math

Then again…

I taught 2 fractions lessons, and they went really well.  I came away from them thinking, “This is going so well I could probably stop right here!  They get this!!!”

And then we did a few things, like had a long weekend, and when we got back to our math I realized they didn’t have it after all.  I mean, they probably could have done a few worksheets, and probably would have coloured them in correctly, but I am not much of a worksheet teacher.

Instead, I asked them to do some drawing. I wrote this on the board.  A decided to just give them paper and set them to it for two reasons:  1)  I thought they thought fractions were easy-peasy. 2) I wanted them to do more of the thinking that I did.

I wrote this on the board:

The bottom line, circled in green, was there as a challenge for anyone who actually did find the work easy-peasy and needed a challenge.

The little people were my feeble attempt to get them to focus on the math, not the art.  (When will I learn?!  Everything is art, right?)

Here are some examples of what happened:

Most did not take the challenge.  It was too challenging.  Fine. I can live with that.  We haven’t done that much work with equivalent fractions, so it makes sense.

The picture with the  pink dresses belongs to a child who wasn’t sure how to sort out the eye issue.  He kept repeating, “I just don’t get it.” and he kept repeating the question: 1/4 blue eyes. Finally I said, “Well, you know, 1/4 have blue eyes, and the remaining 3/4 have eyes that are a different colour.  Like green eyes, or brown eyes.”
That was all it took.  I felt like that was a good prompt that didn’t give away any of the answer.  I know him as a mathematician, so I knew that what he was really trying to say was, “This question doesn’t make sense to me.” And I knew that he probably didn’t need a math hint.  Luckily, the first thing I said to him, that green and brown are also eye colours, happened to be the thing he needed.  If only I was always this lucky!

As a follow up, I sat everyone on the carpet the next day.  We were in a big circle, and I put 4 people in the middle.  I started describing them using fractions.  “Half of these people are boys. 1/4 of them have a pink shirt.  2/4 of them have long pants.”  Then I invited others to come up with fractions to describe the group.  After a few, we switched to new people in the middle.  After 4 or 5 things were said about this group, we spotlighted a different group, and so on until the whole class had been in the middle once.  They started to get more creative with their noticing as we went along, and after the first group, I didn’t give any more answers.

I sent them back for more drawings.  I gave them three options, and told them to pick two.  Some picked all three.

Draw a group of people.

1. 1/2 are tall.
2. 1/3 have hats.
3. 3/4 have long hair.

They did OK.

My point:

Here are the grade 2 math expectations that mention fractions:

• –  determine, through investigation using concrete materials, the relationship between the number of fractional parts of a whole and the size of the fractional parts (e.g., a paper plate divided into fourths has larger parts than a paper plate divided into eighths) (Sample problem: Use paper squares to show which is bigger, one half of a square or one fourth of a square.);
• –  regroup fractional parts into wholes, using concrete materials (e.g., combine nine fourths to form two wholes and one fourth);
• –  compare fractions using concrete materials, without using standard fractional notation (e.g., use fraction pieces to show that three fourths are bigger than one half, but smaller than one whole);

I felt like we had done that on the first day.  I felt like all of my grade 2s and most of my grade 3s were good with this.  Except when I looked closer, especially after the second time around, I realized they didn’t understand “the relationship between the number of parts of a while and the size of the parts.”  I mean, they seemed to get that when it came to fractions of a whole, but NOT for fractions of a set.

Here are the grade 3 expectations that mention fractions:  (I am looking at number sense only.  There is related stuff when you look at time and money.)

•  divide whole objects and sets of objects into equal parts, and identify the parts using fractional names (e.g., one half; three thirds; two fourths or two quarters), without using numbers in standard fractional notation;

I feel like we have this now.  If I took them outside and said, “Fill this bucket half way up with water.”  They’d do it.  And if I said, “Do you want 1/4 of the pizza, or 2/4 of the pizza?”  They could give me a reasonable answer like, “2/4.  I am starving!” or “1/4. I have other stuff in my lunch.”