connect jan-feb 07 r8 - synergy learning

January/February 2007 VOLUME 20 ISSUE 3

TEACHERS’ INNOVATIONS IN

K–8 SCIENCE,MATH AND

TECHNOLOGY

THIS ISSUE’S FOCUS

Good Science,Good Math

Connect®

I N S I D E

Constructing KnowledgePia Hansen-Powell 1

How Do We Teachand Assess for Ownership?Charlotte Stetson 5

The Power to Make PredictionsJulie King 8

Using HistoricalExplanations to Teach How Science WorksJean Beard 11

Planning for UnderstandingBob Coulter 14

Literature Links 16

Resource Reviews 18

To Teach or to Test: That is theQuestion!Michelle Kirk 20

A Funny ThingHappened on theWay to the TestMary Cowhey 23

Innovators in Our Classrooms 26

Good Science, Good Math

Are we maintaining the levels of creativity and innovation in math and science teachingthat can provide motivation and encourage ongoing inquiry and problem solving? Ananswer to this question may involve national and state initiatives, school and classroomgoals and demands, teacher skills and energy, along with a host of minor factors thatoften become major, including such things as adequate funds for manipulative materials

or funds for a field trip.Some authors in this issue of Connect identify trends

that may point to erosion of creativity in the face ofmultiple challenges to schools. Two articles address thehazards of high stakes summative assessments. Yet,every author cites examples of outstanding work on thepart of teachers and students engaged in exemplary,inquiry-based investigations. The newest challenge maybe to integrate effective practices with recent demandsfor greater accountability.

We hope these articles will provide points of discus-sion for many educators.

Ed Press Awards for Connect: 1990, 1992 Distinguished Achievement Awards1996 Winner: Honor Award for Excellence in Educational Publishing

Editor: Casey Murrow

Associate Editor: Heather Taylor

Circulation: Susan Hathaway

Design and Production: Judy Wingerter

Connect (ISSN: 1041-682X) is published five times per year, September, No-vember, January, March and May, by Synergy Learning International, Inc.,PO Box 60, Brattleboro, VT 05302. Tel. 800-769-6199, Fax 802-254-5233,email [email protected].

©2007 by Synergy Learning International, Inc. Published as a non-profitservice. All rights reserved. Special permission is required to reproduce inany manner, in whole or in part, the material contained herein. Call 800-769-6199 for reprint permission information.Periodical postage paid at Brattleboro, VT 05301 and additional mailing offices. (USPS 005-389).

POSTMASTER: Send address changes to Connect, PO Box 60, Brattle-boro, VT 05302-0060.How to subscribe (print version): send $25.00 for a one year subscription($31.00 Canada/Mexico, $44.00 other foreign, in U.S. funds). Volume dis-counts are $20.00 each for five or more subscriptions sent to the same ad-dress. Back issues are $6.00 each, ten or more issues $4.00 each postpaid.Mail to Synergy Learning, Inc., PO Box 60, Brattleboro, VT 05302-0060.Or call 800-769-6199.How to subscribe (PDF version): Send $20.00 and your email address forthe PDF version, available worldwide at this price. Same content and illus-trations. Email us for details on our annual site license, allowing you to dis-tribute Connect to others in your school or district.For more articles and focus topics, see the Connect archive on our Web Site:http://www.synergylearning.org.Printer: Vermont Graphics

Synergy Learning International, Inc. is a 501(c)(3) not-for-profit corporation, engaged in publishing and professional developmentfor educators, pre-k to middle school. We are dedicated to supporting schools, teachers and families with challenging science, mathand technology learning for children.

Connect©

published by S Y N E R G Y L E A R N I N G ™

Connect is published five times per year (bi-monthly through the school year) and offers a wide range of practical, teacher-written articles. Each issue isthematic and supports hands-on learning, problem solving and multidisciplinary approaches.

Connect is printed on Chorus Art paper, 50% recycled;manufactured acid and elemental chlorine free.

PHOTO CREDITS: Cover, p. 26, Synergy Learning; inside front cover, p. 5, 6, 7, 11, 21, 23, Heather Taylor; p. 1, 2, The Math Learning Center; p. 4, PiaHansen-Powell; p. 24, 25, Mary Cowhey.

ILLUSTRATIONS: Page 13, Heather Taylor.

http://www.synergylearning.org

www.synergylearning.org • JANUARY-FEBRUARY 2007 Connect • PAGE 1

Highly qualified teachers all across thenation are interpreting the national

standards for math and science, unpackingthe essential learning for their grade level,and using a variety of authentic, perfor-mance-based assessments to guide theirclassroom practice. At the same time, theyare preparing students to take standardizedhigh-stakes, multiple-choice tests that askstudents to recall basic facts and informa-tion. Which approach will leave studentsbehind? What content is worth knowing?

We all agree that our students should bescientifically literate and mathematicallyfluent. The business community needsworkers who can learn, reason, think cre-atively, make decisions, and solve prob-lems. The goals of the National ScienceStandards are to educate students who areable to:

• Experience the richness and excitementof knowing about and understandingthe natural world;

• Use appropriate scientific processes andprinciples in making personal deci-sions;

• Engage intelligently in public discourseand debate about matters of scientificand technological concern;

• Increase their economic productivitythrough the use of the knowledge,understanding, and skills of the scientif-ically literate person in their careers.

The goals of the National MathematicsStandards are to educate students who:

• Build new mathematical knowledgethrough problem solving;

• Make and investigate mathematical con-jectures;

• Develop and evaluate mathematicalarguments and proofs;

• Communicate mathematical thinkingcoherently and clearly to others;

• Recognize and apply mathematics incontexts outside of mathematics;

• Create and use representation to orga-nize, record, and communicate mathe-matical ideas.

These national goals and standards arealigned with best practice, brain research,and instructional strategies that increasestudent achievement. The National Sci-ence Teachers Association and NationalCouncil of Teachers of Mathematicsdefine who (every child) should learn what(the content standards) and how (theprocess standards). The verbs in thesestandards are experience, engage, build,apply, create, and communicate. One ofthe goals of the No Child Left Behind Actwas to focus instruction and assessment onthese standards and state benchmark out-comes. Therefore, in theory, teachers who

Constructing KnowledgeRECONSTRUCTING NO CHILD LEFT BEHIND

by Pia Hansen-Powell

Taking a keen interest inworking with manipulatives

In primary grades, students useinsect and spider manipulatives tohelp with their investigations.

PAGE 2 • Connect SYNERGY LEARNING • 800-769-6199 • JANUARY-FEBRUARY 2007

implement these standards help their stu-dents learn science and math by activelyengaging in inquiries that are interestingand important to them. The students con-struct a knowledge base for understandingscience and mathematics in relationship toother disciplines.

Just three, forty-hour weeks

Are we teaching for understanding or forachievement on standardized tests, and dowe have to choose one or the other? Withso much to teach and so much to learn, thetask of achieving both goals for our stu-dents is daunting. If the school year has180 days, and we allocate time for testing,announcements, assemblies, and profes-sional development days, this leaves usabout 160 days of teaching. If we schedule45 minutes for each subject, each day, itwould add up to 120 hours of instructionin each subject, each year. Given three,forty-hour workweeks, would we teachersbe able to master complex subject matter?How much can we reasonably expect allof our students to learn in that time?

Coherence is the key to helping stu-dents build the conceptual frameworks

that help them retrieve knowledge andtransfer knowledge to new contexts. Cur-riculum that allows adequate time for in-depth learning that is connected to “bigideas” in several content areas offersteachers a sound approach to teaching tothe standards, increasing student under-standing, and making “adequate yearlyprogress.”

Towards this goal, many teacherssought a standards-based curriculum thatintegrates math, science, language arts,and even social studies, while promotingboth conceptual understanding and basicskills. Bridges in Mathematics, publishedby The Math Learning Center in Salem, Oregon, was developed with initial sup-port from the National Science Founda-tion. This problem-centered curriculumincludes a variety of manipulatives andvisual models that facilitate the develop-ment of students’ mathematical thinkingand reasoning. Some of the investigationsgrow out of ventures into the everydayworld—reading stories, creating models,playing games, making quilts—while oth-ers delve more deeply into the world ofmathematics itself. Students at all gradelevels are encouraged to explore, develop,test, discuss, justify, and apply mathemati-cal ideas.

Skills mastery within a context

The units in Bridges focus on related top-ics across multiple mathematical strands atonce, for example, by having studentsmaster the basic facts in the context ofinvestigating the algebraic patterns amongthem, or by developing strategies for mul-tiplying larger numbers while exploringthe volume of rectangular solids. The cur-riculum integrates fiction and non-fictiontrade books and science and social studiesthemes throughout the year. Several lifeand physical science units integrate math-ematics, literacy, and social studies stan-dards at each grade level. The emphasis inprimary classrooms is generally readingand writing literacy, and teachers fre-quently feel like they don’t have enough

By remaining opento new ideas andstrategies, these

students areencouraged to do

their best work andfocus on being acommunity of

learners.


time to teach mathematics, not to mentionscience. By integrating multiple subjectareas into problem-centered mathematicalinvestigations, teachers and students canaddress a variety of content standards in asingle unit of study, and often within a sin-gle lesson.

