HowMathematics

CountsFractions and algebra represent the most subtle, powerful,

and mind-twisting elements of school mathematics.But how can we teach them so students understand?

Lynn Arthur Steen

Much to the surprise of ihosewho care about such things,mathematics has hecorae theSOO-poLind gorilla in U.S.schools. High-slakes testing

has forced schools to push aside subjects likehistory, science, music, and art in a scramble toavoid the embarrassing consequences of notmaking "adequate yearly progress" in mathe-matics. Reverberations of the math wars of the1990s roil parents and teachers as they seek firmfooting in today's turbulent debates about mathe-matics education.

Much contention occurs near the ends ofelementary and secondary education, wherestudents encounter topics that many Rnd difficultand some tind incomprehensible. In earlierdecades, schools simply left students in the lattercategory behind. Today that option is neitherpolitically nor legally acceptable. Two top ics -fractions and algebra, especially Algebra !I—areparticularly troublesome. Many aduks, includingsome teachers, live their entire lives flummoxedby problems requiring any but the simplest offractions or algebraic formulas. It is easy to seewhy these topics are especially netdesome in

today's school environment. They are exemplarsof why mathematics counts and why the subjectis so controversial.

Confounded by FractionsWhat is the approximate value, to ihe nearestwhole number, of the sum 19/20 + 23/25? Giventhe choices of 1, 2, 42, or 45 on an intemationaitest, more than half of U.S. 8th graders chose 42or 45. Those responses are akin to decoding andpronouncing the word elephant but haWng noidea v '̂hat animal the word represents. Thesestudents had no idea that 19/20 is a numberclose to 1, as is 23/25.

Neither, it is likely, did their parents. Fewadults understand fractions well enough to usethem lluently Because people avoid fractions intheir own lives, some question why schools (andnow entire stales) should insist that all studentsknow, for instance, how to add uncommoncombinations like 2/7 + 9/13 or how to divide1 3/4 by 2/3. When, skeptics ask, is the iast timeany typical adult encountered problems of thissort? Even mathematics teachers have a hardtime imagining authentic problems that requirethese exotic calculations (Ma, 1999).

AS.SOC1M"ION FOR SU i'LUVI.SION AND CURKlCUiUM DUVtLOPMbiNl

Moreover, many peoplecannot properly express incorrect English the fractionsand proportions ihat docotnmotily occur, for instance,in ordinary tables of data. Asimple example illustrates thisdifficulty (Schield, 2002).Even though most peopleknow that 20 percent means1/5 of something, manycannot figure out what thesomething is when confrontedwith an actual example, suchas the table in Figure 1.Although calculators can helpthe innumerate cope withsuch exotica as 2/7 + 9/13 and1 3/4 H- 2/3, they are of nohelp to someone who hastrouble reading tables andexpressing those relationshipsin clear English.

These examples illustrate two verydifferent aspects of mathematics thatapply throughout the discipline. On theone hand is calculation; on the other,interpretation. The one reasons withnumbers to produce an answer; theother reasons about numbers to produceunderstanding. Generally, school mathe-matics focuses on the former, naturaland social sciences on the latter. For lotsot reasons—psychological, pedagogical,logical, motivational—students willleam best when teachers combine thesetwo approaches.

There may be good reasons that somany children and adults have difficultywith fractions. It turns out that evenmathematicians cannot agree on a singleproper definition. One camp argues thatfractions are just names for certainpoints on the number line (Wu, 2005),whereas others say that it's better tothink of them as multiples of basic unitfractions such as 1/3 , 1/4 , and 1/5(Tucker, 2006). Textbooks for prospec-tive elementary school teachers exhibitan even broader and more confusing

array of approaches (McCrory, 2006).Instead of beginning with formal defi-

nitions, when ordinary people speak offractions they tend to emphasize contex-tual meaning. Fractions (like ailnumbers) are human constructs thatarise in particular social and scientificcontexts. They represent the magnitudeof social problems (for example, thepercentage of drug addiction in a givenpopulation); the strength of publicopinion (for example, the percentage ofthe population that supports schoolvouchers); and the consequences ofgovernment policies (for example, theunemployment rate). Every number isthe product of human activity and isselected to serve human purposes (Best,2001, 2007).

Fractions, ratios, proportions, andother numbers convey quantity; wordsconvey meaning. For mathetnatics tomake sense to students as somethingother than a purely mental exercise,teachers need to focus on the interplayof numbers and words, especially onexpressing quantitative relationships in

meaningful sentences. Forusers of mathematics, calcula-tion takes a backseat tomeaning. And to make mathe-matics meaningful, the threeRs must be well blended ineach student's mind.

