Our work with four second graders having difficulties in math.
Luz Maldonado and I worked with four second graders who were having difficulties in math, for about 10 weeks. Each of us got together with them once a week, on separate days. What follows are my reflections on working with these four children. As you will see, our work soon turned to diagnosing and developing base-10 knowledge, a key construct in elementary math.
Table of contents:
Introduction
Why this case study?
What is mathematics?
Why not direct instruction?
Session 1
Problem set #1.
We begin our work together.
Session 2
Problem set #2.
I discover a ‘lack of number sense’ is really a lack of base-10 understanding.
Session 3
Problem set #3.
I make the problems too difficult, the children don’t listen to each other.
Session 4
Problem set #4.
Quick problems to begin session 4.
Developing 10 as a unit.
Session 5
Problem set #5.
Today, we focused.
One dime is 10 cents, and 10 cents is one dime.
Session 6
Session 7
Session 8
Problem set #8.
Extending the children’s thinking.
Session 9
Problem set #9.
True/false number sentences.
Session 10
Problem set #10.
Our last session together.
Alternatively, you can begin here and follow arrows to later posts (in the upper right corner).
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Our last session together.
Today I wrote a mix of problems for the children to solve that would give us some insight into what the children had learned about base-10 concepts and their use in problem solving. I worked with Jack and Sunny, and Luz worked with Emilio and Daniella.
Jack.
Jack seems to have made a great deal of progress. He solved the first problem (8 rolls of candy, 10 in each roll) today by drawing a rectangle-like representation of each roll. At my suggestion, he wrote “10″ above each one. He finished off by drawing the extra 12 candies individually. As he counted them however, he pointed out the extra 10, for a total of 90 and 3, oops, 2 more:
I also asked him to write a number sentence and he wrote “10, 20, 30, 40, 50, 60, 70, 80, 90, 92″ (as before). This strategy is significant because he no longer depends on representing the individual units (each candy) to construct 10. Ten is a unit for him!
In the middle of our session he spontaneously showed me this strategy for subtracting:
It’s an interesting contrast with the rest of his work today. It was as if he was trying to re-call the steps involved in subtraction with regrouping rather than re-construct the steps based on what made sense to him. I didn’t address this mistake with him, because I wanted him to continue to work from what he understood.
I asked Jack if he could solve the second problem (30 pencils, 29 more pencils) in his head; he thought for a moment, said no, and proceeded to draw this:
This time I asked his to write a number sentence using plus and equals to show how he solved it. He wrote “10+10+10+10+10+9=59.” As he was writing the tens, I asked him how many tens in 50. “Five,” he said. He understands the place-value relationship between 50 and 5 10s.
Jack’s solution for the third problem (45 beads, 10 beads per necklace) suggests that his new knowledge of ten as a unit may be somewhat fragile. When I checked in with him, he had written on his paper:
The problem seemed to be solved. He seemed to think the answer was 4. Excellent! But as I questioned him about what he had done and why he had done it, his answer changed … to 5 (pointing to the remainder), then to 40 (the number of beads in 4 necklaces). I continued to ask him questions to help clarify his thinking and to emphasize the context of putting beads on necklaces, and the relationship between beads and necklaces. We finally arrived back at his original answer of 4 total necklaces.
Turning the open number sentences, I again asked Jack if he could solve the problems in his head. “Yes,” he said, for 30+40=__. He counted on by tens from 30 to get 70. I skipped 25+20 in order to see what he would do with another problem that involved only multiplies of 10. “How about 60-20?” I asked. “80,” he replied. I drew his attention to the minus sign (he knew what is was). “So if it’s plus,” I said, wanting to reinforce his mental strategy, “the answer is 80. What if it’s minus?” Jack easily counted back by tens to get 40 and likened the problem to 6 take away 2.
Nice job Jack!
Sunny.
Sunny’s strategies were more concrete than Jack’s, but I noticed that the language she was using to describe these strategies suggested an emerging understanding of base-ten and place-value concepts.
For instance, for the first problem (8 rolls of candies, 10 candies per roll), she direct modeled the problem using unifix cubes in sticks of 10. But when she described her solution she said, “It’s 80, because 8 tens is 80, when you count by 10 8 times, it’s the number 80.” I can see in this explanation that Sunny is making a connection between “counting by tens” a certain number of times and multiples of ten.
