+2 = 5+2x
- 2x = - 2x
x + 2 = 5 + 0
This is very good when you are first learning, but it doesn't have to last long. It's tedious. I remember student-teacher discussions over how many steps can be combined in each change. there was no hard and fast rule, but the teacher said that if you want more partial credit (if it's wrong), show more, rather than less. That was my rule when I taught algebra.
What does the teacher want? Often, that ends the discussion.
3x + 2 = 5 + 2x
2 = 5 + 2x - 3x
This should be an easy next step. I don't think very many would have trouble seeing why this is legal. I do it. Most people do it without any discussion of "rote". And, it saves space.
I would, however, tell students that they can move things in either direction. One direction might make life easier than the other. For this problem, it's better to move the '2x' to the other side to create 'x' rather than '-x', but that's assuming that the kids have already had some practice at it. I remember some students always trying to get the 'x' on the left side of the equals sign. They didn't realize that the equation can be flipped around (a = b; b = a).
Students get into trouble when they start to combine more changes in one step, like
3x+2 = 5+2x
x +2 = 5
After a while, most kids try to do the whole thing in one step. This is understandable, but more prone to mistakes. For complicated problems, I try to limit myself to one move and combine at a time (like x+2=5) if I'm trying to be extra careful. For this problem, many kids can do it in their heads, but this is not helpful when the problems get much more difficult.
"Ed and I are plotting strategy for the state math test & for the rest of the school year."
This sounds like two different things; a state test and the in-class tests. I doubt they are equivalent. What are the consequences of not doing well for either of these things? Is one more important? It could also be (as in our case) that something else matters more, like the material and skills needed for high school math. You can focus too much on gaming the system and miss the bigger picture.
"...my friend Kris says she's begun to think like Ms. K; she's managing to figure out what will be on the test that the kids have never seen before."
She sounds like a wonderful friend to have. My son is already learning that many correct answers only lie in the teacher's head. It's one of of those higher-order thinking skills that students have to develop.
Sit down with your friend and try to come up with a sample test with the right number and mix of problems. Write up multiple variations of the test and change some numbers and problems here and there. The danger is that you miss a big chunk of what's on the test. I think we've all been there.
Of course, parents should NEVER have to do this. This is the teacher's job. Education is not a game or a test of character. I have said before that I used to tell my students that I am on their side. I will do everything I can to help them succeed. The class discussion and homework were focused on the important knowledge and skills and the tests were easy if you did and understood the homework. There was no guessing when it came to the tests. My test reviews would give them a sense of what I thought was important if a lot of material was covered.
Some teachers try to get students to extrapolate their knowledge into new variations of problems on tests. I hate teachers who do this. This skill, even if it can be tested, should not be graded on speed - constructivism gone bad. I remember these tests. Making the extrapolation that was in the teacher's head was more a matter of luck. There was often little correlation between success and best students in class.
I have to be honest. I don't see much time difference between the methods. I would use the first method since as Steveh said, there is less room for mistakes.
After a while, most kids try to do the whole thing in one step. This is understandable, but more prone to mistakes
good point - I see this ALL the time
I see both things: I see Christopher laboriously writing out zillions of steps he could be combining by now - and then refusing to write out steps he still needs to write out
Rory - the time difference is actually substantial when your handwriting is bad and slow
Also, they never, EVER, give the kids enough space to work; nor do they provide lined paper.
They have to do all their calculations in the margins of test papers. (Have I mentioned how much I loathe the public schools recently?)
His handwriting is still HUGE. Every extra figure he writes eats up a lot of the available real estate.
I wonder if crossing out would be helpful.
Ah, I didn't know about the handwriting thing.
Not having scratch paper!!! That is ridiculous... why not just grade penmanship directly. I have learned over the years to use lots of real estate for working out problems. It's a lot easier to track steps.
Once he's mastered the vertical procedure, he can handle some of the shortcuts (as long as he really knows what he is doing) -- whenever something crosses to the other side of the equation, it trips on the equal sign and changes it's sign. Kind of a cute way of remembering what you are doing. If he is comfortable with balancing the equation, these memorable shortcuts can help save time.
hey!
Crossing out is kind of cool!
I make him cross things out in a lot of procedures (as does Ms. K).
I do it myself.
Interesting.
Even if you didn't do it this way, crossing out is good just to keep your eye from lighting on work you've already done...
Thanks!
Education is not a game or a test of character.
That's a perfect way of putting it!
In our middle school education is a test of character.
This is true at the level of official discourse.
For most of Christopher's teachers this year education is education.
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