I observed a lesson in a classroom today and while observing I began to wonder what Parker would do with the problem with a little tweaking. The problem in the classroom was:
You are making reusable grocery bags out of cloth. You have 3.6 yards of cloth. Each bag takes 0.3 yards of cloth. How many bags can you make?
In the class the students were given base-ten blocks to use if they wanted (many did and many just worked symbolically). If you're not familiar with base-ten blocks you can see some of them below. The smallest one can be referred to as a block, the stick of ten blocks put together is called a rod and the largest piece is called a flat. These are sometimes referred to as ones, tens, and hundreds, but as it's important to be able to think of any of these objects as one when starting to think about fractions and decimals.
Base ten blocks: a block, rod, and flat
In the class I observed they treated a flat as one and the "rods" were therefore tenths. So 3.6 yards looked like this:
3.6 yards represented with base-ten blocks
A Revised Problem
Parker could not solve this problem if I just gave it to him as stated, but I did think he would be able to work with the same concepts. So I did the following with him.
Establishing a Common Language
First, I changed the context to blocks of Play-Doh. And to make sure Parker and I were speaking the same language I did the following. I got out one base-ten block and said...
M: Imagine this is one block of Play-Doh and ten kids are going to cut this up and share it. How much Play-Doh would each kid get?
P: Ten [Parker grabbed a one of the rods to show me]
M: Ten what?
P: They'd each get ten little blocks.
M: How did you get that?
P: Well this is 100, so they'd each get ten.
Parker has used base-ten blocks some in the past, so he was used to thinking of the blocks as 1, the rods as 10, and the flats as 100.
M: Okay, but if we imagine this [the "flat"] is one block of Play-Doh, how much of a block does each kid get?
P: A tenth.
I wasn't sure if Parker would remember the term tenth. We have talked about the names of a few fractions before, so I thought he might. If he didn't I would have just said, "we call these pieces tenths. When you cut something up into ten pieces, we call the pieces tenths."
The New Problem
So then I read him the problem, which I had written out. I made the numbers a bit smaller (2.4 instead of 3.6) so that it would require less time/work, but still require breaking up wholes into tenths:
The teacher has 2 whole blocks and 4 tenths of a block of Play-Doh. The teacher is going to give each kid 3 tenths of a block of Play-Doh. How many kids will get Play-Doh?
Parker started by getting out the following:
Parker's representation of 2 whole blocks and 4 tenths of a block
He started by taking three of the rods to make a pile. Then he took some other rods (extra ones off to the side) to start to make an additional pile. After thinking some more he ended up asking for a piece of paper. His final product is a bit messy because the picture is a little messy and because he started crossing things off, so I have recreated some of his steps to make it clearer what he did.
Step 1: A picture of his representation 2 whole blocks and 4 tenths of a block
Step 2: Splitting the wholes into tenths
Step 3: using up 3 tenths at a time and making a pile of 3 rods each time
Each time he did Step 3 he would create a small pile of 3 rods:
Here's P's actual work (his "x's" are just lines through the boxes, so it's hard to follow, plus he originally divided the left box into 12 pieces on accident so he crossed two of them off):
Eventually, he had crossed off all of the tenths in his picture and created the following:
Parker's final product
Parker counted the eight piles and announced that "eight kids can get Play-Doh."
Notation versus Meaning
I've talked before about the power of story problems in supporting young children in doing complex mathematics (especially when it comes to fractions). Too often school mathematics becomes overly focused on notation at the expense of meaning. Notice the difference between "2.4" and "2 whole blocks and 4 tenths." Here I have replaced mathematical notation (2.4) with something that is more meaningful and accessible to Parker.
To be clear: This is not a critique of the lesson I observed. Children need to learn mathematical notation at some point and the lesson I observed was completely appropriate for them. It was appropriate for their current understandings and addressed the standards.
Mathematical notation plays two important roles. One is clarity of communication. If everyone else writes "2.4" you eventually need to know how to make sense of "2.4" if you're going to engage in mathematics with others. However, mathematical notation is not entirely arbitrary—it has been designed to be very powerful. Mathematical notation does a lot of important work for us. Once someone is fluent with the notation it is much faster and easier to work with.
A great example can be found in the history of algebra. Algebra was not always written with x's and y's. For instance, originally people would have written "The thing plus one equals two" instead of x + 1 = 2. (This example is taken from Wikipedia.) You can only get so far in developing algebraic techniques if you write everything out in sentences. Today's symbolic notation for algebraic is a major accomplishment.
But let's go back to the word fluent. It's very important that students develop fluency with mathematical notation. This happens through lots of opportunities to first build meaning for the concepts (for instance, to know what a tenth is, how to represent it visually, how to work with it, etc.), then to be introduced to the notation and have repeated opportunities to move back and forth between the notation and the meaning. For instance children should be asked to represent things like "2.4" visually in ways that make sense to them. They should have to work with decimals in realish contexts like these story problems. They should be allowed to develop their own strategies for solving problems like these. If they do something like "2.4 - 0.3 → 2.1 - 0.3 → 1.8 ..." They should be asked questions about how each step of their work relates back to the context. (All of these are things I saw in the lesson.)
Where Things Go Wrong
The problem is when (a) notation is introduced too soon, and (b) a child's ability to follow memorized rules with the notation is taken as evidence of understanding and sense-making. Unfortunately, both of these things happen far too often in many classrooms.
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