Sherrin B. Hersch

Catherine Twomey Fosnot

Antonia Cameron

H E I N E M A N NPortsmouth, NH

F A C I L I T A T O R ’ S G U I D E

Sharing SubmarineSandwiches,Grades 5–8

Y O U N G M A T H E M A T I C I A N S A T W O R KConstructing Fractions, Decimals, and Percents

A Context for Fractions

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HeinemannA division of Reed Elsevier Inc.361 Hanover StreetPortsmouth, NH 03801-3912www.heinemann.com

Offices and agents throughout the world

Copyright © 2006 by Sherrin B. Hersch, Catherine Twomey Fosnot, Antonia Cameron, andMaarten Dolk. All rights reserved. No part of this book may be reproduced in any form or byany electronic or mechanical means, including information storage and retrieval systems,without permission in writing from the publisher, except by a reviewer, who may quote briefpassages in a review.

This material is supported in part by the National Science Foundation under Grant No.9911841. Any opinions, findings, and conclusions or recommendations expressed in thesematerials are those of the authors and do not necessarily reflect the views of the NationalScience Foundation.

Library of Congress Cataloging-in-Publication Data

Hersch, Sherrin B.Sharing submarine sandwiches, grades 5–8 : a context for fractions (resource package) /

Sherrin B. Hersch, Catherine Twomey Fosnot, Antonia Cameron.p. cm. — (Young mathematicians at work constructing fractions, decimals, and

percents)ISBN 0-325-00898-1 1. Fractions—Study and teaching (Elementary). 2. Ratio and proportion—Study and

teaching (Elementary). I. Fosnot, Catherine Twomey. II. Cameron, Antonia.III. Title. IV. Series.

QA137.H47 2006372.7—dc22

2005024508

Editor: Victoria MereckiProduction management: Renée Le VerrierProduction coordination: Abigail M. HeimComposition: Publishers’ Design and Production Services, Inc.Cover and text design: Catherine Hawkes/Cat & Mouse DesignManufacturing: Jamie Carter

Printed in the United States of America on acid-free paper10 09 08 07 06 VP 1 2 3 4 5

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MATERIALS DEVELOPMENT

Mathematics in the City Freudenthal InstituteCity College of New York Utrecht University, The NetherlandsCatherine Twomey Fosnot Maarten DolkDirector and Principal Investigator Co-Principal Investigator

Sherrin B. Hersch, Co-Principal Investigator Parul Slegers, CD-ROM DeveloperAntonia Cameron, Co-Principal Investigator Chris Rauws, Software Developer

Arian Huisman, Software DeveloperHerbert Seignoret, Staff Assistant Han Hermsen, IT SupervisorJuan Pablo Carvajal, Staff Assistant Logica CMG

Anneleen Post, Secretarial SupportNathalie Kuijpers, Secretarial SupportPeter Croeze, Koen Fransen, Frank Udo, Interns

PARTICIPATING TEACHERS

New York City Public Schools

Region 9 Kara Imm, The Greenwich Village Middle School Joel Spengler, IS 89

City of New Rochelle Public Schools

Kevin Bradford, Isaac Middle School Michael Galland, Isaac Middle SchoolAlyssa Heller, Isaac Middle School Carol Mosesson Teig, Jefferson Elementary

FIELD TEST STAFF

Mathematics in the City, City College of New YorkDawn Selnes, Inservice Director Nina Liu, Inservice StaffChristine Ellrodt, Inservice Staff Suzanne Werner, Inservice StaffCarol Mosesson Teig, Inservice Staff

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NATIONAL FIELD TEST PARTNERS

Victoria Bill Linda CouttsLearning Research and Development Center Columbia, MissouriInstitute for Learning, University of Pittsburgh Public Schools

Cathy Feughlin Cynthia Garland-DoreThe Webster Grove School District Aspen Elementary SchoolSt. Louis, Missouri Aspen, Colorado

