realizing the educational promise of technology · permission to reproduce any part of this...

Kelly Goorevich works withher fourth grade studentsat the Hosmer School inWatertown, MA, on the“Broken Calculator.”

By Joanna Lu and Raymond RoseThe case method is a powerful learning model. For years, schools

of business, law and medicine have used case studies through whichstudents explore real-life principles. By examining critical moments ina case, students enter vividly into the events and can carry the lessonslearned into their professional lives.

The Seeing Math Telecommunications Project has added the forceof audio, video, and interactive computer tools to the already power-ful case study method. Seeing Math is developing nine Web-basedvideo case studies that provide mathematics professional developmentfor elementary and middle school teachers. These case studies use bothreal-life video narratives and guided inquiry to craft a unique learningexperience. By going into real teachers’ classrooms and presenting theproblems they face and the solutions that grow from imperfect situa-tions, Seeing Math provides a rich source of insight that all teacherscan use to develop their own practice.

Creating the CasesEach Seeing Math case study focuses on specific math content that

is widely recognized as difficult to teach. Several months before tap-ing, case developers from the Concord Consortium and Teachscapeidentified a teacher and class of students planning to study the mathconcept. Producers and math specialists talked with the teacher tounderstand the curriculum goals. The day before the lesson, the pro-duction team interviewed the teacher to understand her strategies andexpectations.

Over the course of two or three taped class sessions, pre- and post-lesson interviews with the teacher, and a collection of student work, anumber of stories emerged. The team decided what strands were mostrelevant, determined the storyline, edited the video, and shaped theWeb-based materials.

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Realizing the educational promise of technology www.concord.orgRealizing the educational promise of technology www.concord.org

Seeing Math through MultimediaCase Studies

Seeing Math through MultimediaCase Studies

Volume 7, No. 1 Spring 2003

Editor Robert TinkerManaging Editor Cynthia McIntyre

Design Pointed Communications

@concord is published twice a year by the ConcordConsortium, a nonprofit educational research and devel-opment organization dedicated to educational innovationthrough creative technologies.

Copyright © 2003 by The Concord Consortium. All rightsreserved. Noncommercial reproduction is encouraged,provided permission is obtained and credit is given. Forpermission to reproduce any part of this publication,contact Cynthia McIntyre at [email protected].

The projects described in this newsletter are supportedby grants from the National Science Foundation (REC-0115699, ESI-0242701, EIA-0219345), the U.S.Department of Education (R286A000006), the NoyceFoundation, and the Trust for Mutual Understanding. Allopinions, findings, conclusions, and recommendationsexpressed herein are those of the authors and do notnecessarily reflect the views of the funding agencies.Mention of trade names, commercial products or orga-nizations does not imply endorsement.

If you wish to change your subscription information,please clip the mailing label from this newsletter, indi-cate the changes and mail to:

The Concord Consortium10 Concord Crossing, Suite 300Concord, MA 01742

You also may call 978-369-4367 or fax 978-371-0696.

You can access your account or sign up for a free sub-scription online at www.concord.org/members.

Note: We do not make our mailing list available toother organizations.

As a nonprofit organization, the Concord Consortium hasno ongoing support for @concord. Contributions todefray printing costs would be greatly appreciated.

Visit our Web site at www.concord.org for informationon projects, services, courses, and free software.

The computer has been around for about fifty years now; theInternet, as far as most of us are concerned, for about ten.Together, these inventions have revolutionized the world of

business and changed our lives forever. Yet there has been no compa-rable impact on schools. Why is this? Why has education, surely oneof the more information-intensive sectors of the economy, failed to takefull advantage of the Information Revolution? This is not a problemthat can be blamed on schools or teachers; it is caused by an interlock-ing set of historical, economic, and policy forces.

One critical problem is that there aretoo few convincing real-world demon-strations that computers actuallyimprove learning results – at least whenapplied to the standard curriculum andevaluated in a standard way. We know

that, used well – with innovative cur-ricula built around standards andembedded assessments – computersand software can do this. Indeed,decades of careful research have pro-vided an “existence proof” that, com-bined with research-based pedagogy,

computer use can result in significantimprovement in students’ learning.

But the unfortunate truth is thatmost educational research involvessmall-scale experiments with just a fewstudents and teachers, often in an atyp-ical setting. These results are not real-istic enough to justify major changes inreal schools. So the lack of school-based, large-scale demonstrationsundermines the entire enterprise.

For this reason, we continue to urgelarge-scale applied research projects inschools. The required research mustinvolve innovative technology inte-grated into substantial chunks of cur-riculum – units, courses, or sequencesof courses. The approach should rein-force current teaching and address cur-rent standards, but also should exploitthe power of technology to teach newthings in new and more effective ways.This means developing new software,new authoring environments, and newcomputer-based assessments – all ofwhich will require collaboration andinnovation on a scale that is difficult forany one institution to achieve.

