….The problem with praising kids for their innate intelligence — the “smart” compliment — is that it misrepresents the psychological reality of education. It encourages kids to avoid the most useful kind of learning activities, which is when we learn from our mistakes.…

Tharman Shanmugaratnam, until recently Singapore’s minister of education, explains the difference between his country’s system and that of the United States: “We both have meritocracies,” Shanmugaratnam says. “Yours is a talent meritocracy, ours is an exam meritocracy. We know how to train people to take exams. You know how to use people’s talents to the fullest. Both are important, but there are some parts of the intellect that we are not able to test well — like creativity, curiosity, a sense of adventure, ambition. Most of all, America has a culture of learning that challenges conventional wisdom, even if it means challenging authority.” This is one reason that Singaporean officials recently visited U.S. schools to learn how to create a system that nurtures and rewards ingenuity, quick thinking, and problem solving. “Just by watching, you can see students are more engaged, instead of being spoon-fed all day,” one Singaporean visitor told The Washington Post. While the United States marvels at Asia’s test-taking skills, Asian governments come to the United States to figure out how to get their children to think.

Let’s keep working on teaching our children how to think — and in mathematics, that means solving problems, making arguments, exploring proofs, and creating models, in particular.

I had never thought carefully about exactly what we tell children when teaching the standard algorithm until I heard this discussion. When we teach the standard subtraction algorithm, we tell children to start with the ones. Of course, you don’t have to start with the ones when you subtract. For example, in a problem such as 5,000 – 3,002, it makes much more sense to first subtract 5,000 – 3,000 (the thousands) and then subtract 2 (the ones). We want children to know that 5,000 – 3,000 – 2 is the same as 5,000 – 2 – 3,000. Children with a strong understanding of subtraction know that they don’t need to start with the ones. Understanding how subtraction works will help students solve algebraic equations such as 3x – 48 – x = 49 or 3x – 48 = 49 – x….

What were teachers’ concerns? The majority worried about having enough time to devote to having students solve problems in more than one way and discuss the solutions. Many also were concerned that their lower-achieving students would be confused if they saw more than one way to solve a problem. One teacher confessed, “Sometimes I am scared to put even two strategies up there because [the students] are barely able to get one.” Other teachers believed that higher-achieving students would get bored if a lot of time was spent discussing a single problem. Teachers also had concerns about the limitations of their own knowledge.

As teachers grew to see the value of multiple solutions, they began to wonder about how to choose which solutions to have presented, the order in which they should be presented, and which ones to discuss in depth. One teacher noted that “Explaining is important, but which solutions you focus on have to be tied to the goals of the lesson instead of always sharing everything. I have not considered that before — who I want to share first.” Teachers were divided over whether to present incorrect approaches and how to deal with students’ mistakes. There is a lot of intellectual work involved in making decisions like these and no prescriptions exist. It’s easy to see how the exhortation alone to have students consider multiple strategies is insufficient support for successful implementation.

Beyond Show and Tell

As teachers grappled with these issues in the context of their own teaching, they began to think about how the purposeful selection of students’ solutions for presentation and discussion could help them “steer” the mathematics content.

Moreover, they noted, by attending carefully to the mathematical ideas embedded in students’ responses, a teacher could influence which ideas are likely to be discussed in class, and in what order, thereby improving their chances of meeting their mathematical goals for a lesson…. As one teacher stated, ‘I don’t usually allow multiple strategies, but I now see the possible benefits. [But if you do this], it is also very important to fully understand the mathematical goals of the lesson.’

Silver and his colleagues found that teachers’ instruction shifted over the course of the year to incorporate eliciting and discussing multiple solutions to problems. Teachers made the shift because, through their own discussions and reflections about promoting multiple solutions, they convinced themselves that it helped students’ learning.

But the following story, I assure you, is true. My sister, who lives in Tennessee, recently went to a local restaurant to buy an iced tea to go. She ordered the tea 1/3 sweet, and the rest unsweet. (For you non-Southerners, many restuarants in the South give you a choice of sweetened or unsweetened iced tea. Year round, to boot.) She then heard the clerk turn to the microphone and say, “Tea, 1/3 sweet, and 1/2 plain.”