The emergent nature of her understanding of the base-ten structure of numbers was apparent in her strategy for adding 30 pencils and 29 pencils. Again she direct modeled the quantities. But beyond this, Sunny made little use of base-ten concepts: she counted up by ones from 30. I think she is just arriving at understanding 30 is 3 tens and that applying this knowledge in constructing a solution such as counting on by tens is somewhat beyond her. However, if I were to continue to work with Sunny, I would continue to give her addition and subtraction problems with double-digit quantities to help her develop strategies that made more use of base-ten concepts and processes.
Sunny also did something for this problem that I don’t think I understand. As she was modeling it, she included 2 extra cubes. I asked her what the 2 extra cubes were for. She didn’t say clearly. I pointed to each quantity she had made and asked her what part of the problem it was. She pointed out Jack’s pencils, and Emilio’s pencils and realized that the 2 cubes didn’t have a referent in the problem.
I had to leave at this point (appointment with the dentist) and left the children in Luz’s hands.
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Problem set #10.
1) Sunny has 8 rolls of candy. Each package has 10 candies in it. She also has 12 extra candies. How many candies does she have in all?
2) Jack has 30 pencils. Emilio gives him 29 more pencils. How many pencils does Jack have now? (Is it enough for everyone in second grade to have 1 pencil?)
3) Danielle has 45 beads. She wants to make necklaces and put 10 beads on each necklace. How many necklaces can she make?
4) Use two strategies to solve:
30 + 40 = ______
25 + 20 = ______
60 – 20 = ______
20 + _____ = 31
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True/false number sentences.
I started by asking the children to make up a problem they could solve in their head. I was hoping they would choose numbers that they could mentally manipulate; but as I found out, it requires a degree of metacognitive awareness about what’s easy and what’s hard relative to their own understanding.
My example for them was 10 plus 10 plus 10. I wasn’t subtle at all here: I wanted the children to use tens in their problems. But, as usual, they had other things in mind.
Sunny wanted a story to go with her number problem. So she made up her own: Sunny and Daniella go to the store and buy 90 candies. They eat 8 of them. How many are left? I suspected the problem was too hard for her to solve in her head – and it was. She got 80. Based on her explanation of how she got 8o, it seemed she was trying to do something in her head with the standard algorithm. She mentioned crossing out. But when I suggested she solve it another way using the cubes, she easily modeled everything using tens, and counted what was left to get 82.
We moved on. I had my index cards ready. I wanted to start with a number relationship I knew that they knew: 10 + 10 = 5. I asked the children to write the sentence down then put “t,� “f,� or “?� after it. This one was easy. They all knew it was false.
Then we did: 5 + 5 = 10 + 10. I expected there wouldn’t be consensus on this since many children interpret the “=â€? as “and the answer is…â€? Under this interpretation, the number sentence is true. Sure enough, two of the children said, yes it’s true and one put a question mark. I introduced the language of “is the same amount asâ€? for the equal sign and we talked a little about whether 10 was the same amount as 20. No problem there. But I didn’t necessarily expect the children to have fully assimilated the meaning of “=”.
Next we did: 10 + 10 + 10 + 3 = 33. This one was really designed to see how they figured the sum. Jack counted the 10s then the 3 and said it was true. Daniella used 10 plus 10 is 20, but then counted up by ones. Sunny modeled it with tens to get 33. True.
Next: 22 + 10 = 30. I was so pleased with Sunny’s response on this one! She used relational thinking and I think her prior work with the unifix cubes may have helped. She said that 20 plus 10 is 30, but since the 2 is with the 20 it should “be taken out of� the 10 and 22 plus 8 is 30. It was a flash of insight.
Next: 10 + 10 + 10 + 10 + 10 = 100. Jack counted up by tens. It was easy for him to figure out it was false. Sunny wrote that it was true, but I didn’t find out her reason. Daniella got out 5 tens, said it was 50, but then to prove it, she counted on by ones from 20. It’s just not clear to me how much she understands. She’s right on the cusp.
