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!
Posted in Case Study: Four Second Graders, Emilio, Jack, Sunny, Uncategorized | Comments (0)
Frustration!
I began by posing some quick problems just to check for understanding of the context. I asked how many candies were in 2 rolls, then in 5 rolls. Jack and Sunny both counted by tens to figure these problems out. Emilio too seemed to understand, although looking back, and knowing what he did later in the session, I’m not sure now.
I limit their use of tallies.
Today I asked them to sit at the same table, because I wanted to orchestrate a conversation about their use of tens in their solutions. I began by reminding them how sometimes they solved problems by making single tallies:
I said that today, I didn’t want them to use tallies like this. They could use the unifix cubes in sticks of ten or use numbers written on their paper or solve it mentally. My purpose was to push them to work with groups of tens. Allowing unifix cubes meant that if they needed to count by ones they could; but the structure of groupings of ten would at least be something they had to choose to ignore!
Jack sustains the progess he made last week.
As it turned out, Jack began the problem by drawing the rolls without candies:
He then decided to add the candies in.
Just as he was up to the next to the last roll, I asked him if he needed to show those candies in order to count them. This conversation was just like the one we had last week! He quickly said no and wrote “10� in the last roll. When I asked him later to write a number sentence or write numbers to show how he solved it, he wrote: 10, 20, 30, 40, 50, 60, 70. Progress!
It’s not clear what Sunny undertands about ten as a unit.
Sunny was slow getting started. She seemed to be confusing the idea of 6 rolls (with 10 each) and rolls of 6. She easily modeled the 10 loose candies with 10 single cubes. But for the rolls she had a stick of 6 unifix cubes and described it as “a roll of 6.� I clarified: “6 rolls of 10, not a roll of 6,� and she was off, modeling the rolls with 6 sticks of 10. There was some confusion about how to count the total of 6 sticks of 10 and 10 loose ones; she got 16 at first, but with a discussion in which I asked her to connect it back to rolls and candies, she counted appropriately. I emphasized in my revoicing of what she had done that she could count the rolls – 1, 2, 3, 4, 5, 6 – or count the candies – 10, 20, 30, 40, 50, 60. (Plus the loose ones, which no one has any trouble counting!) Success!
I am frustrated with Emilio!
Emilio solved the first problem and got 16 (adding 6 rolls and 10 candies). I asked him to solve it a second way, and he drew a stick of 6 and a stick of 10, and counted all to get 16. I asked him to talk with Jack about his strategy, and listen to how Jack solved his, but he didn’t talk and didn’t listen. I asked him if the problem was to hard for him, but he didn’t answer (instead, he concentrated on figuring out what time it was and when he could go home). I asked Sunny to explain her (terrific) direct modeling strategy, hoping he would see the difference between 6 rolls (sticks of ten) and 10 loose candies (individual cubes). It seemed like he looked everywhere but at Sunny and, more importantly, her strategy. At each step of Sunny’s explanation, I stopped her to ask Emilio a question, trying to get him to make a connection between the cubes arranged in sticks of ten and rolls of candy, trying to get him to make sense of the problem. It felt like he was resisting. It felt like he was deliberately not engaging.
Maybe. But I decided I didn’t want to assume that he was deliberately avoiding work. Perhaps it was his way of expressing boredom and confusion. So finally, as it was nearing time for our session to be over, I asked him if he wanted me to make him an easier problem. He said he did. So I turned his paper over and wrote “2 rolls, 10 candies, how many candies?â€? “12â€? he quickly replied. So I asked him to use cubes to show the rolls and the loose candies. “How many candies in one roll?â€? I asked him. He put his head down, said he was ready to go home, but, feeling resolute, I told him he couldn’t leave until he solved this problem. I was thinking about the fact that he had solved problems like this in the past. “How many candies in 2 rolls then?â€? I asked. “20â€? he squeaked out, with his head buried under his arms. “So, if you put 10 more candies with them … why don’t you represent those 10 candies with these cubes, any way you want.â€? He picked up about 4 sticks of 10 – just what I was pushing towards him in making my suggestion – and began to put them with the 2 “rolls.â€? “Now,â€? I pressed him, “show me the 10 loose candies.â€? It took a while but, finally, he produced a stick of 10, and put it with the 2 rolls. “How many?â€? I asked. “30,â€? he said, without even counting.
