F i e l d N o t e s...The reason for dividing the way I did is because in my country, students are trained to subtract mentally while dividing. During the first and second grades,

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F i e l d N o t e s

fall 2001 1

TABLE OF CONTENTS

When to Hold, WhenTo Fold: Looking atMath Differentlyby Joan Fournier and Jenifer Mullen

ath phobia. Math anxiety. Fear of math. Whateveryou call it, many of our learners have it. Many educa

tors, too! In fact, chances are there is an official psychiatricdiagnosis for math loathing!

What is it about mathematics that strikes fear in somany people? This question forged the backdrop for our mo-tivation to attend the Teaching Adult Numeracy Certificationpilot course (offered through SABES and the Adult and Com-munity Learning Services at the Massachusetts Departmentof Education.) As instructors wearen’t much different from ourstudents; we bring our ownacademic history with us, and inour case, we also had two totallydifferent per- sonalities, ageperspectives, degrees of enthu-siasm, and levels of numeric comfort. We soon learned thatthe only factor we needed to have in common was the willing-ness to look at math in a new way—to view it as FUN.

Under the guidance of coordinator Esther Leonelli, withassistance by Marilyn Moses, Barbara Goodridge, and MaryJane Schmitt, we actually did learn to look at math differentlyand we did have fun. Each week we were presented with a newlearning technique. Early feelings of intimidation were

Continued on page 4

fall 2001fall 2001fall 2001fall 2001fall 2001

When to Hold, When to FoldJoan Fournier and Jenifer Mullen.........cover

Doing Math the Latin American WayZanna Ebanks........................................3

Mock Elections and Math ClassPat Fina et al......................................... 5

In Everyone’s Good HealthLenore Balliro.........................................6

A Footprint for LearningVeronica Kell........................................ .7

Math in the BasementJoan Reissman........................................9

An Abbreviated Response to StereotypesAmy Park.............................................10

Seeing the Link: GED 2002and the Math Curriculum FrameworksRuth Shwendeman.................................13

Math ActivitiesRuth Estabrook..................................... 14

Math ActivitiesB. Goodridge et al............................... 15

Bringing Reform to Adult EducationNumeracy InstructionEsther D. Leonelli and M. J.Schmitt..... ......11.Remembering a Math TeacherBob Bickerton................................ ......16

MatematikoAlan Tubman........................................18

When Math Becomes a ThrillAlice Levine..........................................19

Fast Math: Straight to the PointMichelle Ede....................................... 21

Understanding NumeracySally Waldron..................................... 23

M

Massachusetts Department of Education

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F i e l d N o t e s

F i e l d N o t e s2

Field Notes is a quarterly newsletterthat provides a place to share innova-tive practices, new resources, andinformation within the field of adultbasic education. It is published bySABES, the System for Adult BasicEducation Support, and funded byAdult and Community Learning Ser-vices (ACLS), Massachusetts Depart-ment of Education.

The Central Resource Center of SABESis located at WorldEducation, 44 Farnsworth Street,Boston, MA 02210.

Opinions expressed in Field Notesare those of the authors and not nec-essarily the opinions of SABES or itsfunders.

Permission is granted to reproduceportions of this newsletter; however,we request appropriate credit to theauthor and Field Notes.

Subscriptions are free to Massachu-setts ABE practitioners. Allothers may subscribe for anannual fee of $8.00. To subscribe,contact Heather Brack,44 Farnsworth Street,Boston, MA 02210. Make checkspayable to World Education.

Submissions are welcome. If you havean idea for an article or wish to sub-mit a letter to the editor, call LenoreBalliro at 617-482-9485.

We do reserve the right to declinepublication.

Editor: Lenore BalliroLayout: Lenore BalliroProduction Assistant: Heather BrackSubscriptions: Heather Brack

Advisory Board for 2000/2001:Bruce Dahlquist, Lee Haller, MarilynMonteiro, Bonniebelle O’Neal, SusanPeltier, Nancy Tariot

n the three years since I’ve been editing Field Notes (and its predeces- sor, Bright Ideas), I have successfully avoided an issue on math. Like many of us, I have suffered from math anxiety most of my life. I agreedwith other SABES staff that it was time to do a math issue. But unlike other topicsin adult basic education where I had interest, experience, and confidence, I hadlittle idea of how to begin with math. So, I decided to enroll in the “TeachingMath in ABE/ESOL for Non-Math Teachers” mini-course at the ALRI, taught byEsther Leonelli and Roberta Froelich and to enlist the help of some ABE mathexperts along the way. As a result of the mini-course, of engaging in discussionsabout the “math wars,” and of researching and compiling material for this issue,I have a new found appreciation of and curiosity about math—and numeracy—and a little more confidence in my abilities to solve problems involving numbers. Exciting changes are happening in math education, largely as a result ofconstructivist learning theory. The math course at the ALRI reflected many ofthese changes: using a discovery method of learning; articulating a line of rea-soning; working in pairs and groups, using manipulatives to reach understand-ing of principles and processes. Learning about “friendly numbers,” exploring avariety of estimation strategies, and trying out a number of ways to solve prob-lems (drawing pictures, using manipulatives), all helped to liberate us from therigidity and abstraction of traditional math and its attending anxieties.

This issue of Field Notes offers stories and lesson plans by creative mathteachers. It also offers math activities and resources for further exploration.Esther Leonelli and Mary Jane Schmitt summarize the history of math reform inMassachusetts adult basic education within a regional, national, and interna-tional context. Their article reminds us how math practitioners in Massachu-setts have been in the foreground of major math reform over the past severalyears—reform grounded in theory, research, and practice. Zanna Ebanksillustrates how math is done in her homeland, teaching us that the NorthAmerican approach to computation is not universal. Alice Levine recounts herpositive experiences teaching math to parents of public school students inBoston. Veronica Kell offers footprints as a way to teach perimeter and area.Joan Reissman encourages bargain basement hunting to teach percentages.Ruth Schwendeman clarifies some of the changes in the new math GED test; shealso makes links to the math curriculum frameworks in Massachusetts. BobBickerton reflects on his years as a math teacher in Cambridge, and SallyWaldron reviews a seminal book for adult basic education math teachers.Finally, Michelle Erde talks about “Fast Math.” I hope this issue helps those ofyou who are math-phobic to relax a bit and those of you who are math-compe-tent to discover new possibilities.

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Lenore Balliro, editor

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F o r e w o r d

Special thanks to Esther Leonellifor her substantial contributions tothis issue of Field Notes and to LouWollrab and Heather Brack for theirvaluable assistance.

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F i e l d N o t e s

fall 2001 3

Doing Math the Latin American WayBy Zanna Ebanks

y first experience with doing math “the American way” was when I had to help my daughter with her homework. I am from Honduras, where I learned to divide one way while my daughter was taughtanother way. It was a very frustrating time. I decided to take the time to learn to divide the way she was

doing it at school, here in the United States.

For example:

The reason for dividing the way I did is because in my country, students are trained to subtract mentally whiledividing. During the first and second grades, students learn to divide using subtractions. For example:

From third grade onward, students are required to do exercises where they practice division using mental sub-traction. Once they have mastered the mechanics of division it is easier to do it mentally and very quickly. An-other difference appears in subtracting and adding fractions. In Honduras, we do it this way:

1.Multiply the denominators among themselves to find the common denominator.2.Divide the common denominator by each denominator and multiply the result by each numerator. Forexample:

The same is done when subtracting fractions. This is just a small sample of how you can obtain the same results inmath by operating different ways.

Zanna Ebanks was an EFL teacher in Honduras, Central America, for 25 years, teaching Basic English, Business Corre-spondence, and TOEFEL Preparation Classes. She now teaches Spanish GED at Concilio Hispano, Inc. in Chelsea. Shecan be reached at 617-889-0888.

See also: Schmitt, M. J. (1991). The Answer is Still the Same: It Doesn’t Matter How You Got It: A Booklet for Math Teachers and Math Stu-dents Who Come from Multicultural Backgrounds.1991. Find this booklet in Number Sense curriculum distributed by the Massachusetts DOE/ACLS. Call 617-338-3850.

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F i e l d N o t e s


quickly replaced by simple suc-cesses, and self-doubts gave way toconfidence inspired by the bondsof camaraderie within our littleclass of five adult learners. Thesocial “ice” was broken in the firstclass when we shared our mathautobiographies. My c0worker,Jen, the younger and more posi-tively energized of our learningduo, wrote of numerous construc-tive mathematical experiences.

I, on the other hand, had re-trieved memories of being taught toplay poker at an early age and en-couraged to compute odds — toknow literally when to hold andwhen to fold. Esther placed equalenthusiasm on each of our experi-ences and proceeded to predictnumeric “greatness” for all of us. Iwas a little less sure of this than mybubbly coworker but gamely hungin there.

Each week we extensivelystudied personal conceptions ofknowledge in regards to math-ematical functions—that is, multi-plication, division, fractions, per-cents, etc.— while never losingsight of individual learning styles. Wewere also required to take the GEDmath to explore our own anxietylevels and shortcomings. Takingthe math test also helped us reflecton our teaching expectations andour use of teaching strategies.

ApplicationsApplicationsApplicationsApplicationsApplicationsThis is where the “fun” be-

gan. Each of our weekly lessonsalso incorporated an “applicationproblems” assignment. Thesesheets were usually twelve prob-lems long and the course requiredus to choose two or three differentproblems, introduce them to ourstudents, and document our expe-

rience in our weekly numeric jour-nals. (These diary-type notationsprovided each of us with a writtencommentary on our readings andour experiences: what we wanted totry, what we did try, what worked,what didn’t, and suggestions of howwe could improve our future per-formance. These notations weretime consuming yet proved to be a

valuable tool in reflecting on andimproving our practice. To help useffectively present these problemsto our students, we were requiredto tackle a similar exercise in ournumeracy class while utilizing avariety of manipulatives.

Hands OnHands OnHands OnHands OnHands OnMy educational background

had not included this hands-onapproach, so as coworker Jen glee-fully played, placed, designed, or-ganized, and, seemingly, masteredblocks, dots, plastic chips, beans,cards, and fractional slices of al-most everything imaginable, Iwondered how I was to introduceall of these techniques while stillremaining within my realm ofcomfort. That answer came as avery pleasant surprise to me. Afteradmitting that I’d rather cleanbathrooms than teach anythingresembling geometry, I was soonshown how to feel confidentenough to say to my class, “I’m notexactly sure what the answer is.Let’s see if we can find it together.”What the best teachers alreadyknew is what we learned: relin-quishing the belief that as an edu-cator one must “know” all of theanswers has its rewards. Our stu-

dents didn’t look at us as being lessof a teacher, but as being more of amodeler. And following that state-ment with an exploration of whattangible items we could use for ourlearning process resulted in a“win-win” situation for all of us.Each week in the pilot class weshared strategies, documentedfeedback from our students, en-

couraged each other, and were en-couraged in return. We tried it all:blocks, graphs, labels, diagrams,magnified rulers, sand, coupons,water, and more. But mainly webattled against the staleness andfutility of the “drill and kill”method of teaching math. We rec-ognized that life-skill relevancyexpands beyond the confines ofcomfort. When a student needs toknow how many square feet of floortiles she should purchase or has theneed to find the congruent anglesfor the stability of a backyard fence,then tidy little worksheets go outthe window.