Ginger Hanlon, Sarah Ryan, Gary Shevell Bill JacobCSD #2, New York, New York University of California, Santa Barbara

Gretchen Johnson Judit KerekesCity College of New York College of Staten Island, New YorkSchool of Education Connie Lewis

Ellen Knudson Tucson, ArizonaBismarck, North Dakota Public SchoolsPublic Schools Wendy Watkins Thomson

Charlotte Stadler Project Construct, University of Missouri, New Rochelle, New York Columbia, MissouriPublic Schools

Sheri WillebrandSanta Barbara, CaliforniaCounty Education Office

EVALUATION TEAM

Joseph Glick, DirectorMara Heppen, ResearcherStephanie Domenici Cabonargi, Researcher

VIDEO PRODUCTION

Cathrine Kellison, Roseville Video, ProducerKiyash Monsef, Co-ProducerJeffrey McLaughlin, Editor

VIDEO CREW

John Bianchi, Tami Evioni, Richard Henning, John Javakian, Michael Kelly, Anthony McGowan, Serafin Menduina, Mark Petracca, Ben Vandenboom, Richard Westlein

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v

Contents

Overview vii

Journey 1

b THE CLASS AT WORK 1

b GROWTH AND DEVELOPMENT 23

Journey 2

b THE ROLE OF CONTEXT 30

b THE ROLE OF THE TEACHER 35

b DEVELOPING A COMMUNITY 39

Epilogue 44

Appendix A: Activity Sheets 45

Appendix B: Submarine Sandwich Posters 50

Appendix C: Transcript of Clip 41, Math Congress: A Deeper Look: Group 1 59

Appendix D: A Handy Guide to the CD-ROM Clips 62

Appendix E: Dialogue Boxes 64

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vii

The CD-ROM Sharing Submarine Sandwiches, Grades 5–8: A Context for Fractions in-vites you into Carol Mosesson Teig’s fourth-fifth-grade classroom where her studentsexplore a problem that arose on a field trip when groups of children were given sub-marine sandwiches to share. They investigate three questions posed by Carol: Wasthe sharing of submarine sandwiches among the different groups of children fair?How much of a sandwich did each child in each group get? Which group would youhave wanted to be in, and why?

Carol does not introduce fractions to her students by using the traditional part/whole quotative context; that is, by shading in slices of a pizza. Instead, she uses a fair-sharing problem to support their development of specific mathematical big ideas:

• Fractional parts must be equivalent in relation to the whole;• Fractional pieces do not have to be congruent, only equivalent;• The connection of multiplication and division to fractions. For example:

° Three subs divided among five children (partitive division) results in threeout of five parts of one sub (quotative division), and

° One-fifth of each sub times three equals 3⁄5 of a sub.

Starting with a fair sharing, the partitive division context avoids common mis-conceptions often found in children who have been taught about fractions from quo-tative contexts. For example, children who think that 1⁄8 is more than 1⁄7 because 8 ismore than 7 may have been asked to think of fractions as rational numbers too soonin their development, and they overgeneralize what they know about whole numbers.When fractions are introduced with fair sharing, students easily understand thatsharing with eighths versus with sevenths results in less.

As Carol’s students start to compare fractions and struggle with these problems,they begin to look for shortcut strategies and are challenged to generalize their find-ings about the denominator (e.g., the denominator is a divisor, and the greater thenumber of pieces, the smaller the amount) and the relationship of the numerator tothe denominator.

Overview

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1

Journey 1

T H E C L A S S A T W O R K

Introduction

S Jefferson School is one of seven elementary schools in the city of New Rochelle, NewYork—an urban area in Westchester County that borders the Bronx. During the2000–2001 academic year the school had an enrollment of 550 children in Kinder-garten to Grade 5.

The diversity of the student population is evident from the following ethnicbreakdown: 42 percent Hispanic; 35 percent Caucasian; 20 percent African Ameri-can and 3 percent American Indian, Alaskan, Asian American or Pacific Islander.Sixty-five percent of the students were eligible to receive free or reduced-price meals;19 percent of the children were categorized as English Language Learners; and 15percent of the students received special education services.

