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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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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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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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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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.
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