
Re: Simplifying Algebraic Expressions with Subtracted Expressions
Posted:
Nov 17, 2013 2:08 PM



On Nov 17, 2013, at 12:19 PM, Pam <Pamkgm@hotmail.com> wrote:
> Using bar diagrams, we come to a quick and easy solution. Draw a bar, label it "sundae", divide it arbitrarily in two sections, label one section "cherry" and the other "$5". Draw an identical bar (or you could add onto the first) but add a cherry to your sundae by adding a section the size of the original cherry. Label the entire bar $6. By now the student may see the solution: 2 cherries equal the difference of $1.
Hold on! I resemble these remarks, that's my approach!:)
I didn't have a problem with diagraming as a way to help students realize the mathematical context of the problem. It was when you were using the bars as a gimmick with which to compute fractions that I had a problem. I said label the bars 3/5 and 1  3/5 (or just 2/5) and let arithmetic and reasoning take its course. You were dividing the bar into fifths and then magically dividing those fifths into halves and shading this piece here and that piece over there. That was what I had a problem with. Yes, an illustration, but not a solving strategy.
Remember those tiles in algebra, used to illustrate factoring and completing the square? What are they for? They are there to provide concrete reassurance that the abstract rationalizations building in your head are real. The tiles are not the means with which to factor and complete a square. Having a lot going on in your head is the means. And having a lot going on in your head is also the point.
Mathematics requires that the student's thinking become progressively more sophisticated. That is not the same as using gimmicks to allow the student to *solve* more sophisticated problems.
On Nov 17, 2013, at 12:19 PM, Pam <Pamkgm@hotmail.com> wrote:
> See? Simply algebra but without having to know how to do algebraic manipulations with variable letters.
I don't understand what you mean? I view this stage (not the gimmick part, the diagraming part) as a precursor to algebra. Another step in that progression of more and more sophisticated thinking. Adopting the notations and conventions of algebra will be a smooth, and as Wayne pointed out, sometimes mechanical, transition. I compare this stage to arithmetic more than algebra. It has some of the elements of algebra if that is what you mean. But technically, *algebra* is letters and manipulation, right?
What is happening in Singapore is that the faster students are making the leap sooner. And seriously, if you have a lot going on in your head, how long does it take you to recognize what is going on in these problems?
I think either of us could recognize a mechanical student in a very short period of time. My method (which I use often with my son) is to change things up on the fly, and I listen for that "thunk" in his head. When I hear it, I know that he is thinking, reanalyzing, refactoring. To Wayne's point, you don't want to do that all the time. There is a lot of mathematics that is just informational and factual and those periods need to be more about focus than rationalization. You don't get all of it all the time but you do need to retain it for later when it does make sense.
Bob Hansen