Yikes! Call the fractions police!

A segrated school in Birmingham. April 4, 1961.

Integration of a high school in Birmingham. September 1963.

If you are white, would you have been among these people protesting the integration of an elementary school? Would you have been this woman? She looks so sure of herself.

Would you have protested the integration of your high school, as these high school students did? How do you know what you would have done?

What is the civil rights issue of our day?

How does a teacher decide when to show and when to not show students how to solve problems? I asked two veteran teachers when, if ever, they show young children how to solve problems. They have been involved in Cognitively Guided Instruction (CGI) for over 15 years each. Both now work as math specialists for the district. Before that, they were classroom teachers. Carrie Valentine taught upper elementary and Mazie Jenkins (sometimes known as Ms. J in CGI writings) taught primary. Both have worked a lot with kids from all kinds of backgrounds, including low income and low achievers in math. It’s certainly a question worth thinking hard about.

Here’s what they had to say:

MJ: The famous question always gets asked. The answer and how you go about getting students to exhibit different strategies is very complex.

Teachers have to learn how to get students to reflect on the strategies that are shared. For example, how is Susan’s strategy like Megan’s strategy? How are they alike and how are they different? Who else solved their problem like Susan’s? Who solved their problem like Megan? Who has a completely different strategy? How is it different than Susan’s? Ask specific questions for the strategy, how are tens used in this strategy? Was this a good way? How do you know? Have we seen this strategy before? When? What kind of problem did we see this strategy used? Did anyone use numbers to solve this problem?

I usually do not show a strategy. I might talk about how I have seen another student solve a similiar problem. (Deborah Ball demonstrates this on a videotape).

I listen and observe to find students who are solving a problem in a different way and build upon that. I ask students to solve problems in at least two different ways (direct modelers show two ways of direct modeling – but they do not know this)

Teachers need to have a good understanding of the development of strategies to know what is developmentally appropriate.

CV: I think it’s possible but requires a great deal of expertise. Skill in problem posing and questioning is essential. I wonder what is meant by strategies? I think of them from a cognitive perspective. Is there something going on internally different in the math understanding? But, I think most people think about the representations that their students use. In that case I definitely see utility in showing ways to represent to hasten the learning. Kids love ‘tools’ such as the empty numberline, arrow language, and ratio tables when they are ready. I would introduce them after some level of understanding emerges. The empty numberline after counting on emerges, arrow language after decomposition and facts of ten are known, and ratio table after kids can double using base ten. There are other ‘tools’ but these seem to capture kids’ attention and are efficient ways to record their thoughts and later to help them solve problems.

You get the gist. It’s actually a complicated question and deserves a nuanced response.

SE: What about upper grades?

CV: Same way. First what is meant by a strategy vs. a tool if there is a difference. Then talk about ways to record thinking.

Consider these two division problems:

1. At a birthday party, 2/3 of a watermelon is left on the table. There are 4 children at the party who want to share this left-over watermelon. They all want the same amount and they want to finish it off. How much can each child have?

2. Okhee has a snowcone machine. It takes 2/3 of a cup of ice to make a snowcone. How many snowcones can Okhee make with 4 cups of ice?

Which one do you think is more difficult for children to solve?

To help you decide, consider one fifth grader’s strategies for each and how the strategies are related to the problem structure.

To figure out how many snowcones Okhee could make (problem #2), he started with the fact that 2/3 of a cup of ice could make 1 snowcone and figured out how many two-thirds-cups were in 4 cups, the total amount of ice Ohkee had.