I don’t know if this was a power stuggle or a cognitive leap. What is the residue (to use a term I like a lot coined by Jim Hiebert and colleagues in Making Sense) of this interaction for Emilio? What did he take away from it? An understanding? A predisposition? A feeling that he CAN do math? A feeling of being forced to do something he didn’t want to do? The answer to that question — which I’m not at all sure of — is much more important than the fact that he answered “30″ in the end.
Posted in Case Study: Four Second Graders, Emilio, Jack, Sunny, Uncategorized | Comments (0)
Progress!
The third problem (11 packages of 10 cookies each) proved interesting and productive. It was a challenge for most of the children.
Jack makes an advance.
Emilio and Jack both started out by adding up the numbers to get 26 for their answer. Jack then decided that wasn’t the correct answer (not sure what changed his mind; he didn’t say why), and easily direct modeled the problem by drawing groups of tens:
As he was drawing all of this out, I asked him if, instead of drawing each cookie in each package, he could represent the cookies in the package by writing “10� in each one. He said that no he couldn’t; it would be too hard. But I noticed when he counted the total he counted by tens, so I again I asked him if he could represent the cookies by “10� instead of drawing each one out. I pointed out to him that he had just counted each group by tens. It seemed to make sense to him this time so I gave him a new but related problem to solve, encouraging him to use this new strategy. The problem was 14 packages of cookies, 10 in each package, and 10 extra cookies. This is what he drew:
He said he forgot to use the strategy we had talked about (and only remembered when his hand started getting tired!), but because he had so easily solved this problem I was sure he could use the more abstract counting approach. So again I posed a new but related problem: him how many cookies would be in 12 packages. When I came back, he had this:
He agreed that this strategy was faster, as well as easier on the hand.
Emilio misinterprets the problem and I try to get him to listen to Daniella’s strategy to change his mind.
Emilio had trouble getting started on this problem. It’s not clear to me why. His initial answer was 26. He told me he got it by adding 11, 10, and 5. I asked him why he decided to add them altogether and whether they were all cookies or packages, but he gave no clear answer.
Daniella, like Jack, direct modeled the entire situation by representing each cookie, but she confounded packages of cookies with single cookies:
Because she had accurately represented the packages of cookies and Emilio had not, I decided to ignore her confusion about the 5 extra cookies for the time being and called Emilio over to compare what he was doing with what Daniella was doing. The first difference he saw was in how each of them had represented the package:
His was more realistic. Daniella saw that he had 6 cookies in his packages and she had put 10 in each of hers. With some prompting from me to address Emilio and not me, she was further able to tell Emilio why she had drawn her packages this way. Emilio decided to start over, and at my suggestion, gathered a bunch of sticks of unifix cubes in tens. He ended up with 34 sticks of 10 arrayed in front of him but didn’t solve the problem before it was time to go.
Sunny thinks really hard.
Like Jack and Daniella, Sunny started out drawing the packages of cookies with each individual cookie represented. I encouraged her to use the cubes in sticks of ten instead, thinking that the ten-to-one structure might support a more sophisticated strategy. She decided she wanted a bunch of sticks of four. I wasn’t sure where she was going with it, but decided to let her create them (and helped her). She then gathered sticks of ten and used the sticks of four to stand for the packages. The number of cubes in each stick was probably irrelevant; perhaps it was the long rectangular shape that reminded her of a package. The she “put� 10 cookies – a stick of ten – in each package, like this:
and counted the total beautifully by tens. During all of this she mentioned that she was thinking so hard she couldn’t even think of anything else. I think it was a keen observation because at the end I had to remind her of the 5 extra cookies. She included them but counted them as tens, as Daniella had at first.
Looking forward.
With encouragement, then, Jack, Daniella, and Sunny were able to move from counting by ones to counting by tens. Jack was able to represent groups of ten by something other than a collection of 10 things – a real advance, if he sustains it. I’m not sure what Emilio can do or how much he understands of problems like this. He has solved them in the past. His focus today seemed split so I think these problems didn’t get his full attention. In fact, he started out wanting to do his spelling homework!
Next time, we’ll work on more problems like the first and third problems and I plan to continue to push the children to represented sets of 10 with the numeral “10�. This may be facilitated by using smaller numbers. We’ll see.
Posted in Case Study: Four Second Graders, Daniella, Emilio, Jack, Sunny, Uncategorized | Comments (0)
Developing ten as a unit.
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):
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.
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.
Posted in Case Study: Four Second Graders, Daniella, Emilio, Jack, Sunny, Uncategorized | Comments (0)
I discover a lack of number sense is really a lack of base-10 understanding.