The Teaching Adult NumeracyCertification pilot course encour-aged honesty in mathematics. If weaccept the fact that there are nu-merous ways in which we learn, it islogical to support the idea thatthere is more than one way toteach.

When to Hold...When to Hold...When to Hold...When to Hold...When to Hold...Continued from page 1

I’d rather clean bathrooms than teachanything resembling geometry...

Joan Fournier and Jenifer Mullen areABE and ESOL teachers at theTaunton Public Schools/Bristol Com-munity College Adult Education Part-nership. The two have combined expe-riences in family literacy, curriculumframeworks, and workplace educa-tion. Jenifer can be reached by emailat <[email protected]>. Joan can bereached by phone at 508-977-9565.

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F i e l d N o t e s

fall 2001 5

The evening pre-GED mathclass did an experiment on presi-dential elections. We held a mockelection for the whole school sothat we could compare the resultsfor the Community Learning Cen-ter (CLC), Cambridge, Massachu-setts, and the United States. Wethought this would be an easy ex-periment, but it didn’t turn out thatway.

Design oDesign oDesign oDesign oDesign of the Experimentf the Experimentf the Experimentf the Experimentf the ExperimentFirst, we made up the ballots.

We decided to focus only on thepresident and vice president. We

n ABE classes, lack of time is always the enemy. For the data analysis, statistics, and probability strand of themath curriculum frameworks, we tend to emphasize graph reading, because graphs can appear in three of theGED exams; we do a class or two on means, medians, and modes, and then it’s on to the next unit of study. Since

our class examples tend to be problems from a textbook, our students learn data analysis, but they don’t really get a feel fordata collection and organization as it occurs in the real world.

One simple, clean, real- life data project manageable in an ABE setting is to have an upper level math class run aschool-wide mock election and report the results of their experiment to the public in writing. This lets students see theentire experimental process from design through data collection, organization,and analysis; moreover, the written reportallows them to hone their math as communication skills. The only unusual class materials are a ballot box (which can beas simple as a recycled photocopier paper box wrapped in colored paper) and enough crepe paper streamers to attract atten-tion to the polling place.

My pre-GED math class at the Community Learning Center in Cambridge ran a mock election in the fall of 2000 andpublished the results in the school newspaper. With permission of the authors, the class essay is reprinted below.

decided to use the honor systemand allow students to vote withoutour class needing to be at the poll-ing place, in the CLC lobby, tooversee the voting. We also al-lowed absentee ballots for stu-dents who would not normallycome to the CLC on Monday, No-vember 6, 2000.

Results oResults oResults oResults oResults of the Experimentf the Experimentf the Experimentf the Experimentf the ExperimentThat night in class we

counted the votes three times tomake sure that they were correct.One hundred seventy -six peoplevoted, but one ballot was disquali-fied because the voter had marked

two X’s. There were three write-invotes: one for Bill Bradley, one forDr. Maya Angelou, and one forTeletubby Tinky Winky. At the endof class we posted the results on thedoor of our classroom, expecting tobe able to compare them with thenational results in our Wednesdayclass.

Then we waited … andwaited … and waited … for sixweeks to learn the results of the realelection. We had to wait until Mon-day, December 18, 2000, to com-plete our project.

We organized the results intothe chart below.

By Pat Fina, Errol Allen, Errol Bannister, T. Campbell, Jean Estivil, Joseph Guerrier, Janetta Quinn,O. Ramsaran, Taysha Rivera, et al.

Mock ElectionMock ElectionMock ElectionMock ElectionMock Election

Mock Elections and Math Class: AnExperiment in Politics

Continued on page 6

I

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F i e l d N o t e s


Pat Fina teaches pre-GED math at theCommunity Learning Center in Cam-bridge, MA. She can be reached at617-349-6363.

Vice President Al Gore wonthe CLC, city, state, and US popu-lar vote. However, GovernorGeorge W. Bush won the ElectoralCollege vote, so he won the presi-dency.

ConclusionsThe 2000 presidential elec-

tion was historical because it tookso long to decide and because thewinner of the election did not winthe majority of US votes.

We are left with some ques-tions about the fairness of the elec-tion system. We suggest that any-one who is upset about the electionwrite to his or her senators andrepresentatives to ask them tochange the voting system. Finally,we congratulate George W. Bush,the forty- third president of theUnited States.

Mock Elections...Mock Elections...Mock Elections...Mock Elections...Mock Elections...continued from page 6

n the spring 2001 issue of Field Notes, the Foreword used a graphicof Chinese characters for “GoodHealth” as an illustration for anarticle about cross-culturalhealth issues. A reader, AmyPark, pointed out that becausethe foreword included an anec-dote about Cambodians, theChinese characters might bemisleading to some readers,who might think Khmer (theCambodian language) is writtenthe same as Chinese. As a re-sult, Amy and I talked about howmuch stereotyping and misin-formation exists in our countryabout the various Asian commu-nities and languages. Wethought this would be a goodidea to address this concernfurther in Field Notes.

I invited Amy, an AsianAmerican who has experienced

In Everyone’s Good Healthby Lenore BalliroSpecial thanks to James MacNeil of World Education for his translations.

stereotyping and racism firsthand,to contribute a piece abouther perspectives. See her article,“An Abbreviated Response to Ste-reotypes,” on page ten.

In addition, we thought itwould be instructive to include anexamination of different Asianwriting systems using the phrase“Good Health.” We have chosenAsian languages in the East andSoutheast regions as examples ofwriting systems often representedby students in ESOL classrooms inMassachusetts.

Look at the samples belowand see if you can match them tothe correct writing system. Don’tworry if you can’t match all of themup; the answers on page ten willhelp fill in your gaps. Now, if wecould only include an audio com-ponent ...

Find answers to the quiz on pageten.

_____Korean

_____Lao

_____Hmong

_____Cambodian (Khmer)

_____Chinese

_____Japanese

_____Vietnamese

_____Thai

.

Noj qab nyob zoo1.

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8.

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F i e l d N o t e s

fall 2001 7

hen I am learning some-thing new, whether it’scolor theory, weaving, or

adult education curriculum plan-ning, doing something visual ormanipulative is important to me. Ihave found that the same is true formy adult learners of mathematics.Traditional paper/pencil/problem/practice methods of teaching math-ematics have a place, but there mustbe something more for the ideas tosink in.

PPPPPerimeter and Areaerimeter and Areaerimeter and Areaerimeter and Areaerimeter and AreaThe concept of perimeter and

area was exceedingly difficult formy students, a class of pre-GEDand GED level learners who meetwith me three mornings a week atthe MWCC/Devens Learning Cen-ter satellite at the Page HilltopSchool in Ayer, Massachusetts. Allof the students are either focusedon passing the GED exam or areattending classes to improve basicmath and language arts skills. Wereviewed the formulas for rect-angles (A=s2, A=lw, P=4s, P=2l+2w)and worked some standard prob-lems. I found that my studentsweren’t really grasping the conceptthat perimeter is the distancearound an object, andarea is theobject’s footprint. There was greatconfusion around when to use theperimeter formulas vs. the area

formulas.I decided to explore the foot-

print concept. Each student re-ceived two sheets of centimeter(cm) square graph paper, a lengthof string, and a ruler with both En-glish and metric measures. Col-ored pencils, scissors, and tapewere in the center of the table.Each of us taped two sheets ofgraph paper together end to end.We then placed our foot on thegraph paper and traced around it.(Keeping shoes on provides for asmoother line and, consequently,an easier estimation task.)

TTTTTracing Fracing Fracing Fracing Fracing FootprintsootprintsootprintsootprintsootprintsThe first task was to find the

perimeter of their footprint (in cm)using the available tools. (I made itclear that the measure they werefinding was an estimate.) The firstreaction was to try measuring thedistance around the footprint withthe ruler. Some students puzzledover the formulas. Someone soonsaw that the string might be help-ful. Before long they were all fit-ting the string around the outlineof their foot, marking the distanceon the string, and measuring thisdistance. We recorded these mea-surements.

The next task was to find thearea of their footprint (in square cm).Two methods were suggested:

1. Try to find a length and width forthe footprint and apply the formula.2. Count each block inside thetracing.

We discussed the advantagesand disadvantages of each. For thefirst method, if you find a lengthand width, you can use the formula.But do you draw the rectangle onthe outside or the inside of thetracing? We agreed that you wouldprobably do both and take the aver-age of the two as the estimate of thearea.

The second method wouldyield a pretty accurate answer, butwould take forever. Through thisdiscussion we came up with a thirdmethod: “rectangle up” the insideof the footprint (i.e., draw largerectangles within the tracing of thefoot, enough to fill up the tracing),then, count the length and width ofeach rectangle, use the formula tofind the area of each rectangle, andsum the areas. We decided thatpartial squares in the tracing wouldbe ignored when drawing the rect-angles since this was an estimate.Each student chose his or her fa-vorite method of finding the area,and we recorded these next to theperimeters. We wound up with atable that looked like the following:

A Footprint for LearningBy Veronica Kell

Footprints

Name Perimeter Area

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F i e l d N o t e s


Veronica Kell is an Adult Secondary Education (ASE) instructor and a computer literacy instructor at the MWCC/Devens Learning Center. She can be reached by email at<[email protected]>.

We compared the perimeter andarea for each foot. We had a grandtime looking at our feet and com-paring their shapes (short, fat feet;long, skinny feet; etc.) and the re-sulting perimeter and area of each.Someone even noted that her totalfootprint (both feet) would bedouble the area of her single foot.

After this activity, we went onto find the perimeter and area ofodd-shaped rectangular figures(those with rectangular cut-outsand add-ons), with much greatersuccess.

The class was lively, and par-ticipatory. When my students takethe GED exam and encounter aproblem dealing with perimeter orarea, I hope they will look to theirfeet and get to work.

A FA FA FA FA Footprint , , ,ootprint , , ,ootprint , , ,ootprint , , ,ootprint , , ,continued from page 7

CurriculuCurriculuCurriculuCurriculuCurriculum Fm Fm Fm Fm Frameworksrameworksrameworksrameworksrameworksand Equipped fand Equipped fand Equipped fand Equipped fand Equipped for the Futureor the Futureor the Futureor the Futureor the Future(EFF) Connections:(EFF) Connections:(EFF) Connections:(EFF) Connections:(EFF) Connections:

Math Curriculum Frameworks:Strand 1: Number Sense Standards 4, 5, and 6 (Esti- mation)Strand 2: Geometry and

Measurement Standards 1, 2, 3, 4, 5, 7, 9

Equipped for the Future (EFF)Strand 1: Communication Skills Standards 3, 4, and 5Strand 2: Decision-Making Skills. Standards 1, 2, and 3Strand 3: Interpersonal Skills. Standards 1, 3, and 4.Strand 4: Lifelong Learning Skills Standard 2

Basic Equipment toKeep on Hand forAn Active MathClassroom:

♦ Scissors♦ Glue sticks♦ Blank 3x5 cards♦ Straight edges (the separators in Lipton tea boxes are good)♦ Rulers (with both English and metric measures)♦ Construction paper (or colored copier paper)♦ Markers♦ String♦ Cups♦ Slinkys TM

♦ Pennies♦ Protractors♦ Colored pencils♦ Unifix cubes

Math ActivityHow Big Is a Million?How Big Is a Million?How Big Is a Million?How Big Is a Million?How Big Is a Million?

