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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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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Problem set #1.
These addition and subtraction problems represent four different problem structures. These structures are based on CGI probem types. (For more information about CGI, see this book.)
1. Maya has 13 jellybeans. Her brother gives her 8 more jellybeans. How many jellybeans does Maya have now? Join Result Unknown
2. Jason has 28 pennies. He loses 13 of them. How many pennies does Jason have now? Separate Result Unknown
3. Eric is putting candles on a birthday cake for his brother. There are 4 candles on the cake so far. How many more candles does Eric need to put on the cake so that there are 7 candles altogether? Join Change Unknown [not given]
4. Sarita has 9 toy rockets. Her brother George has 6 toy rockets. How many more toy rockets does Sarita have than George? Compare Difference Unknown [not given]
Created during instruction: 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?
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Why not direct instruction?
There are two reasons why I am choosing a problem-solving approach over a direct instruction approach in particular for children who are having difficulties in math. The first is that although direct instruction can be effective for learning a specific procedure, it is not an effective way to learn how to decide what procedures to use when — that is, not for learning how to solve problems or develop strategic competence. Second, if direct instruction is effective, it is most effective when conceptual understanding is in place … but struggling students often struggle precisely because they lack certain conceptual understanding. Not understanding makes it harder to remember how a procedure works or when it is applicable.
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What is mathematics?
It’s worth saying something here about what I’m hoping these four children learn. Not the specifics — which I’ll write about later — but, more generally, what it means to engage in math. I like the National Research Council’s definition of mathematical proficiency in terms of five “strands”: procedural fluency, conceputal understanding, strategic competence, adaptive reasoning, and productive disposition. You can read descriptions of each strand here, if you haven’t already. The metaphor of strands is a nice one because it emphasizes how interrelated these competencies are. It is impossible to teach any one in isolation.
To illustrate this point rather dramatically, I’d like to tell you about a fifth-grade girl my research team and I worked with in a previous study. I’ll call her Lemesha. To assess her understanding of math, I gave Lemesha a problem-solving interview. In this assessment, I read problems to her (or she could read them herself), and she solved them using whatever strategy she wanted to. We were assessing how children solved word problems that involved multiplicative situations. Altogether there were about 10 of these type problems. Lemesha didn’t solve a single one correctly. (She attempted to use keyword strategies, which didn’t help because the problems weren’t designed to be solvable by keywords). But when we gave her a multiplication-fact recall test, we found she knew well over half of her multiplication facts by recall. What a contrast! I suspect that having learned her multiplication facts in isolation from problem solving prevented her from understanding how these facts were linked to multiplicative situations.
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Why this case study?
Luz Maldonado and I are working with four second graders who are having difficulties in math. Their teachers have asked us to work with them because, out of all the second graders at their school, these four scored the lowest on a benchmark test.
My research at the University of Texas at Austin focuses on understanding children’s problem solving and the implications of what we know about children’s problem solving for teaching. I have always had a soft spot for people who struggle to understand math, but it hasn’t been until recently that I have begun to seriously research how to help children who are considered struggling or low achieving. I was drawn to work with these four children because I wanted to put my money where my mouth is: I wanted to see whether a problem-solving approach was an effective way to help these children learn. Many people believe that the best way to teach those who have difficulties in math is direct instruction. But I believe that the very things that make a problem-solving approach succesful for middle and high achieving students — building on what children know, integration of concepts, processes, strategies, practices — should hold for low-achieving students too.
My goal is to write about my decision making as I work with these four students and to share with you what I learn about each of them as we interact. Ultimately, I hope that you, the reader, find something useful here — not a formula or script for teaching but a way of thinking about your own work with children learning math.
(You can read about the research base here [pdf file].)
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