The Concord Consortium has been

contributing to this important under-taking through projects described inthis newsletter. Now we propose toexpand these initial efforts into anational center that will bring the bestpeople, theory, technologies, andinstructional strategies to bear on long-term, large-scale research in educa-tional technology. This national center– or Educational Accelerator – willinclude a partnership of outstandingschools and colleges, collaboratingresearch groups, and a support infra-structure. It will foster additional col-laborations between testbed schools,educational researchers, and instruc-tional design experts. It will offerdevelopment, delivery, and assessmenttechnologies that will make it possibleto conduct research with schools andresearchers located anywhere.Educational technology is not a “frill”

Schools simply cannot afford toexperiment with major innovationsthat exploit the power of technology,particularly innovations that requirenew curricula, new teacher skills, ornew assessment strategies. Add to that

2 @concord – Spring 2003 The Concord Consortium www.concord.org

PerspectiveThe Educational

Accelerator

By Paul Horwitz and

Robert Tinker

Now we propose … anational center that will bring

the best people, theory,technologies, and instructional

strategies to bear on long-term, large-scale research in

educational technology.

the fact that teachers and administra-tors still think of computers largely asmachines with which to teach aboutcomputers. Few have a well-formedvision of the longer-term implicationsof the technology for education gener-ally. With only rare exceptions, com-puter applications are seldom an inte-

gral part of texts or infused throughouta course in the current school atmo-sphere. Technology is viewed as anancillary supplement that is usually leftto the teacher to work into instruction.Overcoming these barriers will be acentral role for the Accelerator, whichwill develop innovative curriculumunits for full integration into coursesubjects.

The Accelerator will focus on a sub-set of curriculum themes that can ben-efit most from information technology,including helping students visualizeand experiment with abstract conceptsthat are difficult to convey by passivemedia such as text or static illustra-tions. It will adapt software tools and

models developed by its partner insti-tutions and add them to a commonscripted learning environment. TheAccelerator also will develop morepowerful tools for automated data anal-ysis and report generation. These tech-nology enhancements will introduce apowerful new methodology for educa-tional research – one that combines thecapacity to capture fine details of the

learning process with the large-scaledissemination required for evidence-based studies. To ensure wide adoptionand continued support, all Acceleratorsoftware will be open source and allcurriculum materials will be free toparticipating schools.

Another barrier to more effective useof computers in schools includes thecost of professional development. Thebest uses of computers involve relative-ly independent student explorations insmall groups. This way of learning runscounter to the prevailing culture inmany schools, particularly at the highschool level. Professional developmentaimed at institutional change in thisarea is required, but funding for suchservices is likely to be the first thing cutby school boards and administrators ina time of fiscal constraint. All materi-als produced by the Accelerator will,therefore, include online, self-paced orinstructor-facilitated courses intendedto help teachers to use them effectivelyin the classroom.

To realize the revolutionary poten-tial of educational technologies willrequire large-scale research involvingthe collection of data from hundreds orthousands of teachers, and tens ofthousands of students. A national cen-ter, such as the EducationalAccelerator, can provide the volume ofdata that will further demonstrate thevalue of computers in education for allstudents. With this data, schools willreap the benefits and, at last, join theInformation Revolution.

www.concord.org @concord – Spring 2003 The Concord Consortium 3

Paul Horwitz([email protected])directs the ConcordConsortium ModelingCenter.

Robert Tinker([email protected])is President of theConcord Consortium.

To realize the revolutionary potential of educationaltechnologies will require large-scale researchinvolving the collection of data from hundreds orthousands of teachers, and tens of thousands ofstudents.

An Educational Accelerator could:• support the creation and dissemination of educational technology by

developing, customizing, maintaining, and deploying infrastructuretechnology, including tools for developers

• train future leaders in educational technology through internshipsand summer workshops for master teachers, graduate students, postdocs, and faculty

• recruit and support a network of testbed schools that will use thetechnology and share their experiences and data among themselvesand with researchers

• acquire, store, and transmit to teachers and researchers data from thetestbed schools relevant to technology use and student learning

• develop and share tools and techniques for analyzing such data, andfor automating the generation and dissemination of such analyses

• host national and regional meetings on issues related to educationaltechnology

• offer small “planning grants” aimed at the creation of new education-al projects affiliated with the Accelerator, enabling the Accelerator tobecome self-sustaining

Seeing Math through Multimedia Case Studies

The storyline always integrates two essential elements –a math content strand that is aligned with NCTM stan-dards, and a classroom pedagogy strand (see “Seeing MathVideo Case Studies” on page 5). This integrated approachhelps participants use the case as both a window into anoth-er’s practice, and a mirror for reflection on their own teach-

ing of math. Watchingand analyzing the wayteachers make decisionsabout their teachingleads participants tomake better analyses anddecisions about theirown teaching.Commentary SpursReflection

Video commentariesaugment each case withviews of the featuredclassroom from differentperspectives – that of theteacher reflecting on thelesson she has taught, andthat of a math contentspecialist offering addi-

tional insight. In the teacher commentary, the case teacherdescribes her expectations of the lesson as she envisioned itbefore taping. Following that, she may share her reflectionsabout the classroom experience as it actually unfolds. Didthe students “get it”? What worked and what would shechange next time? Listening to a fellow teacher reflect onher practice offers a way for teachers to identify with anoth-er professional encountering the same problems and mod-eling a path to solutions.