1. Have students work in pairs.2. Each pair will be measuring a million of something. They can decide together what to measure. For example:

♦ How many liters/cups is a million drops of water?

♦ How many days would it take to count to a million if you say one number each second?

♦ If you lived a million days, how old would you be?

♦ If you line up a million pens end to end, how many miles would this be?

♦ How much would a million pennies weigh?

3. After making a guess as to what the answer to their question will be, students conduct experiments and calculations tofind out. For example, if they want to figure out how many liters will be filled by a million drops of water, they can use aneye dropper and count the number of drops in a centiliter; for example, multiply by 100 to get the number of drops inone liter, and divide 1,000,000 by this number to calculate the number of liters. Encourage them to work together tocome up with a plan for solving their problem before they begin.

4. Have each pair make a poster of their results, with their question on the top, an explanation of how they figured it out(along with illustrations), and the answer. They can share/present to the rest of the class.

From <www.literacytech.org/users/estabrook/>

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F i e l d N o t e s

fall 2001 9

Math in the BasementBy Joan Reissman

Math Activity1.1.1.1.1. Background Preparation: Background Preparation: Background Preparation: Background Preparation: Background Preparation: Prepare students ahead of time by practicing how to calculate discounts and sale prices so they can use the automatic mark down system to understand percent discount calculations.

2. Prepare a W2. Prepare a W2. Prepare a W2. Prepare a W2. Prepare a Worksheetorksheetorksheetorksheetorksheet: (See sample below.)

3. Field T3. Field T3. Field T3. Field T3. Field Trip: rip: rip: rip: rip: Take your students to Filene’s Basement or another store that has a similar type of markdown system.

4. Activity: 4. Activity: 4. Activity: 4. Activity: 4. Activity: Tell students to choose several items of clothing they would like to buy. Have them calculate the discount according to the posted dates. Then have them subtract the amount of discount from the marked price. Students will see how the difference in percent disount changes the sale price.

5. Materials5. Materials5. Materials5. Materials5. Materials: Provide worksheets where students will fill out information that includes original price, amount of discount, amount saved, and new price.

Example• Original price: $100• Discount: 25%

•Method 1:100 x .25=$25.00

•Method 2:

100x=2500 x=25

• New price:$100.00 - $25.00 = $75

hen an instructor incorporates practicalmath into the curriculum,

students begin to understand howmath skills can help them in theirdaily lives. This understandinghelps reduce math resistance andanxiety.

Practical math instructioncan start at the ABE level. Althoughmany areas of math serve as a basisfor practical lessons (using recipes,for example), I have found thatpercents are particularly easy towork with because students need

percent skills for many everydaysituations such as department storesales and car buying. Everybodygoes to sales, so students will beanxious to learn how to calculatepercent discounts. A teacher canuse newspaper coupons and salediscounts to teach both simple andsuccessive percents. Boston has aparticularly good resource forteaching percents: Filene’s Base-ment. Filene’s has an automaticmarkdown system, so the clothesbecome increasingly cheaper thelonger they remain unsold. This

system provides an easy way forstudents to compare prices by us-ing percents to compute mark-downs.

Here is a sample lesson planfor students who have been study-ing percents at the GED or pre-GED level.

Worksheet

• Original price:

• Amount of discount:

• How much saved:

• New price:

W

Joan Reissman has been teachingGED for 20 years. She works at Jobs forYouth (JFY)Boston as a math andscience instructor. She can be reachedat <[email protected]>

x100—– ´ –— 25

100

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F i e l d N o t e s


he stereotype that “all Asians are the same” has damaged the identity of AsianAmericans and resulted in the inju-ries and, in some cases, the deathsof Asian Americans. One exampleof mistaken identity occurred in1982 when a Chinese Americanman named Vincent Chin wasbeaten on the head with a baseballbat. Witnesses recalled that theassailant, a white male who workedin an American autoplant in De-troit, verbally blamed Chin for theailing auto industry, assuming thatChin was Japanese. During theheight of anti-Japanese sentimentin Detroit when many workers losttheir jobs due to the economic re-cession and competition fromJapanese automakers, Vincent Chinwas attacked and died because helooked Japanese.

I speak from my personalexperience as a Korean American.My Asian American friends and Iwere physically assaulted by a groupof teenagers who kept calling us“Nomo” (the new Boston Red Soxpitcher who at the time pitched forthe LA Dodgers). They punchedtwo of my friends in the face,kicked the side of my car, andsmashed the back seat window,which shattered over three of myfriends who were seated in theback. None of us was Japanese. Atanother time, a not-so-sober manapproached me saying that hefought in Vietnam to save “mypeople.” Thinking I was Vietnam-ese, he targeted me out of acrowd while waiting for a bus.

The reason for revisitingthese painful experiences is to por-tray a piece of the immigrant ex -experience, which reflects a large

population of adult learners inMassachusetts. Immigrants whocome to ESOL and citizenshipclasses may have questions like,“Why do store workers ignore me,but pay attention to white custom-

ers,” or “What is a ‘chink’?” Howdo adult learners grapple with feel-ings of being victimized fromsubtle and blatant forms of dis-crimination? How should adulteducators discuss issues of diver-sity and racism?

Asians in the United StatesAsians in the United StatesAsians in the United StatesAsians in the United StatesAsians in the United StatesWe are fortunate to have adult

educators who have spent time inforeign countries learning the na-tive language and culture, yet thereis a distinction between learningabout an ethnic group in its nativecountry versus understanding eth-nic groups living in the US. Thelack of an institutionalized systemfor learning about Asians inAmerica nurtures generations ofignorant Americans and places aburden on the individual to learnon one’s own time and interest. Ifwe dig deep into American his-tory, we will find policies drivenby economic and political factorsthat promoted the 1882 ChineseExclusion Act and the internmentcamps of Japanese Americans,who were American-born citizens,

during World War II. Both policiesreinforced the stereotype thatAsians are not Americans, but for-eigners, no matter how long theylive in the US or what contributionsthey make to American society.

One way of preventing theperpetuation of these stereotypes iswriting this article for Field Noteswhich provides background for howa reader could unintentionally in-ternalize stereotypes aboutAsians. I hope this article hashelped to expand your understand-ing of the Asian American immi-grant experience and will assistyour work in the adult educationcommunity. (Note: See “In Everybody’sGood Health” on page 6 for examples of the

differences among Asian languages.)

Amy Park is assistant coordinator ofthe Massachusetts Family LiteracyConsortium at Adult and CommunityLearning Services, MassachusettsDepartment of Education. She can bereached by email at

At another time, a not-so-soberman approached me saying thathe fought in Vietnam to save “mypeople.”

Answer to quiz on p.6Answer to quiz on p.6Answer to quiz on p.6Answer to quiz on p.6Answer to quiz on p.6

T

1. Korean2. Chinese or Japanese*3. Khmer (Cambodian)4. Chinese or Japanese*5. Hmong6. Lao7. Thai8. Vietnamese

*Some characters in Japanese are the same as Chinese,as in the case of “Good Health.” However, the Japaneselanguage is not identical to Chinese.

An Abbreviated Response to StereotypesBy Amy Park

<[email protected]>

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F i e l d N o t e s

fall 2001 11

Bringing Reform to Adult EducationNumeracy Instruction

his year, Massachusetts adult basic education practitioners are engaged inbringing reform to adult numeracyand math instruction on two fronts.Within Massachusetts, revision ofthe ABE Massachusetts Mathemat-ics Curriculum Frameworks is un-derway based on recent researchand prevailing thinking in the field.Nationally, strategic planning toraise awareness of adult numeracyinstruction is ongoing through theAdult Numeracy Network (ANN).Both efforts cap a decade-longstruggle by Massachusetts ABEmath teachers to change how weteach math to adult learners in ourprograms. Both efforts aim tobring numeracy out of the shadowsof adult education and raise it to thesame level as adult literacy and lan-guage learning in the national con-sciousness, policy, planning, andfunding.

Math reform and numeracyinstruction in ABE is almost invis-ible in the national debate. In cur-rent literacy campaigns like theNational Literacy Summit, num-eracy, or adult mathematics lit-eracy, merits barely a mention.1 Incontrast to the US, similar literacycampaigns in other English-speak-ing countries, such as the UnitedKingdom and Australia, speak of anadult literacy and numeracy educa-tional system of learning.

The Development o The Development o The Development o The Development o The Development of thef thef thef thef theABE MassachusetABE MassachusetABE MassachusetABE MassachusetABE MassachusettststststsCurriculuCurriculuCurriculuCurriculuCurriculum Fm Fm Fm Fm Frameworksrameworksrameworksrameworksrameworks

Over the past decade, severalinitiatives set the stage for the Mas-sachusetts ABE Curriculum Frame-

works that are currently under re-vision.

The First VThe First VThe First VThe First VThe First VersionersionersionersionersionIn 1989, the National Council

of Teachers of Mathematics(NCTM) published the Curriculumand Evaluation Standards forSchool Mathematics, a documentthat served as a template for re-forming and improving K-12 math-ematics education across the nation.In 1994, sixteen MassachusettsABE/GED teachers formed the ABEMath Team and studied the K-12standards to see how some of theideas might play out in their adulteducation classrooms. After a yearof action research in their classes,these teachers published two docu-ments, a set of “adult educationmath standards” and stories ofwhat changes looked like in theirclassrooms. Their adult math stan-dards were incorporated into theMassachusetts ABE Math Standards(1995) and were the first set of ABEframeworks to hit the press. Assuch, they served as an early tem-plate for the ABE Frameworks inother subjects in Massachusettsthat were subsequently developed.These early Frameworks alsoserved as a model for other states.In 1996, in the wake of educationreform and a national science andmath initiative in the state (whichincluded ABE), the MassachusettsABE Math Standards were sub-sumed under the MassachusettsCurriculum Frameworks: AchievingMathematical Power (January1996). It was the only adult educa-tion subject included in the statecurriculum frameworks for K-12.

In 2000, the adult ed math frame-works were revised outside of theK-12 framework. The current revi-sion of the Massachusetts ABEMathematics Curriculum Frame-works is a revision of that firstframework, but heavily influencedby developments in the adult edu-cation field since then, both na-tionally and internationally.

National InfluencesNational InfluencesNational InfluencesNational InfluencesNational InfluencesIn 1990, three Massachusetts

teachers joined several others inapproaching the NCTM with a pa-per, A Call to Action, asking thatthe NCTM include adult learners intheir reform agenda. NCTM re-sponded by forming a task force onadult learners and subsequentlyhosted the first national Confer-ence on Adult Mathematical Lit-eracy (March 1994). This confer-ence brought policymakers, re-searchers, and practitioners to-gether to discuss the status of adultnumeracy education and to deter-mine future directions.2 Out of thisconference came at least two sig-nificant events: the formation of anational network of practitionersand the development of the “honestlist: what math we should be teach-ing adults.”