Students gathering data

In primary grades, students investigatebugs, sea creatures, penguins, farm ani-mals, and frogs and toads over the courseof a few weeks. Students gather informa-tion, observe and record their findings,compare and contrast these creatures, readnon-fiction text, and sing about prayingmantises, “hanging upside down, wrig-gling out of old skin and making foam eggcases.”

The emphasis is on creating excitementfor the child’s natural world, and learningthat lasts. My students love comparing theheight and weight measurements of differ-ent kinds of penguins. They learn to use athermometer in order to measure the watertemperatures that represent the oceans ofthe world.

Second graders build courses on whichthey roll marbles to explore the impact ofdifferent variables, including the marbles’mass, tube length, and ramp height. Theyconsider principles of balance and motionto make conjectures, investigate whatactually does happen, and record theirresults in a variety of ways. Then theycompare their data with other students toreach conclusions about the impact of trueexperimental design. Some scientific con-cepts are intuitive, and my students predictthat the marble and wood bead will notroll as far as the steel ball bearing becauseit is heavier.

After each team performs three trials,we collect the data. They are pleased whenthey compare data with their predictions.We place the same marble into tubes ofvarying lengths (toilet paper, paper toweland gift wrap) and send them rolling.Many student predictions are not con-firmed, and they spend their free time try-ing to figure out why. Without pre-plannedunits such as these, many primary class-

rooms would be void of any real physicalscience experiences. My test scores werenot compromised by taking time for theseexperiences because my students knewhow to make connections and think criti-cally about a problem solving situation.

In Bridges, third graders learn aboutdifferent kinds of bridges, build their ownmodel bridges while making conjectures,and draw conclusions about effectivebridges design and construction. My stu-dents are astonished at how many woodcubes can be balanced on a paper beambridge. They challenge each other to makea stronger, longer bridge that could carry alarger load. Students grapple with themany variables that influence the lengthand strength of bridge designs, and quan-tify them by estimating, measuring, andtesting the strength of their own models.By remaining open to new ideas andstrategies, these students are encouragedto do their best work and focus on being acommunity of learners.

Supportive evidence

The National Assessment of EducationalProgress data supports teaching mathemat-ical reasoning and emphasizing complextopics in the curriculum. Teaching thatemphasizes higher-order thinking skills,project-based learning, opportunities to

Students build their own modelbridges, drawing conclusionsabout effective bridge designand construction.

solve problems that have multiple solu-tions, and the application of hands-ontechniques, such as using manipulatives,were all associated with higher perfor-mance on the mathematics NAEP. TheTrends in International Mathematics andScience Study (TIMSS) provides furtherevidence. The NAEP data for scienceachievement suggest that teaching formeaning and focusing on projects inwhich students take on a high degree ofinitiative positively impacts student suc-cess. Completing worksheets and readingfrom textbooks does not.

At a time when we are asking teachersto place more emphasis on understandingby teaching in the context of inquiry,applying technology in a meaningful way,and integrating content areas, a viable cur-riculum can help meet the demands. WithBridges in Mathematics, students investi-gate and analyze questions over anextended period of time, construct originalarguments and explanations, and defendtheir conclusions.

Pia Hansen Powell, a veteran elementaryteacher, now leads professional developmentworkshops in mathematics and coaching andpresents at national conferences. She has taughtat the University of Wyoming and authoredBuddies, a book about cross-grade mentoring.Pia wrote the third-grade curriculum for Bridges2003–2005. With Charlotte Danielson, she hasco-authored, A Collection of PerformanceTasks and Rubrics: Primary School Mathemat-ics.

ResourcesMaranzo, Robert J. What Works in Schools,

Translating Research Into Action. Associa-tion for Supervision and Curriculum Devel-opment (ASCD), 2003.

Bridges in Mathematics K–5. The Math Learn-ing Center, 1999–2007.

Principles and Standards for School Mathe-matics. National Council of Teachers ofMathematics (NCTM), 2000.

National Research Council. National ScienceEducation Standards. National AcademyPress, 1996.

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The Bridges in Mathematics curriculum is available from the Math Learning Center in Salem,Oregon. For more information, including teacher comments, samples, correlations to standards,and resource guides, visit http://www.mathlearningcenter.org. This site also has links toprofessional development workshops and many other products and materials.

Hundreds of articles on exemplary science and math education are included in the Connect archives, online at http://www.synergylearning.org. This service requires no registration and is provided free to educators.

We’ve listed several here:

“I Never Answered a Single Question,” by Sandra Jenoure-Schofield, November/December 1995.

“Digitopolis: a Simulated Math Town,” by Christina Nicholson, November/December 1995.

“Motion, Time, Angles and Beauty,” by Margaret Dale Barrand, November/December 2000.

“When 5 and 6 Year Olds Invent their Own Systems,” by Annette Raphel, January/February 2002.

The ConnectArchives

http://www.mathlearningcenter.org

http://www.synergylearning.org

This issue of Connect challenges edu-cators at all levels to think hard about

how to preserve experiential, hands-on,inquiry-based investigations in the face ofNCLB assessment practices that are morein line with “drill-and-kill” instruction. Intoo many classrooms instructional andassessment practices that have provenover and over to be the most supportive tostudent learning have been abandoned.

I believe one of the most damagingeffects of these changes is that many stu-dents no longer have true ownership ofwhat they know because they haven’tprocessed or experienced the content inways that enable them to apply it in differ-ent contexts. Thus, when a test confrontsthem with a particular situation, they oftenresort to guessing or choosing a formula toplug into the problem, rather than having apersonal arsenal of strategies they’ve usedbefore in varied situations.

Fortunately, there are teachers, schools,programs, and districts that have suc-ceeded in maintaining best practices in theface of this pressure, most often throughstrong leadership, effective professionaldevelopment, and/or high levels of cre-ativity. Instructional practices that enablechildren to engage in realistic, personallymeaningful learning experiences con-tribute to students’ truly owning theirlearning.

From an intrinsic source

How does ownership evolve? I believe theroots of ownership stretch back to the ear-liest years of schooling, when young chil-dren so readily engage in learning.Perceptive early childhood teachers nur-ture and witness the beginnings of owner-ship every day as children immersethemselves in experiential, hands-on,inquiry-based investigations. These teach-

ers know that child-initiated experienceshave the most meaning and the greatestimpact because they come from inside thechild rather than from an external source.In well-planned, purposeful learning envi-ronments where teachers are skilledobservers, children begin to develop own-ership of what they know and can do byhaving multiple and varied opportunitiesto apply content knowledge and inquiryprocesses.

Let’s consider some examples of four-and five-year-olds developing ownershipof content and inquiry processes in someimportant areas of math and science.

NUMBER AND QUANTITY

• As the teacher goes around the circletapping each head, Hattie joins in as thewhole group counts the people at morn-ing meeting. At the end, she exclaims,“I didn’t know we had 17 people here!”

This teacher knows that counting class-mates is more meaningful to young chil-dren that counting arbitrary objects suchas beans or plastic counters. Hattie revealsa developing sense of quantity. She knowsthat 17 is not just a number that comesafter 16, but that it represents a collectionof important people in her life.

• Kenny makes a design with Pattern

How Do We Teach and Assess for Ownership?

by Charlotte Stetson


Pattern Blocks are powerfultools with which childrencan create highly organizeddesigns.

• Connect SYNERGY LEARNING • 800-769-6199 • JANUARY-FEBRUARY 2007

Blocks. He puts an orange square on 3sides of a yellow hexagon, looks atwhat he has done, then takes 3 moreorange squares all at once, and addsthem to the 3 remaining sides.

Kenny’s teacher knows the power ofPattern Blocks to help children createhighly organized designs that have sym-metry and include patterns. As he repre-sents his mathematical thinking throughcreating a design, Kenny demonstratesboth one-to-one correspondence (bymatching one orange square to each sideof the hexagon) and subatizing (recogniz-ing an amount without having to count).

• Barbara is drawing with markers. She isholding 4 markers in her left hand anddrawing with the fifth one with her righthand. She glances over at her neighborwho is drawing with 1 marker and has 3more markers right beside her piece ofpaper. She states softly, “I have moremarkers.”

Some teachers might limit their inter-pretation of this observation as a socialissue, but Barbara’s teacher hones in onthe considerable math skill demonstratedin this everyday occurrence. Barbara firstadds up the markers in her hands, thenadds up her friend’s markers, and thencompares resulting quantities.

GEOMETRIC SHAPES

• During snack, Mike folds his napkin indifferent ways. He folds it once to makea rectangle. He folds the rectangle inhalf to make a small square. Then he

unfolds first the square and then the rec-tangle and re-folds the napkin diago-nally, creating a triangle.

In this observation, Mike demonstratesmuch more geometry knowledge thansimply naming shapes. From a singlesquare napkin, he creates, takes apart, andre-creates multiple shapes.