Algebra for All?Conventional wisdom holdsthat in Thomas Friedmansmetaphorically flat world, allstudents, no matter theirtalents or proclivities, shouldleave high school prepared forboth college and high-techwork (American DiplomaProject, 2004). This implies,for example, that all studentsshould rnaster Algebra II, acourse originally designed asan elective for the mathemati-cally inclined. Indeed, more

than half of U.S. states now requireAlgebra II for almost all high schoolgraduates (Zinth, 2006).

Advocates of algebra advance severalarguments for Lhis dramatic change ineducation policy:

• Workforce projections suggest agrowing shortage of U.S. citizens havingthe kinds of technical skills that buildon such courses as Algebra 11(Committee on Science, Engineering,and Public Policy, 2007).

• Employment and education datashow that Algebra II is a "thresholdcourse" for high-paying jobs, in partic-ular, five in six young people in the topquarter ot the income distribution havecompleted Algebra II (Carnevale &Desrochers, 2003).

• Algebra 11 is a prerequisite forCollege Algebra, the mathematics coursemost commonly required for postsec-ondary degrees. Virtually all collegestudents who have not taken Algebra 11will need to take remedial mathematics.

• Students most likely to opt out ofalgebra when it Is not required are those

10 tUUCATlONAL LEADERSHI 1*/NO V BMBHR 2007

whose parents are least engaged in iheirchildren's education. The result is aneducation system that magnifiesinequiLics and perpetuates socioeco-nomic differences from one generationlo ihe next (Haycock, 2007),

Skeptics of Algebra II requirementsnoie that other areas of mathematics,such as data analysis, statistics, andprobability, are in equally short supplyamong high school graduates and aregenerally more useful for employmentand daily life. They point out that thehistoric association of Algebra II witheconomic success may say more aboutcommon causes (for example, familybackground and peer support) thanabout the usefulness of Algebra II skills.And they note that many students whocomplete Algebra 11 also wind up takingremedial mathematics in college.

Indeed, difficulties quickly surfacedas soon as schools tried to implementthis new agenda for mathematics educa-tion. Shortly after standards, courses,and tests were developed toenforce a protocol of "Algebra IIfor all," it became clear thaimany schools were unable toachieve this goal. The reasonsincluded, in varying degrees,inadequacies in preparation,funding, motivation, ability, andinstructional quality The resulthas been a proliferation of "fake"mathemalics courses andlowered proficiency standardsthai enable districts and states topay lip service to this goalwithout making the extraordi-nary investment of resourcesrequired to actually accomplishit (Noddings, 2007).

Several strands of evidencequestion the unarticulatedassumption thai additionalinstruction in algebra wouldnecessarily yield increasedlearning. Although this may betrue iti some subjects, it is far

High school mathematics is the ultimateexercise in deferred gratification. Its payoffcomes years later, and then only for theminority who struggle through it.less clear for subjects such as Algebra IIthat are beset by student indifference,teacher shortages, and unclear purpose.For many of the reasons given, enroll-ments in Algebra 11 have approximatelydoubled during the last two decades(National Center for Education StatisticsINCES], 2005a). Yet during that sameperiod, college enrollments in remedialmathematics and mathematics scores onthe 12th grade National Assessment ofEducational Progress (NAEP) havehardly changed at all (NCES, 2005b;Lutzer, Maxwell, & Rodi, 2007). Some-thing is clearly wrong.

Although we carmot conduct a

FIGURE 1.The Challenge of ExpressingNumerical Data in Ordinary Language

Percentage Who Are Runners

Female

Male

Total

Nonsmoker

50%

25%

37%

Smoker

( 2 0 ^

10%

15%

Total

40%

20%

30%

Source: From Schield Statistical Literacy Inventory: Reading andInterpreting Tables and Graphs Involving Rates and Percentages.by M. Schield, 2002. Minneapolis, MN: Augsburg College,W. M. Keck Statistical Literacy Proiect. Copyright 2002 byM. Schield. Available: http://web.augsburg.edu/~schield/MiloPapers/StatLitKnowledge2r.pdf. Reprinted with permission.