Specialists from the field provide expert commentary onthe case study teacher’s classroom management skills, andhelp participants see beyond this single experience tounderstand other mathematical approaches to the sameproblem. Inquiry-Based Professional Development

While the video narrative is the starting point, there isalso a rich surround of supporting materials. The support

materials answer questions about the school location,demographics, and how the featured lesson fits into the cur-riculum. There are examples of student work with guide-lines for assessment, as well as the teacher’s lesson plan. Amath “diving in” activity – often using an interactive tool –helps participants understand the mathematics from thestudents’ point of view. Teachers thus must wrestle with thesame problem with which their students wrestle.

In addition, User Guides for both the course facilitatorand the participants lay out a path for the course and poseactivities to guide reflection on the case.Open-Ended Presentations

Often student thought is highly original, but in real time,in a classroom, it may be difficult to understand. A videocase offers the luxury of multiple chances to listen, review,and even study transcripts of what a student says. Read thetranscript below and see if you can figure out what the stu-dent is talking about as he tries to explain why he conclud-ed that 5/18 is closer to 1/4 than 1/3.

Teacher: So, how did ... tell me about the tally marks. How didyou use those tally marks? How did you come about 1/4?

Student: Because first from five, you have to multiply three andit’s fifteen, and then we try four ... five times four, which is twen-ty. So, um, twenty, is, um, two above from eighteen and fifteen isthree less than eighteen so we decided 1/4...

Teacher: Oh, all right. From Number and Operations: Fractions

Jennifer Bradley’s 4th grade class Timmerman Elementary School, Pflugerville, Texas

What was the student’s strategy? Did it lead to the cor-rect answer? Why would it work … or not work?

As any good storyteller knows, sometimes the greatestimpact on learning comes not from what is told, but what isnot told. Sometimes the strongest way to encourage reflec-tion is not to resolve a problem shown in the case.Therefore, sometimes a video episode ends without a tidyresolution, and the participant is asked in the surroundingmaterials to reflect on how, in a similar situation, she mightencourage her own students to move to a deeper under-standing. In these cases, the video is a starting point for theteacher to think “what if this were my class and these weremy students, what would I do?”


The Broken Calculator case study encourages alternativeproblem-solving strategies.

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ARTICLE LINKS & NOTESSeeing Math Telecommunications Project – http://seeingmath.concord.orgTeachscape – http://www.teachscape.com

Joanna Lu([email protected])is Project Manager of the Seeing MathTelecommunicationsProject.

Raymond Rose([email protected]) isVice President of theConcord Consortium.


Numbers and OperationsDivision with Remainders For 4th and 5th graders, division with remainders and the inverse nature of division andmultiplication are key ideas. Nancy Horowitz and Mary Beth O’Connor present the same les-son. Students in each class build understanding about the nature of the division by creatingand solving their own story problems.

Content: Division with remainders, problem solving, inverse nature of divisionand multiplication, division as sharing or partitioning

Pedagogy: Using effective questioning strategies, encouraging students to graphically represent abstract ideas

Location: Springfield, MA

Fractions Jennifer Bradley’s 4th grade students expand their understanding of fractions as parts of awhole to understanding fractions as numbers. They use multi-link blocks, folded numberstrips, and number lines to compare magnitudes of “familiar fractions” with fractions theygenerate from data about the classroom.

Content: Number sense, fractions, part/whole relationships, comparing fractionsPedagogy: Using linear models of fractions to build understanding of fractions as

a quantity, understanding different meanings of fractionsLocation: Pflugerville, TX

Broken Calculator Stacy Riggle’s 3rd grade class explores division of largenumbers using a “broken calculator” strategy: studentsmust use handheld calculators that are limited by adivision key that does not work. Kelly Goorevich’s 4thgrade class uses the Broken Calculator program – withvarious keys and operations disabled in the software –to explore addition and multiplication problems.

Content: Number sense, grouping, relation-ship of multiplication and division,relationship of addition and multi-plication, place value

Pedagogy: Alternative problem solving strate-gies, communicating mathematicalideas, using technology to supportlearning in mathematics

Locations: Pittsburgh, PA; Watertown, MA

Pre-Algebra Pan Balance EquationsAudrey Soglin’s 5th grade class manipulates concrete representationsof equivalence, using pan balances to understand the nature of equations and operations on equations.

Content: Algebra, equivalence, operations on equationsPedagogy: Using 2D and 3D models to represent

abstract processes, value of a challengingproblem

Location: Evanston, IL

FunctionsLeyani von Rotz’s 4th grade students develop their ideas about func-tions, patterns and predictions by exploring linear growth patterns oftile arrangements. Using T-charts, students compare stages of growthwith the number of tiles used at each stage. They generate rulesthat permit them to make predictions.

Content: Early algebra, functions, patterns, graphing,prediction, generalization

Pedagogy: Using patterns to support inference, under-standing the difference between deductive andinductive inference

Location: Emeryville, CA

Geometry2D and 3D FiguresStudents in Jeanine Airesman’s 4th grade class build rectangularprisms using straws and connectors, then describe and comparetheir constructions. They use a workbook, along with wooden poly-hedral solids and other shapes, to solve riddles about faces andvertices. Students also draw 2D representations of their 3D shapes.

