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Problem set #5. »

Developing ten as a unit.

March 1st, 2005

I asked the children to begin by giving a story to the first number sentence (30 – 12 = ___). The problem they came up with went like this: “Jack and Daniella went to the candy store and bought 30 pieces of Valentine’s gum. Jack ate 1 piece and Daniella ate 11 pieces. How many pieces did they have left?”

The children use a variety of incorrect and correct strategies.

Emilio solved the problem by counting back by ones (no miscount this time). Jack solved it by, as usual, direct modeling by ones — he made 30 tally marks, and crossed out 12 of them. Both boys got 18.

Daniella solved it like this, using a common buggy algorithm (her original answer, erased, was 22):

Daniella's work for 30-12 (with first answer erased).
But when she heard that Emilio and Jack had gotten 18, she erased her “22″ and wrote “18.” When I asked her how she got 18, she said that 0 take away 2 was 8 … which seemed to me an example of making the reason fit the answer! I emphasized how important it was for whatever she did in math to make sense to her. I don’t want her to assume that because two other people got an answer different from hers, they, and not she, must be right. (Not a very productive disposition.)

Despite the story frame that she helped create, Sunny first added 30 and 12. But when I reminded her of the story she easily figured she should subtract.

Listening to each other’s strategies.

I decided to have the children listen to each other’s strategies as a way to move their thinking forward.

So far, it seems the biggest problem these children have is limited understanding of base-ten concepts and processes. Although they can count by tens and can identify groups of tens, they do not, for the most part, use this knowledge to solve problems. It is not very flexible knowledge for them. So my goal was to use the group discussion to help them begin to make connections and develop this base-ten understanding.

I had a big piece of newsprint that we could all easily see (and reach, if needed). I asked Jack to share his strategy first, because it was basic direct modeling. I represented his strategy using tallies. Sunny had the idea of grouping the tallies into tens to make them easier to count. I grouped the tallies and everyone said it was 30. This was consistent with the very first quick activity we had done with the unifix cubes and it seemed to be a good way for the children to develop an understanding of the ten-ones-is-one-ten relationship.

But when it came to subtracting 12 from this group of 3 tens by subtracting 1 group of ten and then 2 ones, the difficulty of applying knowledge of tens became apparent. None of the children knew spontaneously what 30 take away 10 was. Daniella said 29 but it seemed to be a guess. I told them to use their unifix cubes to figure it out. They did, easily.

Mental strategies versus concrete strategies.

The ease with which the children solved this problem using manipulatives suggests a clear cognitive distinction between modeling with tens, as they had done, and working with ten as a unit mentally, which they could not do. I plan to continue working with the children to make connections like the ones they made today between ones grouped into tens and ten as a unit. It is difficult conceptual work for them but I believe that repeated opportunities to use ten as a unit in their strategies will pay off.

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