Wednesday, February 1, 2017

Ball Don't Lie

A phrase commonly used by former professional basketball player Rasheed Wallace; once famously yelled by the late coach Flip Saunders
"Ball don't lie" is said when a player misses one, two or all three of his free throws after a questionable (read as: unwarranted) foul call is made by an official. The ball is, essentially, the unbiased judge who will not reward the player by going in if the apparent foul was indeed unwarranted.
Recently I visited a fourth grade classroom where the teacher was conducting a lesson on comparing fractions.  She explained that the task would be made much easier if the fractions in question had common denominators, and she was reviewing the method they were to use:

After several worked examples, the students were divided into two groups.  One group was directed to work on Chromebooks.  Their task was to look at two fractions written side by side and choose a comparison symbol from a drop down menu to make the expression true.  The second group had a similar task.  They were sent to a table with a stack of laminated cards.  Each card had two fractions with a blank box between them.  They were asked to copy the fractions onto a worksheet and select the correct comparison symbol.
After giving the kids a few minutes to settle in, I started to circulate.  I happened upon a student working on a Chromebook.  The screen displayed two fractions, 7/8 and 3/4.  She had selected the correct sign and was ready to click to the next screen when I asked her to take a minute to explain how she arrived at her answer.  That's when she pulled a whiteboard from her lap.

 She restated the teacher's explanation almost word for word.

I decided to leave the multiplication error aside and press her understanding a little.  I wanted to see of she could compare the fractions by using 1 as a benchmark.

Me: Do you know how many eighths make a whole?
Student: Eight.  Eight eighths.
Me: Good!  And so do you know how far away 7/8 is from 1 whole?
Student: One.
Me: One?
Student: One eighth?
Me: What about fourths?  Do you know how many fourths you need to make a whole?
Student: Four.
Me: And so how far away is 3/4 from a whole?
Student: One.
Me: One?
Student: One fourth?
Me: And so which fraction is closer to 1 whole?
Student: They both are.  They're both one away.

I decided to try another line of reasoning.

Me: Let's look back at 7/8.  You said that eight eighths makes one whole.  So is 7/8 less
than, more than, or equal to a whole?
Student: Less.
Me: What about twenty-eighths?  How many of those would make one whole?
Student: Twenty-eight.
Me: (pointing to the 7/8 and 28/28) So then these two fractions are equal?
Student: Yes.  See?  I did the butterfly multiplication.

Leaving the student to continue her work in peace, I crouched down next to a student looking at a card with the fractions 4/6 and 8/18.  He had implemented the butterfly method, and written the expression 72/108 > 48/108 on his worksheet, and was ready to move on to the next card.  Not so fast.  I wanted to see if he could compare them using 1/2 as a benchmark.

Me: Nice!  You did some fancy multiplication there!
Student: (no response)
Me: I want to talk about these fractions for a minute. Let's look at 4/6.  Is that more or less
than 1/2?
Student: (silence)
Me: Well do you know how many sixths you'd need to make 1/2?
Student: (more silence)
Me: (picking up a nearby pencil) Well, say I drew a rectangle and divided it into sixths...

Me: ...Could you color in half the rectangle?

Me: Great!  Now how much of the rectangle is colored in?
Student: Three thirds?
Me: Carry on.

A teacher, looking at the Chromebook data and the turned in worksheet, might conclude that both students have a firm grasp of the relative size of fractions.  Like the basketball referee making the questionable call, he sees what he sees.  But the truth about what these students understand about fractions won't be found in the colorful charts and impressive graphs generated by the computer program.  And it's not on the worksheet.  So where is it?  Sometimes the truth is on a whiteboard. Sometimes it's scribbled in the margin.  Sometimes it's written on a piece of scrap paper, and if you look hard, sometimes you can even see it in the faint trace left after it's been erased.  The truth is in the minds of our students, sometimes out in the open, sometimes hiding in a dusty corner. That's where we need to look.  So let's keep our eyes on the ball, because ball don't lie.

1. Thank you for this post. My love for fractions is uncontrollable but only since I've become a teacher. The amount of sensemaking that can happen in a well thought out fractions lesson/task is mind blowing (for me and for students!