The Adult NuThe Adult NuThe Adult NuThe Adult NuThe Adult NumeracymeracymeracymeracymeracyNetworkNetworkNetworkNetworkNetwork

The Adult Numeracy Practi-tioners Network was formed by theadult education practitioners at this1994 Conference on Adult Math-ematical Learning. In 1997, theANPN board voted to change the

By Esther D. Leonelli and Mary Jane Schmitt

Continued on page 12

T

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F i e l d N o t e s


name of the Network to the AdultNumeracy Network (ANN) after itbecame officially affiliated withNCTM. The ANN has since held anational conference for adult edu-cation practitioners in conjunctionwith the national meeting of theNCTM ; this year, ANN held itsannual meeting at COABE 2000 inMemphis, Tennessee. In additionto publishing a newsletter for itsmembers three times a year, ANNsponsors an electronic discussionlist that has an internationalreach.3

The Adult NuThe Adult NuThe Adult NuThe Adult NuThe Adult NumeracymeracymeracymeracymeracyFFFFFrameworksrameworksrameworksrameworksrameworks

In 1995, the ANN wasgranted one of eight planninggrants for system reform and im-provement funded by the NationalInstitute for Literacy (NIFL) aspart of the Equipped for the Future(EFF) project. World Education,in cooperation with five state lit-eracy resource centers, acceptedthe grant on behalf of the AdultNumeracy Practitioners Network(ANPN). Over the course of a year,through teacher-led focus groups,and an online virtual study group,the ANN expanded upon the “hon-est list” developed from the confer-ence. See: A Framework for AdultNumeracy Standards: The Math-ematical Skills and Abilities AdultsNeed to be Equipped for the Fu-ture(1996).4 The teacher teamsstudied and discussed other docu-ments and developed seven themesthat serve as the foundation foradult numeracy standards: Rel-evance/Connections; Problem-Solving/Reasoning/Decision-Mak-ing; Communication; Number andNumber Sense; Data; Geometry:Spatial Sense and Measurement;

frameworks are written to addressthe purposes for learning math-ematics and do not proceed from aschool-based mathematics curricu-lum model. Rather, the frameworkslook at the mathematics that is usedin the context of adult lives.

The numeracy framework inthe United Kingdom is organizedby mathematical topic rather thanby function. The UK frameworkalso shows examples of whereadults use numeracy skills, andincludes, at every level, numberwork, geometry, measurement,and data and statistics.

Many countries now partici-pate in an international researchforum called Adults LearningMathematics (ALM). The NationalCenter for the Study of AdultLearning and Literacy (NCSALL)hosted their conference in July2000. Several US practitionersparticipated.

An international survey ofthe numeracy abilities of adults willbe part of the Adult Literacy andLifeskills (ALL) Project in 2002.The framework created in the de-velopment of the survey definesand describes the complex natureof numeracy. The revised Massa-chusetts ABE Curriculum Frame-works for Mathematics andNumeracy incorporate some ofthese ideas in the current revision.(Source: Gal, I., van Groenestijn,M., Manly, M., Schmitt, M. J., &Tout, D. (1999). Adult Literacy andLifeskills Survey Numeracy FrameworkWorking Draft. Ottawa: StatisticsCanada)

Influencing National PInfluencing National PInfluencing National PInfluencing National PInfluencing National PolicyolicyolicyolicyolicyIn April 2001, ANN held a

strategic planning meeting to con-sider recent developments in na-tional literacy policy through thelens of numeracy practitioners.

and Algebra: Patterns and Func-tions.

As a result of this work,mathematics was included in theEquipped for the Future ContentStandards: What Adults Need toKnow for the 21st Century (Stein,2000).However, of the 16 EFFstandards, only one specificallyaddresses numeracy or mathemat-ics; listed under Decision-MakingSkills, it is Use Math to Solve Prob-lems and Communicate.

International Influences:International Influences:International Influences:International Influences:International Influences:Defining NuDefining NuDefining NuDefining NuDefining Numeracymeracymeracymeracymeracy

International influences havebegun to find their way into USnumeracy practice through frame-works from other countries, in-cluding Australia, the United King-dom, and the Netherlands.

Since the 1980s, work by adult edu-cators in Australia, the UnitedKingdom, and other countries hasexpanded the definition ofnumeracy. Current thinking sug-gests that numeracy includes morethan the ability to perform basiccalculations. In the Australian cur-riculum frameworks, numeracydenotes the ability to perform awider range of math skills, such asmeasuring and designing, inter-preting statistical information,giving and following directions,and using formulas. Moreover,numeracy and literacy are pre-sented as interconnected and on anequal footing. The Australian

Bringing RefBringing RefBringing RefBringing RefBringing Reform ...orm ...orm ...orm ...orm ...continued from page 11


See the following Web sites for adefinition of “numerate behavior:”

<www.nces.ed.gov/surveys/all/resources.asp>

<www.nces.ed.gov/surveys/all/documents/num.pdf>

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F i e l d N o t e s

fall 2001 13

s most ABE practitionersknow, the GED TestingService rolls out its latest

version of the new test in January of2002. The last version of the testwas famous for one very majorchange: the addition of the essay.The newest form has its own drama:the use of the scientific calculator.But that’s not the whole math story,and it might be useful for GEDpractitioners to see the broaderclimate change in both math testingand instruction. This shift is notonly profiled in the GED test butalso in another developing docu-ment, the ABE Mathematics Cur-riculum Frameworks.

Basic TBasic TBasic TBasic TBasic Test Fest Fest Fest Fest FactsactsactsactsactsFirst, some basic facts about

the 2002 math test itself. Accord-ing to press announcements fromthe GED Testing Service, here’ssome of what’s new:

♦ Increased emphasis on graphic stimuli.

♦ More attention to data analysis, statistics, and probability.

♦ A two-part test, equally weighted; one part will allow use of a scientific calculator, one part will not.

CalculatorsCalculatorsCalculatorsCalculatorsCalculatorsThe general ra-

tionale for the use ofthe calculator is thatgraduating highschool seniors areexpected to haveexperience with it,

and employers count on compe-tence in that area. Obviously, whatthis means for us as instructors isthat we, too, must upgrade or evendevelop our own scientific calculatorskills.

Beyond the teaching andlearning implications of the calcu-lator in our classrooms, otherchanges are in the wind. Test ques-tions will be “reality-based,” withemphasis placed on natural math-ematical tasks rather than purelyacademic problems. The context ofmath questions will involve moresituations found in the world ofwork, consumer issues, and thefamily. The use of more than onemath concept per test item will bestressed, which seems appropriateconsidering that many real lifemath situations usually do involve arange of these concepts. Higherorder thinking skills will be re-quired.

Math CurriculuMath CurriculuMath CurriculuMath CurriculuMath CurriculummmmmFFFFFrameworksrameworksrameworksrameworksrameworks

In the face of these imminentchanges, some math practitioners Ihave spoken with in my role as GEDliaison for SABES express real con-cern about how and what they mustteach. Fortunately, a concurrentfocus on these areas is underway inthe developing ABE Math Curricu-lum Frameworks, which shouldgive professionals some neededguidance and concrete instruc-tional ideas. Here’s an excerptfrom the document’s “GuidingPrinciples” (currently in DRAFTform):

♦ [In the ABE classroom] real-life context for mathematical concepts and skills across mathematical content areas ... [should] drive curriculum development.

♦ Mathematics instruction[should] mirror real-life ac-

tivity through the use of hands-on as well as printed instructional materials… [and] experience with a broad range of technological tools….

♦ Adult mathematics instruction [should be] more than textbook-driven computation practice…it[should] include experience in understanding and communicating ideas mathematically, clarifying one’s thinking, making con- vincing arguments and reach- ing decisions individually and as part of a group.

Once completed, the ABEMath Frameworks will connectteachers to substantive ways foraddressing the “sea change” re-flected in the new test. And beyondthe exam, students, too, will be thewinners as they gain real-lifemathematics skills essential tomoving forward.

Seeing the Link: GED 2002 Math andthe Math Curriculum Frameworks

Ruth Schwendeman is the GED Liai-son at the SABES Central Region anda writer/researcher for the ABE Math-ematics Curriculum Frameworks 2001Revision Team. She can be reached at508- 885 -6255 or by email at

By Ruth Schwendeman

A

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F i e l d N o t e s


Student Centered Algebra Ruth Estabrook (See Ruth’s Web page: <www.literacytech.org/users/estabrook>.)

WWWWWriting Algebraic Expressionsriting Algebraic Expressionsriting Algebraic Expressionsriting Algebraic Expressionsriting Algebraic ExpressionsAn alternative approach to teaching the concept of algebraic expressions is through the use of paper bagsor paper cups. This makes the lesson visual and hands-on, and it involves a degree of critical thinking. (The followingactivity is based on a lesson in Impact Mathematics published by Everyday Learning, 2000.)

Tell the students that each bag contains the same number of blocks, but you don't know how many, so they can call itx. If they have two bags containing the same number of blocks, plus 4 additional blocks, how many blocks wouldthey have? This can be written as 2x + 4. Then when you tell them how many are in the bags, they can evaluatetheir expression. After this brief discussion, have your students work in groups of two or three to discuss and answerthe following questions.

1. Suppose you had three bags, each containing the same number of blocks, plus two extra blocks. Write an algebraic expression for this situation. Use n for the number of blocks in each bag.

2. If n = 2, what is the total number of blocks? If n = 0, what is the total number of blocks? If n = 25, what is the total number of blocks?

3. Complete this table.

4. Could the total number of blocks in this situation be 18? Explain.

5. To represent this situation with the expression 3n + 2, you need to assume all the bags contain the same number of blocks. Why?

6. Now suppose you have 5 bags and 4 extra blocks.

a. What is the total number of blocks if each bag contains 3 blocks? If each bag contains 10 blocks b. Using n to represent the total number of blocks in each bag, write an algebraic expression for the total number of blocks c. Find the value of your expression for n = 3 and n = 10. Do you get the same answers you found in part a?

7. Write an expression to represent 7 bags each with the same number of blocks, plus 5 extra blocks.

8. Rebecca wrote the expression 3b + 1 to describe the total number of blocks represented in this picture.

a. What does the variable b stand for in her expression?b. What does the 3 stand for?c. What does the 1 stand for?

Math Activities

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F i e l d N o t e s

fall 2001 15

Guess and checkCorey watches TV from 4:30pm to 8:00pm with a short break of 20 minutes. How long does Corey watch TV?

(a) 2 hrs and 20 minutes(b) 3 hours and 20 minutes(3) 3 hours and 10 minutes(d) none of theseAnswer:

Draw a picture or diagramMike wants to paint his bedroom, which is15 ft by12 ft. The walls are 8 ft high. How many square feet will he paintif he paints the walls and the ceiling? (Ignore the window and door.)

Make an organized listHow many sheets of paper will you have if you tear the paper in half six times?

Make a table or chartWhat possible combinations of coins equals 25 cents?

♦ Guess and check

♦ Draw a picture or diagram

♦ Use logic and arithmatic computation

♦ Make an organized list or table

By B.Goodridge, E. Leonelli, M. Moses, M. J. Schmitt, from the Math/Numeracy Pilot, April 2000

Draw a PictureRead the problem all the way through before starting to work it out!

1. Eva spends 1/2 of her income on food and rent, 1/4 of her income on clothing, 1/12 on entertainment, and saves $1,200 per year. What is her yearly income?

2. Maria spends 1/2 of her income on food and rent, 1/3 of her income on clothing, 1/2 of her income on entertainment, and saves $1,200 per year. What is her yearly income?

After you read #2 but before you solve the problem, ask yourself the following questions:♦What changed from #1 to #2?♦Do you think Maria earns more or less money than Eva? Why?♦Now solve the problem,using pictures.