As a result of the school’s partnership with the Math in the City program andthe comprehensive inquiry-based Math Initiative advanced by the district, the per-centage of Jefferson’s Grade 4 students at Performance Levels 3 and 4 on the NYSMath Test has increased from 46 percent in 2001 to 87 percent in 2004.

The class is a fourth-fifth grade with fifteen students. Carol Mosesson Teig isthe teacher.

Developing the Context

S By clicking on the video icon below, youcan see Carol introduce the context ofsharing submarine sandwiches. Solve theproblems the children are about to investi-gate yourself.

By clicking on the Palette icon to theright of the Notepad, a drawing pad opens.Represent your thinking with a drawing.Note: You can return to the text field atany time by clicking on the Notepad iconthat appears where the Palette icon was.

Please refer to the Professional Development Overview Manualincluded in this kit, which is a helpful guide for facilitators new to

working with the technology of this CD-ROM. The section “Using the DigitalLearning Environment,” pages 3 to 7, suggests materials for interactive groupuse, shows how to create your digital learning environment, and also givesdescriptions of possible journeys.

FACILITATION TIP 1

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Carol Mosesson Teig introduces the context by telling her current students about afield trip that she had taken with former students when she taught in New York City—a trip that they are still complaining about because they think that some groups gotmore to eat for lunch than others.

This page gives participants their first opportunity to explore the drawing tool andto represent their own thinking about the fair sharing of the sandwiches, if they wishto stay in the digital environment. Later, in their working groups, some of these initialideas can be shared. A facilitator may, however, decide to do the investigation out of thedigital environment using large poster paper and markers. Activity 1 is written with thatapproach in mind.

2 Journey 1

Facilitators have found it crucial at this juncture to have participants solve theproblems themselves before they continue with the CD-ROM because partici-

pants’ own knowledge of fractions is too weak for them to understand what they arewitnessing when they explore children’s work. This is also an opportunity for facilitators tomodel math workshop and a subsequent math congress.

FACILITATION TIP 2

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When you click on the Palette icon, the crayon appears.

Explore clicking on the colors and line widths to see what happens. Use your mouse toactivate the crayon. It takes some practice to improve your control of the crayon.

TECH TIP 1 Exploring the Drawing Tool

U

You have a choice of colors

and of

line widths.

Click Clear to erase all.

The Palette icon is replaced by the paper icon.

Click on it to return to the Notepad.

Use the white color to erase something.

The fractions were written using the thin line and crayon.

To type, click on this box.

Click here to send your drawing to the clipboard.

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Math Congress: A Deeper Look

S For Math Congress, Carol asks two groups to share their work. Click on the pull-down list Math Congress: A Deeper Look to get a deeper look.

The probing discussions and activity thathave been engaging participants’ thoughtson the possible journeys that the students’strategies can take them to a crossroad inMath Congress: A Deeper Look.

As participants work on this page besure to encourage them to think about themathematical reasons Carol might havehad for picking these two groups of stu-dents and for presenting them in thisorder. Be sure to also ask participants tothink about the connections and differ-ences in the students’ work. If participantsfocus on superficial aspects (e.g., “theposters of both groups are clearly orga-nized”), redirect them to consider whatmathematical ideas each piece of workcontains, whether these ideas are con-nected and, if they are, how they are.

Math Congress: Group 1

S Carol starts the math congress with Ashleigh, Gabrielle, and Michael. Describe theirthinking.