• A child is adding triangular blocks tothe top edge of a block building. Sheuses all of the triangles around her andasks Lori, “Can you pass me somemore triangles?” Lori hands the childone of the tall, skinny triangles. Thechild rejects it, saying, “Not that kind,the other kind.” Lori puts the first trian-gle back on the shelf and hands thechild one of the small equilateral trian-gle blocks.

Lori’s teacher notes that she knowsabout different kinds of triangles andsolves a problem by thinking flexibly.

• Paul and three other children are help-ing the teacher label circular things inthe classroom with round sticker dots.He points to the clock on the wall, thewastebasket, the holes on the pencilsharpener, and a plate on a shelf.

This teacher takes advantage of chil-dren’s acute observation skills and theirstrong desire to sort and classify things intheir environment. Paul makes connec-tions among multiple circular shapes inthe environment, thus demonstrating afirm grasp on the concept of a circle.

LIVING THINGS

• Levalle spots some ants on the groundwhile playing outside. He gets down onhis knees so his face is very near theants. After watching for a few minutes,he crawls away on all fours saying,“Watch out, here comes a giant ant!”

Levalle is not just being silly. He hasobserved and internalized the anatomy andmotions of ants and then communicateswhat he has learned through dramatization.

• Olivia cradles a baby doll in her armsand holds a cylindrical block up to itsmouth as if giving it a bottle. After awhile, she puts the block aside, holds

Using blocks and workingtogether, students canpractice identifying shapesand creatively solvingproblems.

the baby over her shoulder, and pats itsback. After several pats, she makes aburping sound, and returns to feedingthe baby.

As adults, we often take children’s pre-tend play for granted, but in this scene,Olivia demonstrates some importantunderstandings about infants—they eatdifferently than older people, they aredependent on other people to feed them,and it is important to help babies burp!

• Ian draws a large tree-like form on apiece of paper. He adds squiggly linescoming out of the bottom. When askedabout the lines, he responds simply,“little roots.”

Not only does Ian make a representationof a living plant, he includes parts of thetree that are usually underground andtherefore, not visible. Hopefully, Ian’steacher will try to find out how he gainedthis knowledge.

EARTH AND SPACE

• Winona is picking up rocks on the play-ground. She examines each one, putssome in her pocket, and returns othersto the ground. When asked how shedecides which ones to keep, she says, “Ionly want the ones with sparkles.”

Natural objects are much harder to sortand classify than commercial sorting mate-rials such as plastic teddy bears. The dif-ferent attributes of rocks are less obvious,so it is quite impressive that Winonaobserves, sorts, and classifies them.

• After watching the teacher blow bub-bles from a plastic wand for severalminutes, Roberto asks to try. The firstfew times, he blows hard and fast. Nobubbles come out of the wand; hefrowns. He watches the teacher somemore and tries again. He blows softlyand slowly. Many bubbles float into theair; he grins from ear to ear!

Roberto observes and then experimentswith different ways to blow air to producebubbles. He uses a specialized piece ofequipment (the bubble wand) and adjusts hisown breath to achieve the desired results.

• On field trip day, Tanika enters the

classroom looking very sad. Whenasked about the frown on her face, sheresponds, “I saw the clouds. It’s goingto rain. We can’t go to the farm today.”

In the busy life of a teacher, it might beeasy to miss the scientific connectionsTanika weaves together. She observes theweather, makes a prediction based on herobservations, and then draws a conclusion.

Reading the signals

In all of these examples, young childrenare demonstrating a budding ownership ofmathematical and scientific knowledge andinquiry skills in ways that could not beassessed on standardized tests. Perceptiveteachers see these demonstrations andtranslate them into future instructionalsteps that will deepen and broaden chil-dren’s knowledge and understandings.

Effective teachers of young childrenexpose their students to knowledge, skills,and behaviors. They provide many engag-ing opportunities for children to indepen-dently apply what they are learning, andthen watch and listen very carefully. Thisis how teachers know which children needmore attention to certain concepts andwhich children are ready to extend theirlearning further. Children applying knowl-edge, skills, and concepts independentlysignal to teachers that they are taking own-ership of their learning. �

Sorting rocks on theplayground is a greatexercise in identifying andcommunicating differencesand similarities.

Charlotte Stetson is theprincipal author of TheCreative Curriculum StudyStarters (Teaching Strate-gies, Inc.), co-author ofparenting guides entitledWinning Ways to Learn(Pearson Early Learning),and contributing author toVermont Center for theBook’s Mother Goose pub-lications. She is a seniormember of The CreativeCurriculum Staff Develop-ment Network and TheWork Sampling SystemNational Faculty.http://www.charlottestetson.com.


http://www.charlottestetson.com


Using a simple activity that children loveto do, draw stars, Julie King presentsquestions and gentle guidance that takethe investigation to a sophisticated level. This article was first published in Connectin March of 1990.

In the last few years the study of patternshas become a distinct topic in the math-

ematics curriculum, from counting to alge-bra. Stringing beads, clapping outrhythms, making designs with PatternBlocks, and studying symmetry havegained the status of legitimate mathemati-cal activities. Many teachers wonder why.What have these things to do with math?Will the ability to recognize or constructgeometric patterns be helpful to children?

“Yes,” the textbooks say. But unlessyou have experienced the excitement ofexploring patterns yourself and discover-ing some of the remarkable interconnec-tions and relationships which exist, it maybe hard to see how. To get a sense ofwhere patterns fit in, we will take a singlekind of pattern and examine it in manyways: how it can be described and pro-duced, where it is found, what happens ifyou extend it, what other kinds of patternsit resembles, and so on.

Let’s consider a star

Children learn very early to draw five-pointed stars. Make five strokes; don’t liftyour pencil; come back to the beginning.“Look, I drew a star!” The stars have fivepoints. Some are “perfect.” The perfectstars have identical points in a circle,equally spaced. If you turn one part wayaround you see the same star again. This iscalled symmetry (five-fold rotational orradial symmetry).

Young children can be happy for a long

time just drawing stars. A bulletin-board-sized piece of paper is a wonderful invita-tion. After the impulse to make those fivemotions over and over has been satisfied,children may begin to experiment withother methods. Can I draw a star back-ward? If I start at the top of the rightinstead of the left, will it look the same?What if I started inside, with the penta-gon? Is there another way?

To the disappointment of some chil-dren, real stars don’t look like the starsthey draw, but there are plenty of thingswhich do: starfish, star grass flowers (andmany other kinds), starfruit slices. . . . Ifyou’re investigating stars with a class,start a collection of objects with star pat-terns. Not many of the natural items inyour collection will be perfect stars. Thelines will be curved or the points not quitethe same size. But they are all star-like.They all have something in common. It’sthe five-fold symmetry we mentioned.Turn each a fifth of the way around, and itwill look the same as when you started. Ifyou look for other things which have thisstar-like quality, you will find apples(sliced across), sand dollars, and straw-berry blossoms.

Children may notice that some radiallysymmetric things have more than five sec-tions, or points. There are stars with sixpoints: snowflakes, the Star of David, astar made with one yellow Pattern Blockand five green ones, the star on a Chinesecheckers board. You can collect six-pointed stars, too, and cut out somesnowflakes.

Drawing six-pointed stars

Can we draw a six-pointed star in thesame way we can draw a five-pointed star,without lifting pencil from paper? Notice

The Power to MakePredictions

by Julie King

that if you mark the points first, the patternfor drawing the five-pointed star was toskip a point each time, going to the secondpoint.

Skipping to the second point doesn’twork for the six-pointed star. Could wedraw it by skipping to the third or fourthpoint each time? Try it.

The answer appears to be no. We candraw a six-pointed star easily in two steps,lifting the pencil once, but not in one step.

New questions pop up. Could we havedrawn our five-pointed star by skipping tothe third point instead of to the second?Try it.

Is the five-pointed star the only onewhich can be drawn without lifting thepencil? What about one with seven points?

Is there more than one way to do it? (Infact, there are two ways to draw a five-pointed star, and four ways to draw aseven-pointed start with one continuousline.) Do both five-pointed stars look thesame? (Are they congruent?) How aboutseven-pointed stars?

Other forms of star patterns

A more physical way to approach star-making is to stand with people in a circleand throw a beanbag around instead ofdrawing lines. Skip over to the second per-son each time. If everyone gets a turn andthe beanbag comes back to the starter, youhave traced a star in the air. Try skippingto the third person, then the fourth, etc. Tosee the stars as they are created, substitutea ball of yarn for the beanbag and let itunwind as it is thrown, each thrower hold-ing on to the yarn.

Another exciting means of exploringthe same idea is to hammer a circle ofround-headed brass nails into a piece ofplywood, tie a colored thread or piece ofyarn to one, and wrap the thread aroundthe nails, skipping to the second nail eachtime. Use another color to test skipping tothe third nail, and another for skipping tothe fourth, and soon you will have boththe answer and a lovely string design.