Math Activities

Solving Problems

Problem Solving StrategiesProblem Solving StrategiesProblem Solving StrategiesProblem Solving StrategiesProblem Solving Strategies

♦ Relate to real-life experience

♦ Make a model and act it out

♦ Work backwards

♦ Look for a pattern

♦ Write an equation

♦ Use a system of equations

The turkey diet problemYou’re on a diet and can only eat 1/4 lb turkey breast for dinner. The butcher gives you three slices and marks thepackage 1/3 lb. How many slices can you eat? What strategies did you use?

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F i e l d N o t e s


By Bob Bickerton

Remembering A Math Teacher

was cleaning out a corner of the attic and came across a box that hadn’t beenopened in years. Out came papersand notes that instantly broughtstudents back to life. I remem-bered Roberta, who kept mumbling“I’m dumb at math” as we tried torelearn subtraction with borrowing.I bumped into Roberta several yearslater— teaching math at a commu-nity college! There was Marie, whotook one look at fractions andmoaned, “Oh no, not those again.”She desperately wanted to stopemptying bed pans and become anurse. I also recalled Miguel, whowas doing a little carpentry on theside, but wanted to be a computerprogrammer. Finally, there wasBrian, who dropped out of highschool and needed a quick brush upbefore taking the GED and movingon to the stage. A few years later Iwould attend his wake after anoverdose.

There were more, manymore. Each of their lives is woventhroughout the journal I kept formy second and third years as a full-time math teacher. I’d alreadytaught math as a part-time teacherduring the prior three years. Did Isay “part time?” Actually, it wasthree part-time jobs. I’d made themistake of taking on a total of 36hours of teaching at three differentprograms in the Boston area andwith all the preparation I had to do,there was nothing that felt part-

time about it. Now I had afull- time job in Cam-

bridge, twelve hours ofclasses in the morningand twelve in the

evening. This was reallygreat! I could bicycle my

daughter to school in the morningand pick her up after school. Eat anearly dinner with my family andback to class. Friday’s were staffmeeting days and I was learningwhat it was like to be part of alearning community. And Fridaynights were mine!

Reflection JournalReflection JournalReflection JournalReflection JournalReflection JournalThe idea for a journal came to

mind as I reflected on my first yearas a full-time math teacher. Whileworking part-time (that is, jugglingthree part-time teaching jobs), Ihad been troubled by the lack ofreal progress by some of my stu-dents. As a full-time teacher withpaid time devoted to preparation,planning, curriculum, and materi-als development, I had a betteropportunity to reflect on why I washaving such limited success inreaching these students. I knew itwas because they didn’t really un-derstand what was going on withthe math. I knew that no one reallylearns math by memorizing abunch of rules. In my classes wespent more time developing con-cepts than on mechanics, and thiswas working well for most. But itwas not working for all, and thatjust wasn’t good enough.

As I started my second year, Idecided to document every aspectof one six-hour-a-week math class:planning, implementation, andstudent interactions. Lesson plansincluded detailed notes from con-ceptual underpinnings to the mostminute subskills. I summarizedhow every class progressed. Idocumented how students, to-gether and individually, engaged inthe work—conceptually, operation-ally, and emotionally.

Look-ing back, Irealized howmost of my lesson plans (all typed!)were too ambitious; many objec-tives and activities rolled over to thenext class. I also recalled what itfelt like each time I “got it”— eachtime I understood where and why astudent was getting stuck. It usuallyhad to do with my overlooking somepiece of the path that led to under-standing and mastery. Many stu-dents could make the leap over amissing piece, but for others, whatseemed like a very small omissionturned out to be a chasm. Thesestudents weren’t going to leap for-ward without filling in some moreof the puzzle.

Owning the MathOwning the MathOwning the MathOwning the MathOwning the MathBack to the students. Roberta

didn’t have a good grasp of quan-tity. Once we figured this out, ithelped explain why none of the“operations” meant to manipulatequantities made sense. She neededtime to count out the totals, to learnfor herself how multiplicationworked as a fast way to add and thatdivision could be thought of as ei-ther “grouping” or “partitioning” –and how each means somethingquite different. It also meant thatonce we tackled the hard work ofmastering quantity, she was free totake off and fly. She did and alongthe way she found great joy inmathematics.

Miguel did fine with frac-tions—after all, he was measuringwood with a high degree of preci-sion almost every day. But heblocked on algebra. He explainedthat failing algebra had helped his

I


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F i e l d N o t e s

fall 2001 17

decision to drop out of high school.So, we started into algebra shortlyafter he got settled into the class. Ibrought in a balance beam scaleand presented him with a series ofprojects where he had to balanceknown and unknown quantities(items in a “black box”) on thescale. One day he got so excited itwas impossible to contain him—and who would even try? The ideaof what it meant to do “inverse op-erations” to both sides of the equalsign had burst into his conscious-ness. And none too soon. He hadbeen troubled with “how is thisgoing to help me pass the GED?”Once he understood basic con-cepts, he was poised to tear throughalgebra much faster and with a levelof understanding that would staywith him for life.

Marie tried to rememberrules about manipulating fractions,but she really didn’t know whatthey represented. In addition tousing the many manipulatives wehad created, I also tried using ca-pacity measurement to explainfractions—and more! She had mea-suring spoons, cups, beakers, and

test tubes to use, and a load ofprojects to undertake. Once thisaspiring nurse understood the re-

lationship of capacity measurementto medication doses, her frustra-tion was replaced with a burningdesire to learn. She wanted to besure she had mastered every pos-sible aspect of fractions— after all,patients’ lives would be at stake.

Practitioner ResearchPractitioner ResearchPractitioner ResearchPractitioner ResearchPractitioner ResearchI spent seven years at this

program, and the two years I docu-mented teaching and learningmade these some of the most re-warding years of my life. Docu-mentation and reflection pushedme to study more of my subject andcraft. I needed to fill in my own“gaps” in understanding before Icould be more useful to students. Ispent hundreds of hours at the li-brary researching math analysisand learning disabilities. This

pushed me to enroll in a mastersprogram where my two years ofdocumentation and some addi-

tional research led to a new mathdiagnostic instrument for theLearning Center.

We didn’t have a name for thisapproach to the teaching andlearning process back then, butthose who pursue their craftthrough “practitioner research”projects today have my full supportand admiration. What a gloriousway to live a life and make a living!

Bob Bickerton has been the director ofAdult and Community Learning Ser-vice at the Massachusetts Departmentof Education for the past 13 years. Hestarted in ABE 30 years ago and hasworked as a teacher, teacher trainerand program director. He can bereached at 617- 338-3850 or by emailat <[email protected]>.

Remembering...Remembering...Remembering...Remembering...Remembering...continued from page 16

I needed to fill in my own “gaps”in understanding before I couldbe more useful to students.

Upcoming Issues of Field NotesWe welcome submissions on the topics below. Please call Lenore Balliro, editor, at 617-482-9485 or email<[email protected]> by the date indicated below to discuss your idea for an article.,

Winter 2002Managing ABE ProgramsFull

Spring 2002Youth in ABE ProgramsCall by: December 1Submit by: December 15

Summer 2002How I Entered the Field of ABEand Why I Stay TherePlease tell us your story inabout 500 words.Call by: April 1Submit by: April 15

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F i e l d N o t e s


Framework for AdultNumeracy Standards

Matematiko

started developing a math practice program several years ago as a supplement to

the materials that were available inmy GED classroom. I was fortunateto receive a donation of three orfour IBM compatible computersthat were discarded from organiza-tions that were updating to newertechnology. These were of coursetext-only computers that would runonly DOS programs.

I began writing programs fordrilling multiplication facts and astime permitted, I added othermodules until finally I had about ahundred modules ranging fromsimple operations with whole num-bers to basic algebra and geometryformulas. These modules were notintended as a self- study program,but as a tool to offer students extrapractice with material taught in theclassroom.

I then started sharing theseprograms with colleagues in Mas-sachusetts. At first the programswere all text (DOS). Then I decidedto switch to WINDOWS 2000/95/98/NT4.0. I began with the math-ematics tables and continued withbasic math, fractions, decimals,mixed numbers, etc.

For more information, con-tact: Alan Tubman, 26 Quissett St.,Worcester, Massachusetts 01602

Alan Tubman is a GED instructor atthe Worcester Community ActionCouncil (WCAC) in Worcester. teach aclass of youth between 16 and 21.Hecan be reached by email at<[email protected]>.

FFFFFrom htrom htrom htrom htrom http://literacynet.org/sciencelincs/teachertutor-nutp://literacynet.org/sciencelincs/teachertutor-nutp://literacynet.org/sciencelincs/teachertutor-nutp://literacynet.org/sciencelincs/teachertutor-nutp://literacynet.org/sciencelincs/teachertutor-num.htmlm.htmlm.htmlm.htmlm.html

"A Framework for Adult Numeracy Standards"is based upon a study bythe Adult Numeracy Network (ANN). In this study, instructors andlearners identified the mathematical skills and abilities adults need tofulfill their life roles. These topics are similar to the subjects that will beemphasized in the 2002 GED Math test. For a more in depth examina-tion of the study and description of the frameworks, see A Frameworkfor Adult Numeracy Students, The Mathematical Skills and AbilitiesAdults Need to Be Equipped for the Future, by Donna Curry, Mary JaneSchmitt, and Sally Waldron at <www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/framewkTframewkTframewkTframewkTframewkTOC.html>.OC.html>.OC.html>.OC.html>.OC.html>.

NuNuNuNuNumber and Number and Number and Number and Number and Number Sensember Sensember Sensember Sensember Sense This skill (being able to handle numbers comfortably and competently)needs to be explored using whole numbers, fractions, decimals, per-cents, ratio, money, and estimation. Estimation, mental math, compu-tation, and calculators are all tools that develop number and numbersense.

Problem Solving: Reasoning and Decision MakingProblem Solving: Reasoning and Decision MakingProblem Solving: Reasoning and Decision MakingProblem Solving: Reasoning and Decision MakingProblem Solving: Reasoning and Decision MakingProblem solving includes seeking to understand the problem thenfiguring out what information and math skills are important to use andsolve the problem. While problem solving is embedded in mathematics,there are specific skills and strategies that help greatly.

DDDDData Analysis, Probability and Statiata Analysis, Probability and Statiata Analysis, Probability and Statiata Analysis, Probability and Statiata Analysis, Probability and Statistics, and Graphingstics, and Graphingstics, and Graphingstics, and Graphingstics, and GraphingReading charts and graphs, interpreting data, and making decisionsbased on information are key skills to being a successful worker andinformed citizen.

Geometry: Spatial Sense andMeasurementGeometry: Spatial Sense andMeasurementGeometry: Spatial Sense andMeasurementGeometry: Spatial Sense andMeasurementGeometry: Spatial Sense andMeasurementMeasurement, a foundation skill for geometry, is an essential life skill.Awareness of acceptable tolerances, margins, and upper and lower lim-its critical to measurement competence. Today, much is computerized,but the results are only as good as the information inputted. Visualiza-tion and concrete models help reasoning in this area.