This group’s strategy was used by many other groups: that is, dividing the subs in asmany halves as could be shared fairly (e.g., for the Planetarium, cutting each of the3 sandwiches in half) and then dividing what is left over into the number of childrensharing the subs (e.g., for the Planetarium, the remaining half is cut into fifths).Their findings are represented with unit fractions. “Each person gets 1/2 and 1/5 ofa half, or 1/10,” says Gabrielle (Clip 39). Jackie’s questions to Gabrielle indicate thatshe is familiar with their strategy and is using the congress to probe for clarification:Jackie first asks, “When you said or a tenth, does that mean altogether or a half and atenth?” Gabrielle uses the drawing to have the class look at the last sandwich, wherehalf of it is divided in fifths: “this all right here, each child got a tenth because if theseare fifths, it would be a tenth.” Later Jackie asks, “So you mean 1/2 and 1/10? or 1/2and 1/5 of a half?” This time Michael answers, “one-fifth of a half and one-tenth isthe same thing,” and Christina corroborates that “they gave each person a fifth of thelast half, which is 1/10 of the last sub.”

Ashleigh describes the same strategy for sharing the sandwiches with the Mu-seum of Natural History group: “We had 3 subs so we cut 2 subs in half because wehad 4 children and that was 4 halves. And for the last sub, um, we cut that into quar-ters . . . fourths” (Clip 40). Jennifer comments, “So they each got 1/2 and 1/4.” WithMichael’s interjection of “or 3/4” he moves the discussion to the idea of equivalence:“a half is 2/4 and then if you add a fourth then it’s 3/4.” In fact, Michael even sug-gests to Jennifer, who asks him to repeat his statement, that she say it in her ownwords. Their dialogue convinces the class that 1/2 + 1/4 is equivalent to 3/4.

For the Statue of Liberty–Ellis Island group (7 sandwiches shared among 8 chil-dren) Gabrielle (Clip 42) continues to use the group’s strategy, and describes howthey cut 4 of them in half (making 8 halves, one for each child) and had three leftover. “So we cut two of them into fourths” (making 8 fourths, one for each child)“and the last one we cut into eighths, and each child got 1/2, 1/4, and 1/8.”

The Class at Work 17

Should some participants express disappointment that the groupsthey might have chosen for the congress if they were Carol were not

the same as those Carol asked to share, be sure to emphatically dispel thenotion that there is “an ideal group” to choose. There is no one right way tostructure a math congress; indeed other experienced teachers may well havechosen altogether different groups. More important is participants’ reasoningfor what groups they would choose. Listen for their beliefs about learning.Are they taking into consideration how

• the groups’ strategies are connected to the work of other pairs?

• big ideas are likely to come up for discussion?

• confusions might arise that may cause healthy disequilibrium?

• both groups’ strategies might engender questioning by the class atlarge?

FACILITATION TIP 18

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It is interesting to note that when they shared the sandwiches for the Museum ofModern Art (Clip 41), Ashleigh’s group changed their strategy of first dividing thesubs in half. Michael quickly says, “there were 4 subs with 5 children and we cut itinto, um, 4/5. Each, um, sub was fifths and everyone got 4/5.” The answer is correct,but the group’s drawing provokes a lively exchange with the class, which pusheseveryone to defend their thinking (see the next section, Math Congress: A Spirited Dis-cussion). The children finally conclude that this group got 1/2 + 1/5 + 1/10.

To determine which group got more, Ashleigh’s group looked at what they have:

1/2 + 1/101/2 + 1/41/2 + 1/4 + 1/81/2 + 1/5 + 1/10

First they set aside the halves because each child in each group got 1/2 of a sand-wich. They noticed that the fourth scenario had 1/5 more than the first scenario sothey eliminated the first. Continuing in this vein, they eliminated the second groupbecause the third group had 1/8 of a sandwich more. What remained to be comparedwere the third and fourth groups. When Ashleigh proposes that the Statue of Lib-erty–Ellis Island group got the most to eat because “each child would get 1/4, 1/2,and 1/8,” Jackie asks for proof of how it is more. Here they use what they have con-structed about denominators to compare the other fractional pieces: in fractions 1/4is bigger than 1/5 (as Michael says, “we’re dividing it for more people so the piecesare smaller because they’re cut into more pieces.” Jackie confirms, “The more pieces,the smaller the size.”). What is more, “if you know that 1/8 is bigger than 1/10 thenyou automatically know that 1/4 is bigger than 1/10 because 1/4 is bigger than 1/8”(Ashleigh).