Content: Geometry, 3D description, 3D to 2D represen-tation, reasoning and proof

Pedagogy: Using models to support close interpretationof text, building understanding by translating3D models to 2D representation

Location: Pittsburgh, PA

Calculating the Area of a TriangleTraditionally students learn to find the area of geometric shapes byusing formulas. Noreen Winningham guides her 5th grade studentsas they build a foundation for understanding the area of a triangleand methods to calculate it before learning the standard formula.

Content: Geometry, measurement, symmetry, represen-tation, reasoning and proof

Pedagogy: Using personal experience to inform a concept of area, applyingmultiple strategies tosolve problems, studentwriting in mathematics Location: Evanston, IL

Data Analysis and ProbabilityUsing Data to Make PredictionsThis case presents ways to support NCTM standards for grades 3-5 that invite students tocollect, analyze, and make predictions from data. The video shows two lessons fromRhonda Singleton’s 5th grade class. The first explores mathematical fairness. The secondshows the relationship between sample size and accuracy of predictions about a population.

Content: Collecting data, making graphs, data analysis, making predictions,probability

Pedagogy: Collaborative learning, games to support learning, addressing misconceptions

Location: Myrtle Beach, SC

Data Sets and Measures of CenterStudents collect, organize and analyze data to determine maximum, minimum, range, mode,median, and mean. Thirty percent of Lala Sahakian’s 4th grade students have arrived in theU.S. during the past 18 months. Teaching in an ESL class presents unique challenges. Notonly must a teacher ensure students understand math content, but she must also assessunderstanding when students lack strong language skills for communicating their ideas.

Content: Data sets, data analysis, measurement, representationPedagogy: Teaching mathematics to ESL and ELL students, using concrete

experiences to ground terminologyLocation: Glendale, CA

Fourth gradersDenzel, Eddie andStephanie use alter-native strategies forsolving a problemwith the BrokenCalculator.

Seeing Math Video Case Studies


By Cara DiMattia and Daniel Cogan-DrewReflecting on one’s practice is an important part of teach-

ing. The classroom, however, is a constantly moving place– it is very difficult to isolate particularly puzzling momentsor review questions raised by students without stopping theflow of a lesson. Video is an excellent way to capture thesefleeting but important moments. This Monday’s Lessonfocuses on what you can do professionally as a teacher-prac-titioner to reflect on your teaching.

VideoPapers facilitate the use of video for reflection onclassroom experiences, and the exchange of ideas throughan accessible and engaging medium. VideoPaper Builder 2(VPB2) is software designed to enable computer users ofaverage ability to create multimedia documents, which

closely link textand video, therebygrounding com-mentary directlyin classroom data.

These primarysources of data –video from the class-room and a teacher’swritten reflections –become the basis fordiscussion amongprofessional col-leagues. The videofootage is supple-mented by the useof still images cap-tured from the

video itself (notes from the board, a student’s facial expres-sion), scanned content (student work, teacher handout), orother digital images of interest (explanatory diagrams,graphics).

VideoPapers may be published via the Internet or onCD-ROM. Educators are thus able to observe each other’sclassrooms and to discuss their responses, both to the videosand to the author’s reflections in a meaningful way.Local Teacher Professional Development

Unlike numerous professionally produced teacher devel-opment videos, the VideoPaper is produced and distributedlocally within schools or across neighboring districts. Thecontent is shot and edited by the teachers themselves, for

use and exchange with each other; the video clips containimages of educators and students in the community. Thecontext for the filming is immediate, and the strategies thefilm illustrates are applicable to the intended audience. Thevoice and message of the author is trusted; it speaks from acommon experience. The authenticity of this voice earns itrecognition and regard not often accorded the professionaldevelopment “expert” who istypically invited to address anunknown audience of teach-ers.

A vehicle for presentinguseful strategies to her col-leagues, the VideoPaper alsoprovides its author with anequally significant opportu-nity to reflect on her ownteaching. The need to explainfully the theory and practiceof their teaching techniquesrequires that teachers have aprofound knowledge of theirown content. The act of creating a VideoPaperbecomes a process of profes-sional self-development.

Download VideoPaperBuilder 2

VideoPaper Builder 2, anew version of software orig-inally developed at TERC, is a tool for users with moder-ate technology experience. VPB2 is free, open source soft-ware; it is both Windows and OSX compatible.1. Point your Web browser to: http://vpb.concord.org2. Click on “download,” then choose the appropriate file

for your system:VPB2 for Windows (14.6 MB) VPB2 for the Mac (4.1 MB)

3. The software will download to your desktop and expandautomatically. Click on the Installer and follow the step-by-step directions to install the software.

To create a VideoPaper, you will need three types of files:digital video files (.mov, .mpg), text pages converted to.html format, and images (.gif, .jpg).