He added 2/3 twice and got 1 1/3. He then remembered (from a previous problem) that there were 3 one-and-one-thirds in 4, which meant that there were 6 two-thirds in 4. He concluded Okhee could make 6 snowcones. Besides the last bit of relational thinking which is not integral to the approach he took, he treated the division problem primarily in terms of addition of like units (here, thirds). To figure out how much watermelon each of the 4 children would get if they shared 2/3 of a watermelon equally (problem #1), he drew a circle, divided into thirds. Then he blacked out one of the thirds to show the part of the watermelon already gone. Next he decided to cut each of the remaining thirds into 4 parts to share with the 4 children. He knew this would make 8 parts altogether and that each child would get a total of 2 of those parts, so he marked those off:

He confidently reported the answer as “2/8 of 2/3.” (Decide for yourself: Does this make sense?) Yes, but … we really want to know how much of the whole watermelon that is, don’t we? He quickly drew in the missing third divided into 4 parts:

and reinterpreted his answer in terms of the whole watermelon. He concluded 2/8 of 2/3 was “2/12.” See how his unit of reference for the whole shifted, from 2/3 of a watermelon to the entire watermelon? Similarly, the same piece was first interpreted as an eighth of 2/3 then as a twelfth of the whole. The ability to describe a part and its relationship to different unit wholes is a hallmark of more advanced fraction thinking. So although sharing a quantity among 4 children is a fairly straigthforward kind of problem, in the context of part of parts of wholes, interpreting the resulting share for each child as a fractional quantity is not.

Ms. S teaches fifth grade in a small district just outside of a large college town, not unlike Austin, Texas. A couple of years ago, her district adopted Everyday Mathematics, a program developed by the University of Chicago School Mathematics Project and lauded by many. Now, Ms. S has been teaching for many years. Her tried and true approach — whether it’s first grade or fifth grade — is to pose problems that students can solve using a variety of strategies, then help her students to express the concepts behind the strategies they use. She doesn’t show kids how to solve problems; she writes problems in such a way that her students can use what they know to construct their own strategies. In fact, she insists when her students come to her using procedures they don’t understand that they not use those procedures, for the time being. Her highest priority as a teacher is to understand her students’ thinking and to build on it.

As it happened, the year her district adopted Everyday Math, her principal decided to place a large number of students in Ms. S’s class who were considered struggling.
When her district delivered the boxes and boxes of Everyday Mathematics materials to her room, Ms. S took a look at the curriculum. She realized that she could write problems that were better tailored to her students’ needs and understanding and also met district curriculum objectives. So she pushed the boxes aside, closed her door, and began teaching.

Every once in a while, her principal would inquire about what page she was on. Ms. S would tell him, truthfully, “We’re learning a lot.”

All the students in Ms. S’s district take a standardized test at the beginning and end of the year. Students are expected to gain 7-9 points on this test. 10 points is considered quite a respectable gain. At the end of the year, about a week after Ms. S’s students had taken the test, she was surprised to find the principal knocking loudly on her classroom door. When she answered, he told her, with a great deal of excitement, that out of all the fifth-grade classes in the district (about 15 sections), hers was the ONLY class where every single student had met the passing standard for the test. Moreover, he continued, many of these students had gained an astounding 30 or so points between the beginning and the end of the year tests. He asked her, almost rhetorically, “You don’t even use Everyday Math, do you?” “Nope,” she answered. And then she took the opportunity to suggest that perhaps all the money that had gone into the Everyday Math materials may have been better used to support teachers to take the time to learn about their children’s mathematical thinking. Richard Elmore, writing in Harvard Magazine, agrees: “You can’t improve a school’s perfomance, or that of any teacher or student in it, without increasing the investment in teachers’ knowledge, pedagogical skills, and understanding of students” (p. 37).

This year, for the first time, Ms. S  has a preponderance of students from the high end of the achievement spectrum. They are challenging in a different way, she says, because they know how to execute procedures such as subtraction with regrouping and long division very well, but often have little understanding of why the procedures work, and almost no flexibility in their choice of strategies. Even with this group, she says, she has to begin the year by posing many types of problems with smaller numbers than one would normally expect at fifth grade in order to help her students develop some flexibility and variety in the strategies they use. It’s worth it; they surpass expectations in the end and their understanding is much, much deeper.

Thank you, Ms. S, for reminding us what teaching is all about.

REFERENCE: Elmore, Richard. (2002). Testing Trap. Harvard Magazine. October, 35-37+.

