Pam
Posts:
1,507
Registered:
12/6/04


Re: Simplifying Algebraic Expressions with Subtracted Expressions
Posted:
Nov 16, 2013 4:33 PM


Posted: Nov 16, 2013 2:33 PM by Bob:
> > 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.
Consider me riddled. This seems so obvious, I don't know how to respond.
> > 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.
How do you suppose you know, Bob, where to mark "approximately 3/5" on a line, with any kind of accuracy?
> 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.
Still riddled. This is as easy to solve with a bar diagram as the original. But your point is well taken: some problems are cumbersome to solve with a bar diagram. And what a fantastic introduction into algebra that makes! "We could solve this using a bar diagram, as you have practiced earlier, but let me show you a more generaly applicable method."
> > 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,
It really is quite routine among the problems solved in 5th grade.
> 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." > >
If you understood how bar diagrams develop throughout the curriculum, you would, well, understand that you indeed have it backwards.
> 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.:) >
Yes.
Pam