Which of the following correctly describes the 20%circled in the table above?

a. 20% of runners are female smokers.b. 20% of females are runners who smoke.c. 20% of female smokers are runners.d. 20% of smokers are females v̂ /ho run.

randomized controlled study of schoolmathematics, with some studentsreceiving a treatment and others aplacebo, we can examine the effects ofthe current curriculum on those who gothrough it. Here we find moredisturbing evidence:

• One in three students who enter9th grade fails to graduate with his orher class, leaving the United States withthe highest secondary school dropoutrate among industrialized nations(Barton, 2005). Moreover, approxi-mately half of all blacks, Hispanics, andAmerican Indians fail to graduate withtheir class (Swanson, 2004). Although

mathematics is not uniquely toblame for this shameful record, itis the academic subject thatstudents most often fail.

• One in three students whoenter college must remediatemajor parts of high school math-ematics as a prerequisite totaking such courses as CoUegeAlgebra or Elementary Statistics(Greene & Winters, 2005).

• In one study of studentwriting, one in three students at ahighly selective college failed touse any quantitative reasoningwhen vioiting about subjects inwhich quantitative evidenceshould ha\'e played a central role(Lutsky, 2006).

• College students in the ; ,,natural and social sciencesconsistently have troubleexpressing in precise English themeaning of data presented intables or graphs (Schield, 2006).

AS.SOCIATION rOH SUPLRVLSION AND CURKiCULUM DtVtLOPMENI 11

One explanation for these discour-aging results is that the trajectory ofschool mathematics moves from theconcrete and functional (for example,measuring and counting) in lower gradesto the abstract and apparently non-functional (for example, factoring andstmplilying) in high school. As manyobservers have noted ruefully, highschool mathematics is the ultimate exer-cise in deferred gratification. Its payoflcomes years later, and then only for theminority who struggle through it.

In the past, schools offered thisabstract and ultimately powerful main-stream mathematics curriculum toapproximately half their students—-thoseheaded for college—and little if anythingworthwhile to the other half. Theconviction that has emerged in the lasttwo decades that all students should beoffered useful and powerful mathematicsis long overdue. However, it is not yetclear whether the best option for all isthe historic algebra-based mainstreamthat is animated primarily by the powerol increasing abstraction.

Mastering Mathematicsl-ractions and algebra may be among themost difficult parts of school mathe-matics, but they are not the only areas tocause students trouble. Experienceshows that many students fail to masterimportant mathematical topics. What'smissing from traditional instruction issufficient emphasis on three importantingredients: communication, connec-tions, and contexts.

CommunicationColleges expect students to communi-cate effectively with people fromdifferent backgrounds and with differentexpertise and to synthesize skills frommultiple areas. Employers seek the samethings. They emphasize that formalknowledge is not, by itself, sufficient todeal with today's challenges. Instead oflooking primarily for technical skills.

today's business leaders talk more aboutteamwork and adaptability Interviewersexamine candidates' ability lo s)Titbesizeinformation, make sound assutiiptions,capitalize on ambiguity, and explaintbeir reasoning. They seek graduateswho can interpret data as well as calcu-late with it and who can communicateeffectively about quantitative topics(Taylor, 2007).

To meet these demands of collegeand work, K-12 students need exten-sive practice expressing verbally thequantitative meanings of botb problemsand solutions. Tbey need to be able towrite fluently in complete sentences

ence show just how naive this belief is.If we want students to be able tocommunicate mathematically, we needto ensure that they both practice thisskill in mathematics class and regularlyuse quantitative arguments in subjectswhere writing is taught and critiqued.

ConnectionsOne reason that students think mathe-matics is useless is thai the only peoplethey see who use it are mathematicsteachers. Unless teachers of all subjects—both academic and vocational—usemathematics regularly and significantlyin their courses, students will treat math-

Oil the one hand is calculation; on the other,interpretation. The one reasons with numhersto produce an answer; the other reasons aboutnumhers to produce understanding.

and coherent paragraphs; to explain themeaning of data, tables, graphs, andformulas; and to express tbe relation-ships among these different representa-tions. For example, science studentscould use data on global warming towrite a letter lo the editor about carbontaxes; civics students could use datafrom a recent election to write op-edcolumns advocating for or against analternative voting system; economicsstudents could examine tables of dataconcerning tbe national debt and writeletters to their representatives aboutlimiring the debt being transferred tothe next generation.

We used to believe that if mathe-matics teachers taught students how tocalculate and English teachers taughtstudents how to write, then studentswould naturally blend these skills towrite clearly about quantitative ideas.Data and years of frustrating experi-

ematics teachers' exhortations about itsusefulness as self-serving rhetoric.