Billy started first grade six weeks late, having never been to kindergarten. He couldn’t count or recognize numerals. His teacher, Ms. J, helped him learn to count and as soon as he could count, she gave him word problems to solve and asked him “How did you figure that out?” He spent the majority of the year solving and discussing word problems and, in the process, learning a great deal about himself as a mathematician. Here’s how it happened:

Ms. J. and the other children helped Billy learn to count objects, first to five and then to ten. …When he continued to have great difficulty recognizing numerals, Ms. J. gave him a number line with each number clearly identified. Billy carried the number line with him continuously, and if he needed to know what a numeral looked like, he would count the marks on the number line and know that the numeral written beside the appropriate mark was the numeral he needed. As soon as Billy could count, Ms. J. began giving him simple word problems to solve. On a sheet of paper, she would write a word problem such as, ‘If Billy had two pennies, and Maria gave him three more, how many would he have then?’ … Either Ms. J. or another child would then read the problem to Billy, who would get some counters and patiently model the problem. In this problem, Billy made a set of two cubes and a set of three cubes and then counted all the cubes. Ms. J. would then ask Billy to explain how he got his answer. He would tell her what each set meant, and how he had counted them all and gotten five and then counted up his number line to know what five looked like.

During mathematics class, Billy might solve only two or three of these simple problems, but he knew what he was doing, and he was able to report his thinking so that Ms. J. could understand what he had done. When Ms. J. was sure he understood the simple problems, such as the joining and separating result-unknown problems, she moved on to somewhat harder problems and to somewhat larger numbers. She encouraged Billy to make up his own problems to solve and to give to other children. Almost all of Billy’s time in mathematics class during the year was spent in solving problems by direct modeling or in making up problems for other children to solve.

When we interviewed Billy near the end of the year, he was solving problems more difficult than those typically included in most first-grade textbooks. Billy had become less reliant on his number line, and he could solve result-unknown and change-unknown problems with numbers up to twenty. Although at that point Billy was not yet able to recall basic arithmetic facts, he nonetheless understood conceptually what addition and subtraction meant, and he could directly model problems to find the answer. Billy was no less proud of himself or excited about mathematics than any other child in the classroom. As he said to the school principal, ‘Do you know those kids in Ms. J.’s class who love math? Well I’m one of them.’ (Peterson, Fennema, & Carpenter, 1991, pp. 89-91)

Excerpted from: Peterson, P., Fennema, E., & Carpenter, T. P. (1991). Using children’s mathematical knowledge. In B. Means (Ed.) Teaching advanced skills to educationally disadvantaged students (pp. 68-111). Menlo Park, CA: SRI International.

Update: Sometimes people want to know what happened to Billy after his first-grade year. Ms. J, who stays in touch with her former students as much as possible, reports that he moved away the next year and she lost track of him.

The question insists that a child respond with something like, ‘First I tried to …’ In other words, it requires the student to position himself as a story teller with himself as the protagonist in the story. ….Such a narrative invites a sense of agency as part of the child’s literate [or mathematical or insert-what-you-wish] identity.

This ‘how did you’ invitation to an agentive role is particularly important. We hear a lot about teaching children strategies, but we often encounter classrooms in which children are being taught strategies yet are not being strategic… Teaching children strategies results in them knowing strategies, but not necessarily in their acting strategically and having a sense of agency. …. Teaching for strategies requires setting children up to generate strategies, then reviewing with them, in an agentive retelling, the effectiveness of the strategies they generated, as in, ‘You figured out that tricky word by yourself. How did you do that?’ As children do this, they are in control of the problem-solving process and are asked to consciously recognize that control in an agentive narrative. (p. 31)

Don’t underestimate the power of the cumulative effects of asking this question of all of your students, as often as possible. The story of Billy, a first grader, illustrates it well.

Paradoxically, overall mean achievement in math is higher for private schools than public, an example of Simpson’s paradox. This counterintuitive result appears to be due to a higher concentration of high SES students in private schools rather than better teaching. Lubienski and Lubienski’s data suggest that, in fact, public schools do a better job of teaching math than private schools.

Just a couple of days ago, I ran across a folktale called “A Fair Division” in the October 2003 issue of Highlights for Children. It shows very cleverly why “fair” is not always “equal” and it makes a good math problem too. Here’s my paraphrase of it (or you can view a jpg of the entire text, minus the hints the author gave for figuring it out. Yes, that’s right. I took out the hints!).

