> Bob, let me try again; I think I expressed myself poorly the first time. I will accept that your son understands the concept completely. I was curious (and I still am) what picture he would have drawn.
The problem was to convert 8/3 to a mixed number. The book does not develop the math of that conversion, instead it draws pictures of 8/3 as 3 groups of 3 with only 2 items colored in the third group. I suppose he was going to draw something like that. I was fooled somewhat because the chapter on multiplication defined multiples and factors and I suspected it was for the later developments in fractions. I suspected wrong. Evidently, these books just mash things together to meet state standards requirements and don't tie them together in a systematic development. I know, me, reviewer of a couple dozen textbooks, just figuring that out. Well, it really hits home when your son is trying to explain the math in his head and you are baffled by his inability to communicate. You go to the book, thinking you are going to say to him "Here, look, this is what you are trying to explain, right? Didn't you guys cover this?" only to realize what he is trying to explain he made up on his own, there is nothing but pictures in the book, if you filtered for math they would be blank pages!
> I think a young student (not your son though) could understand the concept of 'fraction' and not necessarily get the concept of converting between fraction formats.
Yeah, many young students obviously, this is what we mean when we say "Not being able to rub two fractions together and produce a sum."
The ability to extract fractions from pictures is a necessary skill, it leads to extracting fractions from real world contexts. But that is not the math of fractions. In fact, technically speaking, that is not math at all. To say that the student is successful with fractions, that student must be able to walk forward and backward through the arithmetic of fractions and that student must be able to apply that to problems, whether abstract or real. An abstract problem is "What is 1/2 of 1/4?" and a real problem is "Look at the picture, what fraction of ...?"
And let's get two things straight on "memorizing algorithms"...
1. You can't succeed at math by just memorizing algorithms. 2. If you are successful in math you will memorize algorithms.
Memorizing algorithms is unavoidable. The problem isn't algorithms it is the student's approach to math. Memorizing pictures is no fix! Remember the story of catching and how in the beginning you throw the ball to the kid's glove? That is what you are referring to when you say "memorizing algorithms". That is easily fixed. Instead of throwing the ball to his glove, you throw it right at him. That will cure him of "memorizing algorithms". This works in math to. They are not going to succeed if you don't challenge them. And the ball is going to hit them more than a few times.
On Feb 28, 2012, at 10:45 AM, Richard Strausz wrote:
> Bob, let me try again; I think I expressed myself poorly the first time. I will accept that your son understands the concept completely. I was curious (and I still am) what picture he would have drawn. > > I think a young student (not your son though) could understand the concept of 'fraction' and not necessarily get the concept of converting between fraction formats.