Monday’Using VideoPaRRRReeeefffflllleeeeccccttttiiiinnnngggg OOOOnnnn

VideoPaper Builder 2System RequirementsWindows

Windows 98, NT, 2000, or XPQuickTime 6.0 Full Install (including QuickTime for Java)Java Runtime Environment 1.3 or greaterWeb Browser (Netscape 4.x or Internet Explorer 5.x)

Mac Macintosh OS X 10.1 or greaterQuickTime 6.0 Full Install (including QuickTime for Java)Java Runtime Environment 1.3 or greaterWeb Browser (Netscape 4.x or Internet Explorer 5.x)

This classroom episode is part of an AdvanHigh School in Boston, Mass. Because theCreole and English, the students' gestures cussion and explanations. This VideoPaper


The technology skills required to use VideoPaperBuilder 2 are easy to learn and useful in many other con-texts. For instance, the process of shooting and editingvideo requires the author to become familiar with digitalvideo cameras and to learn how to use a basic video editingsoftware, like iMovie.

The text pages can be produced in a word processor andconverted to HTML format.Word processors such asMicrosoft Word have anautomatic “save as HTML”function; learning to authorHTML (web page) docu-ments may be another valu-able skill.

Images used in theVideoPaper (as “slides” thatare synchronized to thevideo) can be captured as stillsfrom the video itself, scannedimages of handouts, student

work, photographs, or other graphics. The author maylearn new skills with a scanner, a digital camera, or a digitalimage editing software (such as Adobe Photoshop).

VPB2 generates menus, links, framesets, and image slideshows in order to interconnect the author’s video and text.These elements are then organized into a single multi-media presentation. The final product is viewable on a Macor a PC, using an Internet browser, such as InternetExplorer or Netscape Navigator.Tutorial

For a quick “10 Easy Steps” tutorial, point to the follow-ing address in your Web browser:

http://vpb.concord.org/help/tutorial.htmlThe VideoPaper Builder Web site also includes a com-

plete User Guide, sample VideoPapers, and a Communityarea for exchanging information and ideas regarding the useof VPB2.

When you create your VideoPaper, share it with us. Posta URL to your VideoPaper in the Community area and wecan all reflect on Monday, Tuesday or Friday’s lesson.

Cara DiMattia ([email protected]) is a multimedia developer forthe Bridging Research &Practice project at TERC.

Daniel Cogan-Drew([email protected]) isAdministrator of the CurriculumResource Center at TuftsUniversity.

’s Lessonaper Builder 2

ARTICLE LINKS & NOTESTERC – http://www.terc.eduVideoPaper Builder 2 – http://vpb.concord.orgVPB2 Tutorial – http://vpb.concord.org/help/tutorial.htmliMovie – http://www.apple.com/imovieAdobe Photoshop – http://www.adobe.com/products/photoshop/main.htmlThe VideoPaper Builder 1.0 software was originally developed as part of the Bridging Research & Practiceproject at TERC and funded by the National Science Foundation (Grant #9805289). VideoPaper Builder 2.0was developed in partnership with TERC by the Seeing Math Telecommunications Project at the ConcordConsortium through funding from the U.S. Department of Education (Grant #R286A000006).

Through a Teacher’s LensBy Chris Mainhart and Maggie Woodcome

“Looking at the video with someoneelse was the most important part of theprofessional development experience.”Maggie has expressed this sentiment overand over again as we talk to other educa-tors about the VideoPaper case we devel-oped together.

After collecting video in her 7th grademathematics classroom, Maggie and Irevisited the footage several times in aneffort to tell a mathematical story. We haddifferent purposes for collecting and view-ing the video. Maggie saw a genuine valuein the reflective process that comes from

viewing the same video several times. Asthe math staff developer, I also relished theopportunity to share my comments andquestions. I could emphasize differentaspects of her teaching that contribute tostudent success. Trust was an essential ele-ment that allowed us to work in this man-ner. We devoted many hours to buildingthe VideoPaper in an effort to analyze theteaching and instructional strategiesimplemented in her classroom.

The classroom story will vary depend-ing upon the focus of the teachersinvolved. The issue that spoke to us wasstudent voices – Maggie created opportu-

nities for students to be seen as mathexperts. Building the VideoPaper allowedus to highlight the importance of givingstudents time to share their mathematicalthinking. Other teachers might see differ-ent storylines.

Teachers can use this process to reflecton their own practice and to discuss thepedagogy and content with other teachers.We gained immeasurably from havingfocused our attention on teaching andencourage other educators to useVideoPapers to do the same.

Chris Mainhart and Maggie Woodcome are at the Hudson Public Schools in Massachusetts.

nced Algebra course at the Jeremiah Burkee class is bilingual, speaking Cape Verdeanare important in understanding their dis-focuses on their use of gestures.

How do teachers engage with pro-fessional development video case stud-ies and what do they learn from them?These have been some of the drivingresearch questions of the Seeing MathTelecommunications Project (see also“Seeing Math through MultimediaCase Studies” on page 1).

Using spring and fall 2002 data fromteachers in South Dakota, Vermont,Washington, D.C., and Massachusetts,we have identified two critical aspectsthat make a significant difference in thedepth and quality of the professionaldevelopment outcomes:

• Grounding the discussion in thespecifics of the video case.