Two farmers, Ram and Shyam were eating chapatis. Ram had 3 pieces of the flat, round bread and Shyam had 5. A traveller who looked hungry and tired rode up to the two men. Ram and Shyam decided to share their chapatis with him. The 3 men stacked the 8 chapatis (like pancakes) and cut the stack into 3 equal parts. They shared the pieces equally and ate until nothing was left. The traveller, who was a nobleman, was so grateful that he gave the two farmers 8 gold coins for his share of the food.

After the traveller left, Ram and Shyam wondered how they should share the 8 gold coins. Ram said that there were 8 coins and only 2 people, so each person should get an equal share of 4 coins. “But that’s not fair,” said Shyam, “since I had 5 chapatis to begin with.” Ram could see his point, but he didn’t really want to give 5 of the coins to Shyam. So he suggested they go see Maulvi, who was very wise. Shyam agreed.

Ram and Shyam told the whole story to Maulvi. After thinking for a long time, he said that the fairest way to share the coins was to give Shyam 7 coins and Ram only 1 coin. Both men were surprised. But when they asked Maulvi to explain his reasoning, they were satisfied that it was a fair division of the 8 coins.

What do you think Maulvi’ reasoning could have been? What makes it fair?

We also had an amazing revelation by a first grader on equivalent fractions. The class had done a great deal of playing with the fractions and were pretty able in terms of equivalence. But this first grader blew our minds. The problem was 8 [people] sharing 14 [of something, e.g., pancakes]. So we had the different responses but one little girl said that 14/8 was equivalent to 7/4 because 2/8 was equal to 1/4 so you could double the 7/4 and get 14/8 or you could half 14/8 to get 7/4. It was too close to 7/4 x 2/2 idea to be an accident…. I am not sure how far you would take this with a first grader but it felt like a teachable moment to explore. Curious…would you have just let that big idea slide or done something with it?

I wonder how she arrived at that conclusion. I think she might be reasoning by analogy — a form of reasoning not specific to mathematics — rather than from mathematical principles — such as multiplication by 1 = 2/2. I’m also curious whether she can relate her ideas back to the sharing situation (e.g., a child in a group of 8 people sharing 14 pancakes gets as much pancake as a child in a group of 4 people sharing 7 pancakes) and to what kinds of other fraction relationships she can extend this reasoning. My guess is it would be harder for her to reason about equivalence relationships that involved tripling or “thirding” the numerators and denominators, and so on. But all of this doesn’t make her insight any less fascinating or powerful in the context of learning fractions. And yes, I’d definitely want to explore it!

The darkened part shows what part of this brownie Jackie ate. How much of the brownie was eaten?

The son who was in third grade said it was an impossible amount, because the pieces were not all the same size and it therefore could not be “one out of three.” The first grader said the portion was half of a half. How surprising that the third grader, who had been in school longer, gave a non-sensical response!

So I am curious about what other children would do. I have heard other children describe that piece of brownie as “one third,” but I don’t know how widespread this kind of response is. Similarly, I’ve seen other children describe it as half of a half, and I wonder how common this more intuitive response is.

If you’re curious too, give this question to your own students and see how they respond. How many children say the amount is one third? half of a half? one fourth? an impossible amount?

Table of contents:

Introduction

Why this case study?
What is mathematics?
Why not direct instruction?

Session 1

Problem set #1.
We begin our work together.

Session 2

Problem set #2.
I discover a ‘lack of number sense’ is really a lack of base-10 understanding.

Session 3

Problem set #3.
I make the problems too difficult, the children don’t listen to each other.

Session 4

Problem set #4.
Quick problems to begin session 4.
Developing 10 as a unit.

Session 5

Problem set #5.
Today, we focused.
One dime is 10 cents, and 10 cents is one dime.

Session 6

Problem set #6.
Progress!

Session 7

Problem set #7.
Frustration!

Session 8

Problem set #8.
Extending the children’s thinking.

Session 9

Problem set #9.
True/false number sentences.

