A publication of the National Council of Teachers of Mathematics …oldmoodle.escco.org/file.php/1/MATH/Groping_Hoping.pdf · 2006-01-05 · A publication of the National Council

Dialogues

Volum

e2,

Issue

3

A publication of the National Council of Teachers of Mathematics

May/June 1999

M a t h e m a t i c sE d u c a t i o n

Inside this issue …

• Let’s Abolish Pencil-and-Paper Arithmetic 2

• Do We Need Calculators? 3

• How Our Readers Feelabout Calculators 4

• The Research Backs Calculators 6

• Ensuring That All Children ArePowerful Technology Users 7

• A Calculator Tour aroundCanada and the United States 8

• Questioning the Use ofCalculators in theElementary Grades 10

• Calculators at the ElementarySchool Level? Yes, It Just Makes Sense! 11

• Tools for MathematicalUnderstanding in Middle School 12

• A Revolution in My High School Classroom 13

• An Attempt in Sweden at Consensus 14

By the early 1980s, those deterrents lostforce because of the appearance of solar-powered, hard-case, four-function and sci-entific calculators costing less than $10 and$15, respectively. And so the new genera-tion of calculators began to be used.

In 1985, the first user-friendly calculatorsappeared that could graph functions. Liketheir simpler counterparts, these calcula-tors were too expensive to be widelyadopted when they first appeared, buttoday many high schools require them forall or virtually all their students. The use ofthese calculators in secondary school hasnot generated as much controversy as theuse of simpler calculators in elementaryschool, and they are required on many col-lege-entrance tests.

More recently, user-friendly calculatorshave appeared that can solve literal alge-braic equations, manipulate algebraic ex-pressions, differentiate and integrate, andsolve systems of equations. Just as theirearlier counterparts raised questionsabout the amount of paper-and-pencilarithmetic a person needs to know, thesesymbol manipulators force an examina-tion of the amount of paper-and-pencilmathematics a person needs in algebrathrough calculus and beyond.

The issues relating to calculator usereach the very core of mathematics in-struction. What type of understandingdoes one obtain through repeated appli-cation of algorithms? What new under-standings, if any, can arise from calculatoruse, and what understandings, if any, maybe lost? How important are speed and ac-curacy with paper and pencil when a cal-culator is usually faster and more accu-rate? What becomes obsolete because ofthe existence of calculators?

Though it has been more than a quar-ter-century since these questions werefirst raised, we still seem to be a long wayfrom developing a consensus on the useof calculators in the classroom. We hopethe discussion has moved away from thesimplistic yes-no responses of yesteryearto a more sophisticated analysis. Resolu-tion requires an answer to the same ques-tion asked by the mathematicians of Eu-rope 400 years ago, when algebra as weuse it today was first developed: Howshould we make use of this extraordinarytechnology to further the mathematicseducation of our students?

Zalman Usiskin, EditorFor the Editorial Panel

The first four-function and scientific handheld calculators appeared in the early 1970s. Their appearance gave rise to simplistic yes-no articles in newspapers and magazines. Yet thoseopinions had little effect because cost, fragility, and short batterylife limited calculator use.

Groping and Hoping for aConsensus on Calculator Use

Mathematics Education DialoguesMay/June 1999

A paper of mine with the title above will be published laterthis year in the Journal of Computers in Mathematics and ScienceTeaching (see www.doc.ic.ac.uk/~ar9/abolpub.htm). In it Ipropose that the elementary school mathematics curriculumshould not attempt to achieve any level of proficiency what-ever in pencil-and-paper arithmetic.

Such a proposal may seem peculiarly perverse at a timewhen there is grave concern about the poor showing ofAmerican students in TIMSS (Third International Mathe-matics and Science Study)and when there is a feelingamong many university mathematicians that their studentsare ever more poorly prepared in mathematics. But Imean it.

Although the problems with precollege education in theUnited States are multifaceted, with curriculum being onlyone of the possible causes of those problems, many ascribethe present situation in mathematics to a progressive “dumb-ing down” of the curriculum in recent years. Calculatorusage, particularly in elementary school, is often seen as thechief culprit. The solution often proposed—for example, inCalifornia—is a “back to basics” approach with a ban on cal-culators in elementary school.

Nevertheless, my proposal is to allow full use of calcula-tors from kindergarten on, with instruction in pencil-and-paper arithmetic replaced by a greatly increased stress onmental arithmetic for all one- and two-digit calculations.The aim would be for a curriculum at least as demandingas any currently in use or proposed, one in which stu-dents would develop their own algorithms for mentalarithmetic—guided by their teachers—and in which cal-culators would be used not just for arithmetic but also forcreative explorations and problem solving. I haven’t thespace here to discuss the details further, but I believe thatsuch a curriculum, appropriately elaborated, not onlywould serve elementary school children better than anycurrent one but also would prepare them better than atpresent for secondary school and college mathematics.

It is important to under-stand that there is no significantresearch that suggests that calculator use at any level is harm-ful to mathematical development or that pencil-and-paperarithmetic, a skill with rapidly declining practical value, isnecessary or even particularly useful for later mathematicaldevelopment. Moreover, the lack of technique, so often foundin secondary school and university students and rightly de-plored by university mathematicians, is not at all the result ofcalculator usage but rather the failure to develop number sensein elementary school and symbol sense in secondary school.Both number sense and symbol sense can, I believe, be betterdeveloped with a calculator-and-mental-arithmetic–based cur-riculum than with current curricula.

What stands in the way of developing and implementingan elementary school mathematics curriculum based onmental arithmetic and calculator usage? One barrier mightbe the ability of current elementary school teachers to teachsuch a curriculum. For this reason (among others) I advo-cate the use of mathematical specialists in elementary schoolfrom third grade on, if not earlier.

Then there are the political and parental barriers, which Icannot discuss here. Finally, there is the research mathe-matics community, which, despite some notable exceptions,understands little but inveighs a lot about the elementaryschool curriculum. In any event, my prediction is that untilwe abolish teaching pencil-and-paper arithmetic in Ameri-can elementary schools, the hand-wringing at the poor per-formance of American students in international compar-isons will be a continuing phenomenon of the Americaneducational scene.

Tony Ralston is professor emeritus of computer science andmathematics at the State University of New York at Buffaloand an Academic Visitor in the Department of Computing, Im-perial College, London. He has a long-standing interest in,and has written widely about, mathematics education.

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The purpose of Mathematics Education Dialogues is to provide a forum through which NCTM members can be well informed about com-pelling, complex, timely issues that transcend grade levels in mathematics education. The opinions expressed in this publication are those ofthe writers and do not constitute an official position of the NCTM. Mathematics Education Dialogues is published as a supplement to theNews Bulletin by the National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1593. NCTM Reston staff:Charles Clements, Copy Editor; Sherry Grimm, Editorial Assistant; Debra G. Kushner, Graphic Designer; Andy Reeves, Staff Liaison. Pagesmay be reproduced for classroom use without permission.

Let’s AbolishPencil-and-PaperArithmeticby Anthony Ralston


We all use calculators in daily life; why should we forbidthem in school mathematics classes, hobbling our childrenas we would never consider hobbling ourselves?

