Modeling Mathematics – Developing the Need

Today we were talking about things we noticed as we worked with finding a fraction of 1/2. Students are noticing things I expected: that the denominator doubles each time, the numerator is staying the same as the fraction you are dividing the half up into, some are starting to notice that the numerators are multiplying and so are the denominators, and some are just flat out complaining that they have to model it on the fraction the bar.

So the fraction of 1/2 was pulling some great noticings, however I wanted the students to feel the value of being able to model the mathematics, to show what was happening, so I asked them what would it look like if I had 3/4 of a candy bar and wanted to split it with two friends. What fraction of the whole bar would I get?

I was excited that some labeled the 3/4 on a fraction bar with 6/8 and then split that in half and labeled 3/8. They said they “knew half of 6/8 was 3/8” I asked why didn’t they work with 3/4, they explained that splitting the three was not working and 6 was easier because it was even.

Some said they had split the 3/4 in half and it looked about like a 1/3, so it was 1/3. I appreciated the estimations, but looking for them to dig further after they estimated.

I would say that a third of the class had written 3/4 of 1/2= 3/8 with a fraction bar 3/4 shaded and then split in half and labeled 3/8. When asked how the fraction bar modeled their answer, they told me that they didn’t “need” the fraction bar to find the answer, they noticed that you multiply the numerators and denominators. “Can’t we just give you the answer?” “It’s the right answer, right?”

We don’t “need” the fraction bar. Huh.

Then an interesting thought hit me….they see pictures as a tool they don’t need rather than a model of a mathematical situation. It almost seemed as if they viewed the bar as “baby-ish” to use. You know how certain things hit you as WoW?!? That is completely what that comment did. I immediately started to reflect on how I had made the fraction bar sound…did I just make it sound like a way to solve? Did I even use the word model? Am I placing too much emphasis on the modeling piece?

I can see why students view diagrams as a way to solve….when they learn to add, they draw pictures. When they first work with “groups of” they draw pictures. When they first work with arrays, they draw things in row and columns. Once they have learned how to add, the pictures aren’t necessary. When they have learned how to multiply, the arrays and groups of aren’t seen as a necessity.

In that moment, I wanted the students to appreciate how important (and difficult) modeling is in mathematics.  I pushed them to explain how to name that line drawn at half of the 3/4 and we had some great conversations about why this was more difficult than a fraction of 1/2.

In the end, I assured them that I sit with adults all of the time and we struggle (and find MUCH enjoyment) in making models of mathematical situations. They felt ok with knowing it wasn’t just tough for them and I felt ok that they could see a “need” for their fraction bars!

-Kristin

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