Algebra: PAlgebra: PAlgebra: PAlgebra: PAlgebra: Patatatatatterns and Functionsterns and Functionsterns and Functionsterns and Functionsterns and Functions Algebra includes more than formal methods of equation solving, ageproblems, and lots of X’s and Y’s. Conceptual understanding, algebra as a means of representation, and algebraic methods are all problemsolving tools. Algebraic reasoning allows us to think about and expresspatterns, relations, and functions which ultimately give us access totechnology.

www.matematiko.com

By Alan Tubman

I

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fall 2001 19

ve been teaching adult basic education in a variety of settings for over 20 years,and though I had taught some mathhere and there, I had never seenmyself primarily as a math teacher.In the old days, I remember sayingthat I liked teaching a few mathclasses a week because it was so

neat and concrete, the progress wasso clear, and it provided a nice con-trast to the complexities of teachingESOL. However, in my currentwork with parents in a Family Lit-eracy Program at the David EllisSchool (a public school in Boston),I have fallen in love with the newways that math is being taught andlearned in elementary school class-rooms and have found it thrilling toshare these new approaches withmy adult students.

Until I started teaching par-ents at the Ellis, I’d never workedwith adults who were so focused onreally understanding what theywere learning (with no interferingagendas like preparing to take astandardized test or obtain a cre-dential). The Ellis parents come tothe class initially because they wantto be able to help their kids, butsomewhere along the way, they getso excited about math makingsense to them for the first time it’sclear they are also learning forthemselves.

Once the parents believe thatthere isn’t just one right way tosolve a problem, they are reallymotivated to share their mathstrategies and to articulate theirmathematical thinking with eachother. Because they want to be ableto explain what they’ve learned inclass to their children, they have a

natural interest in the communica-tive aspects of math learning.

Responding to NeedsResponding to NeedsResponding to NeedsResponding to NeedsResponding to NeedsAssessmentAssessmentAssessmentAssessmentAssessment

When we started the FamilyLiteracy Program at the Ellis,Spanish-speaking parents clearlywanted ESOL classes; it was not soclear how we could serve the En-glish-speaking population at theschool. After getting no significantresponse to a flyer about ABE-typeclasses, we put out a survey. Weasked parents what they’d like tosee in parent classes, and we pur-posely included items that empha-sized how they could help their kidsas well as ones that focused moreon the adults’ own skills. One ofthe most frequently checked itemswas “Understanding How ChildrenAre Learning Math Now: Why DoesIt Look So Different from When WeWere in School?” Based on thisclear interest, we started a six-weekpilot math class for parents in May

1998. The response of the parentswho participated was so positivethat we ran the class throughout theschool year for the next three years.During the 2000–2001 school year,we added a second section of theclass taught in Spanish.

Classes meet just once a week.They are purposely scheduledduring the day while the childrenare in school. Although I alwaysbring in interesting math prob-lems, the direction of the classtakes unexpected turns as parentsask questions, bring in sampleproblems, and share their ownideas. Our class sessions usuallyinvolve one or more of the follow-ing activities:

♦ Trying out sample problemsfrom the 4th grade MCAS test andthen discussing what kinds of mathactivities students need in lowergrades to prepare them for the kindof mathematical thinking on thetest;

♦ Working through problemsfrom children’s homework thatparents bring in to class (or that Iget from K-5 teachers at the school);

♦ Experimenting with the samemanipulatives the children use inclass;

♦ Adding to our running list ofParent Math Vocabulary. Parentsnote that math terminology haschanged from when they were inschool. For example, they knowwhat “borrowing” is, but wonderabout the meaning of “regrouping”or “trading;”

♦ Doing lots of collaborativework and talking about differentways of solving problems;

♦ Generating ideas for thingsparents can do at home with house-


For example, they know what“borrowing” is but wonder about themeaning of “regrouping” or “trading.”

I

When Math Becomes a ThrillBy Alice Levine

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hold materials, through daily ac-tivities, and through games, tohelp their kids develop importantmath concepts and to develop aconfidence in and enjoyment ofmath.

As parents become enthusi-astic about math and start seeingmath problems as positive chal-lenges, they convey their new feel-ings about math to their children.Parents have told me that helpingtheir children with math home-work is no longer drudgery; ithas become something they lookforward to. Many of the mothersand grandmothers who attendthe class talk about creativeways they have found to en-gage their children in math-ematical thinking—on theplayground, in the kitchen,or as they plan graduationparties, barbecues, or familyreunions.

Math AmbassadorsMath AmbassadorsMath AmbassadorsMath AmbassadorsMath AmbassadorsThe Parent Math Class has

created a core of parents who feelcomfortable in the school and whobegin to take on leadership rolesthere. As their confidence aboutmath increases, they become lessintimidated about engaging in dia-logue with the K-5 teachers about

issues of curriculum and instruc-tion. They start to spread the wordto other parents—and the schoolbegins to get the reputation of beinga more welcoming, accessible place.We have a group of informal “mathambassadors” who spread positivefeelings about math and math re-form—to their children, to otherparents in the school, and to peoplein their communities.

Parents in the math class

naturally progress from thinkingabout their own children to think-ing about the performance of stu-dents in the school as a whole. Theybegin to want to help other Ellisparents understand the new mathstandards and approaches. Thus,our initial strategy—of providingintensive math classes to smallgroups of parents—has expanded;

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we now reach out to larger numbersof families without losing our em-phasis on helping parents gain adeeper understanding of the el-ementary school math curriculum.

Family Math NightsFor two consecutive years, we

have run Family Math Nights at theschool, where 200–250 parents andchildren have engaged in inter-generational, hands-on math ac-tivities. It’s wonderful to see theconfidence of my students as theystaff the math stations—alongsideEllis teachers or alone—sharing

their own understanding ofmath with other Ellis parentsand children.

Teaching math is nolonger just a break from thecomplexities of teachingESOL. Now, I’d prefer to

teach math all day long. Andparent participants who

used to hate math now tellme that they see math every-

where and seek out opportuni-ties to learn more. For all of us,math has become something muchdifferent from the old-style rotelearning; math, at its best mo-ments, has become a thrill.

This article was based on a presen-tation at the Adults Learning Maths7th AnnualConference held inMedord, MA. , July, 2000. Herabstract can be found online at<www.alm-online.org/>.

Alice Levine is a family literacy coor-dinator for Boston Excels, aprogram of The Home for Little Wan-derers. If anyone is doing similarwork with parents, or is interested inknowing more, Alice would be pleasedto communicate with him or her. Shecan be reached by email at<[email protected]>.

Good mathematics is NOT howmany answers you know...buthow you behave when youDON’T know.

— author unknown

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fall 2001 21

By Michelle Ede

ve been an adult educationteacher for eight years.Like many in the field, I’ve

taught a variety of subjects. I’veworked with lots of different stu-dents, all struggling with their ownpersonal trials and tribulations.Nothing seems to compare, how-ever, to the fear and loathing ofmathematics that so many GEDlearners bring to class. “Does thetest have word problems? I can’t dothose.” “Why does the science havemath stuff in it?” and “Can’t I justskip the math?” were all questions Iheard regularly.

Many math books weren’tworking very well for the learnersin my class, especially learningdisabled students. Even the bestbooks were wordy, visually unap-pealing, and dry. I used a variety ofapproaches, such as manipulativeuse and group work, but because oftime constraints, multi-level/multi-subject classes, and differingskill levels, students still neededbooks for independent study. Thetrouble was, learners were oftenunable to use them without becom-ing de-motivated and requiringsignificant one-on-one assistance.I found that the students faringworst were the ones with ADD andLD, the ones who were not strongvisualizers, and the ones who hadreflexive thoughts of, “forget it!”whenever math was mentioned.

Math is real. It is not sup-posed to be scary or disconnected

from its real-life pur-poses. Math, like sci-ence and art, is natu-rally rooted in ourphysical existence—

that is, until somebodyplucks it up, strips it of its

images and causes, and drops itlifelessly onto a clean white page.Often, despite the tiny size of theletters and cryptic examples given,there is still little space left inwhich students can stretch outcognitively. This didn’t make senseto me. We all know that a classroomwithout enough space is uncom-fortable, and that crowded citieshave high crime rates.

Clearly, our brains need space, andwhen it isn’t present, we’re un-happy. If the learners in my classwere going to be successful in math,they would first have to be happy.

Hey WHey WHey WHey WHey Wake Up!ake Up!ake Up!ake Up!ake Up!Students also needed a visu-

ally stimulating book that explainedmath in the same conversationalstyle I used in class. They needed abook that got straight to the math-ematical point, sidesteppingexcessive verbiage and shoutingout, “Hey, wake up and try this!Math is great, you can be good at it,and whoever has told you any dif-ferent is wrong!” I knew the bookwas unlikely to materialize; in fact,it never did. My students and Isettled for my repeatedly sayingthose types of things as often aspossible.

This past year, however, I gotthe chance to develop a student-centered GED math book for the

Fast Math: Straight to the Point

Math is real. It is not supposed to bescary or disconnected from its real-life purposes.

I Northeast YALD (Young Adults withLearning Disabilities) Project, forwhich I’ve been a practitionertrainer since 1995. The book wouldaddress the math learning difficul-ties observed in GED classrooms,using information gathered frommy work.

I visualized the book Iwanted learners to have. I put whatI knew about ABE teaching into it. I

wrote it in the teaching style mystudents have helped me develop byshowing me what works for them. Ileft space here and there, and putin plenty of interesting graphics. Inamed it “Fast Math” because of itssimplified style and pointed ap-proach. When it was finished, Ibrought it to my classroom for afield test. It has since become afavorite, and my students keep ask-ing me if I’ll make a similar bookfor the other GED subjects. I don’tknow if I will, but the questionsounds much better to me than“Can’t I just skip the math?”

Michelle Ede is a GED/ESOL teacherfor Haverhill Community Action. Shecan be reached by email at<[email protected]>.

If your learning center would like aIf your learning center would like aIf your learning center would like aIf your learning center would like aIf your learning center would like afree, reproducible copy ofree, reproducible copy ofree, reproducible copy ofree, reproducible copy ofree, reproducible copy of “Ff “Ff “Ff “Ff “FastastastastastMath,” contact Michelle Ede at North-Math,” contact Michelle Ede at North-Math,” contact Michelle Ede at North-Math,” contact Michelle Ede at North-Math,” contact Michelle Ede at North-east Yeast Yeast Yeast Yeast YALD 978- 373–1971, ext. 251.ALD 978- 373–1971, ext. 251.ALD 978- 373–1971, ext. 251.ALD 978- 373–1971, ext. 251.ALD 978- 373–1971, ext. 251.