Even so, I believe that calculators are often detrimental tothe teaching of mathematics. As the mathematician RalphRaimi has written, “Education is not imitation of life; it is anartificial process designed to put ideas into the mind andnot answers on paper.” At the grades K–6 level, and giventhe knowledge base of our average teacher, calculators pro-duce only answers.

From the TIMSS results it is clear that mathematical com-petence at the grades K–6 level does not require calculators.Two of the highest-achieving countries at the fourth- andeighth-grade levels, Singapore and Japan, use calculatorssparingly in elementary schools.

At the grades 7–16 level we might do well to heed the voicesof those with the most experience with calculator usage. JohnDuncan, a mathematics professor at the University of Arkansasand an early advocate for technology in a college setting, wrotethese cautionary words in 1995 (American Mathematical Monthly102, p. 194): “Some of us who were very early to use technologyto alleviate drudgery, to visualize graphs and surfaces, to con-duct helpful experiments, etc., are now alarmed at its use as asubstitute for thinking. It even seems to deter problem solversfrom producing general mathematical proofs by holding theirfocus to computing a few numerical examples.”

Overseas, Great Britain has been at the forefront of usingcalculators in both elementary and secondary school settingssince the 1982 Cockcroft Report. Anthony Gardiner, a math-ematician from the University of Birmingham, has gradedthousands of competition papers and has developed an ex-cellent sense of the shift in mathematical capabilities of stu-dents in his country over the past fifteen years. According toGardiner (personal e-mail), the worst effects of calculatorusage are the following:

1. The loss of experience in simplifying and the consequent lossof the student’s (and the teacher’s or examiner’s) expectationthat expressions should have any meaning.

2. The destruction within half a generation of a hard-won, effectivealgebraic symbolism, developed and proved over centuries,capable of being manipulated as a “calculus” for exact numer-ical and symbolic calculations, and its replacement by slavishverbatim copies of what appears in calculator displays.

3. The collapse within ten years of arithmetical fluency withinthe very best students, with the resulting loss of meaning forsymbolic generalizations of numerical expressions.

4. The loss of all attempts to teach pupils to present solutions informs that others can make sense of, and the decline intomere personal jottings en route to an answer, which is related—I suspect—to the next effect.

5. The two most damning outcomes of post-Cockcroftianinnovations:

a) The astonishing switch from solving simple problems (i.e.methods) to caring only about answers (i.e., things that ap-pear in the display of a calculator)—exactly the oppositeeffect of what the innovators claimed they wanted—and sohorribly widespread that no one can pretend not to knowwhat has happened;

b) The inability of students to solve two-step problems be-cause teachers and examiners have learned to accept mereanswers, since psychologically it is almost impossible totrain students who are expected to use a calculator to writeanything else down on paper.

One might assume from what I have written and quotedthat I am an antitech Luddite who forces his students to dothousands of long-division problems by guttering candlelight.Not so. I have a dozen computers in my room and ten TI-92sfor students’ use. But I think too much of my students and themathematics they need to learn to condemn them to a black-box paradise of mindless button pushing merely for the sakeof being on the cutting edge of the mathematics reformmovement.

Kim Mackey has taught mathe-matics and science in Alaska fortwelve years. In 1994 he receiveda Distinguished Teacher awardfrom the White House Commis-sion on Presidential Scholars,and in 1998 he was honored asthe National Academic DecathlonCoach of the Year.

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Correction: The quote from Judy Sowder in the “Reactions toTracking” section on page 14 of the April 1999 issue of Dialoguescontains an error. Here is the quote as it should have appeared,with the corrected word in boldface.

An article in the November 1998 issue of the Journal for Researchin Mathematics Education (JRME) offers some interesting perspec-tives on tracking students based on ability levels. The Israeli researchers, Liora Linchevski and Bilha Kutscher, investigated theeffects of placing students in mathematics classes that weretracked by ability levels (homogeneous groups) and those inwhich students of all ability levels were in the same class (hetero-geneous groups). They found that the achievements of high-abil-ity students in heterogeneous classes were not compromised butthat the achievements of average- and low-ability students weresignificantly lower in homogeneous classes.

Judith T. SowderUniversity Teacher, California; Editor, JRME

by Kim Mackey

How Our ReadersFeel about Calculators


How Our ReadersFeel about Calculatorsby Cynthia Ballheim, For the Editorial Panel

In anticipation of this issue of Dialogues, we invited our read-ers to share their “preactions” concerning the way equity con-siderations affected how their schools (or districts) dealt withcalculators. Of the more than 360 responses received, only oneperson thought that equity issues did not apply in his school,because the school was against calculator use. The rest of therespondents informed us that their schools supplied calcula-tors to all students who needed them and, in fact, issued themalong with textbooks. They agreed, however, that some stu-dents in the same class did have more powerful calculatorsthan others and that this is where inequity existed.

Personal views on the use of handheld calculators were verysimilar. No one suggested that calculators should not be usedin schools. Most recommended that the mathematics that istaught should be revised to acknowledge the power of calcula-tors but that calculators should be used only after teacherswere knowledgeable about the equipment and students hadreceived appropriate instruction. The majority believed thatcalculators should be used only after students had learnedhow to do the relevant mathematics without them. If grade re-strictions were to be put on calculator usage, our readers weresplit on just when this was to occur. One-third of those who re-sponded to this question thought that calculators “should al-ways be available” for student use without any kind of restric-tion. If we interpret the response “always, with no restriction”as an “anytime after kindergarten” response, we have the re-sults from our respondents shown in figure 1.

In a nutshell, our readers told us that calculators are power-ful tools when used appropriately and that they should beused in all grades where problem solving is the main focus.They further stated that calculators allow students to exploreand try new ideas and that they relieve students of cumbersome

Fig. 1

computation, allowing them to concentrate on more meaning-ful mathematical activities. Readers said that as a tool, a calcula-tor can enhance and enrich students’ competencies. Readersadvised caution, however, and strongly advocated education onhow to use calculators. Some of the responses follow.

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There needs to be a balance between mental, paper, and calcula-tor work.

Cyndy MontesGrade 8 Teacher, New Jersey

They should be allowed for visualdemonstrations and should serve toexpedite calculations that studentscan already do.

Maria J. VlahosGrades 9–12 Teacher, Illinois

Calculators and other technologicalresources can be incredibly powerfultools when used in the classroom forprediction and demonstration. It isimperative, though, that teachersgive careful thought about how theycan be used most effectively as a toolrather than a crutch.

Lora C. PitmanGrades 9–12 Teacher, New Jersey

Calculators are tools that studentsneed to know how to use.

Janice JamesGrades 1–3 Teacher, Kentucky

Calculators can be used to teachbasic concepts to students beforegetting into the details of how.

Ronda ColletteGrades 9–12 Teacher, Colorado

I have little tolerance for the equityissue. If kids can prance aroundwith sculptured nails and piercedbody parts, they can afford a calcu-lator. In addition, if they paid fortheir calculators themselves, they’dtake better care of them.