To make mathematics count in theeyes of students, schools need to makemathematics pervasive, as writing nowis. This can best be done by cross-disciplinary planning buili on acommitment from teachers and admin-istrators to make the goal of numeracyas important as literacy. Virtually everysubject taught in school is amenable tosome use of quantitative or logical argu-ments that tie e\'idence to conclusions.Measurement and calculation are part ofall vocational subjects; tables, data, andgraphs abound in the social and naturalsciences; business requires financialmathematics; equations are common meconomics and chemistry; logical infer-ence is fundamental to history andcivics, if eacb content-area teacher iden-tifies just a few units where quantitativethinking can enhance understanding.

12 EDUCATIONAL LEADER^HI p/NovtMBhH 2007

students will get the message.The example of many otherwise well-

prepared college students refraining fromu.sing even simple quantitative reasoningto buttress their arguments shows thatstudents in high school need much morepractice using the mathematical resourcesintroduced in the elementary and middlegrades. Much of this practice should takeplace across the curriculum. Mathematicsis too important to leave lo mathematicsteachers alone.

ContextsOne of the common criticisms of schoolmathematics is that it focuses toonarrowly on procedures (algorithms) atthe expense of understanding. This is aspecial problem in relation to fractionsand algebra because both represent alevel of abstraction that is significantlyhigher than simple integer arithmetic.Without reliable contexts to anchormeaning, many students see only ameaningless cloud of abstract symbols.

As the level of abstraction increases,algorithms proliferate and their links tomeaning iade. Why do you invert andmultiply? Why is (a + hY ^ a^+ b^l Thereasons are obvious if you understandwhat the symbols mean, but they aremysterious if you do not. Understand-ably, this apparent disjuncture of proce-dures from meaning leaves manystudents thoroughly confused. Therecent increase in standardized testinghas aggravated this problem becauseeven those teachers who want to avoidthis trap find that they cannot. So longas procedures predominate on high-stakes tests, procedures will preoccupyboth teachers and students.

There is, however, an alternative tomeaningless abstraction. Most applica-lions of mathematical reasoning in dailylife and typical jobs involve sophisticatedthinking witb elementary skills (forexample, arithmetic, percentages, ratios),whereas the mainstream of mathematicsill high school (algebra, geometry,

trigonometry) introduces students toincreasingly abstract concepts that arethen illustrated with oversimplifiedtemplate exercises (for example, trainsmeeting in the night). By enriching thisdiet of simple abstract problems v/ithsophisticated realistic problems thatrequire only simple skills, teachers canhelp students see that mathematics isreally helpful for understanding thingsthey care about (Steen, 2001). Global

wanning, college tuition, and gas pricesare examples of data-rich topics thatinterest students but that can also chal-lenge them with surprising complica-tions. Such a focus can also help combatstudent boredom, a primary cause ofdropping out of school (Bridgeland,Dilulio, & Morison, 2006).

Most important, the pedagogicalactivity of connecting meaning tonumbers needs to take place in

My "Aha! " MomentDouglas Hofstadter, Distinguished Professor of Cognitive Science,Indiana University, Bloomington.

I first realized the deep lure of mathematics when, at about age 3, I thought upthe "great idea" of generalizing the concept of 2 x 2 to what seemed to me tobe the inconceivably fancier concept of 3 x 3 x 3. My inspiration was that since2 x 2 uses the concept of two-ness twice, I wanted to use theconcept of three-ness thrice\ It wasn't finding out theactual value of this expression (27 obviously} thatthrilled me—it was the idea of the fluid conceptualstructures that I could play with in my imaginationthat turned me on to math at that early age.

Another "aha" moment came a few years later,when I noticed that 3̂ x 5̂ is equal to (3 x 5)^ Onceagain I was playing around with structures, nottrying to prove anything. (I didn't even know thatproofs existed!) It thrilled me to discover this pattern,which of course I verified for other values and foundmystically exciting.

I believe that teachers should encourage playfulness with mathematicalconcepts and should encourage the discoveries of patterns of whatever sort.Any time a child recognizes an unexpected pattern, it may evoke a senseof wonder.

A S S O C I A T I O N I - O H S i i i ' t R V [ s i o N A N D C i i R R t ( _ L i L i f M 13

authentic contexts, such as in history,geography, economics, or biology—wherever things are counted, measured,inferred, or analyzed. Contexts in whichmathematical reasoning is used are bestintroduced in natural situations acrossthe cuniculum. Otherwise, despitemathematics teachers' best elforts,students will see mathematics as some-thing that is useful only in mathematicsclass. The best way to make mathematicscount in the eyes of students is for themto see their teachers using it widely inmany different contexts. Si

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Lynn Arthur Steen is Professor of Math-ematics at St, Olaf College. 1520 St. OlafAve., Northfield, MN; steen@stolaf.edu.

14 2007