Then, a controversial one. (Sunny wanted to know what “controversial� meant.) 12 + 10 = 10 + 12. At first, they all said it was false, although Jack wavered a little and wanted to put a question mark. So I told them I was going to make up a story problem to help them think about it. It went something like this: Sunny had 12 cents. Then she got 10 cents for her birthday. Daniella had 10 cents. Then she got 12 cents for her birthday. Do they have he same amount of money or not? The answers were interesting! Daniella added both sides up (counted on by ones from the first numbers for each), got 22 for each. Jack said you could add the numbers in either order, it didn’t matter. As we talked, I wrote this for them:
But the next number sentence brought a nice surprise from Daniella: 10 + 2 = 5 + 5 + 2. Sunny and Jack though it was false but Daniella argued it was true and her reason was just beautiful. She grabbed the card from me and wrote on it:
For next time: what does Daniella understand about grouping tens? Sunny? Can Jack use invented strategies for adding double digit numbers? We’ve got one more session together.
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Problem set #9.
I decided to carry on with true/false number sentences, to help the children begin to use mental strategies involving tens, and to help them become comfortable working with symbols.
True or False?
10 + 10 = 5
5 + 5 = 10 + 10
10 + 10 + 10 + 3 = 33
33 = 10 + 3
10 + 10 + 10 + 10 + 10 = 100
22 + 10 = 32
12 + 10 = 10 + 12
10 + 2 = 5 + 5 + 2
2 dimes = 4 nickels
4 dimes = 10 pennies
As Megan Franke has described doing in her own work with elementary students, I wrote these number sentences on index cards so that I could easily shuffle through them as I was working with the children.
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Extending the children’s thinking.
As the kids were having their snack, I introduced the idea of true/false number sentences, based on ideas in Thinking Mathematically, by Tom Carpenter, Megan Franke, and Linda Levi.
First I wrote on a big piece of paper with a marker: “2+3=5â€? and asked them if it was true or false. Jack and Sunny said it was true; Emilio wasn’t sure, perhaps because, I figured out later, he wasn’t sure what “trueâ€? and “falseâ€? meant, especially in the context of math. Then we looked at: “2+3=6â€?. Catching on, Emilio decided it was false. The number sentence that generated the most controversy was “10 = 10â€?. Sunny declared it false, because, Sunny said, a number cannot equal itself. (So, you see, the reflexive property isn’t axiomatic for everyone!) Emilio agreed with Sunny. Jack declared it was true, because it was like 10 and nothing added was 10. I wrote underneath to clarify: “10+0=10â€?. Sunny agreed that this one was true but continued to disagree that 10 could be “equal to itself.” I promised we would revisit this debate and moved on to the first problem.
Emilio does something different.
As the children worked on the new Candy Factory problems, they fell into their usual patterns, with the exception of Emilio. I sat with him first to get him started. He read the problem to himself then I asked him rephrase it in his own words outloud. No problem; he remembered the quantities in the Candy Factory. I asked him how many candies in a roll; he said 10. So he understood the context and the problem parameters. “So,â€? I asked him, “how many candies does Dr. E. have?â€? His first reponse was 40 because, he spontaneously gave the reason, there’s candy in the four rolls. When I asked him about the 11 loose ones, he got 52, at first, because he added 10 on to the 40 (nice work!), and counted up somehow to get 52. (Being off by one in his counting reminds me of the counting mistake he made the first day we worked together. I’m not sure what’s going on.) When I asked him why he added the 10 on, he didn’t really say, and ended up solving the problem by counting up from 40 by ones. What a terrific solution! In contrast to his thinking last week, he did not seem to have any problem distinguishing groups of 10 from singletons; and once he understood the context, he had no problem applying his knowledge of multiples of 10.
I must admit, I am puzzled by how easily this strategy came to Emilio, considering the struggle last time we met. I wonder how much of his success, or lack of it, is based on whether he is preoccupied with something more important or perhaps just more interesting than the problem at hand. After all, when our attention is split, our capacity to reason is compromised, as this research suggests.
Jack uses numerals to represent his thinking.