Alice Hess, I.H.M.Grades 9–12 Teacher, Pennsylvania

Calculators should be available after grade:

K 69 Grade 6 14Grade 1 2 Grade 7 21Grade 3 2 Grade 8 38Grade 4 16 Grade 9 4Grade 5 8 Grade 10 6


Students will attempt a multistep,difficult problem with a calculatorbut not without one.

Mary M. SwartGrades 4–9 Teacher, New York

Calculators make higher math ac-cessible to many students who havetrouble with arithmetic.

Laura ReedGrades 9–12 Teacher, Vermont

It is important that children de-velop math sense and understandthe reasonableness of an answer.

June Lange PrewittKindergarten Teacher, Missouri

Use the technology!

Don CardinalGrades 10–12 Teacher, Pennsylvania

A waste of my time. NCTM shouldconcentrate on promoting qualifiedmath teachers in all classrooms andless on this nonsense.

David Detje, FSCGrade 7 Teacher, New York

We must put technology into thehands of our children. Then wemust teach them how and when touse calculators.

Kim P. LoucksGrade 8 Teacher, New York

Calculators should be given to stu-dents before paper and pencils.

William J. HewittGrades 9–12 Teacher, New Jersey

People who lack good grammarskills can be misled by the correc-tions offered by computer grammarcheckers. People who lack goodmath skills (including interpretingresults) can be misled by what theircalculators tell them. Calculatorsmust be used thoughtfully.

Wayne A. WilliamsGrades 6–8 Teacher, North Carolina

Numeracy and conceptual under-standing are developed through avariety of modalities.

Muriel AynanabaGrade 5 Teacher, California

I have high school students who lit-erally can’t tell me 7 × 8 without acalculator.

Jill ThompsonGrades 9–12 Teacher, Missouri

This issue has been the burning onefor centuries. Calculate or com-pute? That is the question.

Robert FooteGrades 6–7 Teacher, Illinois

There are times when calculatorsneed to be held back temporarily—for example, matrix inverses, graph-ing techniques.

Raymond WhippleGrades 9–12 Teacher, Massachusetts

The power of calculators is chang-ing so rapidly. Teachers must up-grade the curriculum to keep up.

Charles MitchellCommunity College Teacher, Illinois

Calculators allow for the explorationof number patterns and use of real-world data. Calculators are part ofeveryday life. We cheat students ifthey are not allowed to use them.

Lucy HahnGrade 2 Teacher, Idaho

It is necessary to learn a variety of al-gorithms as well as use a calculator.

Sylvia Linda CotterGrades 4–5 Teacher, Ohio

Calculators synthesize concepts andapplications that are both vital com-ponents to internalize a “sense” ofmathematics.

Grace CavalloSupervisor, New York

Lower-ability math students need tohave a greater availability of calcu-lators than others.

Steve PetersonGrade 6 Teacher, Minnesota

Certain skills that should not requirea calculator are still useful, but manyother skills are antiquated and obso-lete.

Steve LiebermanGrades 8–12 Teacher, California

Our rural county still does notallow calculators to be used, be-cause of local objections to theiruse on state tests.

Mata J. BanksGrade 8 Teacher, Tennessee

“Students must be prepared for thetwenty-first century” is the currentmantra in support of calculators inthe classroom. Using technology iseasy to learn and certainly does notrequire the hours of practice thatwe give our children. Creating tech-nology is the real challenge. Thiscreativity springs from an under-standing of mathematical ideas andalgorithms, which the prevailingover-reliance on calculators stifles.

Diane HunsakerTutor, California

The resourceful teacher can assessa student’s understanding of con-cepts and even computational skillwithout denying calculator usage.

Craig RussellGrades 8–12 Teacher, Illinois

To educate math students withoutcalculators and computers is to de-prive them of a rich experience incomplex problem solving. However,two things must be remembered.First, basic skills such as mental cal-culations and pencil-and-paper cal-culations must be developed andmaintained. Second, teachers mustreceive training so they become cal-culator literate before calculatorsare used in the classroom.

Holt ZauggGrades 7–12 Teacher, Alberta

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Calculators are an essential technologicaldevice in our society. Schools have the dutyto provide instruction for the appropriateand effective use of calculators. Most teach-ers agree that after students master computa-tional skills, calculator use aids in checkingcomputation and facilitates problem solving.We, however, maintain that calculatorsshould be an integral part of mathematics in-struction including the development of con-cepts and computational skills, and the re-search supports our position.

Hembree and Dessart (1992) reported thefindings of a meta-analysis of the effects ofprecollege calculator use. This research ana-lyzed results from eighty-eight studies fo-cused on students’ achievement and atti-tude. Each study involved one group ofstudents using calculators and anothergroup having no access to calculators. Fromtheir analysis, Hembree and Dessart con-cluded that the calculator did not hinder stu-dents’ acquisition of conceptual knowledgeand that it significantly improved their atti-tude and self-concept concerning mathe-matics.

Smith (1997) conducted a meta-analysisthat extended the results of Hembree andDessart. Smith analyzed twenty-four researchstudies conducted from 1984 through 1995,asking questions about attitude and achieve-ment as a result of student use of calculators.As in the Hembree and Dessart study, test re-sults of students using calculators were com-pared to those of students not using calcula-tors. Smith’s study showed that the calculatorhad a positive effect on increasing concep-tual knowledge. This effect was evidentthrough all grades and statistically significantfor students in third grade, seventh throughtenth grades, and twelfth grade. Smith alsofound that calculator usage had a positive ef-fect on students in both problem solving andcomputation. Smith concluded that the cal-culator improved mathematical computationand did not hinder the development of pen-cil-and-paper skills.

A recent large study examined effects overa longer term. The purpose of the project,Calculators in Primary Mathematics, fundedby the Australian Research Council, DeakinUniversity, and the University of Melbourne,was to have primary and elementary schoolstudents explore and develop number sense

using calculators before standard algorithmswere taught (Groves and Stacey 1998). It in-volved one thousand students and eightyteachers over a three-year period. The per-formance of students in the project wascompared with that of a control group forthe same schools using a written test, a cal-culator test, and an interview. The resultsshowed that the project students performedbetter overall on a wide range of items in-cluding place value, decimals, negative num-bers, and mental computation. No detri-mental effects of calculator use wereobserved.

Until curricular innovations such as thattried in Australia are implemented, we be-lieve that students and teachers should dis-tinguish among three tools of computation:mental arithmetic, pencil and paper, and cal-culators. For example, we would chastise anystudent who reaches for the calculator tofind 3 × 4; we would suggest pencil andpaper for calculating 27 × 340; and we wouldinsist on using the calculator for 2.7568 ×345.8972 after the student estimates mentallyan answer of 900 (3 × 300).

In conclusion, we recommend thatschools strongly encourage the use of calcu-lators in all aspects of mathematical instruc-tion including the development of mathe-matical concepts and the acquisition ofcomputational skills. We believe that calcula-tor education is an obligation of schools toour society where calculators are in com-mon, daily use.