• Integrating the events of thevideo case with actual ongoingevents in the classrooms of theparticipating teachers.

To illustrate these points we selectedthe following excerpts from onlinepostings in the Seeing Math profes-sional development courses. We startwith a facilitator’s “seed” and tworesponses.

Facilitator: How can the questioningstrategies in this case study serve all thestudents, not just those in the middle butthe strongest and the weakest ones as well?Do the strategies offer a way to strengthenteaching and learning for all these studentssimultaneously?

Teacher: I think that the questioningstrategies and students’ responses allow thehigher level thinkers to rethink their pro-cess and allows the lower level students togather more information on different pro-cesses that were used.

Teacher: I think that using a variety ofquestioning strategies strengthens all stu-dents. Students need to hear the way otherstudents think and understand the process-es. When a variety of ways are discussed,all students reap the benefits.

Notably, these messages are devoidof specific references to the video case.Rather, the teachers respond with gen-eral statements reflecting their alreadyheld beliefs about the value of “ques-tioning strategies” whose nature is notspelled out. The video cases play onlya marginal or superfluous role in theexchanges. Further, we could not iden-tify instances in which the teachers hadchanged or enriched their views.

In order to deepen the discussionsand the rich potential of the video casesfor teacher professional development,we realized the need to “ground” thediscussions in the particulars of thevideo case and the actual experiences ofthe teachers in their classrooms. Thisshift of facilitator strategies was appar-ent during the fall 2002 course.

Facilitator: What statements did youhear in the introduction that caused you to

begin reflecting on the traditional methodof teaching division? Did anything MaryBeth or Nancy said “ring a bell” with you?

Teacher: When they presented the arraysfor division facts and then discussed usingmanipulatives, I realized that I had reallyreinforced the notion that mastering thealgorithm for division was the be-all andend-all of division, rather than to help mystudents to develop a solid numericalunderstanding of what it means to divideand how to fluently describe division withlanguage.

Teacher: Now I am starting to utilizethe ideas presented with arrays – manipu-latives and writing of problems. I amteaching multiplication now and will havethe kids write their stories tomorrow.

Here, the facilitator requests com-mentaries on particular utterancesincluded in the video case. Teachersrespond to examples in the video casewith reference to the practices in theirown classroom, regarding their ownteaching.

The Seeing Math project has learnedthat sustaining rich, grounded discus-sions – grounded in the video case aswell as in actual ongoing stories fromthe participants’ classrooms – is thefacilitator’s major role. This role needsto be made explicit and actively sup-ported.

Video Case Studies:Grounded Dialogue Matters Most

By Alvaro Galvis and Ricardo Nemirovsky


Seeing Math’s VisionVideo cases should:

• Be analyzed, not imitated• Initiate shared inquiry about the object of study, rather than transmit a given teaching model• Be authored by a professional group of producers and educators• Adhere to a production standard, while allowing flexibility for the uniqueness of each case• Exhibit a clearly defined pedagogy, as chosen by the creator of the video case

Teacher Professional Development using video case studies should have the following characteristics:• Facilitator moderates from the side, not from the center – in either face-to-face or Web dialogue• Discussions are driven by both professional and personal interests and needs• Discussions are focused around both local and shared, “supralocal” issues• Participants are both intrinsically and extrinsically motivated and rewarded• Discussions are grounded in video and classroom experience – not merely in participants’ ideas and opinions• Video case study and classroom practices are integrated

Alvaro Galvis([email protected])is Senior Researcher

at the Concord Consortium.

Ricardo Nemirovsky([email protected]) is SeniorScientist at TERC.

CC Resource Center Offers Free Software and Curriculum

By Robert Tinker

We have developed exciting software, curricula, and toolsthat are available to you – for free! We hope you will usethese materials, distribute them, critique them, and tell usabout your experiences.

You do not need to be a member of the ConcordConsortium to get these materials, but we encourage you tojoin at our Web site. Joining is free and easy, and you willnot be added to any of our mailing lists unless you sorequest. We just like to know who is using our software. Asa member, you’ll also have access to a community discussionarea for sharing stories of your successes or for trou-bleshooting technical glitches.

To access these materials, go to our Web site and click onthe CC Resource Center. You’ll find additional informationand detailed instructions for downloading software andother tools. There are several kinds of material from whichto pick and choose, according to your subject area orresearch interest. Whatever interests bring you here, wehope you’ll find something you can use.

Genetics Modeling Activities. The BioLogica softwareconsists of 13 stand-alone activities that enable students tolearn classical genetics through guided exploration. Thereare also assessments and an open-ended version.

Molecular Modeling Activities. There are 14 activitiesand associated curricula that use molecular dynamics mod-els to allow students to explore the ways atoms andmolecules determine the basic properties of matter.Activities cover diffusion, phase change, compression, heattransfer, the gas laws, dissolving, and osmosis.

Dynamics Modeling Activities. Five activities focus onvectors and elastic collisions, topics addressed in middleschool physical science and physics.

