Numbers and operations

Alcaro, Patricia C., Alice S. Alston, and Nancy Katims. “Fractions Attack! Children Thinking and Talking Mathematically.” Teaching Children Mathematics 6 (May 2000): 562–67.

Copyright © 2000 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. For personal use only. This material may not be copied or distributed electronically or in other formats without written permission from NCTM.

How do we know when children are think-ing mathematically? How do we establish a setting that will elicit mathematical thinking?

What do we learn about the children as a result?

This article explores these questions through two videotaped vignettes in which fourth-grade students think and talk mathe- matically while tackling a complex real-life investigation called Snack Attack (from the PACKETS Program for Upper Elementary

Mathematics, developed by Educational Testing Service [1998] with support from the National Sci- ence Foundation). The investigation addresses state and national content standards involving pro- portional reasoning with whole numbers, fractions, and decimals, as well as the process standards of problem solving, reasoning, communication, and connections (NCTM 1989).

The students worked on the investigation dur- ing several class sessions, beginning with an introductory activity that set the context. As part of this activity, the students received a brochure called Food Matters that contained the following information:

• The meaning of calorie • The way in which calories are burned through

exercise • The fact that the number of calories burned dur-

ing exercise varies according to the type and duration of the exercise

562 TEACHING CHILDREN MATHEMATICS

Patricia C. Alcaro, Alice S. Alston, and Nancy Katims

Pat Alcaro, [email protected], teaches fourth grade at Point Road School in Little Silver, NJ 07739. Alice Alston, [email protected], teaches mathematics education at Rutgers Uni- versity in New Brunswick, NJ 08901. Nancy Katims, [email protected], is the director of assessment, research, and evaluation for the Edmonds School district in Edmonds, WA 98026. She was formerly the PACKETS project director with Educational Testing Service.

Fractions Attack! Children Thinking and Talking Mathematically These students were videotaped during workwith the vignettes.

• The number of calories burned in ten minutes of doing various exercises (shown as a graphic; see fig. 1)

After the introductory activity, the students began working in small groups on the investiga- tion. The children’s task was to develop a method to figure out how much time people must exercise to burn off the calories in snacks that they eat. Each group had a chart listing the five exercises from the brochure and the number of calories in seven dif- ferent snacks (see fig. 2).

Students used a variety of mathematical ideas and strategies to find the exercise times needed to burn the calories in a snack of their choice. Then they tested their methods using a second snack. The groups worked to explain and justify their solutions in the format of letters. Each group pre- sented its solutions, approaches, and explanations to the class. The teacher facilitated lively discus- sions throughout the activity’s several days as stu- dents questioned one another’s solutions, com- pared different approaches, and worked together to understand the mathematics involved in the differ- ent solutions.

Transcripts of videotapes of the sessions docu- mented the students’ mathematical activity and dis- course. Two vignettes offer particularly interesting illustrations of students’ emerging understandings, as well as their confusion and misconceptions, con- cerning proportional reasoning and fractions.

Understanding Ratio: Vignette 1 On the second day of working on the problem, one group (Allan, Keely, Sarah, and Paul) expressed some disagreement among themselves and asked for assistance. They had chosen the chocolate cookie (55 calories) as their snack and discovered that less than 10 minutes of jumping rope would be required to burn the 55 calories because jumping rope used 60 calories in 10 minutes. Keely and Allan decided that 9 1/2 minutes would be needed and made a table to explain their thinking. Paul was not convinced. The teacher enlisted the entire class in helping the students think through their dilemma. Keely came to the chalkboard and drew the table for the class to review (see fig. 3a).

The students knew that 10 minutes of jumping rope would burn 60 calories, and they assumed that 9 minutes would burn 50 calories. They reasoned that half of the 10-calorie difference, or 5 calories, would correspond to half a minute. So 9 1/2 min- utes would burn 55 calories. Their table showed a continuation of this logic for 8 and 7 minutes.