ReferencesGroves, Susie, and Kaye Stacey. “Calculators in

Primary Mathematics: Exploring Number be-fore Teaching Algorithms.” In The Teaching andLearning of Algorithms in School Mathematics,edited by Lorna J. Morrow, pp. 120–29. Reston,Va.: National Council of Teachers of Mathe-matics, 1998.

Hembree, Ray, and Donald J. Dessart. “Researchon Calculators in Mathematics Education.” InCalculators in Mathematics Education, edited byJames T. Fey and Christian R. Hirsch, pp.23–32. Reston, Va.: National Council of Teach-ers of Mathematics, 1992.

Smith, Brian A. “A Meta-analysis of Outcomesfrom the Use of Calculators in Mathematics Ed-ucation.” Dissertation Abstracts International 58(1997): 787A.

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Donald J. Dessart has beena professor of mathematicsand mathematics education atthe University of Tennessee—Knoxville (UTK) since 1962.Charleen M. DeRidder is anadjunct professor of mathe-matics education at UTK andsupervisor of mathematics forgrades K–12 in the KnoxCounty Schools of Tennessee,and Aimee J. Ellington is aninstructor of mathematics atthe University of Tennessee atChattanooga.

The Research Backs Calculatorsby Donald J. Dessart, Charleen M. DeRidder, and Aimee J. Ellington

Donald Dessart

Aimee J. Ellington

Charleen M. DeRidder


A calculator is not only an essential tool in today’s world buta bridge for young children into the world of scientific tech-nology. Our reluctance, as educators, to teach children to usecalculators as a mathematical tool promotes inequities that willlater affect students’ access to col-lege courses and to work. When wedon’t structure equitable opportuni-ties for children to use technology,we allow social forces to take over,leading to more comfort with tech-nology on the part of, and usage by,males than females and more com-fort on the part of those studentswhose families are willing and ableto buy and use technology.

Historically, the use of tools hasempowered tool users to gain con-trol over their environment and tobecome experts, with associatedeconomic benefits. In fact, the useof tools, along with language, is theessence of being human. The use oftechnologically advanced tools, es-pecially those associated with paid work rather than house-work, has been the province of males. We will need to workhard to make sure that these cultural factors do not preventfemales (and other historically disenfranchised groups) fromhaving equal access to important modern-day tools. TheAmerican Association of University Women (AAUW), in itsrecent publication Gender Gaps: Where Schools Still Fail OurChildren, has identified the use of technology as havingamong the most significant gender gaps of any area inschool.

Lest you are about to jump up and toss calculators into the classroom, let me

caution you that this iswhere things gettricky. What if calcula-tor use does not pro-mote a “girl friendly”learning environ-ment? According to

the AAUW again,“Girls have developed

an appreciably differ-ent relationship to tech-nology than boys … andtechnology may exacer-bate rather than diminish

inequities by gender as itbecomes more integral to

the K–12 classroom.” Moreboys have and use technological

tools and toys. Boys more often perceive themselves as goinginto careers, such as engineering, that require technology likecalculators and computers. They can envision a payoff for learn-ing to use these tools. We need to be sensitive to past and pre-

sent inequities when structuring opportunities for all children.

In our own program, Summer-Math, a four-week program for adiverse group of high school girlsof varied mathematical experi-ences, girls enthusiastically take totechnology—both calculators andcomputers. But we are not satisfiedwith girls using technology onlywhen others say to do so. Our ob-jective is for girls to be powerfulusers of technology—initiators oftechnology use. This summer weare recognizing the importance ofthe ownership of tools by buyingeach student a graphing calculatorto take home.

Unstructured experiences with technology will simply pre-serve the social order—opportunities need to be structuredin order to be equitable. The AAUW has many suggestionson this topic. It is incumbent on us as educators to ensurethat these structured opportunities occur in school. Al-though mathematics educators argue about whether or notto teach and encourage children to use calculators from anearly age, some children—-more males than females—willbe learning to use them anyway and thus will be ready for awider array of college andcareer opportunities.

Charlene Morrow, [email protected], is the co-directorof the SummerMath program at Mount Holyoke College, SouthHadley, MA 01075-1441, whereshe is a faculty member in thePsychology and Education Department. She is a past chairof the NCTM Committee for aComprehensive Mathematics Ed-ucation of Every Child and acoauthor of the new book Notable Women in Mathe-matics. Her current work in-volves the exploration of mathe-matical ideas in quilts and inorigami.

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Ensuring That All Children Are Powerful Technology Usersby Charlene Morrow


by Johnny Lott, For the Editorial Panel

Although there may be controversy in some segments of the popula-tion about student use of calculators in learning and doing mathe-matics, that controversy is not apparent in most policy documents atthe provincial or state level. Most policies are very similar, only withsome having more specificity than others. The adapted samplesbelow are representative of the genre.

Canadian ViewpointsThe intelligent use of a graphing calculator should be em-

phasized at all times. Students should think about the bestmethod of solution before they reach for their calculators.Up to 15 percent of the provincial exam will contain ques-tions that are dependent on a graphing calculator. Using agraphing calculator, a student should be able to—

1. produce a graph within a specified viewing window;2. determine an appropriate viewing window to examine a

graph and change the window’s dimensions;3. use the zoom features of the calculator;4. find zeroes and intersection points (usually to two decimal

places);5. find maximum or minimum points.…

Most of the currently available graphing calculators have the built-in capabili-

ties to do all the skillsabove; however, some ofthe older models mayrequire programs.

C a l c u l a t o rm e m o r i e swill not becleared atthe time ofe x a m i n a -tion. Many

g r a p h i n gca lcu la tor s

do much morethan these five

specified calculatorskills. Teachers are en-

couraged to use othergraphing calculatorfeatures, such as para-metric or polargraphs, to enhancethe learning ofmathematics in the

classroom. However,

teachers and students should be assured that only the spe-cific calculator skills listed above will be required for theprovincial examination.

British Columbia Graphing Calculator Resource Package for Principles of Mathematics 12, p. 5

Emphasis is placed on the role of technology and the ap-propriate concepts and skills related to its use. Changes intechnology and the broadening of the areas in which mathe-matics is applied have resulted in growth and changes in thediscipline of mathematics itself. The new technology notonly has made calculations and graphing easier but has alsochanged the very nature of the problems important tomathematics and the methods mathematicians use to investi-gate them.

Atlantic Canada Mathematics Curriculum Teachers Draft Guidelines, p. ii

Scientific calculators, graphing calculators and relatedprobes, dynamic graphing software, and accessing the Inter-net are becoming commonplace in our students’ lives. Tech-nology makes students more powerful learners by allowingthem to visually explore mathematical concepts more easilyand quickly. This allows fundamental ideas to be studied ingreater depth, giving students more time for exploration inthe areas of data collection, data analysis, simulations, andcomplex problem solving. Whereas investigators once reliedon their creativity and the sophistication of known mathe-matical models to guide them in the solution of problems,technology now provides capabilities that alter both theform and the means of solution. Calculators save time inperforming complex arithmetic calculations. Graphing utili-ties enable students to visualize relationships and test hy-potheses.

Ontario—Secondary Policy Document for Mathematics,Grades 9 and 10 (in draft form), p. 4

Overall expectations in the Ontario Curriculum, Grades1–8, include the following:

• Use a calculator to solve number questions that are be-yond the proficiency expectations for operations usingpaper and pencil.