At the end of our last session, I decided that I wanted to find out more about what the children understood about groups of 10 and base-10 concepts. I had a hunch that what seemed like a lack of number sense for some of them was in fact little to no understanding of base-10 concepts. Each of the problems for this session was designed to assess and help develop base-10 understanding.
Most central to base-10 concepts is understanding 10 as a unit. Children who understand 10 as a unit find the Valentine’s Day problem easy; those who do not, find this problem just as difficult as any other grouping problem (e.g., they would solve 4 groups of 7 in the same way as they would solve 4 groups of 10).
Emilio’s thinking about the Valentine’s Day problem — in contrast to his strategy last week — showed no base-10 understanding. First he interpreted the context to mean he should add 10 and 4 to get 14 rolls altogether. After questioning him, unsuccesfully, about why he added, I described a context where he was the candy maker and had to put 10 candies into each of 4 boxes. The librarian helpfully handed us a roll of sweet tarts to help Emilio visualize the 10 candies to 1 roll relationship. I then left him to solve the problem and he solved it by modeling each box with cubes (arranged in a squarish shape) with 10 single cubes on the interior of each. He counted his answer by twos and got 40.
Success, of a limited sort then: he used his knowledge of the context to construct a solution. But he didn’t use any of the knowledge of 10s that was in evidence last week.
The other children performed similarly. I have a hunch that much of their current difficulties in math may be traced to under-developed knowledge of 10 as a unit. I decide that we will focus on developing this understanding over the next few weeks. At the same time, because of the nature of my work with them, these children will also be working on developing strategic competence and productive dispositions towards math.
Posted in Case Study: Four Second Graders, Emilio, Uncategorized | Comments (0)
We begin our work together.
The four children we are working with are Sunny, Daniella, Jack, and Emilio. I decided to spend the first session finding out about what they could do, what they understood, how they expressed it — both orally and in writing — how confident they were, and so on.
Not knowing exactly what to expect, I start with fairly small numbers.
The first problem was a Join Result Unknown (13 jelly beans, get 8 more). The children either counted up from 13 to solve it or direct modeled it with cubes or pictures. Nobody used any derived or recalled facts.
Emilio used a counting up strategy but his answer was off by one. I noticed when he counted up he started with the number 13 and counted it as one of the 8 he wanted to add on. He ended up with 20. When he heard that the other children got 21 for their answer, he changed his answer from 20 to 21.
I wanted to explore this counting mistake with Emilio. So I asked Sunny to explain her direct modeling strategy using cubes. My idea was that Emilio would be able to see the 13 as a separate set and that the 8 to be joined to the set should be counted beginning with 14. No such insight occurred! But I believe nonetheless that it will be useful for Emilio to continue to attempt to relate counting strategies to more concrete direct modeling strategies so that he can figure out how counting works.
The second problem we worked on was a Separate Result Unknown (28 pennies, lose 13). Daniella solved this problem by writing 28 – 13, vertically:
She separated the tens and ones into two columns, and subtracted the ones first (got 5) then the tens (1). Her use of this algorithm made me curious about what base-10 concepts she understood. I began listening for evidence of base-10 understanding among all the children and noticed that, even though the problem involved double-digit numbers, none of the children had used base-10 concepts in their strategies.
I therefore created a third problem, on the spot. After ascertaining that they all knew about and liked soccer, I posed this problem: “You’ve got 3 big bags of soccer balls. Each bag has 10 balls in it. You’ve also got 2 loose balls. How many balls do you have?� I used hand gestures to indicate the bags were BIG and repeated the problem to be sure the children heard it. They set to work. Everyone but Emilio was direct modeling the problem by drawing all the balls individually:
No use of tens! Emilio wasn’t doing anything, so I repeated the problem for him. “Oh,� he said. “10 plus 10 is 20.� I was so pleased with this insight that I emphasized to him (and for the benefit of the others) that he didn’t even need to draw any pictures to figure it out. It prompted Jack to remember that he too knew that 10 plus 10 was 20.
Still the children struggled. Sunny wasn’t sure whether to add or subtract the two loose balls. I told her she had to decide for herself what made sense. She subtracted because, she said, the two “loose� balls could roll away. (Interesting point!) Emilio ended up with 30, then when I asked him how he was going to count the 2 loose balls, he changed his answer to 31. And Jack ended up with 28, because one of his bags had the wrong number of balls in it.
Wow. There was so much to talk about, but it was time to go, so I made a note to myself to return to problems like this one next time.
Posted in Case Study: Four Second Graders, Daniella, Emilio, Jack, Sunny, Uncategorized | Comments (0)