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Understanding NumeracyA Review of Adult Numeracy Development: Theory, Research,Practice by Iddo GalHampton Press, 2000

dult Numeracy Develop-ment is part of the Series

on Literacy: Research, Policy andPractice published by the NationalCenter on Adult Literacy at the Uni-versity of Pennsylvania. Iddo Gal,editor and the author of severalchapters, served as the director ofthe Numeracy Project at NCAL,where he began work on this book.He currently teaches in the Depart-ment of Human Services at theUniversity of Haifa, Israel. Inhis preface, Dr. Gal states: In developing this vol-ume …We sought colleaguesable to address both theoreticalaspects and classroom realitiesand to reflect on the challengesand dilemmas involved inimplementing new as well asproven teaching methods whenworking with diverse types ofadult students. (p. ix-x )

Adult Numeracy Devel-opment is a terrific resourcefor math teachers and otherpractitioners; it succeeds inbalancing research and prac-tice, and innovation andproven approaches. Whilemany of the participating au-thors come from the UnitedStates (and some from Massa-chusetts), contributors also comefrom the United Kingdom, Nether-lands, Israel, Australia, Canada,and Malaysia. And while there aremany differences in the contextsfrom which each author writes,there are also many similarities:adult educators who teach mathoften lack prior training in math

teaching and mathematics itself;they are challenged to “meet themath-related goals and needs ofadult learners, as well as to satisfythe increasing demand by publicagencies, community organiza-tions, or business organizations toimprove adults’ numeracy or math-ematical literacy.” (p. 1)

Adult Numeracy Develop-ment is divided into four sections:

Perspectives on Numeracy, Ap-proaches to Instruction, Reflectingon Practice and Learning, and As-sessment. Each section has chap-ters written by practitioners andresearchers, and by US and inter-national authors. For this review,I am focusing on two chapters:“Journey Into Journal Jottings:

Mathematics as Communication”by Donna Curry and “Assessment ofAdult Students’ Mathematical Strate-gies” by Mieke van Groenestijn.

WWWWWriting About Mathriting About Mathriting About Mathriting About Mathriting About Math“Journey Into Jottings” really

bridges the research and practicedivide. The author, Donna Curry,provides straightforward descrip-

tions of how to implementjournal writing in a math classand what she and her studentslearned by asking them towrite about math.Donna’s brings years of expe-rience teaching math in a vari-ety of ABE contexts to her writ-ing. This chapter reflects herexperiences as a member of theMassachusetts ABE Math Teamwhere she explored the Na-tional Council of Teachers ofMathematics (NCTM) stan-dard—Mathematics as Com-munication—while teaching ina workplace education programand doing teacher research.Donna wanted to know if jour-nal writing would increase herstudents’ understanding ofmathematical concepts, confi-dence in tackling math prob-lems at work and at home, and

comfort with speaking the languageof math. Students in Donna’s classwere preparing to pass a requiredstandardized test for the workplace.She chose journal writing because“it would address the writing skillsfor communicating mathematically


AReview by Sally Waldron

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fall 2001 23

and provide opportunities for stu-dents to reflect on and clarify theirown thinking about math.” (p. 241)She felt that journal writing wouldbe easy to implement in a struc-tured class.

Having students keep dailymath journals led to four changesin Donna’s classroom:

♦ Students increased their use ofmath language in the classroom.

♦ The teacher was able to engagein ongoing student assessmentmore easily.

♦ Students gave Donna morefeedback on the classroom tech-niques she used.

♦ Students reported to Donnaexamples of their application ofmath learning to everyday life.

Research also led Donna toother questions: How can I en-courage students to write more?How do I make sure students aren’tjust writing what they think I wantto hear? How can I tell which spe-cific activity helped students gainconfidence and learn new skills?How do I help students who areafraid of being wrong?

Math AssessmentMath AssessmentMath AssessmentMath AssessmentMath Assessment“Assessment of Adult Stu-

dents’ Mathematical Strategies”describes the math interview pro-cess, the “Supermarket Strategy,”used nationally in the Netherlandsto assess ABE students. Written byMieke van Groenestijn, a professorin adult and special education, thechapter summarizes the growth ofRealistic Mathematics Education(RME) in the Netherlands in theearly 1980s. RME “emphasizes theuse of realistic context problems,representations of reality, andmodels to relate classroom instruc-tion and learning to the student’s

real environment and real experi-ence.” (p. 336)

With the adoption of RMEcame the need for assessment toolsthat would let teachers analyzelearners’ mathematical strategiesand skills. The resulting Super-market Strategy was developed tobe used nationally for placement,ongoing assessment, and post as-sessment. Considerations in de-veloping this tool included that it“must consist of functional prob-lems in an everyday-life contextthat can be solved in different ways.The student must be free to use hisor her own methods … A good an-swer is important, but the processby which the student solves theproblem gives more informationabout his or her way of thinkingand the quality of his or her calcu-lations.” (p. 337)

The Supermarket StrategyThe Supermarket StrategyThe Supermarket StrategyThe Supermarket StrategyThe Supermarket StrategyThe Supermarket Strategy

tests three skills areas (basic skills,proportions, and measurement andgeometry further subdivided intoseven fields of knowledge) acrosssix levels of complexity (elemen-tary, intermediate and advanced,each broken into two levels). Theassessment process consists of a30–45 minute interview/observa-tion with each student using a spe-cial advertising leaflet developed incollaboration with a large super-market chain and a selection of 60“context-rich” problems. Threetypes of questions are asked oneach problem: “How are you goingto do it? How are you calculating/what are you doing now? How didyou do it?” Since observing theprocess by which students arrive atan answer is so critical to under-standing mathematical reasoning,interviewers are allowed (actuallyencouraged) to help students whenthey get stuck in certain ways. The

interviewer can structure the task,give a similar but easier task, pro-vide concrete materials, or try cal-culating together with student.Parts of a sample assessment inter-view conducted with an ABE stu-dent from Morocco are included inthe chapter and provide an oppor-tunity to really understand what theSupermarket Strategy is and howthe process works. The descriptionparalleled my experiences withMassachusetts students and re-minded me of how much more Ilearned working and talking withmath learners than I did by admin-istering a paper-and-pencil com-putational test.

The Netherlands found that“well-trained teachers using ap-propriate adult assessment instru-ments such as the SupermarketStrategy are the staring points forquality in math education in ABE.”(p. 100) Unfortunately Adult BasicEducation services have mergedwith Further Education and Voca-tional Education services in theNetherlands as part of a nationalscheme to reduce unemployment,and a new assessment is being de-veloped. This new written testyields a standardized score and agrade level. Since the new test has anumber of problems, however, theuse of the Supermarket Strategy isagain increasing. Maybe there’shope for the US, too.

Sally Waldron works as theDirector of the SABES Cen-tral Resource Center atWorld Education. Shetaught math for more than 15 years.she can be reached at 617-482-9485or by email at<[email protected]>

Adult Numeracy Development is avail-

able from <www.amazon.com>.

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ANN members and invited guestsconsidered four major policy docu-ments and the role of numeracy inthese documents. As a result of thismeeting and as part of its plan fornational visibility, ANN submittedthree commitments toward work-ing on goals of the National Lit-eracy Summit Initiative for thecoming two years:1. ANN will draft and circulate aposition paper urging policy mak-ers to include numeracy as well asliteracy and language as a focal area;2. ANN will develop and deliverpresentations for national and stateadult education conferences thathighlight the recommendations inthe ANN Framework, a nationalstandard for numeracy instruction;and3. ANN will continue to connectUS practitioners to the work of ALMand other relevant researchthrough its newsletter and Web siteand will encourage national re-search agendas on adult educationand literacy to include numeracyissues.

SuSuSuSuSummarymmarymmarymmarymmaryThe past decade has seen

teachers in Massachusetts adultbasic education organize to rethinkand reform numeracy instruction.Certainly, our work would havebeen much less rewarding had we

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Esther Leonelli has been an ABE and GEDmath instructor at the Community Learn-ing Center in Cambridge, MA since 1985and a teacher-writer for theEMPower(Extending Mathematical Power toAdults) Curriculum Project at TERC,Cambridge, Massachusetts, for the lastyear. She also moderates the NumeracyList ([email protected]), Shecan be reached at 617-349-6363 or byemail at <[email protected]>.

Mary Jane Schmitt works at TERC as co-director of the EMPower Project, a math-ematics curriculum development projectfor adults and out-of-school youth. Shehopes all ABE/GED teachers who teachmath check out the ANN and ALM Websites and join the numeracy list discus-sion. She can be reached by email at<mary_ jane_ [email protected]>.

not looked to others outside of Massa-chusetts: to our energetic and talentedcolleagues in Adult Numeracy Networkand the Adults Learning MathematicsResearch Forum. Taken together, weare confident these grass roots effortshave laid a solid foundation to helpguide policy, research, and large-scalecurriculum frameworks and assess-ment for adult literacy and numeracyinstruction in all 50 states.

AdultNumeracyNetwork(ANN)

NotesNotesNotesNotesNotes

1 See From the Margins to theMainstream: An Action Agenda forLiteracy (2000) Report of the TheNational Literacy Summit Initiative.“The National Literacy SummitInitiative is a field-driven effort toimprove our nation’s system of adultliteracy, language, and lifelonglearning services. Its Action Agendawas developed through grassrootsconsensus building and serves as ablueprint for community action.”<www.nifl.gov/coalition/summit/index.html> While “compute andsolve problems” is included in thedefinition of literacy, and“calculating” is included in thebroad range of essential skills,nowhere do the words numeracy ormathematics appear in the goals ofthe initiative or in the document.

2 Proceedings of the Conference onAdult Mathematical Literacy,editedby Iddo Gal and Mary JaneSchmitt, and published by theNational Center on Adult Literacy(Philadelphia, PA: 1994). For anelectronic Summary of Discussions,go to <www.std.com/anpn/virtual.html>

3 [email protected] listis archived on both the NIFL LINCSdiscussion lists <www.nifl.gov/lincs/discussions/numeracy/numeracy.html>and the Math Forum<http://forum.swarthmore.edu/epigone/numeracy.>

4 Available at <www.std.com/anpn/framewkTOC.html>.

ANN, an affiliate of the National Council of Teachers of Math-

ematics (NCTM), is a volunteer group of math teachers interested in

promoting good math practices in adult basic education. In addition

to drafting several important documents, they also sponsor an elec-

tronic forum, The Numeracy List <[email protected]>. To

learn more about ANN, go to their Web site at <www.std.com/anpn/

anpnmission.html>.