• Justify the choice of method for calculations: estimation,mental computation, concrete materials, pencil andpaper, algorithms (rules for calculations), or calculators.

Ontario—The Ontario Curriculum, Grades 1–8: Mathematics, Grade 8, p. 26

U.S. ViewpointsIn Illinois, Mississippi, New Jersey, and Colorado, there are specific

statements about calculator use in the state standards. These standards

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A Calculator Touraround Canada and the United States


are more generic than some of the ones seen in the Canadian provincesabove, but they mirror comparable state standards in the rest of theUnited States.

General Mathematics Standards• Identify and describe patterns and relationships in ac-

tual data, as well as solve problems and predict resultsusing algebraic methods and symbols, tables, graphs,calculators, and computers.

• Analyze, categorize, and draw conclusions about objectsand spatial relationships using geometric methods anddrawings, sketches, graphs, models, symbols, calculators,and computers.

Adapted from Illinois Academic Standardsin Learning Areas (Draft, 1996)

Grade 8 Algebra• Solve equations and inequalities containing rational co-

efficients; include real-life problem-solving situations;use manipulative materials and calculators or computerswhere appropriate.

Adapted from Mississippi Mathematics Curriculum Structure (1995)

Grade 8 Mathematics• Explore linear equations through the use of calculators,

computers, and other technology.

Taken from New Jersey Core Curriculum Content Standards (1996)

General Mathematics Standards• Students link concepts and procedures as they develop

and use computational techniques, including estimation,mental arithmetic, paper and pencil, calculators, andcomputers, in problem-solving situations and communi-cate the reasoning used in solving these problems.

Benchmark for Mathematics Standards: Grades 5–8• Solving simple linear equations in problem-solving situa-

tions using a variety of methods (informal, formal,graphical) and a variety of tools (physical materials, cal-culators, computers).

Adapted from Colorado Model Content Standards for Mathematics (1995)

(Other state standards like those above can be found atwww.ccsso.org.)

In stark contrast to other state and provincial calculator policystatements is the Mathematics Framework for California PublicSchools K–12. Selected pertinent quotes follow:

The Mathematics Standards were written with the beliefthat there is a body of mathematical knowledge—indepen-dent of technology—that every student in K–12 ought toknow, and know well. More importantly, the STAR assess-ment program—carefully formulated to be in line with theStandards—does not allow the use of calculators all throughK–11.

More to the point, it is imperative that students in earlygrades be given every opportunity to develop a facility withbasic arithmetic skills without reliance on calculators.

For these reasons, this Framework recommends that calcu-lators not be used in the classroom before grade 6.

Beyond the anecdotal, there is also the input from theThird International Mathematics and Science Study(TIMSS). For the 8th grade assessment, the majority (>50%)of the students from three of the five nations with top scores(Belgium, Korea, and Japan) never or rarely (once or twice amonth) used calculators in mathematics classes. In contrast,the majority of students (>65%) from 10 of 11 nations, in-cluding the United States, with scores below the interna-tional mean, used calculators almost every day or severaltimes a week in mathematics classes (Beaton, Mullis, Martin,Gonzalez, Kelly, and Smith 1996) While such data do notprove that calculator usage is damaging to the developmentof mathematical skills, it would be folly to ignore this.

Taken from Mathematics Framework for California PublicSchools K–12 (draft, 1998)

All the standards above relate to precollege mathematics. TheAmerican Mathematical Association of Two-Year Colleges has takena strong stand on the use of calculators and technology, as seen inthe following:

Basic Principle• The use of technology is an essential part of an up-

to-date curriculum.

• Faculty and students will make effective use of appropri-ate technology. The technology available to studentsshould include, but not be limited to, that used by prac-titioners in the field. Faculty should take advantage ofsoftware and graphing calculators that are designedspecifically as teaching and learning tools. The technol-ogy must have graphics, computer algebra, spreadsheet,interactive geometry, and statistical capabilities. (p. 5)

Standard I–6: Using Technology• Students will use appropriate technology to enhance

their mathematical thinking and understanding and tosolve mathematical problems and judge the reasonable-ness of their results. (p. 11)

Standard P–1: Teaching with Technology • Mathematics faculty will model the use of appropriate

technology in the teaching of mathematics so that stu-dents can benefit from the opportunities it presents as amedium of instruction. (p. 15)

Crossroads in Mathematics: Standards for IntroductoryCollege Mathematics before Calculus (1995)

9


I was probably one of the first ele-mentary school students ever to use acalculator. As a second grader in ele-mentary school, I remember mar-veling at my father’s Hewlett PackardHP-35 electronic calculator. I believethis calculator was the first commer-cially available pocket scientific calcu-lator. The HP-35 was introduced withmuch fanfare in 1972 and was sonamed because it had 35 keys. My fa-ther, an associate professor of civil en-gineering at the Rensselaer Polytech-nic Institute (RPI) in Troy, New York,bought one of the calculators for$350, no small sum considering my fa-ther’s annual salary was only $15,000.

Prior to purchasing the HP-35, my fa-ther owned a large, cumbersome me-chanical calculator. Even though hehimself invested a significant amountin these labor-saving devices, my fatherstill insisted that I learn basic computa-tional skills through labor-intensivemethods: through memorization and,yes, drill. He didn’t, however, forbid myuse of the calculator.

I was allowed to use the calculatoronly to check answers to my home-work. Being able to hold and use thisprecious and unique device, albeit ina very controlled setting, was powerfulmotivation for me to do my mathe-matics homework.

My present opinions on calculatorsstem from these early experiences.Therefore, at the elementary schoollevel, I believe that the use of calcula-tors should be restricted and con-trolled. It is not until the high schoolgrades that I believe students shouldbe allowed to use calculators in an un-controlled setting (meaning that stu-dents have calculators readily avail-able and exercise their own discretionon when and how to use them).

Do I think calculators have value inthe classroom? Yes. I think calculatorscan serve two main purposes: (1) re-ducing the time spent performing te-dious calculations and (2) illustratingconcepts. For example, I don’t objectto students using a calculator to calcu-late to several decimal places the areaof a circle. A typical scientific calcula-tor uses up to 10 decimal places in acalculation. I also think calculators areuseful in illustrating concepts—for ex-ample, the concept of exponentialgrowth: repeated multiplication by 2can rapidly result in a large number.The calculator can show this growtheasily and efficiently.

Note that both examples cited aresituations that arise in the later ele-mentary school years. My concernabout calculators belies a muchgreater concern of mine: the teachingand practice of very basic mathemati-cal concepts. I repeat here one cele-brated example cited by former Assis-

tant Secretary of Education DianeRavitch in her book National Standardsin American Education. Ravitch notedthat on the 1986 NAEP (National As-sessment of Educational Progress), no9-year-olds, fewer than 0.5 percent ofthe 13-year-olds, and only 6.4 percentof the 17-year-olds could solve prob-lems like this one:

Christine borrowed $850 for oneyear from the Friendly FinanceCompany. If she paid 12% simpleinterest on the loan, what was thetotal amount she repaid?