The Concord Modeling Workbench. This large col-lection of activities uses molecular dynamics along with anauthoring environment that allows you to make your ownstudent activities. The content is similar to the modelingactivities listed above, but also includes chemical reactionsand protein conformation. (See also “Experimenting withAtoms and Molecules” on page 10.)

Sustainable Future Applications. Three applicationsand curriculum units help learners think about the future.For example, the simple “What If? Builder” allows studentsto create and share branching stories that have alternativeendings. Also available are the “Ecological FootprintCalculator” and the “Community Planner.”

Video Case Studies. We currently have a video casestudy of the use of a pan balance for developing a sense ofequivalence. This case study features professional video seg-ments linked to student work, class notes, standards, andcommentary. This can be used as a stimulus for discussions

and reflection on elementary or middle school mathematicsteaching practice. (Due to the nature of these materials, wewill need to verify your educator status before you can viewthis case study.)

VideoPaper Builder 2 (VPB2). This application simpli-fies the creation of classroom video case studies. The result-ing case studies can be used to stimulate reflection on teach-ing strategies. (For additional information on VPB2, see“Reflecting on Monday’s Lesson Using VideoPaper Builder2” on page 6.)

This list will expand, so please return to the CC ResourceCenter regularly for new software. Additional materials –including NetLogo activities, molecular biology models,and a system dynamics package – are under development.Members can request email notification when additions orupdates are available.

All our software is open source, which means there willnever be a charge for it and others are welcome to improveon it. As a result, many people will be able to support thesoftware into the future.

Please use these materials and share them with your col-leagues. They are all based on solid instructional theory andmost have been tested in classrooms. The materials share aneducational vision that uses student exploration of modelsand tools to revolutionize learning. Help us realize thisdream.


Robert Tinker([email protected]) is

President of theConcord Consortium.

With this dynamics modeling activity, students learn vector addition while helping thepirate find the treasure.

Atoms and their properties determine our world. Theway atoms and molecules bounce, stick, stretch, bend, com-bine, fall apart, and interact determines what you feel, touchand see, how every living thing grows and dies, and why weuse certain materials in our environment. Most of theimportant modern technologies – molecular biology, chem-

istry, nanotech-nology, electron-ics, cryogenics,lasers, and com-posites – can beunderstood onlyby consideringatoms and theirinteractions. Thecommon thread ofatomic-scale inter-actions could inte-grate the sciencesand technologyaround a few coreideas that runthrough the entirecurriculum.

One of the rea-sons that the con-nections betweenmacroscopic andmicroscopic prop-erties are not welladdressed in thetypical curriculum

is that the order of subjects is wrong. The basic ideas ofatomic-scale energy conservation, temperature, heat, pres-sure, electrostatic forces, and electrons belong in physics.But physics is usually taught after chemistry and biology,which need to build on these ideas. Worse, physics coursesusually fail to apply mechanics and electrostatics to atomsand molecules. Physics or physical science at the ninthgrade should address atoms and molecules to support sub-sequent chemistry and biology courses, but they rarely do.

Enter computers. Now it is a simple matter to model lotsof atoms that collide according to Newton’s Second Lawapplied to the kinds of forces that actually exist betweenatoms. This is called a molecular dynamics (MD) model.Students can understand the forces that apply to each atom

in such a model and experiment with ensembles of lots ofthem. The gas laws are an experimental result of such a sys-tem. Students can even observe non-ideal behavior in a verydense gas of attracting atoms or molecules.

This is an example of the many macroscopic propertiesthat “emerge” from simple laws governing the microscopicworld. It is not at all obvious that the gas laws will emergefrom classical mechanics applied to interacting atoms, but amolecular dynamics model can convincingly demonstratethat this is so.

Molecular dynamic models can help students discover: Energy conservation. Put any number of atoms ina “box” and start them each off at any velocity. Thetotal energy of the system will stay constant. Temperature. Each atom in a mix of atoms in abox – whether small, large or part of a molecule –will have the same average kinetic energy, definedas temperature. Thermal conduction. Put a hot gas of fast atomsin contact with a cold gas of slow atoms. The hotgas will cool and the cold one will warm until thetwo are the same temperature. Entropy. Start a model with atoms arranged insome distinct pattern; the atoms always end uplooking essentially the same, bouncing around atrandom. Phase change. At high temperatures, a collectionof atoms resembles a gas. Remove energy and it willcondense into a disordered, dense swarm – a liquid.Continue removing heat and the atoms becomeordered – a solid.

These examples are only the beginning. The ideal andnon-ideal gas laws, vapor pressure, diffusion, dissolving (seeFigure 1), osmosis, filtering (see Figure 2), breaking, andmany other basic phenomena can be understood by explor-ing MD models.

Molecular dynamics provides another way to understandrelationships such as the gas laws. “Laws” such as these areno longer seen as the mysterious result of an abstract math-ematical derivation, but can now be understood in the lightof something new: simple, quick computer experiments.Because these experiments are more accessible to younglearners, it should be possible to use them to make new con-nections between subjects and to give students a far betterunderstanding of large segments of science. New three-dimensional, interactive models make it possible to visual-

Experimenting with Atoms and MoleculesMolecular Dynamics lays the foundation for science reform

By Robert Tinker


Robert Tinker([email protected]) is

President of theConcord Consortium.