Math Congress: A Spirited Discussion

S When Ashleigh, Gabrielle, and Michael move on to the Museum of Modern Artsituation—four subs, five kids—a spirited discussion starts as Ernie says: “I don’t un-derstand the picture; you just have numbers splattered all over; I don’t understand.”What do you think is causing this discussion?

Halfway through the very animated discussion in Clip 12, Michael’s telling statementcan hardly be heard. He says, “It’s reversed. We reversed it.” What did they reverse?What did Michael mean?

Their drawing shows four sandwiches, each of which they have divided in fivepieces as follows:

1, 2, 3, 4, 1 for the first sandwich2, 3, 4, 1, 2 for the second sandwich3, 4, 1, 2, 3 for the third sandwich4, 1, 2, 3, 4 for the fourth sandwich

The labeling of the pieces in this manner may be puzzling to participants, and itcertainly threw these fifth graders into great disequilibrium and stimulated a rain-storm of questions and statements, e.g., you’re giving them a whole piece! They don’t geta whole piece. You didn’t put the fifth person; you only went up to four. There’s 2 ones in thefirst, then 3 more; that’s 5 pieces! Like if each person gets 4/5 wouldn’t there be 1/5 left over?

Having gone through their struggles in Children at Work, Group 1 is able to ex-plain their thinking—with help from Carol’s suggestion, why don’t you circle each per-son’s share with a black magic marker—until the class sees that each of the 5 childrendid get 4/5 of a sandwich. The word reversed in “It’s reversed. We reversed it,”meant that Group 1 showed their answer of 4/5 instead of how they figured it out.

18 Journey 1

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Also perplexing to the class is that Group 1 wrote: “each person would get 1/2,1/5, and 1/10,” but their drawing shows the common fraction of 4/5, not these unitfractions. Group 1 has clearly wandered off the beaten track on their equivalencejourney. “Where’s the half a sandwich?” someone asks. The drawing below repre-sents Michael’s explanation—another reverse—of how 4/5 of a sandwich is also 1/2+ 1/5 + 1/10:

The Class at Work 19

4/5 of the sandwich

1/2 of the sandwich + 1/10 + 1/5

1/5 1/5 1/5 1/51/10 1/10

Each of the three sandwiches has been cut into fourths. The pair’s use of color-coding shows each person’s share of 3/4 of a sandwich.

Jennifer and John used the same strategy for all the groups, and listed their findings under 1A on their poster.

The group divided the sandwich into fifths and then marks 21⁄2 fifths as the half.The halfway mark is drawn right through the third slice, leaving 1/2 of 1/5. They callthis 1/10. It is interesting to ponder whether they are using the commutative prop-erty and realize that 1/5 of 1/2 is equivalent to 1/2 of 1/5.

Math Congress: Group 2

S Carol asks Jennifer and John to share next. What strategy do they share?

Jennifer and John’s strategy was to cut each sandwich into the number of childrensharing it. “We, um, did the 3 sandwiches and since there was 4 people, we cut it in,um, 4 pieces so then each person would get 3/4 of a sandwich.” They representedtheir findings throughout with common fractions. Following is a close-up of theirstrategy for the Museum of Natural History group and their findings for how mucheach person in each of the groups got to eat:

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Jackie notices that “like, 3 sandwiches and 4 people, so it’s 3/4. There’s 4 sandwiches,5 people; it’s 4/5. There’s, um, 7 sandwiches, 8 people, so it’s 7/8. It’s like, the num-

ber of sandwiches goes on top and the . . .[number of children goes on the bot-tom].” It is interesting that although Jen-nifer and John noticed this pattern, theirstrategy throughout was to multiply: e.g.,each person got 3/4 because 3 × 1/4 = 3/4.This is the important big idea underlyingfractions as division. 3/5 is 3 × 1/5.