More recent national and interna-tional tests reveal that students todayare still unable to solve such prob-lems. Would having a calculator havehelped students solve the problem? Icontend not.

The problem is not whether ele-mentary school students should be al-lowed to use calculators. The problemis that our elementary school studentscannot solve problems involving basicmathematical concepts such as per-cents, ratios, and proportions. Ifsomeone can show me that using cal-culators can help elementary schoolstudents achieve a mastery of thesebasic mathematical concepts, I wouldheartily welcome them; otherwise, Isuggest the same sort of caution andcare with calculators that my fatherexercised with me.

Questioning the Use of

Calculators in the

Elementary Grades

Frank Wang is president of SaxonPublishers. He received his Ph.D. inpure mathematics from MIT in 1991.

by Frank Wang

Mathematics Education DialoguesMay/June 1999 11

Thanks and anAnnouncement

Dialogues began in the sum-mer of 1997 as an idea of somemembers of the NCTM Boardof Directors who were con-cerned that there was noNCTM periodical in which im-portant controversial issuesthat cover all the grades couldbe discussed in depth. TheBoard immediately approvedthe idea of two trial issues inthe 1997–98 school year and—in an unprecedented move—placed three Board memberson the six-member task forceto bring the idea to fruition.The strong positive reactionsto those issues led the Boardto convert the task force intoan editorial panel and to ap-prove three issues for 1998–99and succeeding school years.

I have had the honor of chairing the task force andbeing editor for this year’s is-sues. It has been a pleasureworking with fellow Boardmembers Peggy House andJohnny Lott and also with Cyn-thia Ballheim, Hung-Hsi Wu,and Barbara Marshall on thetask force and then workingwith Peggy, Johnny, Cynthia,and Hyman Bass on the Edito-rial Panel. It also has beenwonderful to have the assis-tance of the NCTM staff withdesign and other publicationissues. And we have all verymuch appreciated the re-sponses that you, the Dialoguesreaders, have made. Withoutyour contributions, there canbe no dialogue and so therecan be no Dialogues.

We are pleased that JohnnyLott was asked—and hasagreed—to be the editor ofDialogues for the next twoyears. He will be assisted nextyear by an editorial panel con-sisting of Cynthia Ballheim,Paul Zorn, and Frank Lester.

Zalman Usiskin, Editor

Calculators at the Elementary School Level?Yes, It Just Makes Sense!by Randall I. Charles

Randy Charles is a professor in the Department of Mathematics and Computer Science, San Jose State Univer-sity, San Jose, California. He has been involved in curricu-lum development at the elementary grades for nearly twentyyears. Developing a vision and a plan for calculator use atthe early grades has been part of his professional effort duringthese years.

Calculator use in the ele-mentary grades does makesense. In fact, it is essentialto attaining key goals forthe mathematics educationof young children.

We use calculators in theelementary grades becausenot using them is almostcertain to lead to the devel-opment of habits that arecounterproductive to thedevelopment of numbersense, problem solving, andpositive dispositions.

Number sense is a founda-tion for early success withmathematics. Calculatorscan be used as a tool to

help in developing the con-ceptual understandings andabilities that underlie strongnumber sense. Calculatorsare particularly powerful inenabling children to makeand test conjectures andgeneralizations related tonumbers and operations.For example, making andtesting conjectures aboutcounting patterns helpschildren understand num-ber relationships, developsflexibility with numbers,and promotes the develop-ment of mental and paper-and-pencil computationalstrategies.

Problem solving is a prin-cipal reason for studyingmathematics. The use of cal-culators allows realistic datato be used as problem con-texts, problems whose solu-tions are within the concep-tual grasp of children butwhose computational de-mands are not. The use ofrealistic data is motivationaland helps children see con-nections between schoolmathematics and the mathe-matics used in the world.

Positive dispositions are fun-damental for success inmathematics. Children’s life-long beliefs about calculatoruse are shaped to a great extent by their mathemati-cal experiences in the earlygrades. At the same time we are helping children develop accurate and effi-cient mental strategies andpaper-and-pencil calculationstrategies, it is also our re-sponsibility to help childrenbecome responsible users ofcalculators. Children mustlearn that there are timeswhen it makes sense to use acalculator and times when itdoes not.


Long before the high-tech calculators of today, Mesa SchoolDistrict conducted research that showed how calculators werenecessary tools, like rulers and compasses, and how they couldassist students in the process of solving complex mathematicalproblems. A threefold plan was developed that would providetraining for specialists and teachers, assist in the implementa-tion of calculators in the classroom, and support ongoing useof calculators in the classroom to promote students’ mathe-matical thinking.

First, the district-based specialists in mathematics were trainedon the use of various calculators (four-function, fraction, andgraphing calculators). These specialists, along with the directorof mathematics, trained teachers at schools on a specific calcula-tor along with appropriate activities where calculators could beused to promote and enhance students’ mathematical thinking.Teachers were taught that if they were teaching a mathematicalconcept where they did not want the computation to interferewith learning the concept, then it was appropriate to use the cal-culator to facilitate the instruction and learning. As the teacherswere trained, they received a class set of calculators along withan overhead projection calculator.

Second, teachers receive support as they implement the use ofcalculators. This is done through many models of professional de-velopment, such as miniconferences on Saturdays, workshopsafter school, specialists providing demonstration lessons in theclassrooms, and the mathematics leadership at each site receivingadditional training in the use of calculators throughout all stan-dards-based lessons. The mathematics leadership can then fur-nish on-site support to all mathematics teachers. We have foundthat the key to appropriate calculator use in the classrooms is ex-cellent professional development for teachers and access to calcu-lators at any time throughout the day.

Third, support for the ongoing use of calculators is always ev-ident. Currently, we are training our teachers in grades 7–9 onthe TI-73 calculator, and at the end of the training they will re-ceive a classroom set of calculators along with the overheadprojection calculator. Future purchases will include Rangerprobes so that “real time” data may be observed, recorded, andmanipulated through investigations. In our science classrooms,teachers are using the TI-82 calculators with probes to experi-ment with data involving such investigations as heart rate, pres-sure-volume relationship in gases, acid-base titration, light in-tensity, and motion. When students work with data analysis andthe resulting statistics, the calculator is used as a tool for reason-ing and for analyzing and interpreting data. Students can seethe connections between the mathematics and the scienceclassroom. There are always ample opportunities for all teach-ers to receive introductory or advanced training on using the

calculator as atool to promote mathematical thinking.

Calculators give students an opportunity to manipulate largenumbers while solving higher-level thinking investigations. Wehave found that calculators help students to think flexibly.They aid in the exploration of various techniques for solvingand evaluating different situations. Also, calculators assist stu-dents with organizing and storing data as well as graphing con-clusions. Students learn not only how to use the calculatorscorrectly but when calculator use is appropriate. Students alsolearn when an exact answer is appropriate and when an esti-mate is appropriate. To simulate real life, calculators are read-ily available in the classroom, allowing students to focus on themathematics rather than on the novelty of the tool. In fact, stu-dents learn when use of mental mathematics, paper and pen-cil, or a calculator is appropriate.

