Observing Student Representation
 Introduction | Estimating Blocks | Problem Reflection #1 | Counting Blocks | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Reflection Questions | Your Journal

Now that the students have made estimates, they need to count the number of cubes in their containers, and they need to show the process they're using to count the cubes. They have previously been working on grouping items by tens, and the teacher is interested to see if they use any grouping strategies.

The group we visited in Problem 1 is counting the blocks and putting them in piles of 10. They have represented their thinking in the following way:

Teacher: How many cubes were in your container?
Vanessa: Eighty-one.

Teacher: Explain to me how this shows 81.
Richard: We have eight groups with 10 blocks, so that's 80. There was one extra block, so that means we have 81 all together.

Teacher: That is a good way to show 81 with the blocks. How could you show your work on paper?

The group members go back to work to show their thinking using paper and pencil. They come up with another way to represent their thinking:

Teacher: What do you have here?
Jose: Well, we used tallies -- we made a 5 and a 5 for each pile of 10 blocks. We kept doing that until we had tallies for all of the blocks.

Teacher: Show me how you would count them.
Group: (counting by fives, pointing to each group as they count) Five, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80. . .
Vanessa: No wait -- that should only be 71.

Teacher: Why do you think that?
Vanessa: Because the last one is all by itself, so that's only 1.

Teacher: (pointing to the groups of 10 blocks) How many blocks did you say you had here?
Jose: Eighty-one.

Teacher: But you said there are 71 tallies. Which is correct?
Jose: We better count them both.

After counting, the students realize that they did not record one set of blocks, and they add 10 more tallies to their record sheet:

Here is how another group of students recorded their thinking (they began drawing individual blocks in the first circle but then erased them):

Teacher: Can you explain what you have here?
Marsha: We have eight sets and one left over.

Teacher: What does each set mean?
Jim: Each one is 10 blocks.

Teacher: (pointing to the first set) What is this?
Anthony: We started to draw the cubes, but it was too much work.

Teacher: That is a lot of work -- but how can you show me that each circle has 10 if you don't want to draw all of those cubes?
Marsha: We could use tallies or something easier to draw.

Teacher: How else can you show how many are in the circle?
Cathy: We can just put a 10 in each one.

Teacher: (pointing to the extra square) Okay -- now, what is this?
Cathy: That's the extra block. We could just mark that with a 1.

Teacher: Sounds like a good plan. Call me back when you finish.

The students set to work. Their completed representation looks like this:

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