Math Congress: Generalization

S In the last part of the congress Carol generalizes the division of the submarine sand-wiches. How does she do that? And why do you think she does it at this moment?

An animated dialogue unfolds with Carol’s question (Clip 9 ) about the pattern thechildren are noticing—will it always work?—revealing uncertainties and convictions:

Carol: Will that work always?Students: Maybe. You can’t tell. I don’t know.

John: You see, it works when you cut ’em up in fractions.C: When wouldn’t it work?

Student: When you’re not using fractions.J: If you had 8 sandwiches and 8 children, then it wouldn’t work.

Nicole: Yes, it would . . . if there’s 8 children and 8 sandwiches, yes.Jackie: I think it will always work, too.

Carol is pushing toward the generalization of fractions as division in Clip 10,where she has just written 12/3 = ? on the board.

Carol: You’re saying that there are 3 children, so how much would each per-son then get?

20 Journey 1

Many participants may be enthusiastic, too, and may comment, “They got it.” Butwhat is the “it” that they got? What is the big idea on the landscape? The children

seem to be latching on to the patterns that emerged from Group 2’s poster, in an algorith-mic way: put the subs on the top and the people on the bottom and you get your answer.But do they know why it works? The dialogue in the clips indicates that they are struggling.

Ask participants in their small groups to discuss whether they think the children reallyunderstand why this pattern is happening. Suggest that they think back to the drawingsthat they made early on in their own investigation of submarine sandwiches and talkabout what in their drawing convinced them of the connection of multiplication to divi-sion, that 3 × 1/5 = 3/5.

FACILITATION TIP 20

NN

Kids

Subs

521 3 4

It is important that you examine with participants the connectionbetween multiplication (3 × 1/5) and division (3/5) in the work of

Jennifer and John. This is a big idea on the landscape of learning for fractions,decimals, and percents.

FACILITATION TIP 19

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Students: Four.C: (completes the equation 12/3 = 4) Four what?

Student: Four whole subs.C: And 3 subs shared by 4 people? (writes 3 ÷ 4 ) How much does each per-

son get?S: I don’t know.S: Three-fourths.S: Then it must work.C: So now you’re saying it works. What works?

Jackie: That the subs go on the top and the people go on the bottom.

Math Congress: Comparing Strategies

S Compare the strategies shared by the two groups. How does the work of these twogroups and the discussion in the community of learners enable the children to grow?

As part of this discussion, have participants consider how the structure of Carol’smath congress (where she begins and where she ends) had the potential to supportstudents’ informal strategies as well as help them to develop more efficient ones.Here, participants need to think about how Carol’s choice to initially work withinthe range of most learners helped scaffold mathematical development. Since manystudents cut sandwiches in half and represented their sharing of the sandwiches withunit fractions, they understood Ashleigh, Gabrielle, and Michael’s work because itwas not that far removed from their own. The lively discussion that was provoked byGroup 1’s representation of how they shared the sandwiches for the Museum ofModern Art group brought forth many struggles and puzzlements children were de-veloping about equivalence that were subsequently supported by the work of Group2, Jennifer and John. Group 2’s strategy for sharing—to divide each of the sandwichesamong the number of children sharing them—related to the work of other groupswho used this strategy for sharing the sandwiches. More important, the color codingin Jennifer and John’s expression of each group’s fair share with common fractionsvery clearly showed the connection of multiplication to division.

What Next?

S You have seen the children investigate sharing submarine sandwiches. You have ob-served the children sharing and discussing their strategies, and you have made noteson their mathematical development. One of the hardest parts of teaching is to decidehow to proceed the next day to ensure that the development of each child continues.What investigation would you plan next? Design one and explain the thinking be-hind your choice.