The teacher asked the students how they deter-

mined that 7 minutes burned 30 calories and that 8 minutes burned 40 calories. The group replied that its calculation was based on “10 calories for every 1 minute.” When the teacher asked the students to begin with 1 minute and show their idea, the chil- dren constructed another table on the chalkboard, shown in figure 3b.

The students moaned when they realized that according to their new table, 6 minutes of jumping rope burned 60 calories, but that according to the brochure, 10 minutes was supposed to burn 60 calories. The group’s first table had shown 30 calo- ries burned at 7 minutes, but this table showed 30 calories burned at 3 minutes. Looking back at the group’s original table, the class determined that the only fact they knew for certain was that 10 minutes of jumping rope burned 60 calories. Allan and Keely agreed that their other information repre- sented guesses, and they knew that they were sup-

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E 1 Exercise data from the brochure

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E 2 Exercise-times chart

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posed to check all mathematical guesses. When they checked the logic of their two tables, they realized that their guesses did not work.

Allan then posed an alternative idea. He said, “Maybe every odd number of minutes burns calo- ries ending in 5” and began exploring this idea by setting up a third table, shown in figure 3c. Allan quickly realized, however, that this notion did not work either because 10 minutes of exercise accounted for only 50 calories in this scenario. The class sat silently for several seconds. Then Allan excitedly called out, “I think it counts by sixes!”

When the teacher asked him to explain this idea, Allan thought quietly for some time, then said that he could not explain it. All the children appeared to be thinking intensely about these ideas. Jeff, another member of the class, eagerly

joined the group at the chalkboard and set up a table in which the calories increased by sixes. The students in Allan’s group later used this table, shown in figure 4, in their written solution.

When the teacher asked Allan to explain Jeff’s table and why he knew it worked, Allan said that he had first counted by eights and could not get to 60. Then, he had counted by sevens and could not get to 60 that way either. When he counted by sixes, the solution worked.

Referring to Jeff’s table, the group found that 9 minutes accounted for only 54 calories. Allan quickly said that if 1 minute was 6 calories, then half a minute would be half of 6, or 3 calories. The whole class agreed that 9 1/2 minutes of jumping rope would burn 57 calories.

The groups then continued working on their own solutions, leaving Allan’s group to think about the time that would be needed to burn the 55 calories in a chocolate cookie. Allan’s group wrote a final estimate of 9 1/4 minutes on its solu- tion chart.

The Mathematics Revealed in Vignette 1 In this vignette, the students were building a solu- tion based on equal ratios. In the process, they showed evidence of logical mathematical reason- ing. Their challenge was to find the pattern that would allow all the numbers to make sense, given

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E 3 Students’ tables for determining calories burned by jumping rope

Minutes of Jumping Rope 7 8 9 1/2 10

Calories Burned 30 40 55 60

The group’s first table (a)

Minutes of Jumping Rope 1 2 3 4 5 6

Calories Burned 10 20 30 40 50 60

The group’s second table (b)

Minutes of Jumping Rope 1 2 3 4 5 6 7 8 9 10

Calories Burned 5 10 15 20 25 30 35 40 45 50

Allan’s table (c)

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E 4 Final solution table for calories burned while jumping rope

the one piece of information that they knew was true—that is, that 10 minutes of jumping rope burned 60 calories. While solving the problem, the students demonstrated their ability to use whole- number multiplication (e.g., “. . . it counts by sixes”), multiplication of fractions (e.g., figuring half of 6), and appropriate number sense about fractions (e.g., estimating 9 1/4 minutes as the solution because 9 minutes would account for 54 calories and 9 1/2 minutes would use 57).

Understanding the Meaning of Fractions: Vignette 2 The next day, Jackie and Takitha asked the class for help in figuring out how many minutes a per- son would have to run to burn exactly 10 calories. When the teacher asked the girls to explain what they were thinking, Jackie pointed to the table that her group was constructing and said, “We needed 10 calories to burn and we needed to find one-half of 2 1/2. You know in running how it says for 10 minutes, we burn 80 calories. We kept going down to see what we’d get” (See fig. 5).