As stated in the NCTM Professional Standards for TeachingMathematics: Executive Summary, “Today’s students will be citi-zens of the twenty-first century, a century that promises to bedramatically different from the one we have known. The ef-fects of technological innovation will continue to permeateevery aspect of life.… [C]omputational skills alone do not suf-fice.… By the turn of the century, the need to understandmathematics in order to succeed in all walks of life will bewithout precedent”(p. 3). We in Mesa use calculators as a toolto help develop students’ deeper understanding of the mathe-matics that will be needed in the future.

12

Tools for MathematicalUnderstandingin Middle Schoolby Perry Montoya and Vicki Graber

Vicki Graber is a mathematics specialist and Perry Montoya is an instructional-technology specialist in the Mesa School District, Phoenix,Arizona. The district consists of sixty-one elementary schools, thirteen ju-nior high schools, and five high schools serving more than 70000 studentsin a city of 350000 people. They work collaboratively with many other dis-trict-based specialists to provide comprehensive professional development inusing calculators as tools for mathematical understanding.


From the very first day in the late1980s when I gave my students graphingcalculators, my classroom changed. Thechange was gradual—I started by givingtests where students turned in a “withoutyour calculator” page before receivingthe rest of the test, where a calculatorwas permitted. But I found myself ques-tioning whether it was really a signal ofstudents’ ability to do mathematics ifthey could find the sine of π/12. Whatdoes it tell me about students’ under-standing that I cannot get by askingthem how the sum and difference for-mulas can be used to derive the formulafor sin 2φ? The mathematics had tocome from the questions I asked, notthe tools students use to answer.

What is different? My focus in algebrahad been on routine procedures forsolving equations or systems of equa-tions, such as the linear combinationmethod. Calculators allowed me toswitch my focus to thinking and reason-ing about, and with, the mathematicalconcepts we were studying and as a mat-ter of fact even about some of the proce-dures. Access to calculators allowed us todo interesting problems we could not dobefore with real and messy numbers,and it allowed students who had troublewith computation or procedures to haveaccess to another way to do those thingsso they could move on to the mathe-matics we were studying. Calculatorschanged the nature of my courses andthe way I taught.

Calculators allow me to do old thingsbetter. Moving freely among tables,graphs, and symbols helps students un-derstand relationships previouslytaught as separate entities. Studentsonce spent a whole class filling in atable for y = sin x, y = sin 2x, y = sin 0.5x,y = 2 sin x, where finding sin (2π)/3 be-came the focus; the effect of the vari-able on the range or on the shape ofthe curve received a quick “look” for

those who waded through the compu-tation. With a calculator, studentsquickly generate enough specific exam-ples to see and discuss the effect. Thelesson is on the mathematical idea Iwant to teach, not on how to set it up.“List” functions help students developthe notion of variable and work withformulas in a different way.

Calculators made it possible for my stu-dents to learn new concepts that were notpossible to teach before. We now study re-gression, correlation, and modeling inearly high school mathematics. Loga-rithms are used to handle scaling effectsand transform data, not for computation.Force fields are now part of calculus. Stu-dents use matrices to manage informationand mathematical situations and to solvesystems of equations. They discuss the lim-itations and conditions on the process,freed from routine calculations to gener-alize about solving systems.

Calculators also make some of thethings we used to do unnecessary. Anexample is standard deviation. Statisticstexts presented the formula for stan-dard deviation, a statistic that describesthe variation of a data point from themean, by first using the definition thathighlights the difference of each datapoint from the mean,

In practice, to avoid tedious compu-tation, students were instructed to usean equivalent formula,

where the difference is not obvious. Asa result, many students never reallycame to understand the concept.

Although the content has changed, amore important change has occurredin my students. They are ready to inves-

tigate any option, they accept a chal-lenge, and they also make mistakes.They do not always use their calculatorswisely, but then most of them are doingthings they never would have done be-fore. Many of them would not evenhave been in that mathematics class. Mystudents think about mathematics dif-ferently from the way I do. They reasonfrom graphs about differences in func-tional values; they reason from tablesabout relationships; they use the replaykey to sort their thinking; they reasonnumerically; and they confront proof ina different way from what is presentedin their geometry book. They use a cal-culator when I least expect it on prob-lems where I cannot see how one willhelp.

As in any revolution, upheaval, con-cern, and dissension must be ex-pected; so, too, in the mathematics ed-ucation reform movement upheaval,concern, and dissension about the useof calculators must be expected. But asa teacher, I have a responsibility to pre-pare my students for a world wheretechnology is dominant and to makeuse of tools that will enable them tolearn more mathematics and to learnit in a deeper way. And it is essential asI move with my students into the nextgeneration of calculators that I remainfocused on the mathematics I wantthem to learn, not on buttons to push.Giving students a state-of-the-art calcu-lator gives them access to the power ofmathematics.

Gail Burrill, the immediate past presi-dent of NCTM, is a classroom teacherfrom suburban Milwaukee, Wisconsin.She is currently on staff at the Center forEducation Research at the University ofWisconsin—Madison and is on loan tothe Center for Science, Mathematics, andEngineering Education at the NationalResearch Council, where she is a seniorprogram officer.

13

x x

ni −( )∑ 2

.

x x

ni2 2−∑ ,

A Revolution inMy High SchoolClassroomby Gail Burrill

An Attempt in Sweden at Consensus


Excerpted by Zalman Usiskin from a paper given by Lars-EricBjörk and Hans Brolin presented at the University of Chicago Sci-ence and Mathematics Program’s Fourth International Conferenceon Mathematics Education in August 1998.

This report is from the ADM (theSwedish acronym for “analysis of the

consequences of the computer for mathe-matics education”) project at the Depart-

ment of Teacher Training at the Uni-versity of Uppsala, Sweden. Wepresent examples of what routineskills in the topics of algebra,trigonometry, derivatives, and func-

tions and integrals students who intend to major in a mathematical sci-

ence at university level should be able todo by hand. We also give examples ofwhen it is appropriate to use such techno-logical devices as graphing calculators.The task of listing important topics wascarried out jointly with a large number ofsecondary school and university teachers.

Algebra and trigonometryStudents graduating from secondary schools

today do not have the same level of skill in alge-bra as their counterparts of twenty or thirty years

ago did. This can be seen both in internationalstudies and in the diagnostic tests given at the uni-versities. This is partly because of broader student

recruitment and partly because algebraic manipula-tion has been postponed ever later in the Swedish school sys-tem. There are mainly two reasons for this. First, algebraic ma-nipulation without understanding and application is of littlevalue, and second, the first course, Mathematics A, which iscommon for all streams, contains very little content of a dis-tinctly secondary school nature. In addition, high skill in alge-bra requires repeated review, which is not easy to arrange be-cause of time shortage and modular courses. Deficientalgebraic skills often constitute a severe impediment in thestruggle for higher achievement levels in the understanding ofconcepts and in formulating and solving problems.