Organizing discoveries

The variations can get a bit confusingunless we stand back, take a breather, andtry to organize our discoveries somehow.Perhaps there is a pattern here. We hopeso, or all this star-drawing will have noend. So far we have found that stars withfive and seven points can be drawn with-out lifting the pencil, but stars with sixpoints cannot, no matter how many pointsare skipped.

1

1 1

0 0

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www.synergylearning.org • SEPTEMBER–OCTOBER 2005 Connect • PAGE 9


Could it be that odd-numbered stars canalways be drawn this way and even-num-bered stars cannot? To look at our workmore systematically we will number thepoints, 0 being the starting point. To testthis conjecture we will also try drawingeight-pointed stars.

Then we can make a table of the suc-cessful and unsuccessful tries.

SUCCESSFUL

Points in star: 5 5 7 7 7 8 8

Number of first point

Drawn to: 2 3 2 3 4 3 5

UNSUCCESSFUL

Points in star: 6 6 8 8 8

Number of first point

Drawn to: 2 3 2 4 6

Now indeed, some patterns begin toemerge. Our odd and even guess will haveto go, since some eight-pointed stars canbe drawn. But look! In the “unsuccessful”list the first four numbers on the bottomare divisors of the numbers above them.Unfortunately, the last entry does not fol-low this pattern—another good ideademolished. However, 8 and 6 are bothdivisible by 2. No pair of numbers in the“successful” list has a common divisor,but all in the “unsuccessful” list do. Per-haps the secret is that we will succeedonly if the number of points in the star andthe number of the first point you go to donot have a common factor.

We haven’t seen all there is to see byany means, but we’ve come a long way.We began by simply drawing stars, thenlooked for stars and star-like things aroundus. We learned about rotational symmetry

and then returned to search for patterns indifferent kinds of stars. We ended up talk-ing about the numbers and divisibility.We’re on the verge of being able to makesome predictions. I can say now, withouthaving tried it, that I will be able to draw atwelve-pointed star without lifting mypencil if I go to the fifth point each time.Do you agree?

This is what patterns can do—help us tosee order in shapes and numbers and theworld around us. They suggest new waysto explore it. They give us the power tomake predictions. They enable us to joinour knowledge of number and shape.

Don’t be afraid to start such an explo-ration in your classroom, even though youdon’t know exactly where it will lead.Some of the most memorable learningexperiences are those in which teacher andstudent explore together.

Julie King taught for many years at Antioch-NewEngland Graduate School in Keene, NewHampshire. She has authored several books onteaching mathematics, including, ExploringEveryday Math (1993), co-authored with MajaApelman.

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Patterns . . . giveus the power to

make predictions.

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11

As science has become more dependenton magnification and sophisticated

chemical and physical analysis of the naturalworld, more of our conclusions and explana-tions of how things work must be acceptedbased on the authority of others. Much sci-entific knowledge has become abstract. Adanger in teaching science as a body ofknowledge is that such acceptance of whatothers have figured out encourages goodminds of all ages to ignore their own obser-vations and intuition.

We all make observations that don’t jibewith currently accepted knowledge. Forinstance, both the sun and the moon appearto rise in the east and set in the west. So theyboth appear to orbit the Earth, but among thefirst things we teach about the solar systemis to make the point that only the moonorbits the Earth. Children, who are oftenvery literal, may struggle with counterintu-itive ideas or simply ignore them. In one ofmy literal periods at age four or five Iremember being disappointed when I real-ized that it could rain on Sun-day.

Many ancient explanations are based onnon-technical, but careful, observations.These phenomena are still observable andindeed are part of the first knowledge gath-ered by young people. A number of theseobservations can be used as the basis forlearning how science works.

Imposed limits on learning

Children are inherently curious. What theylearn about their world comes from theirown perceptions and it is combined withinformation added by others. The others aremostly adults who, from the child’s perspec-tive, also provide a variety of rewards andpunishments. For the child, it becomesimportant to agree with the authoritativeadults, parents and caregivers, and later,teachers in school. Students often explore to

find patterns, but stop searching when theirnew explanation seems to please an adult.Many times the adult explanation does notsatisfy the young person’s curiosity nor pro-vide a satisfactory end to the search. Thiscan be an early source of misconceptions.

In early schooling, students are taughtthe prevailing language and its communica-tion patterns. The symbols—letters andnumerals—are precise. Letters are combinedto make words with correct spellings andthey are strung together in appropriategroupings to convey limited meanings.Numerals are manipulated to gain specificresults. And by this regimentation primarygrade students are persuaded to accept adultdirections and explanations and to ignoreother possibilities that may occur to them.This results in acceptance, without question,of adult products such as print and othermedia.

Schooling in science

If science is presented in primary grades asan exploratory activity, students should beallowed to arrive at their own conclusions.However, what often happens is that stu-dents mess around with the provided materi-als long enough to figure out what the

Using Historical Explanationsto Teach How Science Works

by Jean Beard


Much of earlyschooling relies onstudents acceptingadult directions andexplanations, as is thecase in learning theconventions of print.

teacher wants them to “learn” from theactivity. All too often the problem posedcomes from the teacher, and would not haveoccurred to the students. And so curiosityand science are separated for most studentsby the time they reach intermediate gradesand middle school.

The incoming middle school student usu-ally accepts what is in the science text,because it is in the book, not because itmakes sense. Too often the texts and lessonsfrom earlier grades provide isolated facts tobe memorized. These specific facts need tobe associated with the concepts they supportto foster understanding of how the naturalworld works. In order to teach how scienceis done and to create more scientists andengineers, students’ schooling needs to fos-ter observation, curiosity, and to require suf-ficient evidence from authorities forconceptual learning.

It became apparent to me years ago whileteaching junior high school science that usingpreviously accepted explanations of naturalphenomena can provide a useful review func-tion. The ancient explanations were based onobservations available to all because theyrequired neither visual magnification, normolecular analysis.

No air

Early in my teaching career, as a briefreview before the unit test on air, I proposedto seventh graders that if I had come froman ancient time I would not understand theirconversation about air. The text chapter wasabout air and its components. In classdemonstrations we had chemically gener-ated oxygen, carbon dioxide and hydrogenas described in our textbook. (Note: Thesedemonstrations are no longer consideredsafe in a classroom without a ventilatinghood.) So the unit test was to be on thechemical components and physical proper-ties of air that they had studied.

To be consistent in my conversation withthe students, I adopted the old four-elementexplanation, but I mentioned only earth, fireand water. My premise to the students wasthat I didn’t understand how they thoughtthere was something that I couldn’t see,smell, taste or feel in the empty spaces

between us. My rebuttal to each of theirexplanations was based on this premise orwas composed of a lie so bold that theyshould have had the information to refute it.

To my surprise, and joy, the studentsbecame seriously engaged in trying to provethat I was wrong. They begged for time thenext day to demonstrate to me that there isair. And the following day they brought inmaterials to demonstrate the force of airmovement by various means, but with ourequipment had difficulty making that casethat air had mass. Later I learned fromteachers and parents that they had askedmany others for help in making their case.

The nature of science

The Flat Earth is another example of an oldconcept that I have used. Intuitively, ourown observations support a flat rather than aspherical earth. The earliest recorded expla-nations for the earth included water on alledges and pillars to hold up the sky. Suchdescriptions fit our own early, less preciseobservations. Travel to other land massesand better observations of stellar move-ments make such explanations less accept-able. There are two sources available todemonstrate how this might work.

The first source is designed to help stu-dents decide what makes informationacceptable in science. It was assembled toassist high school biology teachers developstudent skills for sorting science from alter-native explanations that purport to be sci-ence, such as creationism. ENSIweb is anonline resource for teachers that containslesson plans and resources for teachingabout the nature of science and about evolu-tion. (ENSI was a series of NSF supportedinstitutes called the Evolution and Nature ofScience Institutes.) The Web site was devel-oped and is tended by an ENSI alum afterthe institutes (1989–1997) were completed.For the first time we have a common formatfor lessons from many teachers working indifferent institute groups. We found thatmany of the nature of science lessons weresuitable for younger students. These lessonplans can be found athttp://www.indiana.edu/~ensiweb/.


Skepticalcuriosity andinsistence onevidence are

essential to theprocess ofscience.

http://www.indiana.edu/~ensiweb/

Web lesson

The Flat Earth lesson on ENSIweb asks stu-dents, based on their own information, todemonstrate that the Earth is not flat. In theprocess they sort their own observations fromsecondary sources. Secondary sources forthem include pictures from space that areusually printed on flat surfaces. They alsorecheck their own observations to determinewhich do and don’t make the case for a non-flat Earth. For instance, the Earth’s shadowon the moon could be from a flat disk that isround but not spherical. As it turns out moststudents have not observed a ship appearingor disappearing on the horizon, and so per-sonally have no valid primary observation ofa non-flat Earth.

The final part of this lesson encouragesstudents to develop a short list of terms thatcharacterize science. These can vary depend-ing on the class, but must include such thingsas repeatable and/or testable, natural and ten-tative. Use of such lists can help students andthe class judge the reasonableness of newsstories and other public information.