The girls had started with the given informa- tion, which was that 10 minutes of running burns 80 calories. They then divided each number by 2 to find that 5 minutes of running burns 40 calories and repeated this procedure to find that 2 1/2 min- utes of running burns 20 calories. They easily divided the 20 calories by 2 to arrive at 10 calo- ries and knew that they should do the same to the 2 1/2 minutes. This requirement, in essence, was their dilemma.

Jackie and Takitha decided to build a concrete representation of 2 1/2 using folded strips of paper. They cut paper strips of equal sizes to represent whole-number units, which they referred to as “wholes.” They then folded and cut some of the strips in half to form “halves.” As they generated numbers for their solution, they taped the appropri- ate combinations of paper strips to the board.

The class was now ready to think about Jackie and Takitha’s problem: “What is half of 2 1/2?” Again the class engaged in considerable discus- sion, leading to the following demonstration:

Bonnie. This is 2 1/2, right? So we’re going to take away one-half [of it]. So we’re going to take one-half away from the 1/2. That is a fourth. [She took the half strip from the board, tore it in half, and removed one of the pieces.] If you take half away from this, . . . and it equals 1/2. You take half away from this, and it’s another 1/2. We cut everything in half.

As she spoke, Bonnie tore one whole strip of paper

into two equal parts and removed one part. Then she repeated this action with the second whole strip, leaving two half-strips and one fourth-strip on the chalkboard.

Teacher. What is one-half of 2 1/2? Class. 1 and 1/4. Teacher. How did you get 1 and 1/4? I see two

halves and one fourth. Class. Two halves makes a whole and then

there’s one fourth, so that’s 1 and 1/4. Teacher. All right then, can you make an equa-

tion up on the board with the 2 1/2? Class. 2 1/2 – 1/2 = 1 1/4. [As the class spoke,

Jackie wrote the equation on the board.] Teacher. 2 1/2 – 1/2 = 1 1/4?

The teacher asked the stu- dents to build the model of 2 1/2 again and think carefully about what they were doing as they acted out the problem. When the children removed half of the 2 1/2 a second time, she challenged them to rethink their equation.

Paul. I mean, 2 1/2 minus half of 2 1/2 equals 1 1/4.

Bonnie. I see. Teacher. Bonnie, what equation do you see up

there? Bonnie. 2 1/2 . . . half of everything . . . and this

is what we came up with, right? But what we had left . . . what we took away from the 2 1/2 . . . was two halves and a fourth.

Bonnie reconstructed the 2 1/2 using strips of paper representing two wholes and one half. As she explained her idea, she ripped each strip in half and removed one of the pieces.

Lara. Two halves is a whole, Bonnie. Bonnie. Two halves and a fourth. If you put

these together like that and this like that, you have 2 1/2. But you took half away. So we take 2 1/2 minus one whole and a fourth. ☛

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E 5 Jackie and Takitha’s incomplete table

Minutes of Running 2 1/2 5 10

Calories Burned 10 20 40 80

The students used

folded strips of paper

as concrete

representations

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As she continued her explanation, Bonnie reassembled the strips once more to show the 2 1/2. This time, as she removed the halves from the two units, she pieced them together to show that the remaining amount was indeed one whole and one quarter. Takitha, standing beside Bonnie as she modeled the problem, wrote the following equa- tion on the chalkboard:

2 1/2 – 1 1/4 = 1 1/4

The girls’ written solution, shown in figure 6, described how they consistently used this strategy for other snacks and exercises. The girls success- fully demonstrated their ability to generalize their

solution strategy to other sets of numbers, with the exception of a careless calculation error in figuring the number of min- utes for biking.