Session 10

Problem set #10.
Our last session together.

Alternatively, you can begin here and follow arrows to later posts (in the upper right corner).

Jack.

Jack seems to have made a great deal of progress. He solved the first problem (8 rolls of candy, 10 in each roll) today by drawing a rectangle-like representation of each roll. At my suggestion, he wrote “10″ above each one. He finished off by drawing the extra 12 candies individually. As he counted them however, he pointed out the extra 10, for a total of 90 and 3, oops, 2 more:

I also asked him to write a number sentence and he wrote “10, 20, 30, 40, 50, 60, 70, 80, 90, 92″ (as before). This strategy is significant because he no longer depends on representing the individual units (each candy) to construct 10. Ten is a unit for him!

In the middle of our session he spontaneously showed me this strategy for subtracting:

It’s an interesting contrast with the rest of his work today. It was as if he was trying to re-call the steps involved in subtraction with regrouping rather than re-construct the steps based on what made sense to him. I didn’t address this mistake with him, because I wanted him to continue to work from what he understood.

I asked Jack if he could solve the second problem (30 pencils, 29 more pencils) in his head; he thought for a moment, said no, and proceeded to draw this:

This time I asked his to write a number sentence using plus and equals to show how he solved it. He wrote “10+10+10+10+10+9=59.” As he was writing the tens, I asked him how many tens in 50. “Five,” he said. He understands the place-value relationship between 50 and 5 10s.

Jack’s solution for the third problem (45 beads, 10 beads per necklace) suggests that his new knowledge of ten as a unit may be somewhat fragile. When I checked in with him, he had written on his paper:

The problem seemed to be solved. He seemed to think the answer was 4. Excellent! But as I questioned him about what he had done and why he had done it, his answer changed … to 5 (pointing to the remainder), then to 40 (the number of beads in 4 necklaces). I continued to ask him questions to help clarify his thinking and to emphasize the context of putting beads on necklaces, and the relationship between beads and necklaces. We finally arrived back at his original answer of 4 total necklaces.

Turning the open number sentences, I again asked Jack if he could solve the problems in his head. “Yes,” he said, for 30+40=__. He counted on by tens from 30 to get 70. I skipped 25+20 in order to see what he would do with another problem that involved only multiplies of 10. “How about 60-20?” I asked. “80,” he replied. I drew his attention to the minus sign (he knew what is was). “So if it’s plus,” I said, wanting to reinforce his mental strategy, “the answer is 80. What if it’s minus?” Jack easily counted back by tens to get 40 and likened the problem to 6 take away 2.

Nice job Jack!

Sunny.

Sunny’s strategies were more concrete than Jack’s, but I noticed that the language she was using to describe these strategies suggested an emerging understanding of base-ten and place-value concepts.

For instance, for the first problem (8 rolls of candies, 10 candies per roll), she direct modeled the problem using unifix cubes in sticks of 10. But when she described her solution she said, “It’s 80, because 8 tens is 80, when you count by 10 8 times, it’s the number 80.” I can see in this explanation that Sunny is making a connection between “counting by tens” a certain number of times and multiples of ten.

The emergent nature of her understanding of the base-ten structure of numbers was apparent in her strategy for adding 30 pencils and 29 pencils. Again she direct modeled the quantities. But beyond this, Sunny made little use of base-ten concepts: she counted up by ones from 30. I think she is just arriving at understanding 30 is 3 tens and that applying this knowledge in constructing a solution such as counting on by tens is somewhat beyond her. However, if I were to continue to work with Sunny, I would continue to give her addition and subtraction problems with double-digit quantities to help her develop strategies that made more use of base-ten concepts and processes.

Sunny also did something for this problem that I don’t think I understand. As she was modeling it, she included 2 extra cubes. I asked her what the 2 extra cubes were for. She didn’t say clearly. I pointed to each quantity she had made and asked her what part of the problem it was. She pointed out Jack’s pencils, and Emilio’s pencils and realized that the 2 cubes didn’t have a referent in the problem.

I had to leave at this point (appointment with the dentist) and left the children in Luz’s hands.

Back to TABLE of CONTENTS.