Figure 1. An ionic crystal dissolving in water. The ions are labeledwith plus and minus signs. All the ions are surrounded byuncharged water molecules represented here by unlabeled balls.When this model is run, the ions tend to clump, but if there is suf-ficient water, they become completely surrounded with water“snowballs,” one of which can be seen on the right.

The Concord Consortium invites interested high schools to join Modeling Across the Curriculum (http://mac.concord.org) as con-tributing schools. This NSF-funded project is a three-year longitudinal study of the effects of modeling technology on sciencelearning. Implementations in three Massachusetts high schools began in 2002-2003 and ten additional U.S. schools were select-

ed to participate in 2003-2004. We are observing students closely in order to measure cumulative gains in science content areas. Ourmain concerns are to determine whether computer-based modeling helps students learn to use mental models as explanatory devices, andif so, whether student modeling ability is transferable between content areas.

Contributing schools will be provided with two-week replacement curriculum units in Physics, Chemistry, and Biology that includeembedded assessments and teacher manuals.

Subject Replacement Units Physical Science Kinematics, Newtonian Mechanics Chemistry Gas Laws, States of Matter Biology Genetics, Population Dynamics

Benefits to teachers include access to information in the project database. Student work is recorded and organized, allowing teachersto see how students approach and find solutions to problem-solving activities. By logging answers in each activity, we generate individ-ualized reports for each student with information such as:

• Answers to multiple-choice questions• Relevant student actions as they run models and manipulate variables to solve problems• Numbers of trials needed to arrive at target outcomes successfully • Readouts of hypotheses and conclusions• Activity scoresFurther data analysis provides teachers the ability to compare answers to each question across the class, giving an overview of class

needs and strengths.For more information contact:

Paul Horwitz, Principal Investigator – [email protected], 978.371.5856Joanna Lu, Project Director – [email protected], 978.371.1339

ize cause and effect, turning abstract,theoretical ideas into concrete phe-nomena.

Indeed, our initial classroom experi-ences indicate that middle and highschool students can learn throughexperimentation with moleculardynamics models; they can understandatomic-scale models, relate them tomacroscopic phenomena, and transfertheir learning to new situations.Encouraged by our initial findings, weare in the process of extending ourmodels to encompass new content:

chemical reactions and protein confor-mation.

The molecular dynamics work at theConcord Consortium is laying thefoundation for major reform in intro-ductory science teaching that focuseson atoms, molecules, their interactions,and how these determine macroscopicproperties. We encourage you to useand modify the software and curricu-lum materials. Please see “CCResource Center Offers Free Softwareand Curriculum” on page 9 for addi-tional information.


Figure 2. A solid, liquid, and gas encounter a sieve. Here a sieve or filter is modeled bytwo barriers shown as brick rectangles. When the model is run at a low temperature(top), the balls form a rigid crystal that is trapped on the right side of the sieve. Whenthe temperature rises (middle), a liquid forms, but it still cannot squeeze through, becausethe atoms attract one another. This illustrates surface tension. Further heating (bottom)creates a gas that can get through. (To appreciate fully these images requires seeing themodels evolve dynamically and interacting with them.)

1 Seeing Math through Multimedia Case StudiesFollow the production of a multimedia video case study for math teacher professional development.

2 Perspective:The Educational AcceleratorWith an Educational Accelerator, schools can join the Information Revolution at last.

6 Reflecting on Monday’s Lesson Using VideoPaper Builder 2Teachers can use this software to create their own video cases to reflecton their teaching practices.

8 Video Case Studies: Grounded Dialogue Matters Most

Two key areas make a significant difference in professional developmentoutcomes with video case studies.

9 CC Resource Center Offers Free Software and Curriculum

Become a member of the Concord Consortium and receive free software,curriculum, and tools.

10 Experimenting with Atoms and MoleculesMolecular dynamics modeling helps students understand atoms, molecules, their interactions,

and how these determine macroscopic properties.

11 Modeling Across the Curriculum The MAC project seeks contributing schools for three-year longitudinal study of the effects of modeling technologyon science learning.

The Virtual High School – Seven years ago, the ConcordConsortium, along with Hudson Public Schools, created theVirtual High School®. Although there have been many imitators,the Virtual High School remains the innovative leader, offeringthe highest quality courses for the least costs using a coopera-tive model. A book that chronicles its development and qualityhas been published recently by the project evaluation team:Zucker, A., et al. (2003).Teaching Generation V:The Virtual HighSchool and the Future of Virtual Secondary Education. NewYork:Teachers College Press.The VHS is now a separate non-profit dedicated entirely to providing excellent online secondary

courses.This is an outstanding example of how grant-supportedR&D can be converted into a permanent service post-funding.The Concord Consortium is proud that VHS is now standingon its own and has been for two years.http://www.goVHS.org

Online Learning Services – Concord Consortium offers onlinecourses based on our two books: Facilitating Online Learning:Effective Strategies for Moderators and Essential Elements:Prepare, Design, and Teach Your Online Course.http://www.concord.org/courses

Inside for Spring 2003

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