Solutions generated by other students in the class also demonstrated learning from the discussion about using paper strips. For example, Bonnie and Lara’s group also included a set of paper strips in

the group’s solution. Their explanation and chart indicated that they used these paper strips to help them add the fractions 1/2, 1/4, and 1/8 to find

what portion of a minute of running is required to burn exactly 7 calories.

Mathematics Revealed in Vignette 2 Just as in the first vignette, the students here were building on their intuitive understanding of ratio and proportion, but the essence of the vignette focused on the meaning of fractions. The students were dividing a mixed number into two equal parts. This mathematical concept is difficult for students to handle, and it is frequently taught as a mechani- cal rule that they memorize and apply without true understanding. In this instance, because the stu- dents decided to use a concrete representation to help them solve their problem, a meaningful solu- tion was within their grasp. In the process, they demonstrated their understanding of such concepts as “two quarters make a half,” “two halves make a whole,” and “half of 5 is 2 1/2.” Equally important, students were able to use the paper strips as tools to help them solve similar problems.

Closing Observations In assessing the students’ performance in this activ- ity, the teacher was struck by several important observations. For example, she observed that although the children appeared to understand the concept of ratio when using a table, they did not attempt to check whether their initial answers made sense. If the students in the first vignette had been left unchallenged, they would have been quite con- tent to submit their untested tables to the teacher. When the teacher asked the class to think together about the entries in the first table, however, the stu- dents engaged in a thoughtful reasoning process that resulted in solutions that made sense.

An important insight gained from the second vignette was the fragile nature of the students’ grasp of particular concepts. Even after asserting and proving concretely that one-half of 2 1/2 is 1 1/4, the students had difficulty expressing this action with an equation. In this example, although it looked as if the students had mastered the con- cept, their first symbolic expression showed that their understanding was not yet complete. It is important for educators to help students connect concrete, verbal, and symbolic representations in ways that build meaning and help students develop precision in their use of mathematical language.

The two vignettes share interesting characteris- tics, some of which reflect student behaviors and others, teacher behaviors. Both vignettes illustrate powerful mathematical thinking on the part of the students. The students—

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E 6 Jackie and Takitha’s written explanation

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• posed and solved their own mathematical questions;

• were undaunted by the challenging nature of some of the questions;

• persevered through a series of attempts to a suc- cessful solution;

• used basic calculations and skills, coupled with logic, reasoning, and higher-order thinking;

• respected one another’s ideas and worked together to build a solution, showing true coop- eration rather than competitiveness in their interactions; and

• were willing to admit when they did not under- stand something.

The teacher clearly played an important role, also, by—

• bringing the whole class together to help an individual group grapple with a challenging problem;

• using the simple technique of saying “I don’t understand” or “I’m confused” to encourage students to explain their thinking more clearly;

• making no assumptions and insisting that the students clarify their statements every step of the way; and

• ensuring that all the students understood each step of the process, asking other students to explain an idea rather than explain it herself.

Admittedly, for this class, certain concepts involving fractions were still fragile, but the stu- dents appeared to be constructing powerful mental images of how these mathematical processes work. As teachers make decisions about how to cover the prescribed curriculum, this type of task may become increasingly helpful. Such activities may be used both for instruction and as assess- ment tools to reveal and document how students build models of mathematical ideas that help them make sense of basic skills and procedures, then use these models in unfamiliar mathematical situ- ations that call for higher levels of thinking (Lesh and Lamon 1992).

References Educational Testing Service (ETS). PACKETS Program for

Upper Elementary Mathematics. Princeton, N.J.: ETS, 1998.

Lesh, Richard, and Susan J. Lamon. “Assessing Authentic Mathematical Performance.” In Assessment of Authentic Performance in School Mathematics, edited by Richard Lesh and Susan J. Lamon, 17–62. Washington, D.C.: Amer- ican Association for the Advancement of Science, 1992.

National Council of Teachers of Mathematics (NCTM). Cur- riculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989. ▲

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