Conceptual Gaps

Often no matter what modality you try, a student just won't get it because there are conceptual gaps in his learning.

A precious student comes to mind. He had trouble with fractions. He just didn't understand the concept. He could recognize fractions, write fractions, read fractions properly. He could even (through tough perseverance) add, subtract, multiply, divide fractions. But the concept of a fraction was fuzzy. Very fuzzy.

His learning center teacher and I and other previous teachers, I'm sure, had him cut pies in half, divide groups into equal parts, but he continued to have those gaps. Something was just not clicking and none of us knew what it was, or wasn't.

I say all of this because that "conceptual gap" showed itself, not in computation of fractions, which he became very good at. It showed it's ugly head in the middle of a word problem, that most 5th graders could handle with very little trouble. A specific example comes to mind, which I will share now.

There is a problem in Saxon 6/5 something like this one:

Joe walked 288 feet to the end of the pier and back. How long was the pier?

This student had no idea how to go about solving this problem. When he asked for help, I realized that it was because he really didn't understand what a fraction was, and how finding a half of 288 feet would solve his problem. I think he knew that there were two distances, but he didn't see them as halves.

To begin with the number was too huge for his mind to grasp. So I picked up his pencil and drew a line on his paper and said, "OK, here is another pier, but it's a very short pier. And when Joe walked on this pier to the end and back, he had walked 10 feet. How long was the pier?

He immediately, said, "Five feet."

I said, "Good for you. How did you know that?"

His answer was, "because 5 + 5 = 10". Notice how his mind was working. He still didn't see it as half of 10. That was why he couldn't solve the 288 feet problem. He didn't know two numbers that equaled 288. But he did know 2 numbers that equaled 10. And he picked 5 because he knew the 2 numbers had to be equal. But he didn't see it as halves.

So I knew we were only a part of the way there.

So I said to him, "OK, now, let's think abouat how we could work that problem so we could start with the 10 feet and know that he had walked 5 feet one way? Let's see if you can do another one and maybe that will help. Let's make another pier and make it shorter. (I'm drawiang the pier.) OK, Joe walked to the end of the pier and back and he had walked 8 feet. How long is the pier?"

He immediately said "4 feet".

And so I said something like, "OK, when Joe walked 10 feet to the end and back, the pier was 5 feet long. (And we wrote that information down by the pier I had drawn.) And when Joe walked 8 feet to the end and back, the pier was 4 feet long (and we labled that pier also)."

Now, my question: "OK, how could we work that problem to figure out that answer?"

And bless his heart. He said, "2 divided into 10". (Now I would have preferred that he say, "ten divided by two" but I was not going to quibble at this juncture.)

"Good for you," I said. "Now let's try that on another one. (drawing another pier) If Joe walked 20 feet going to the end of this pier and back, how long is the pier? How could we work that problem?"

And he understood the answer, and he smiled and wrote it.

"Now," I said. Let's look at our problem in the book. Joe walked 288 feet to the end of a LOOONNGG pier and back. How can we figure out the length of this long pier?

A HUGE, HUGE GRIN burst all over his face, as he said, "2 divided into 288".

It took getting down into smaller numbers of which he had some concept. He knew 8's and 10's. He could grasp numbers that size. And from there, he was able to know "how" to do the problem. He didn't understand fractions any better. He was just so happy that he knew how to get the right answer. And he felt successful.

That story happened two years ago. I don't know when he will fully grasp fractions. In a new story problem, in a new setting, he may have to be led all over again, step by step, from something small and "graspable" to something larger. And he may need that help for many years. I just hope he has teachers who understand him.

-- CarolynMorgan - 15 Jul 2005

Comments

"Joe walked 288 feet to the end of the pier and back. How long was the pier?"

I'm embarassed to say I had trouble figuring this problem out. I thought he walked to the end of the pier -- 288 feet -- and back, another 288 feet. How long was the pier? Duh, 288 feet. Wrong.

I missed the point, didn't I? So did your student. But your phrasing of the question using 10 feet as the new problem was different than Saxon's.

-- BarryGarelick - 15 Jul 2005

It's an example of the non-associativity of language:

(288 feet to the end of the pier) and back

versus

288 feet (to the end of the pier and back)

-- CarolynJohnston - 16 Jul 2005

Oh I'm glad I saw this.

I'm so tired today I couldn't figure out how to punctuate this.

I put a comma in.

-- CatherineJohnson - 18 Jul 2005

Back to: Main Page.