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Wicked Good Math Web Sites

literacynet.org sciencelincs/literacynet.org sciencelincs/literacynet.org sciencelincs/literacynet.org sciencelincs/literacynet.org sciencelincs/teachertutor-nuteachertutor-nuteachertutor-nuteachertutor-nuteachertutor-num.htmlm.htmlm.htmlm.htmlm.htmlThis highly recommended site isbased on frameworks developed bythe Adult Numeracy Network(ANN) and is sponsored by NIFL.Don’t miss it.

hub1.worlded.org/docs/hub1.worlded.org/docs/hub1.worlded.org/docs/hub1.worlded.org/docs/hub1.worlded.org/docs/maththing/ny1home.htmmaththing/ny1home.htmmaththing/ny1home.htmmaththing/ny1home.htmmaththing/ny1home.htm“Meaningful Activities That HelpMath.” Includes workshop plans,activities, and experiences to helpdevelop a positive atmosphere forlearning math.

www.literacytech.org/users/www.literacytech.org/users/www.literacytech.org/users/www.literacytech.org/users/www.literacytech.org/users/estabrookestabrookestabrookestabrookestabrook“Math Beyond Worksheets”writtenby Ruth Estabrook.Contents in-clude: creating student centeredlessons; turning a worksheet intoan activity; brain teasers and num-ber tricks; resources and links.

ffffforuoruoruoruorum. swarthmore. edu/m. swarthmore. edu/m. swarthmore. edu/m. swarthmore. edu/m. swarthmore. edu/Includes problem of the week, AskDr. Math, teacher 2 teacher,Weblessons,discussion groups andmore. Math resources by subject.

ffffforuoruoruoruorum. swarthmore. edu/m. swarthmore. edu/m. swarthmore. edu/m. swarthmore. edu/m. swarthmore. edu/teachersnonenglish. htmlteachersnonenglish. htmlteachersnonenglish. htmlteachersnonenglish. htmlteachersnonenglish. htmlMinorities and math, non-Englishlanguage resources.

www.west.asu.edu/www.west.asu.edu/www.west.asu.edu/www.west.asu.edu/www.west.asu.edu/achristie/math/achristie/math/achristie/math/achristie/math/achristie/math/mathconst.htmlmathconst.htmlmathconst.htmlmathconst.htmlmathconst.html“Dr. Alice Christie’sConstructivist Math Page.” Devel-oping mathematical literacy,online math manipulatives, stu-dents as information producers,finding online collaborativeprojects, and more.

www.nctm.org/www.nctm.org/www.nctm.org/www.nctm.org/www.nctm.org/National Council of Teachers ofMathematics (NCTM), the profes-sional organization for math teach-ers. Log on here to look at mathstandards, review publications, getinformation about joining.

www.net1plus.com/users/www.net1plus.com/users/www.net1plus.com/users/www.net1plus.com/users/www.net1plus.com/users/devenslc/mathexercises.devenslc/mathexercises.devenslc/mathexercises.devenslc/mathexercises.devenslc/mathexercises.htmlhtmlhtmlhtmlhtml“ABE/GED Hands on Math Activi-ties.” These notation games weredeveloped for adult learners by theMount Wachusett CommunityCollege in Massachusetts. Richcollection of math activities forteachers and learners in groupsettings.

www2.wgbh.org/MBCWEIS/www2.wgbh.org/MBCWEIS/www2.wgbh.org/MBCWEIS/www2.wgbh.org/MBCWEIS/www2.wgbh.org/MBCWEIS/LLLLLTTTTTC/CLC/nuC/CLC/nuC/CLC/nuC/CLC/nuC/CLC/numintro.htmlmintro.htmlmintro.htmlmintro.htmlmintro.htmlBoston adult numeracy practitio-ners homepage.

projects.edte.utwente.nl/projects.edte.utwente.nl/projects.edte.utwente.nl/projects.edte.utwente.nl/projects.edte.utwente.nl/hotmath/home.htm)hotmath/home.htm)hotmath/home.htm)hotmath/home.htm)hotmath/home.htm)Practical, hands-on math activities.

gseweb.harvard.gseweb.harvard.gseweb.harvard.gseweb.harvard.gseweb.harvard.edu/~ncsall/fedu/~ncsall/fedu/~ncsall/fedu/~ncsall/fedu/~ncsall/fob/2000/ob/2000/ob/2000/ob/2000/ob/2000/fffffobv4ib.htmobv4ib.htmobv4ib.htmobv4ib.htmobv4ib.htmSeptember 2000 issue of Focuson Basics is devoted to math. Forhard copies, contact NCSALL by

email at <[email protected]>.

www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/Adult Numeracy Network. Find outhow ANN is a leader for makingnumeracy a priority in adult basiceducation.

www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/www.std.com/anpn/framewkTframewkTframewkTframewkTframewkTOC.htmlOC.htmlOC.htmlOC.htmlOC.html

The Mathematical Skills and Abili-ties Adults Need to Be Equipped forthe Future, by Donna Curry,Mary Jane Schmitt, & SallyWaldron. The Adult NumeracyPractitioners Network System Re-form PlanningProject, July,1996.

www.shianet.org/www.shianet.org/www.shianet.org/www.shianet.org/www.shianet.org/~reneenew/mathLD~reneenew/mathLD~reneenew/mathLD~reneenew/mathLD~reneenew/mathLD.html.html.html.html.htmlMath and learning disabilities.www.alm-online.org/www.alm-online.org/www.alm-online.org/www.alm-online.org/www.alm-online.org/

Adults Learning Mathematics

(ALM) is an international re-

search forum bringing together

researchers and practitioners in

adult mathematics/numeracy

teaching and learning in order

to promote the learning of

mathematics by adults.

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F i e l d N o t e s


Resources fResources fResources fResources fResources for Tor Tor Tor Tor Teacherseacherseacherseacherseachers(Includes all areas of math)

Alcoze, T., et al. (1993). Multiculturalism in Math, Sci-ence, and Technology: Readings and Activities CA: Addison Wesley.

Cohen, S. (1991).Figure It Out, Thinking Like a MathProblem Solver (Books 2 & 3). North Billerica, MA: Cur-riculum Associates.

Goddard, R., Marr, B., & Martin, J. (1996). Strength inNumbers: A Resource Book for Teaching Adult Numeracy.Melbourne:ARIS/Language Australia. Available fromPeppercorn books, title # 8069.

Helme, S. & Marr, B.(1999). Mathematics: A New Be-ginning : A Resource Book for Teachers of Adults Re-turning to Study. Melbourne:Language Australia,

Good SGood SGood SGood SGood Series feries feries feries feries for Students (Books)or Students (Books)or Students (Books)or Students (Books)or Students (Books)

Akers, J., et al. (1981). Investigations in Number, Data,and Space. Palo Alto, CA: Dale Seymour Publications,book 1. (This book was developed at TERC in Cam-bridge, MA).

Lappan, G., et al.(1998). Connected Mathematics.Menlo Park, CA: Dale Seymour Publications.

AAAAAlgebralgebralgebralgebralgebra

Goodridge, B., Leonelli, E., Moses, M., Steinback, M.,& Tierney, C. (1999). Foundations for Algebra: ABEMath Curriculum Frameworks Unit . Malden, MA:Massachusetts Department of Adult Education andCommunity Services.

Meader, P., & Storer, J. (1998). Math for All Learners,Algebra. Portland, ME: J. Weston Walch Publisher.(There is also a Pre-Algebra edition.)

Marr, B., & Helme, S. (1995). Resources for Teachers:Higher Math Some Beginnings in Algebra.Melbourne: ARIS/Language Australia.(Available from Peppercorn Press, title 8086.)

Math Resources

NewsletNewsletNewsletNewsletNewsletters, Journalsters, Journalsters, Journalsters, Journalsters, Journals

Math Literacy News, Ohio Literacy Resource CenterResearch 1-1100, Summit St., Kent State University POBox 5190. Kenyt, OH 44242.email: <[email protected]>Phone: 330-672-7841, 800-765-2897

As a member of NCTM, you can subscribe to the fol-lowing journals: Mathematics Teacher (MT), Journalfor Research in Mathematics Education (JRME),Journalfor Research in Mathematics Education Online (JRMEOnline). Go to <www.nctm.org> for more information.

1999. Available from Peppercorn Press, Title #8052.

Herr, T., & Johnson K. (1994). Problem Solving Strategies:Crossing the River with Dogs. Berkeley, CA: Key Curricu-lum Press.

Stenbmark, J.K,, Thompson, V, & Cossey, R. (1986).Family Math. Berkeley: Lawrence Hall of Science.

Huntington, L., Leonelli, E., & Merson, M. (1998). ABEPriority Math Curriculum—Number Sense Measurement,Data. Boston, MA: Adult Literacy Resource Institute.

Note: Peppercorn Books, PO Box 693,Snow Camp, NC 27349, distributes manyinternational numeracy titles. Write for acatalogue.

Learning Differences and Disabilities inLearning Differences and Disabilities inLearning Differences and Disabilities inLearning Differences and Disabilities inLearning Differences and Disabilities inMathematicsMathematicsMathematicsMathematicsMathematics

Bley, N., & Thornton, C. (1995).Teaching Math-ematics to Students with Learning Disabilities(Third Edition). Austin, TX: Pro-Ed Publishers.

Marolda, M., & Davidson. P. (1994). “Assessingmathematical abilities and learning approaches.” InWindows of Opportunity: Mathematics for Studentswith Special Needs. Reston, VA: National Council ofTeachers of Mathematics.

Sharma, M. (1994). Learning Problems in Math-ematics—Diagnosis and Remedial Perspectives.Framingham, MA: Center for Teaching/Learning ofMathematics.

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F i e l d N o t e s

fall 2001 27

October 12–13October 12–13October 12–13October 12–13October 12–13MATSOL 2001: Telling Our StoriesLocation: Sturbridge Host Hotel andConference Center<www.matsol.org>

October 17–20October 17–20October 17–20October 17–20October 17–20Literacy Volunteers of America (LVA),Annual Conference2001: A Literacy OdysseyLocation: Albuquerque, NMContact: Peggy May, (843)671-2008<www.literacyvolunteers.org/conference>

Mark Your CalendarOctober 24–25, 2001October 24–25, 2001October 24–25, 2001October 24–25, 2001October 24–25, 2001

Massachusetts Coalition for Adult Education(MCAE), Annual Conference

Network 2001Location: Marlborough, MAContact: MCAE, 800-339-2498,<www.SABES.org/mcae2001.htm>

December 2–4, 2001December 2–4, 2001December 2–4, 2001December 2–4, 2001December 2–4, 2001Institute for Work & the Economy/NorthernIllinois Univ., Workplace LearningConference, 6th Annual

Location: Chicago, ILContact: NIU, 888-619-9215,<www.workplace-learning.net/>

lassroom teachers often use a constructivist approach in their teaching (in literacy aswell as numeracy) without explicitlynaming it as such. Simply put,constructivism, a theory with rootsin Dewey, Piaget, and Vygotsky,posits that learners actively con-struct meaning about the world;they are not merely passive recipi-ents of some stable body of knowl-edge that is somehow “out there”independent of how knowledge isexperienced by the learner. Profes-sor George E. Hein (1991) hasnamed a number of learning prin-ciples we can derive fromconstructivist thought. I havedrawn from his principles to clarifyhow many current approaches tomath reflect a constructivist frame-work.

1. Learning involves hands on ex-perience plus reflection about thatexperience. Constructivist mathinvolves manipulatives and other

concrete approaches to lead to adeeper understanding of mathprinciples.

2. Learning is social. Constructivistmath encourages working in groupsor pairs.

3. Learning involves language: thelanguage we use influences learn-ing. So, thinking aloud or talkingwith others is a critical componentof learning and understanding. Aslearners in math groups work to-gether, they are talking aloud aboutwhat they are doing. As they presenttheir answers to the class, they arearticulating their process and deep-ening their understanding of mathprinciples.

4. Learning is contextual: we learnin relationship to what else weknow, believe, and feel.

5. We need previous knowledge tobuild new knowledge. (Many teach-

ers will recognize this principle inteaching reading through develop-ing scaffolding or building schemasto increase comprehension, thenspiraling new knowledge from es-tablished knowledge.) It takes timeto learn: We need time to reflect,revisit ideas, and think about themfor true learning to take place.

6. Motivation is essential forlearning. We need to understandthe reason why we are doing some-thing or we will not be very in-vested in it or remember what welearn. Constructivist math is com-mitted to helping learners under-stand reasoning processes, prob-lem solving, and other math strate-gies rather relying on drill-and-kill approaches.

George E. Hein, paper delivered atthe International Committee ofMuseum Educators ConferenceJerusalem, Israel. Oct 15–22, 1991.

What Is Constructivism?By Lenore Balliro

C

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F i e l d N o t e s


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F i e l d N o t e s...The reason for dividing the way I did is because in my country, students are trained to subtract mentally while dividing. During the first and second grades,