At this point in their journeys, participantshave had experiences analyzing students’strategies and thinking about how the fairsharing context supported mathematicalgrowth and development. Now partici-pants are asked to think of an investiga-tion that would continue to support anddeepen students’ mathematical thinkingand why they would do this.

Although designing a next investiga-tion in itself may not be difficult for par-ticipants, creating a rich context that

The Class at Work 21

Before having a whole-group discussion, ask participant pairs orthreesomes to share their next teaching steps with another small

group. Ask them to reflect on these questions: How are the plans you createdsimilar or different? Where on the landscape of learning are you working?How will your plan support growth and development? When you bring par-ticipants back together for a whole-group conversation, ask one participantfrom each new working group (the two pairs who shared) to share not theirideas for where to go next, but those that were presented to them in thesmall-group conversations.

ACTIVITY 4 What Next?

©©

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builds on the mathematical ideas children have constructed thus far and supports fur-ther development of mathematizing* will be. This is to be expected in a beginningjourney, for many reasons.

Participants may never have thought about teaching as an ongoing process ofplanning or that the direction this planning takes is deeply connected to students’mathematical development (where they are on the landscape of learning for fractions)and to the big ideas, strategies, and models they need to construct. Certainly, if whatparticipants have done in their own classroom practice has been antithetical to whatthey have experienced on the CD-ROM (e.g., if what they do next is solely deter-mined by where they are in a curriculum), it is not surprising that planning a lessonthat builds upon the prior lesson and supports students’ mathematical developmentwill be difficult for them.

Even if participants have a limited or superficial understanding of how Carol’suse of context is connected to and supported by her working knowledge of a landscapeof learning for fractions and how learners develop mathematical ideas, it is importantfor them to envision themselves in the role of the teacher and to plan the next in-vestigation. Whatever investigations they create—some may design contexts thatare trivialized word problems or contexts that are activities for “practice”—these canbe used to help participants think about the mathematical ideas their new investiga-tion would bring up for discussion, and how these ideas build upon and support chil-dren’s previous learning.

No matter how difficult it may be to consider a next investigation, it is importantfor participants to develop the ability to plan in this way if they are to deepen theirown practice. No matter what they plan—and the range here can be great—this is acritical next step in their journeys.

Backburner: The Class at Work

S This is the last page in the folder “The Class at Work.” However, you may haveother questions on this topic that you would like to investigate. Go to the TOOLSmenu above and add them to your Backburner notes.

The Backburner page can be a much-usedtool for participants and facilitators. Asparticipants work with the materials, theymay raise many questions that they cannoteasily or immediately answer. The Back-burner page offers them a place to keeptheir questions for another time. Thesemay be answered or evolve as participantswork more deeply with the CD-ROM.

This page in the digital learning envi-ronment can be the context for further in-vestigations stemming from learners’ ownquestions. A facilitator can use them toform study groups; participants with simi-lar questions can research them (e.g.,“What more can I find out about the de-velopment of proportional reasoning?How can I learn to know what to ask next?What are the power relationships in pairgroupings?” Some additional reading ma-terials are provided under INFO in themenu bar).

22 Journey 1

After participants have finished working with the folder “The Class atWork,” ask them to review their questions about students’ strategies

that they posted on the Backburner page of the CD-ROM and think aboutthese questions:

• What were your original questions?

• Have they been answered? If so, what in your CD-ROM experiencemight have helped you answer them?

If participants still have unanswered questions about students’ strategies,ask them to share these in a small-group discussion (four to six people) andsee if the group can answer these questions. At the end of the small-groupdiscussion, bring the whole group back again and chart the questions thathave not been answered. These can be used by participants to help themanalyze an individual student’s work over the course of the investigation inthe folder “Growth and Development.”

ACTIVITY 5 Reexamining Questions About Students’ Strategies

©©

*For more information on mathematizing, refer to the companion book, Young Mathematicians at Work: Con-structing Fractions, Decimals, and Percents, Chapter 1: “Mathematics” or “Mathematizing?”

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