Another concept from this lesson is thatscience explanations must change whennewer and more complete evidence requiresa newer and better explanation of naturalphenomena. So while the Earth was oncethought to be flat, we now know that it isspherical. The early “Flat Earthers” wereusing all of the available information. Thecurrent “Flat Earthers” are ignoring validavailable observations.

Video of Flat EarthIn Search of the Edge was produced as a“fake science” video in defense of the currentFlat Earth explanation: that it is a disk-shaped earth with the North Pole in the cen-ter, and an ice wall on the outer edge. The icewall is compatible with observations ofAntarctica. The video format is similar to theone used in science programs such as Nova.“Experts” are interviewed to explain the prin-ciples of the Flat Earth. The human interestelement of the program is the story of AndreaBarnes who is intent on traveling to the icewall to prove that the Earth is flat, contrary towhat she was forced to learn in school. The

Flat Earth “experts” and their explanationsare humorous to those who understand thetongue-in-cheek presentations.

While the video is a wonderful spoof, acareful debrief is essential for an audience ofpeople of any age who take what they see ontelevision as fact. The unintended conse-quences of an incomplete debrief may be sell-ing the Flat Earth concept, a lesson I learnedwhile teaching a collegegeneral education sci-ence class a fewyears ago.

A full debrief should include discussion ofthe evidence provided in the story by the“experts” and the inferred credentials of the“experts.” Discussion should get to a listingof qualifications for acceptable authoritiessuch as profession or position appropriate to“expertise.” Further discussion could provideauthenticity-screening criteria for Internet,TV and other media presentations.

In the “No Child Left Behind” era of moretesting, we often only demonstrate “learning”of factoids rather than understanding of con-cepts. In other words, it is now more accept-able to teach about what scientists havefigured out than to teach how science is doneand why it is fun.

Skeptical curiosity and insistence on evi-dence are essential to the process of science.Young people who are more dependent onmedia need to develop skills for being skepti-cal of all information sources. Giving stu-dents practice in assembling evidence forconcepts they have accepted is a good reviewtechnique. Using examples from explana-tions of less sophisticated times has muchpotential. I have offered two examples here.More old concepts and explanations could bedeveloped for science reviews. I’d be glad tohear about others that work. �

Jean Beard is ProfessorEmerita in the BiologyDepartment at San JoseState University inCalifornia. At SJSU shewas director of theScience EducationProgram, and taughtgeneral educationscience to collegestudents. She hasconcentrated onimproving evolutioneducation, and educationabout the nature ofscience.

Recent images of theFlat Earth are of adisk with the North

Pole in the center, theIce Wall on the outeredge and the Equator

circle half waybetween. Some claim

the United Nationslogo is such an

image.


Planning

for Under-

standing

by

B O B C O U LT E R


It’s no secret that recent accountabilitymeasures have presented challenges forteachers seeking to maintain rich and deeplearning environments for their students.Those who do manage to achieve this areoften put on the defensive, expected tojustify why they aren’t “following theplan.” On the theory that the best defenseis a good offense, planning a strong unitand being able to affirm why your curricu-lum is a better choice for students canhead off the nay-sayers. To that end, on-line planning tools such as the Collabora-tive Curriculum Design Tool can be avaluable part of your technology tool belt.This online resource, offered by the Har-vard Graduate School of Education,allows you (and colleagues, if you wish)to develop and share plans for a stronglearning environment that is based inwhatever academic standards you areaccountable for.

After completing a free registration (athttp://learnweb.harvard.edu/ccdt), you canbegin creating a curriculum unit usingeither of two major curriculum frame-works: Teaching for Understanding orDimensions of Understanding. The exam-ples here use the former model since I’veused that a number of times in planningunits. Within the Teaching for Understand-ing model, you will be guided to articulateyour plan using an online template. If youget stuck or need inspiration, help andexamples are only a click away.

Throughlines provide the big, overarch-ing topics that guide your curriculum plan-ning such as, “What causes an ecosystemto change?” While they are usually tooglobal to provide direct planning guid-ance, they can function as a metaphoricalnorth star guiding the more detailed plan-ning you will be doing. The substance ofyour planning follows in the later ele-ments. Given the time constraints that all

teachers face, using your throughlines canhelp in determining what activities aremost relevant to keep and what you can letgo of without taking away from yourlarger purposes.

Generative Topics are the areas ofinvestigation that hook your students’attention, inviting them to inquire, ana-lyze, and reflect on. They are intrinsicallyengaging, and likely to sustain interestwell beyond the time allocated in yourcurriculum. The trick here is to make thetopic invitational. The same topic can beframed quite differently, drawing a differ-ent response from your students. Forexample, water chemistry will have only alimited appeal; investigating how waterquality impacts organisms in your localpond or creek is more likely to engage andsustain interest.

Understanding Goals allow you toarticulate what students should get out oftheir involvement in the unit. This is yourchance to define publicly the learning ben-efit students will gain as a result of theirparticipation. The key here is multi-faceted understanding, not simply recita-tion. Students who understand ecosystems,for example, will be able to see how indi-vidual organisms fit into the ecosystem,understand roles such as producer, con-sumer, and decomposer, and evaluate theimpact of various changes in an ecosystemon its inhabitants.

Note that underneath this understandingis a range of more specific bits of knowl-edge (such as the difference betweenbeing a producer, a consumer, or a decom-poser). The difference is in the movementaway from a bag full of isolated factstoward a larger, more intellectually robustconceptual network. For example, Iworked with my fourth graders in the localcommunity to develop a robust under-standing of biomes, which they were then

Technology for Learning

Bob Coulter is director ofMapping the Environment,a program at the MissouriBotanical Garden’sLitzsinger Road EcologyCenter that supportsteachers’ efforts toenhance their science cur-riculum through use of theInternet and geographicinformations system (GIS)software. Previously, Bobtaught elementary gradesfor 12 years. [email protected]

http://learnweb.harvard.edu/ccdt


able to transfer from the first-hand workwe did in the temperate deciduous forestto other more distant biomes. While thespecifics changed in terms of climate andspecies present, their conceptual under-standing of biomes stood up and proved tobe useful in new contexts.

Performances of Understanding pro-vide opportunities for students to showwhat they know. More than just getting agood grade on a test, the Teaching forUnderstanding framework calls for a“flexible performance capability” thatallows students to show what they havelearned in an authentic context. Forexample, in a unit engaging students instudying optics, students could apply whatthey have learned by combining lenses tomake a telescope.

More generally, the Teaching forUnderstanding framework suggests thatstudents be able to display their knowl-edge, engage in focused inquiry, be awareof different domains of understanding, andbe able to communicate their understand-ing to specific audiences. Collec-tively, this suite of skills goes wellpast the usual short answer, artificialresponse solicited by most standard-ized assessment tools. Obviously,not every learning opportunity inyour classroom will build toward amajor, culminating performance, butthese opportunities should be pur-sued where possible, since they aremotivating and provide an opportu-nity for students to integrate whatthey are learning. Practically speak-ing, being able to document and dis-play students’ learning in this largerrealm can inspire confidence in thequality of learning experiences.

Ongoing Assessment provides away to document how students arebuilding toward their performancesof understanding. All too often,teachers new to more complex,project-oriented learning leaveassessment to the end, and thenwhen projects don’t meet their ini-tial hopes, they write off the idea ofcomplex learning either as some-thing that their students couldn’t do,

or even as a bad idea all together. Big con-structions are made of small pieces, andyour ability to assess how students aredoing en route to their performances willgo a long way in ensuring the success ofthe larger project. Armed with this knowl-edge, you can provide constructive feed-back while there is still time for studentsto improve their performance, and whereneeded, alter the pacing or instruction youare providing.

In addition to these sections in the plan-ning template, you can also articulate thetechnology resources you will use, andmost important for the context of thisissue’s theme, articulate the curriculumstandards you are addressing. By craftinga well-articulated curriculum design that isgrounded in relevant standards, your planscan stand more strongly and less defen-sively if you are challenged. More impor-tantly, your plans will ensure that yourstudents will have a meaningful learningexperience.

Here are suggestions of a few books toencourage creativity, inquiry, observation,and attention to the process of learning andfiguring things out. Some of them also showchildren engaged in everyday activities thatuse math, science or technology.

What Do You Do with aTail Like This?, by SteveJenkins and Robin Page(Houghton Mifflin Com-pany, 2003), captures theimagination by revealingonly parts of animals’ bod-ies (tails, ears, noses, feet,etc.), and asking readerswhat they might do withthem. A format like thisdoes at least two things

simultaneously: first, it asks us to regard iso-lated parts which increases our focus andwidens the capacity for noting diversity.Secondly, it stimulates creative thinking andoffers readers a chance to internalize whatthey are seeing and learning about. What if Ihad a nose like a starry-nosed mole? Or anelephant? What if I could walk on the ceil-ing like a gecko? The exquisite illustrationsare made of cut paper. Factual informationon each animal follows the story.

