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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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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I make the problems too difficult, the children don’t listen to each other.
Today we worked on double-digit addition and subtraction problems involving multiples of 10. Both Sunny and Daniella had some trouble with the pennies problem (Join Change Unknown: 22 pennies, how many more to have 50). What I find fascinating is that Sunny and Daniella both used unifix cubes in sticks of 10 to build 22:
Next time, I’ve decided to write problems with numbers that are more conducive to using tens in the strategy. For example, if the problem is something like 20 pennies and how many more pennies to have 45, it will be easier to build up from 20 pennies to 45 using unifix cubes in sticks of ten. (Of course! I now say to myself.)
Jack solved this problem handily although his strategy made no use of tens. He counted up by ones from 22 using tallies to keep track.
I have noticed several of the children writing vertical double-digit problems for the Separate Result Unknown problem (40 chocolate chips, eat 15) but not making use of tens in their solutions. For instance, Jack wrote a vertical number sentence for 40-15, but actually solved the problem by modeling it with individual “chips”:
I will continue to write problems like these and provide children with materials that are structured in tens (e.g., unifix cubes in sticks of tens or base-10 blocks). I want to urge all of the children to use these materials to solve problems, and – perhaps just as important – to record their strategies using numbers (not words) to help them connect the base-10 structure of the materials with their symbols. I expect that developing an understanding of base-10 concepts in a way that these conepts are usable in their strategies will take some time.
Finally, I must lament that I have not been very successful getting the children to listen to each other’s strategies! Each child is perfectly willing to tell me – and whoever WILL listen for that matter! – about what they have done. There is, truth be told, a lot of noise as they solve problems. So for now, I have the children spread out to solve their problems and I talk to them individually.
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Problem set #3.
I wrote these problems as a follow up to assessing and developing base-10 concepts. Again I included the children’s names for interest. In hindsight, some of the problems were TOO DIFFICULT and I already have some ideas about how I will change them for our next session.
1) Daniella has 40 chocolate chips. She ate 15 of them. How many chocolate chips does Dominuqe have left? (Separate Result Unknown)
2) Sunny has 22 pennies. How many more pennies does she need to have 50 pennies to buy a book? (Join Change Unknown)
3) Jack has 60 cards in his collection. He gave away 26 cards. How many cards does he have left? (Separate Result Unknown)
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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.
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Problem set #2.
I wrote these problems to find out more about the children’s understanding of base-10 concepts.
1) For Valentine’s Day, Emilio got 4 rolls of candy. Each roll had 10 candies in it. How many candies did Emilio get altogether?
2) Sunny has 16 pieces of chocolate. Just to be nice, her friend gives her 20 more pieces of chocolate. How many pieces of candy does Sunny have now?
3) Jack has 30 dollars. How many more dollars does he need to have 45 dollars to buy a new bike?
4) Daniella has 55 tropical fish. She wants to put 10 fish in each bowl. How many bowls does she need for all of the fish?
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