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.
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Problem set #7
Candy Factory
At the Candy Factory, the candy packing machine puts 10 candies in each roll and 10 rolls in each carton.
Then, we solved some problems:
1) Mr. Diaz has 6 rolls of candy and 10 loose candies. How many candies does he have?
2) Ms. Principal has 110 candies. How many rolls of candy can she make?
3) Ms. Teacher1 and Ms. Teacher2 are buying candy for their classes. They want each child to have only 1 candy each. How many rolls of candy should they buy altogether?
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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.
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Problem set #6.
1) Daniella has 42 beads. She wants to make necklaces with 10 beads on each necklace. How many necklaces can she make?
2) Jack has 30 cents. Emilio gives him 52 cents. How much money does Jack have now? What could he buy with this much money?
3) Sunny has 11 packages of cookies. Each package has 10 cookies in it. She also has 5 extra cookies. How many cookies does she have in all?
I’m still using multiples of tens in the problems. (And no one yet as asked me why we work with multiples of 10 every week; a sure sign that we need to keep working with multiplies of 10 because the children don’t differentiate them, as a class of numbers, from other numbers.) I put bigger numbers here hoping that using individual tallies for all the objects would get tedious. And it did, for some.
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One dime is ten cents and tens cents is one dime.
We continued to work on building a flexible understanding of ten as a unit. We read Only One and talked about the big idea that one thing could be many things. They seemed to understand this idea in the abstract, and it gave us a point of reference when talking about tens and one. “Oh, you mean a dime is the SAME AS ten cents!�
Emilio and Jack both solved problems in ways today that showed me they are building this understanding. I was especially happy to see Jack draw this:
to represent 5 dimes with 20 cents subtracted out. In my sessions with him before today, he has been representing tens with ten tally marks or something similar, so to use one circle to represent 10 things was a real advance! I was hoping that a dime would have for the children a “one-nessâ€? and also a “ten-nessâ€?. But the problems I wrote may not help children develop this understanding if they don’t know money denominations
What’s so hard about these problems?
Both Daniella and Sunny struggled with these problems. In fact, Sunny didn’t solve a single problem. She represented the 5 dimes in problem 1 with 5 cubes. We talked about how much 20 cents was; she knew it was 2 dimes. But when I left her to work on the problem, she took all of the tens she had (unifix cubes) and broke them into ones to represent the stars in the sky that Emilio, in the story problems, had decided to buy. Daniella seemed confused about dimes and cents as different units, and how they related. She wrote Emilio “has 0 mony now� bec 5 of something take away 20 of something leaves you with, at most, 0. Interestingly, she represented the dimes in this way:
which shows she knows at some level that 1 dime is 10; but she didn’t seem able to use that knowledge to solve the problem.
I think these problems were just right for Jack and Emilio, but too hard for Sunny and Daniella.
For children whose understanding of dimes and, more generally, multiples of 10, isn’t automatic, these problems must seem like multistep problems with one of the steps left out! Like this: Henry has 4 packages. He eats 6 cookies. How much food does he have left? It doesn’t make much sense without the crucial information of how many cookies in one package.
Looking forward.
What should we do next? It is clear to me that simply telling or showing these children that one ten is the same as ten ones is not enough to help them learn to use this knowledge in problem solving. It is a difficult concept for young children, although of course if you understand it, the fact that it is a sophisticated mathematical idea isn’t obvious. So this is what’s on the agenda: more problem solving and more explicit discussion of how they are using tens. And once we get those norms for listening developed, more comparisons of each other’s strategies and the differences and similarities in how tens are used.
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Today, we focused.
As I picked up the kids from their classrooms, Jack informed me, “Today I’m going to focus!�? Daniella agreed that she too would focus today and showed me the gesture for focusing in sign language. I was relieved to hear this announcement because the last time we met things were very … unfocused. They are excited about the extra attention, excited to see each other, and excited about special snacks. Just like any group, even a very small group like ours, it takes time to come together and develop norms. And they did focus much better today – or at least, Jack and Emilio didn’t get into a verbal sparring match – although we still need to work on what it means to listen!
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Problem set #5.
First we read a book called Only One, which emphasizes the mathematical big idea of thinking of several things as one thing (examples: 1 dozen is 12 eggs; 1 dime is 10 cents). I asked the children to fill in the blanks:
1 egg carton = ____ eggs
1 dime = ____ cents
1 ____ = ?
1 ____ = ?
Then we solved problems:
1) Emilio had 5 dimes to spend. He bought a _____ that cost 20 cents. How much money did he have left?
2) Sunny had 2 dimes. She wants to buy a ____ that costs 40 cents. How much more money does she need?
3) Dr. E has 70 cents. She spent 52 cents on a chocolate bar. How much money does she have left?
I chose these problems to continue to work on helping the children develop ten as a unit. As you will see, the money context posed some special problems of its own.
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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.
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Quick problems to begin session #4.
I began by holding up 3 sticks of 10 unifix cubes, and asking the children how many I had.
All but Emilio said 30. (Emilio was sharpening his pencil.) Then I held up 52, in 5 tens and 2 ones. That was a little harder for them to see, but basically they understood the tens and ones combination. (Sunny saw 42, Daniella saw 51, and Jack saw 52. Emilio was still sharpening his pencil!) Yet their understanding of the base-ten structure of double-digit numbers seems fragile, because they used very little of that understanding to solve these multidigit problems.
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Problem set #4.
Solve in at least two ways.
30 – 12 = ___
20 + ___ = 45
40 + 21 = ___
I wrote these problems without a story context to find out if children could connect numbers and context. It seemed from previous sessions with the children that they understood operations. Because they have been using a lot of strategies based on ones but not tens (for example, using tally marks), I used multiples of ten in these problems. Before the children started working on these problems, I made sure each child had 70 unifix cubes in groups of ten in front of him or her (we’ve been working on keeping them in tens).
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