
Re: Simplifying Algebraic Expressions with Subtracted Expressions
Posted:
Nov 16, 2013 6:37 PM


On Nov 16, 2013, at 4:33 PM, Pam <Pamkgm@hotmail.com> wrote:
> 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.
That was my point. Of course it seems obvious to you. You already know how to solve it so the illustration makes sense.
> >> >> 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?
I don't have to know with any kind of accuracy. It has a value, a label, 3/5. Like I don't have to know if a geometric diagram is drawn correctly or to scale.
> >> 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."
I think they are beyond cumbersome. To a student that only has a crayon and doesn't understand the reasoning and why they are cumbersome, they are impossible.
> >> >> 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.
Then why do you suppose that so many 5th graders fail to solve them, when they get to the end of 6th 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.
I have it backwards. I can solve these in my head. Why would that not qualify me to know what works and what doesn't?
> >> 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.
You understood what I meant by that, right? It is the reality the students in Singapore live in. Do the math anyway you want from grades 1 through 6. At the end you will take a test and those of you that score high enough go to academic school and those that do not go to vocational school. When I look at the top schools, they do not use the bars in this fashion.
Even if I could get my hands on the actual completed PSLE exams and show you the failure rate of students trying to shade their way through the exam, I still don't think that would convince you. You are so enamored by this manipulative that you don't see its serious limitations. Hopefully some of this discussion will wear off on both of us.
Bob Hansen

