TEACHING CHILDREN MATHEMATICS

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, palcaro@aol.com, teaches fourth grade at Point Road School in Little Silver, NJ 07739. Alice Alston, alston@rci.rutgers.edu, teaches mathematics education at Rutgers Uni- versity in New Brunswick, NJ 08901. Nancy Katims, katimsn@edmonds.wednet.edu, 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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