The Squiggle, by Car-ole Lexa Schaefer(Crown Publishers,1996), is a greatexample of creativethinking. A little girlwalks at the end of theline of children. “I amlast. No one else seeswhat I see on the side-walk.” She finds asquiggly piece of stringand imagines it into allmanner of things, likethe curve of a dragon’sbody, the sky trail of

fireworks, the edge of a storm cloud. PierrMorgan’s calligraphic drawings, like theimagery of the text, reflect examples of Chi-nese culture. Lively and playful, this bookinspires children ages five through eight tolet their imaginations fly.

Alexander, Who Used to be Rich Last Sun-day, by Judith Viorst (Atheneum, 1985),chronicles exactly what happened to the dol-lar Alexander was given by his grandparentsa week ago. Part of the success of this bookis the convincing child’s voice that tells thestory, complete with details about the twoolder brothers who each receive their owndollars (and still have most of the money).This is a great accompaniment to studyingcurrency for second through fourth grade,with references to coins, saving, spending,betting, and borrowing money. One couldread it as a class while students practice sub-tracting cents from 100, or use a calculatorto keep a running tally as Alexander spins(and spends) his tale of woe.

The Button Box, by Margarette S. Reid(Puffin Unicorn, 1990), shows a young boyplaying with the buttons in his grand-mother’s special box. Sorting, classifying,building patterns, and noticing similaritiesand differences are some of the conceptsexplored. Colored linoleum cuts by SarahChamberlain appear slightly dated, but stillclearly and simply show a deeply interestedchild investigating a variety of styles andforms of buttons. This is a good book forbeginning readers or to read aloud as a start-ing point for sorting activities with youngstudents.

Ten Mile Day, by Mary Ann Fraser (HenryHolt and Company, 1993), is an example ofa very natural integration of mathematicsand history. This is the story of April 28,1869, the Ten Mile Day, when the leaders ofthe Union Pacific and the Central PacificRailroad crews wagered that they could each

Literature Links


be the first to lay ten miles of track in oneday. With carefully researched illustrationsand text that does not belittle the hardshipsand inequities of the effort, the author out-lines events leading up to and comprisingthat day. Fighting between rival groups ofChinese workers, attacks from Natives whounderstood the consequences of the railwaycutting across their homelands, fatigue,dehydration, and illnesses are mentioned asterrible conditions. But another focus is themagnitude of this effort. Many details areoffered as to the size and number of materi-als, the timing of the event, and the distancecovered. Also emphasized is the Herculeanefforts of the workers, with names of someand references to sources that students canconsult for more information. Fourththrough sixth graders could use this as partof an integrated study. Teachers should notmiss the opportunity to discuss from whoseperspective the history is written andexplore what it might look like fromanother’s perspective.

What’s Up, What’s Down, by Lola M.Schaefer (Greenwillow Books, 2002), is auniquely formatted book that tells readers toread from, “the BOTTOM of the page to theTOP. Then, halfway through, turn the bookaround and let your eyes travel down, read-ing from TOP to BOTTOM.” “What’s up ifyou’re a mole?” “What’s down if you’re awhale?” This richly illustrated book leads usup to the moon and back down, encouragingimaginative readers to see the world through

many different eyes. First through thirdgraders will enjoy this different way of reading.

If You Find a Rock, by Peggy Christian(Harcourt, 2000), poetically describes manyvarieties of rocks: hiding rocks, skippingrocks, sifting rocks, crossing rocks, walkingrocks. Photographs on each page spreadshow diverse children examining, finding,observing, and holding rocks and carefullynoting characteristics of them. The basiclanguage invites readers to take time tonotice, to gatherinformation usingmany senses. Onemight use this bookto help set the appro-priate mood beforegoing on a rock huntaround schoolgrounds or in anadjacent park. Thisbook can best beused with kinder-garten through fourthgraders.

Chapter Books that are fictional can beinteresting additions to studies of measure-ment, habitats, problem solving, and manyother concepts. Charlotte’s Web could beused to explore animals’ roles in the ecosys-tem, for example. Stuart Little or The Bor-rowers series could be used as examples ofscale. The Phantom Tollbooth, J.R.R.Tolkein’s The Ring Trilogy, and My Side ofthe Mountain are examples of literature thatrely on maps and place. Arthur Ransom’sSwallows and Amazons series draws onnavigation, mapping, codes, and problem-solving as it relates the adventures of foursiblings adventuring on their own.

Remember the value of a conversationwith your school or town librarian, who canprovide suggestions of good literature thatencompass whatever theme you are work-ing on.


The Math Coach Field Guide: ChartingYour Course, edited by Carolyn Felux andPaula Snowdy, is a valuable collection ofnarratives advising math support teachers,specialists, resource teachers and coaches.What are the most effective ways of helpingelementary teachers in mathematics? Manyyears of experience back up the suggestionsof these eleven authors. This book takes theplace of sitting down and chatting withexperts and is a useful guide for novices andlongtime practitioners alike. 124 pages.

Math Matters: Grades K–8, Understand-ing the Math You Teach, by Suzanne H.Chapin and Art Johnson, is a comprehensiveguide for all levels of elementary and mid-dle school. Each of fourteen chapters in thissecond edition tackles a math topic (numbersense, algebra, measurement, etc.) and intro-duces basic concepts, defines terms, andoffers explanations that are clear and sup-port confident teaching of that topic. Writtenfor the busy educator, this is an excellentand easy to understand resource that can beused for quick reference questions as well asto fortify overarching ideas and understand-ings of the “big picture.” Web resources areincluded. 348 pages.

The Math Coach Field Guide: ChartingYour Course ($16.50) and Math Matters:Grades K-8, Understanding the Math YouTeach ($32.95) are available from MarilynBurns Education Associates, Math SolutionsPublications, 150 Gate 5 Road, Suite 101,Sausalito, CA 94965. Call 800-868-9092,fax 877-942-8837, online athttp://www.mathsolutions.com.

You’re Smarter Than You Think: A Kid’sGuide to Multiple Intelligences, by ThomasArmstrong, is written directly to children,explaining Gardner’s multiple intelligences.This book outlines eight different kinds ofintelligences or, “smarts,” including wordsmart, music, logic, picture, body, people,self and nature smart. Each chapter exploreswhat it means to be smart in one way, what

to do if you’re not strong in that area, howto support it if it is a strength, and sugges-tions of activities that would be enjoyable orbeneficial in that area. When these ways ofteaching and learning are explored as aclass, a respect and recognition of variousways of processing are emphasized. Chil-dren regard themselves, and others, withnew esteem for talents and skills that mightotherwise go unnoticed. 192 pages.

You’re Smarter Than You Think: A Kid’s Guide to Multiple Intelligences($15.95) is available from Free Spirit Pub-lishing, 217 Fifth Avenue North, Suite 200,Minneapolis, MN 55401-1299. Call 800-735-7323, fax 612-337-5050, online athttp://www.freespirit.com.

Teaching the Best Practice Way: Methodsthat Matter, K–12, by Harvey Daniels andMarilyn Bizar, is a collection of examples ofeducators who use authentic practices. Inseven chapters the authors outline what theythink are the building blocks of goodinstruction. These are described as: teachingas thinking, representing to learn, smallgroup activities, classroom workshop,authentic experiences, reflective assessment,and integrative units. Different teachers areshowcased and each writes a few pages

Resource Reviews


http://www.mathsolutions.com

http://www.freespirit.com

about valuable teaching. The introductionaddresses how these methods stand up tostandardized tests. Written anecdotally, thisbook also provides examples of children’swork, sample activities, and suggestions forhow to start. 343 pages.

Black Ants and Buddhists, by MaryCowhey (see her article on page 23 of thisissue), is a moving and funny book aboutone teacher’s adventures in teaching all sub-jects critically, with rich examples in mathand science. Cowhey encourages her first-and second-grade students to question, toreason, to express and communicate.Through investigating questions of theirown, community involvement, and socialaction, students learn both the basic acade-mics and the greater lessons of being a

human in the world. Simultaneously inspir-ing and challenging, this book is nothingshort of a call to action. 244 pages.Teaching the Best Practice Way: Methodsthat Matter, K–12 ($24.00) and Black Antsand Buddhists ($18.00) are available fromStenhouse Publishers, PO Box 11020, Port-land, ME 04104-7020. Call 800-988-9812,fax 800-833-9164, online athttp://www.stenhouse.com.

http://www.nctm.org/focalpoints/ isNCTM’s online publication describing focalpoints, the most important mathematicaltopics for each grade level. It comprisesrelated ideas, concepts, skills, and proce-dures that form the foundation for under-standing and lasting learning. From thispage, the book is downloadable by chapters.NCTM relates the focal points to their previ-ously published Standards.

http://www.arborsci.com/Resource.htm isthe resource page of Arbor Scientific. It haslots of links to good sites such as theExploratorium, NSTA, etc. Arbor Scientificitself also has a small but useful collectionof science teacher publications and productsin their catalog.