1. Thanks for your comments, Jill. I agree with you about the connection between well designed tasks and sense-making. I think a really good example of that is what Joshua Greene describes in his comment below.

2. I spent the last three weeks (one hour per week) playing a fraction comparison game (from Denise Gaskins) with a 4th grade class and some one-on-one time with two younger students. This game was really effective at distinguishing levels of understanding:
(0) some kids are totally at sea. They don't really understand what this a/b thing means, how a and b are related, etc. These kids struggle with the first round of the game when the target is 0, when the idea is to just want to make their fraction as small as possible.
(1) Some kids have got a basic understanding of the meaning of the fraction and can play confidently when the target is 0 or 1. They might still be weak about equivalent fractions. Trying to play some spot-on equivalents when 1/3 and 1/2 are targets is a give-away.
(2) familiar with some frequent friends: kids who can tell readily whether their plays are larger or smaller than the target for 1/3, 1/2, 3/4.
(3) proficient: have at least one consistent strategy they can work through to make a comparison
(4) fraction black-belts: using multiple strategies, already familiar with many of the most common comparisons.

I don't know precisely what fraction experience the 4th graders had, but their teacher assured me that "they understand fractions." In that class, I saw kids at stages 0-4.

The younger students had a very sound foundation in models of fractions: diagrams of pies, cakes, chocolate bars, number lines and physical experience with baking measures and fractional inches on measuring tapes and rulers. They were also introduced to and practiced four methods for comparing fractions:
(1) common denominators
(2) common numerators
(3) distance to 1
(4) relationship to another benchmark number. Like 1/2 in your 4/6 and 8/18 example, a "familiar friend" that should be relatively easy to see it is larger than one and smaller than another. In practice, 1/2 seems to be the most popular benchmark.

Not surprisingly, they were the only ones at/approaching stage 5.

I wrote up notes in a couple of blog posts: 4th grade class, one-on-one.

Other than the ideas I listed in my post, I would make other changes for the whole class activity (a) lean toward doing this more as a cooperative puzzle, (b) re-order the targets for the rounds as 0, 1, 1/2, 3/4, 1/3, 2 and (c) I also would consider allowing equivalent fractions to the target as winning plays.

1. Thanks for steering me to Denise's fraction game. I really like the way you used it as a formative assessment. I wonder how surprised the teacher who assured you that the students "understood fractions" was. The way you re-ordered the targets makes sense to me. I could see some students lingering at beginning targets while others move on to the later ones. I will give your posts a close read and look forward to trying this out with our students. And I think I'm also going to modify this to use with whole numbers.

3. This is exactly why I show the teachers I work with the Cathy Humphries/Ruth Parker book Making Number Talks Matter. In this resource they have number talks that work specifically on building students fraction sense. Looking at two fractions and comparing them based on benchmarks of 0.1/2,1 and 2. I also love the one where they have you place two fractions on the board and ask the students if they were to add these two fractions would the answer be closer to 0.1/2,1 or 2. These number talks help build fraction sense so well! Throw in some fractiontalks.com and you are away. Your posts also reminds me of why diagnostic interviews are so important from collecting assessment data through conversation, observation and product. Ontario has now made those three weighted the same in our collection of assessment data, unfortunately we still see product being used the most especially in junior/intermediate grades. Thanks for sharing Joe.

1. I wish that we placed more emphasis on diagnostic interviews. Believe it or not, for us it's all paper and pencil from grade 1 on. I know this is well covered ground, but the reading people place so much emphasis on their running records, and we don't have anything comparable. So teachers are left to their own devices and rely on the paper and pencil, sometimes because they don't know what questions to ask or what to even look or listen for.

4. I think there is also a lot of emphasis on hard evidence in the system. The perception of students, parents and oftentimes teachers is that the evidence of doing math has more value than the flimsy evidence of flimsy understanding. I was once told at the parent conference that no one cares about understanding, it will eventually come one day, or if not no one still cares. People in my province were protesting at the legislature that they want "back to basics"math back. Many students in my community are enrolled in worksheet-based tutoring systems that skip the understanding part all together.
Hanging around MTBoS, I sometimes forget that people on my twitter feed still don't represent status quo in how the discipline of mathematics is perceived.
But there is no sense in complaining about the system, starting change from yourself is always the way.