Jack direct modeled by representing the groups of 10, showing each candy. But just as he has been doing, he counted the solution by 10s. He has shown that he doesn’t really need to do this and so I asked him to write a true number sentence that showed how he solved the problem. He wrote: 10, 20, 30, 40, 10, 1 — showing the quantities separately, and not how he combined them by counting tens. I asked him to write another number sentence using plus and equals like we had been doing, and he wrote 51 = 52 – 1; 51 = 53 – 2; 51 = 54 – 3; 51 = 55 – 4 — NOT, as I was hoping, 10 + 10 + 10 + 10+ 11 = 51. I wonder if it would make a difference if I ask him to show the rolls and candies with a number sentence? I want him to articulate (verbally or symbolically) how 51 is related to groups of 10.
Dramatization helps Sunny.
Sunny, like before, seemed to have trouble getting started. She confused rolls and candies, and at one point said there were 10 rolls, instead of 4. She also didn’t combine the rolls and the loose candies at first. Although her strategy wasn’t clear to me, I think she separated out the 11 loose candies from the 4 rolls. I thought that animating the situation for her, and in particular, putting her in the problem with me, might help her visualize the context. So I dramatized the problem with her as a character asking Dr. E. about the candies she had, just as Vicki Jacobs and Becky Ambrose reported teachers do. It worked. She decided that she needed 4 rolls of 10 and the extra 11, and counted them all to get 51.
Making connections.
At this point I decided to gather the children together. They had three different strategies but all of them had in common the use of tens in some way. A number sentence would be a good way to help tie together the ideas that were in each of these strategies. I wantged to create an interplay between representing thinking in abstract ways and their concrete strategies to stretch their understanding. I asked Jack to describe his strategy and as he was talking about the sticks of ten I wrote “10 10 10 10� to represent what he had drawn and to make a connection. Then on big paper, we followed through with: “10 + 10 + 10 + 10 + 11 = 51� to represent the entire situation. In my mind, I was thinking that building up to the number sentence from the concrete strategy to the numbers to stand for the quantities instead of pictures would help children build a connection. We’ll see.
The children write their own problems.
Emilio was pretty engaged throughout and even asked for a piece of paper to write his own problems. Everyone wanted to do this so I let the kids create some problems or number sentences.
Jack wrote a problem: There were 61 cats and then I found 41 more. How many cats? (What nice number choices!). He started to direct model by drawing a stick of 10 and I stopped him and told him to use numbers to solve it, because I knew he could. And he could. Here’s what he wrote: 10, 20, 30, 40, 50, 60, 1; and below it: 10, 20, 30, 40, 1. And so, I asked him, how can you use this to figure out how many in all? And he handily counted 10, 20, 30, 40, 50, 60 (and I was wondering what he was going to do when he started on the 40), 70, 80, 90, 100, 101. (He had forgotten an extra 1, which we talked about.) He loves writing these problems!
Sunny, at my suggestion wrote one true number sentence (10 + 10 = 20) and one false number sentence (35 – 6 = 30). It was interesting to me how she figured out the false number sentence, especially because any number (but the one correct answer of course) would make it a false number sentence. But she carefully solved the problem by counting back by ones to 29, and decided her wrong answer would be just one off – 30. I think there’s an aesthetic to this thinking, akin to a mistake rather than a wildly wrong answer.
Finally, Emilio wrote a bunch of addition and subtraction sentences involving numbers in the hundred thousands which were, he admitted, too hard for him to solve!
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Problem set #8.
We worked with the Candy Factory again and using number sentences to represent the situation.
1) a. Dr. E. has 4 rolls of candy and 11 loose candies. How many candies does she have altogether?
b. Dr. E wants Sunny, Daniella, Jack, and Emilio to share her candies equally. How many candies can each child have?
2)True or False?
10 + 10 is the same as 5
5 + 5 is the same as 10 + 10
10 + 10 + 10 + 3 is the same as 33
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NCTM presentation in Anaheim.
Luz Maldonado and I (and Erin Turner, in absentia) presented at the recent NCTM conference on “Challenging Content, Challenging Students: Supporting Student Engagement in Learning Fractions and Ratios.” In this session we discussed our work teaching a couple of after-school classes with fifth graders from a mix of achievement levels, with a special focus on the students who had been singled out as low-achieving. We’ve posted our handouts here for you to download (Powerpoint and MS Word files):
Sample problems: Sample problems from our fractions unit.
Teaching principles for struggling students: Teaching Principles
Some of the readings that have informed our work: Readings/References
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