Assessment for Learning, by Paul Black,Dylan Wiliam and additional authors,explains their study of formative assessmentas a tool to support positive progress inlearning. The authors critique the effective-ness of on-going formative assessment andpropose ways to improve upon its use inclassrooms. Their research documents thevalue of this type of assessment as a contin-uous feedback loop for teacher and student.In this book, they argue for better imple-mentation of formative assessment, espe-cially to aid in ending the negative impactsof conventional, high stakes assessment onlow-achieving students. The book intro-duces research and also provides actionsteps for schools.135 pages.

Assessment for Learning ($35.95) isavailable from McGraw-Hill Education,online at http://books.mcgraw-hill.com. Tosearch, enter the ISBN: 00335212972.


http://www.stenhouse.com

http://www.nctm.org/focalpoints/

http://www.arborsci.com/Resource.htm

http://books.mcgraw-hill.com

While traveling to Mexico recently, Iencountered many math challenges

as I navigated an airport, checked flightschedules and converted pesos to dollars tofind the price of a sombrero for my class-room birthday hat. I never needed to know,“Which array showed that the correctresult of multiplying a number by 1 is thesame as the number?” Education is assess-ment driven. A determination must bemade as to whether the cart is being putbefore the horse if we allow instructionand curricula to be guided and developedby professional test writers. Do we do adisservice to children when academic pro-nouncements drive the development ofclassroom content without considerationand significant involvement of experiencededucators who are actually responsible forthe communication of this knowledge?

Isolated skills?

The educational pendulum, which hasbecome powered by standardized tests,seems to be swinging away from the the-ory of teaching students the skills they willneed in the real world to teaching studentsthe information they will need to pass stateand national tests. Educators know that

teaching isolated skills is futile unless theyare attached to something which has realmeaning for the student. Yet standardizedtests measure only a command of isolatedskills. Teachers around the country areasking themselves, “How can we continueto make learning exciting, meaningful andfun, despite the pressure of annual highstakes tests?”

Recently my colleagues and I gatheredaround a computer, analyzing the results ofour school’s latest standardized test. Com-ments such as, “He probably guessed toget that one right because I know hedoesn’t understand perimeter very well,”or, “I’m surprised she missed that one.Problem solving is one of her strengths, soshe must have misunderstood the ques-tion,” filled the room. One of my col-leagues turned to me and said, “These arealmost identical to the results we gatheredthe first couple of weeks of school.” Whydoes public education, which is already sodesperately under-funded, spend so muchmoney on expensive tests that we know donot always accurately measure what achild really knows and rarely tell us morethan a good teacher discovers within thefirst few weeks of school? Knowledgeableteachers then must spend time to evaluatewhich students have accurate test resultsand which do not.

Formative assessment

Skilled instructors quickly learn each stu-dent’s strengths and weaknesses throughinformal evaluation that occurs daily. For-mative assessments such as oral question-ing, observing, analyzing written work,listening to student discussions and evenconsidering a student’s self-evaluation, areused to determine which students need addi-tional instruction for specific skills. Ques-tions asked in class, to which students either

To Teach or to Test, That is the Question!

by Michelle Kirk

Test questions such as thisyield little information abouthigher order thinking.

2. Which of these shapes are cylinders?

A) 1 and 2B) 1 and 3C) 2 and 4D) 3 and 4


respond orally or with a visual “thumbsup/down” or similar technique, allow theteacher to make immediate changes inteaching for some or all students.

The teacher’s observation of a child asshe explains the solution to a subtractionproblem using Base Ten Blocks gives theteacher information about the level andcomplexity of the child’s reasoning. Thisobservation is immediately usable andinfinitely more valuable than a singlenumerical score from a standardized test.It allows the teacher to understand theindividual child’s grasp of the concept andto tailor future instruction to the needs ofeach student.

Once a teacher feels confident that herstudents have the basic skills to solvemore complex problems, she can give stu-dents projects that focus on many differentskills at the same time and incorporateinterdisciplinary instruction. For example,after reading a book in which a party wasa central event, the teacher may ask stu-dents to work in small groups to figure outinformation for a real party that the classneeds to plan:

• If each student can invite two guests,how many people will be at theparty? How many guests should weexpect?

• What will we serve and what will bethe size of each serving (in ounces,cups, pounds)?

• What type of container will best holdthe food and how should we constructthe container?

• When purchasing napkins and silver-ware, how many packages will weneed of each?

• If each table seats eight, how manytables will we need?

These questions become important tochildren when they know they are plan-ning an actual party to which real guestswill be invited.

Listening to students

Student discussions provide another effec-tive informal assessment the teacher canuse as she or he circulates and listens to

the thought processes of students whilethey work in a group to solve a problemsuch as:

• If a T-shirt manufacturer charges$0.10 per letter, how much will it costto put your name on the shirt? Howmuch would it cost to buy the lettersfor the whole class? What if the classhad two dollars less than theyneeded?”

Again, this is consequential only if stu-dents are actually designing the shirts.Isn’t there something to be said for a stu-dent who decides that each student mustcome up with a nickname that is five let-ters (or less) to save money? Or a studentwho announces that the craft store she fre-quents usually has coupons for 50% off apurchase? It also provides natural exten-


Planning a party andcooking for guests areauthentic experiences thatare rich in mathematicallearning.

Teachers can watchcarefully as studentsexplain theirreasoning aboutplace value andoperations usingBase Ten Blocks,gaining insight intoexactly whatchildren understand.


sions for a student ready for additionalchallenges. Asking students to evaluatetheir own understanding is helpful to thechild, as well as to the teacher. Rubrics orquestions that require students to evaluatethemselves or their group provide anopportunity for reflection and additionallearning.

Projects such as these use scope andsequence to help teachers provide a foun-dation and then extend learning. They per-mit students at many different levels ofachievement and ability to learn simultane-ously at their own “level” and from eachother. They introduce children to the realworld of teamwork in problem solving.

However, inadequate time has impactedthis method of instruction. Teachers feelmore pressure to cover material that willbe included on state and national assess-ments prior to the scheduled testing win-dow. Less time is devoted to or evenavailable for “real teaching” or real think-ing. To prepare for standardized assess-ments, some teachers try to assess in wayssimilar (or identical) to the actual test,rather than using methods of authenticassessment, such as those described above,in which student learning is an integral partof the assessment process. Accountabilityfor test results has caused some teachers tospend inordinate amounts of time teachingrote memorization of information, usingonly the identical vocabulary and assessingonly the information that will be tested.

State assessments

There are so many more important ideasto learn besides those tested in such a ster-ile way. What happened to the importanceof Bloom’s taxonomy? Why do test devel-opers see more importance in a questionthat asks a child, “Which of these shapesis repeated in the drawing of a rug?” thanan opportunity which allows students tosynthesize or evaluate information, suchas, “Invent a 3-D container that can hold16 ounces of popcorn,” or, “Choose twoways to graph favorite ice cream flavors inyour classroom and explain which graphbest represents the results”?

In the weeks prior to state assessments,

one might overhear teachers commentingto each other such things as, “We’re notready for it, but we’ve got to squeeze inadding fractions so they’ve at least seenthem before the test.” So much for scopeand sequence. In the classroom, one mighthear teachers commenting to students,“No, Sydney, we don’t have time for youto share how you can make a light bulblight with only a paper clip and battery!And no, Ana, we don’t have time for youto share a copy of money printed in the1700’s that your parents came across intheir attic! We haven’t learned how to cal-culate area, and ‘The Test’ is next week!”

So much for, “teaching the wholechild.” Sometimes teachers are forced tomake poor decisions in scope andsequence in order to fit material in beforethe test. Sometimes teachers feel they donot have time to relate to their students aspeople. Learning is so rushed, educatorsmay feel pressure to take out the activitiesthat bond students to teachers and let kidsbe kids.

If educators know that active learning isthe foundation for understanding, shouldn’tthe same be true for assessment? Theremay be a need for some sort of standard-ized test, but the importance of it shouldbe equal to the weight of the information itprovides. Proper preparation of teachersand adequate professional developmentshould be much more important than theresults of a multiple choice test. Not onlyare the tests expensive and inadequate,they are time consuming and they dimin-ish the real learning that should be occur-ring in the classrooms. Albert Einsteinsaid, “Everything that can be counted doesnot necessarily count, everything thatcounts can not necessarily be counted.”Although these standardized tests certainlyhave some value, they cannot representthe totality of the educational experience.An emphasis that is focused only onobjective criteria ignores the fundamentalimportance of the learning experience.

Michelle Kirk teaches third grade at Quail RunElementary in Lawrence, Kansas. She has taughtgrades first, second, third, and sixth at variousschools in Lawrence and at the AmericanInternational School of Rotterdam.

