
Re: Simplifying Algebraic Expressions with Subtracted Expressions
Posted:
Nov 16, 2013 2:33 PM


On Nov 16, 2013, at 1:20 AM, Joe Niederberger <niederberger@comcast.net> wrote:
> R Hansen says: >> If so, then why don't people solve problems this way professionally? > > OK Robert  so in your opinion all pedagogy, even at early grades, is professional training?
In Singapore, from whence these problems and alleged techniques came, it is.:)
> >> How do you teach the students to actually solve the problem, without a picture? > > As opposed to unactually solving the problem with a picture and unobtaining the incorrect answer, but rather coming up with the correct answer?
So riddle me this...
The original tart problem was as follows ...
"Mrs. Chen made some tarts. She sold 3/5 of them in the morning and 1?4 of the remainder on the afternoon. If she sold 200 more tarts in the morning than in the afternoon, how many tarts did she make?"
Pam's steps were...
1. They would draw a horizontal rectangular bar, divide it into 5 equal parts. 2. lightly shade 3 3. divide each of the remaining 2 parts in half 4. (as well as the original 3 so as to continue to have equal portions) 5. By shading 1/4 of the "afternoon tarts" (1/10 of the whole bar) 6. and shading an equal portion of the "morning tarts" 7. what is left of the morning tarts is equal to 200 8. Since there are 5 parts, each part is equal to 40
Where did steps 3 and 4 come from? and step 6? Think like a child that doesn't know how to solve this problem. You can't get those steps without already having rationalized the solution mathematically.
Illustrating a solution is not the same as teaching a kid to solve. The kids that manage to illustrate their solutions in this manner already have solutions and move on past this gimmick. The others suffer because they don't get how you know how many blocks to make. My suggestion (and I know it works) is LAY OFF THE MARKINGS. Use the bar to indicate the whole and go as far as showing the parts, then finish it off with plain old arithmetic. Showing the markings might help a 3rd or 4th grader get comfortable with fractions but there comes the time when you have to leave *counting* behind and move on to arithmetic and reasoning. And suppose that the problem is this instead...
"Mrs. Chen made some tarts. She sold 3/5 of them in the morning and 1?6 of the remainder on the afternoon. If she sold 210 more tarts in the morning than in the afternoon, how many tarts did she make?"
Yes, I know you can solve it if you divide the bar into enough pieces.
But what if it is whole numbers rather than fractions, like coprime whole numbers? The number of markings gets rather large.
What if it is decimals? 100 markings? 1000?
What if it involves multiplication? Something simple, like 2. She sold twice as many tarts in the morning? What then?
What if, god forbid, it involves LETTERS!
Don't laugh, there is a problem with kids trying to learn algebra and they can't do it by counting.
Illustrating a solution requires that you know the solution. You can only know the solution if you can rationalize the solution. I said that the problem was a hard 5th grade problem, and it is, because those problem sums are amongst the most challenging problems on the PSLE, which is taken at the end of 6th grade. And we are talking Singapore challenging, not U.S. challenging. Our six graders would faint and suffer PTSD if they had to take the PSLE. Pam said "Not really. Not if they know how to shade..." I say "No. If they know how to solve the problem they can illustrate the solution if they can shade, but shading won't enable them to solve the problem."
But back to your question, is it always about professional training?
I use the "professional card" in place of the argument that if you actually teach students mathematics they will use it in some capacity in their professional lives. That we don't see even traces of gimmicks like this later on is just to make the point that this is not mathematics. But my real concern, the thing that hits me when I see a gimmick, is why? Why a gimmick? I don't even think the kids are looking for gimmicks. Nor the parents. Just because they don't know any better doesn't mean they want a gimmick.
Where does it stop?
Singapore seems to have an answer, the end of 6th grade.:)
Bob Hansen