The following list of problems provides examples of alge-braic skills that should be expected of students who have stud-ied mathematics courses A, B, C, and D and who intend tomajor in a mathematical science at university level. Studentswith merely a “Pass” in the D course do not satisfactorily fulfillthis level of algebraic skill. The problems should not be re-garded as defining primary goals but rather as examples ofthe manipulative skills needed in order to study and under-stand mathematical text and to be able to solve problems.

Powers and logarithms

Polynomials

Mixed rational expressions

Solving equations and inequalities

Formula manipulation

Trigonometry

22. Trigonometry in right-angled triangles. Be able to useexact function values for 30°, 45°, and 60° and to changeangles readily from degrees to radians and vice versa.

1. Simplify a) b)

2. Simplify a) b)

x x xx x

x⋅ ⋅

( ) ⋅

− ( ) +

316

2 3

7

70 7 1 9 27lg lg ln ln

3. Simplify a)

b)

4. Factor a) –

b)

c) 3

5. Complete the square

6. Simplify if

x x x x

x x x

x x

x x x

x xy y

x x

f x f x f x x x

2 2

2

3

2

2 2

2

2

1 1 3 2

3 1 3 1 4 2 3

4

1 4 1

18 27

6

3 2 3

+( ) −( ) − +( )+( ) −( ) − +( )

−( ) − −( )− +

++( ) − ( ) ( ) = + −

7. Simplify a) b)

8. Simplify

9. Simplify a + h if

xx

xx x x

a aaa

f f a f x x

2

3

2

2

1 1 3 21

11 2

21 2

6 24 1

− + −−

−−

++ +

−( ) − ( ) ( ) = −

10. Solve the equation a) b) 12

c)

11. Solve the equation a) b)

2 3 46 3

2 8 42 0

120 6

1 11

2

2

x x xx x

x x

x x x

+ − = + =

+ − =

= −−

=.

12. Solve the simultaneous equations

a) b) y x

x yy x x

y x

+ =− =

= −= −

8

2 73

2 6

2

13. Solve the equations a) –

b)

x x

x x x

1 9 0

2 3 0

2

3 2

( ) −( ) =

− − =

14. Solve the equations a) g + = g

b) 1n 1n 1n

1 1 1 7

1 2 12

x

x x x+( ) + −( ) = −( )15. Solve the equations a) b) c)

16. Solve the equations

17. Solve the inequality

5 +x

x x

x x

x x= = =

+ = −+ ≥ −

2 2 5 10 4

2 1 1

2 1 4 2

2 1

18. Solve for if + =

19. Solve for if = +1

20. Solve for if =

x ax b cx

C A B C

pa

pc

1+

14

21. Solve for if z AB

c z=

+( )2

Mathematics Education DialoguesMay/June 1999 15

23. Triangle theorems (Area of triangle = ab sin C, law ofsines, law of cosines)

24. Solve basic trigonometric equations completely.For example, find all solutions for cos (3x + π/2) = 0.5

25. Use the unit circle to demonstrate simple relationships. For example, sin (x + 180°) = –sin x

26. Be able to usesin (x ± y) = sin x cos y ± cos x sin y,

cos (x ± y) = cos x cos y + sin x sin y

sin 2x = 2 sin x cos x, cos 2x = cos2 x – sin2 x

sin2x + cos2x = 1to rewrite expressions such as

(sinx + cos x)2 = 1 + sin 2x

a sin x + b cos x = A sin(x + v)

Derivatives, plotting curves, integralsIn the parts of previous secondary school courses dealing

with derivatives, curve construction, and integrals, the em-phasis was largely on algebraic skills and rote procedures.Today, using the graphing calculator, algebraic formulationof problems can be complemented with numerical andgraphical representation, which substantially facilitates stu-dents’ understanding and permits the use of realistic mathe-matical models for real-life situations.

The following level of skill should be required of studentswho have studied courses A–D at secondary school.

DerivativesWithout calculating devices

In various applications, be able to• realize that the value of a certain derivative should be

calculated• interpret the meaning of a certain value of a derivative.

Be able to use basic differentiation rules on elementaryfunctions (except for inverse trigonometric functions)

y = x n, y = e x, y = a x, y = ln xy = sin x, y = cos xy = 1/x, y = x, y = 1/ x , y = x p

as well as linear combinations of these.

Be able to use the chain rule in such examples as y = sin 2x, y = e3x, y = ln(4x), y = 1 – x , y = 3 sin(2x + 4)y = (sin x)10, y = e x2

Be able to use the product and quotient rules in simple functions like

With a graphing calculator

Be able to calculate the numerical value of a derivative.

Curve ConstructionWithout calculating devices

Be able to interpret graphs and understand the graphicalsignificance of f(a ), f '(a ), f(x) = 0, f(x) = k, as well as thezeroes, extreme values, and concavity of f .

Understand the connection between the derivative and theappearance of the graph (the essence of traditional signstudy); be able to state the equation for the tangent at apoint.

Recognize and be able to sketch, with or without a table ofvalues, simple curves for

• linear functions• quadratic functions• third-degree polynomials, for example, y = ax 3 + bx• trigonometric functions, for example,

y = sin x, y = sin 3x, y = 2sin x, y = sin x + 2,y = sin (x + π/3)

• exponential and logarithmic functions such asy = 2x, y = e 2x, y = lnx

• power functions such as y = x – 2, y = 5/x , y = 1/x 2

Great importance is placed on understanding the generaland complete appearance of the function graphs studied.Special attention is given to the linear function and the abil-ity to state the equation for a line through two points.

With a graphing calculator

In applications, graphs are plotted with graphing calcula-tors. Good skill in choosing suitable viewing windows for thecoordinate system is desirable. Interesting points such as ex-treme points, zeroes, and points of intersection with axes arefound using the calculator’s tool-kit.

Integrals

In applications, be able to• set up an integral• interpret the meaning of an integral.

Without calculating devices (analytic integration)

Using an antiderivative, be able to calculate by hand verysimple integrals for

• polynomials, for example, f(x) = 5x 5 – x/4• trigonometric functions, for example, f(x) = 3 sin 4x• exponential functions, for example, f(x) = 4e 3x

• power functions, for example, f(x) = 5/x , f(x) = 3/x2, f(x) = 4 x ,

and to understand that an antiderivative can be verified bydifferentiation.

With a graphing calculator (numerical integration)

In applied problems, the programs in the calculator areused to calculate definite integrals. The essential part is setting up the integral. Exact calculation of integrals is de-emphasized.

Hans Brolin is professor of mathematics in the Departmentof Teacher Training at the University of Uppsala, Sweden. Heholds a doctorate in pure mathematics, but his primary interesthas been research in mathematics education. Lars-Eric Björkholds a master’s degree and is currently at Sunnerboskolan inLjungby, Sweden. He has taught mathematics courses in sec-ondary schools for many years and in 1996 received a doctoratehonoris causa at the University of Uppsala for his work inmathematics education.

sincos2 1 22

xx= −

y e x yx

xy

xx

x= = +−

=2 32 15 3

sin , ,sincos

Documents

A publication of the National Council of Teachers of Mathematics …oldmoodle.escco.org/file.php/1/MATH/Groping_Hoping.pdf · 2006-01-05 · A publication of the National Council