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I’ve been thinking about this question ofhow teachers can meet the challenge of

raising standardized test scores while con-tinuing to engage students with authentic,inquiry-based, hands-on learning. Thisquestion reminds me of how I first taughtscience ten years ago. We weren’t preoc-cupied with standards back then. When Iwas hired, I asked what I should teach andwas told, “Reading, writing, math . . . youknow, first grade.” I had no textbooks. Allthe other teachers kept monarch caterpil-lars in their classrooms, so I went out to afield and found some myself. I brought myclass for walks in the local swamp toobserve seasonal changes. In short, I didnot have a standards-based science cur-riculum. I was clueless and winging it.

The yogurt cup experiment

I used a lot of old yogurt cups in the class-room to pass out math manipulatives likebeans and dice. One winter day Ana Marifilled up a yogurt cup with water andbrought it out for recess. She forgot thecup on the stoop outside the classroom.Ana Mari found the cup the next day anddiscovered it no longer contained waterbut ice. When the children came in fromrecess, they crowded around her, jostlingto see and touch the cup, like this was amiracle. I thought that was weird. I mean,by the time you’re six, you’ve seen icecubes, and in Massachusetts, you’ve seenplenty of frozen puddles. I didn’t get themystery.

Ana Mari asked me if she could share. Isaid sure. The audience could hardly con-tain its excitement. She rose with confi-dence, told the story of the yogurt cupbeing left out and then brought in. For herfinale, she dumped the yogurt cup andgave it a dramatic whack. Voila! The wetround ice slid onto the rug, while the chil-

dren all oohed and wowed. “How’d youmake it do that?” Papo asked.

Ana Mari said, “I just left it on the stepyesterday. I got this today!” More wows.This was like leaving your tooth under thepillow for the Tooth Fairy. I thought tomyself could they not know that the coldtemperature turned the water to ice? I waswaiting for some smart aleck to contradictAna Mari, but instead, Carmen asked ifshe could have a yogurt cup. I said sure. Inthe next minute, every other child askedtoo.

This went on for days. Every time I’dpick them up from recess, they’d be linedup, carrying these old yogurt cups in theirmittened hands, asking if they could share.We put names on the cups, to reduce fight-ing, and began to keep a chart of ourobservations. We’d go around the circleand they’d all dump their cups on the rug.I started to spread newspaper on the rugbefore they came in. The first day that thewater in the yogurt cup froze, it wassunny, clear and cold. The whole weekwas like that, until Friday, when it wascloudy and warmer. At the end of theweek, I asked them what they thought washappening. The initial idea that some fairywas busily switching their water for ice

A Funny Thing Happened onthe Way to the Test

By Mary Cowhey

Students marveled at how acup of water left outsideovernight resulted in ablock of ice the nextmorning.


was nixed because sometimes the dumpedcups revealed a piece of ice and a splashof water. They began using the word“freeze” to describe the change.

The next hypothesis was that the sunwas freezing the water. This hypothesiswas bolstered by Friday’s evidence thatthe water was not frozen on the cloudyday. I was tempted to just tell them thatwater freezes at 32 degrees Fahrenheit andmove on. Something told me not to rush it.Since I wasn’t required to teach about ice,I didn’t feel any pressure to “cover it.”The experiment continued for anotherweek of soggy mittens and soaked rug. Iput some thermometers out as bait. WhenI observed Paulo, a selective mute,covertly trying to break one, I pulled himaside and showed him how to use it. Hewatched me and said nothing. By the endof the second week, all of my first gradersknew how to use thermometers and hadfigured out that it was the cold tempera-ture that made their water freeze.

Connecting to thecommunity

Gretel told the yogurt cupstory to her mother, whoknew a chemist whocould make ice creamwith liquid nitrogen. Sheinvited him to visit ourclass. He wore a white labcoat, goggles, and verylarge safety gloves. Hefroze lettuce leaves andballoons. There was lotsof fog tumbling off thetable. I took photos; mystudents wrote captionsfor the photos to make aclass book. Later, statesof matter became part ofthe second-grade sciencecurriculum. Gretel movedout of the district about amonth after the chemist’s

visit, but all these years later, he stillcomes to make ice cream with liquidnitrogen.

I recently talked to one of my former“yogurt cup experiment” students. Sherecalled her memories of first grade:hatching chicks, getting old shopping cartsand tires out of the swamp, learning aboutthe civil rights movement, and liquidnitrogen. “When I was in middle school,my science teacher asked us if we knewwhat liquid nitrogen was. No one elseknew, but I raised my hand and told them Ilearned how to make ice cream with liquidnitrogen when I was in the first grade. Ifelt proud.”

I look back at the yogurt cup experi-ment with a mix of humility, satisfactionand amusement. Then I think about stan-dards and testing. Today I am teaching sci-ence using a standards-based science kiton states of matter that my district pur-chased for about $500.00. The sciencecurriculum committee recommended thekits over science textbooks, which “cov-ered” a curriculum a mile wide and aninch deep.

At the end of one lesson, the childrenwatched me fill an ice cube tray withwater and predicted it would be full of icecubes the next day if I put it in the freezer.In the next uninspired lesson, I gave eachpair an ice cube in a Ziploc bag and wehad an ice-cube-melting race. I followedthe script. They completed the worksheet.

Will they remember this activity in fifthgrade when they take their first standard-ized science test? Probably not. Howabout in tenth grade, when passing a highstakes standardized test in science andother subjects will determine whether ornot they get a high school diploma? Defi-nitely not. Will the standards-based sci-ence kit lesson turn them on to science?Suddenly the yogurt cup experimentdoesn’t seem so silly.

I appreciate the guidance the standardsprovide in articulating the big ideas I

Liquid Nitrogen hazards:Liquid Nitrogen needs to be handledcarefully. You can find informationand links to other safety data atWichita State University: http://webs.wichita.edu/facsme/nitro/safe.htm.

A visiting chemist followed upon their ice discoveries byfreezing things, and making icecream with liquid nitrogen.

http://webs.wichita.edu/facsme/nitro/safe.htm

http://webs.wichita.edu/facsme/nitro/safe.htm


should teach. I teach child-centered,inquiry-oriented science based on thestandards but not because I think it willraise test scores. My teaching does noharm to test scores, but I don’t really carewhether it boosts them. I am moreimpressed by a kid who shows empathyfor a baby chick, or walks to school toreduce carbon emissions, or writes to themayor to ask why an abandoned car lotcan’t be made into a forest, than I am by akid who scores at the advanced level.Ultimately, our society needs creativeproblem solvers, empathetic doctors, andethical researchers.

Authenticity is key

So what do I do? I teach the required cur-riculum from the expensive science kit,but mostly I look for authentic connec-tions. In a unit on simple machines, thatmeans inviting in an old Puerto Ricanlocksmith who explains how keys are allabout wedges. When a fist fight breaks outat recess about whether to save the trees orthe tent caterpillars, we invite in a Bud-dhist scholar for a lesson on interdepen-dence and email an entomologist in theGalapagos Islands with a list of questionslike, “Would tent caterpillars be such pestsif humans didn’t cut down so many trees?”This year I began the unit on states of mat-ter with a field trip to Snow Farm, home of

the New England Crafts Program, wherewe met a glassblower and a welder andwatched them transform their materialsinto whimsical chickens and colorful fishand birds. And yes, I still leave out yogurtcups and keep my eyes peeled for soggymittens.

Mary Cowhey teaches second grade at JacksonStreet School in Northampton, Massachusetts.She is the author of Black Ants and Buddhists:Thinking Critically and Teaching Differently inthe Primary Grades (Stenhouse Publishers,2006) from which some elements of this articlewere drawn. She has received numerous awardsfor her teaching, including a Milken NationalEducator Award and an Anti-DefamationLeague World of Difference Award.

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Watching thedramatic effects ofliquid nitrogen froma safe distance

Tasting ice cream made with liquid nitrogen

Connect®

January\February 2007

Synergy Learning™PO Box 60

Brattleboro, VT 05302

A PUBLICATION OF

“We need significantly higher investment in innovation throughout society . . .” writes ThomasHomer-Dixon about humankind’s need to invent in order to cope with limited resources in thefuture (New York Times, 11/29/06).

A new report on K–8 science supports innovation in many forms and may cause us torethink how we approach teaching science. Taking Science to School: Learning and TeachingScience in Grades K–8 (National Research Council, National Academies Press, 2006) pro-vides multiple new ideas about instruction and definitions of proficiency in science. Research,reviewed in this report, discredits the commonly held belief that children think only in con-

crete and relatively simplistic ways.The report also contains studies ofseveral curriculum strands. The bestmay be: the study of matter. Thisstrand provides immediately usefulmaterial for curriculum and contentdiscussions among teachers.

Taking Science to School, arguesthat we are still chopping scienceinto confusing pieces. “Science edu-cation as currently structured, “itstates, “does not leverage the knowl-edge and capabilities students bringto the classroom.” This report pro-poses specific changes that can helpus focus learning in classrooms inorder for our students to become theinnovators society needs!

Online and as a bound book,Taking Science to School is availableat: http://www.nap.edu.

Innovators in Our Classrooms!

Our 20th year

http://www.nap.edu

connect jan-feb 07 r8 - synergy learning

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