ASCD

Fractions and algebra represent = the most=20 subtle, powerful, and mind-twisting elements of = school=20 mathematics. But how can we teach them so = students=20 understand?

Much to the surprise of = those who care=20 about such things, mathematics has become the = 600-pound=20 gorilla in U.S. schools. High-stakes testing has = forced=20 schools to push aside subjects like history, = science,=20 music, and art in a scramble to avoid the = embarrassing=20 consequences of not making =E2=80=9Cadequate = yearly progress=E2=80=9D in=20 mathematics. Reverberations of the math wars of = the=20 1990s roil parents and teachers as they seek = firm=20 footing in today's turbulent debates about = mathematics=20 education.

Much contention occurs near = the ends=20 of elementary and secondary education, where = students=20 encounter topics that many find difficult and = some find=20 incomprehensible. In earlier decades, schools = simply=20 left students in the latter category behind. = Today, that=20 option is neither politically nor legally = acceptable.=20 Two topics=E2=80=94fractions and algebra, = especially Algebra=20 II=E2=80=94are particularly troublesome. Many = adults, including=20 some teachers, live their entire lives flummoxed = by=20 problems requiring any but the simplest of = fractions or=20 algebraic formulas. It is easy to see why these = topics=20 are especially nettlesome in today's school = environment.=20 They are exemplars of why mathematics counts and = why the=20 subject is so controversial.

Confounded by Fractions

What is the approximate = value, to the=20 nearest whole number, of the sum 19/20 + 23/25? = Given=20 the choices of 1, 2, 42, or 45 on an = international test,=20 more than half of U.S. 8th graders chose 42 or = 45. Those=20 responses are akin to decoding and pronouncing = the word=20 elephant but having no idea what animal = the word=20 represents. These students had no idea that = 19/20 is a=20 number close to 1, as is 23/25.

Neither, it is likely, did = their=20 parents. Few adults understand fractions well = enough to=20 use them fluently. Because people avoid = fractions in=20 their own lives, some question why schools (and = now=20 entire states) should insist that all students = know, for=20 instance, how to add uncommon combinations like = 2/7 +=20 9/13 or how to divide 1 3/4 by 2/3. When, = skeptics ask,=20 is the last time any typical adult encountered = problems=20 of this sort? Even mathematics teachers have a = hard time=20 imagining authentic problems that require these = exotic=20 calculations (Ma, 1999).

Moreover, many people cannot = properly=20 express in correct English the fractions and = proportions=20 that do commonly occur, for instance, in = ordinary=20 tables of data. A simple example illustrates = this=20 difficulty (Schield, 2002). Even though most = people know=20 that 20 percent means 1/5 of something, many = cannot=20 figure out what the something is when confronted = with an=20 actual example, such as the table in Figure 1. = Although=20 calculators can help the innumerate cope with = such=20 exotica as 2/7 + 9/13 and 1 3/4 =C3=B7 2/3, they = are of no=20 help to someone who has trouble reading tables = and=20 expressing those relationships in clear = English.

Figure 1. The Challenge of Expressing = Numerical Data=20 in Ordinary Language

Not available for = electronic=20 dissemination.

These examples illustrate = two very=20 different aspects of mathematics that apply = throughout=20 the discipline. On the one hand is calculation; = on the=20 other, interpretation. The one reasons = with=20 numbers to produce an answer; the other reasons=20 about numbers to produce understanding.=20 Generally, school mathematics focuses on the = former,=20 natural and social sciences on the latter. For = lots of=20 reasons=E2=80=94psychological, pedagogical, = logical,=20 motivational=E2=80=94students will learn best = when teachers=20 combine these two approaches.

There may be good reasons = that so many=20 children and adults have difficulty with = fractions. It=20 turns out that even mathematicians cannot agree = on a=20 single proper definition. One camp argues that = fractions=20 are just names for certain points on the number = line=20 (Wu, 2005), whereas others say that it's better = to think=20 of them as multiples of basic unit fractions = such as=20 1/3, 1/4, and 1/5 (Tucker, 2006). Textbooks for=20 prospective elementary school teachers exhibit = an even=20 broader and more confusing array of approaches = (McCrory,=20 2006).

Instead of beginning with = formal=20 definitions, when ordinary people speak of = fractions=20 they tend to emphasize contextual meaning. = Fractions=20 (like all numbers) are human constructs that = arise in=20 particular social and scientific contexts. They=20 represent the magnitude of social problems (for = example,=20 the percentage of drug addiction in a given = population);=20 the strength of public opinion (for example, the = percentage of the population that supports = school=20 vouchers); and the consequences of government = policies=20 (for example, the unemployment rate). Every = number is=20 the product of human activity and is selected to = serve=20 human purposes (Best, 2001, 2007).

Fractions, ratios, = proportions, and=20 other numbers convey quantity; words convey = meaning. For=20 mathematics to make sense to students as = something other=20 than a purely mental exercise, teachers need to = focus on=20 the interplay of numbers and words, especially = on=20 expressing quantitative relationships in = meaningful=20 sentences. For users of mathematics, calculation = takes a=20 backseat to meaning. And to make mathematics = meaningful,=20 the three Rs must be well blended in each = student's mind.

Algebra for All?

Conventional wisdom holds = that in=20 Thomas Friedman's metaphorically flat world, all = students, no matter their talents or = proclivities,=20 should leave high school prepared for both = college and=20 high-tech work (American Diploma Project, 2004). = This=20 implies, for example, that all students should = master=20 Algebra II, a course originally designed as an = elective=20 for the mathematically inclined. Indeed, more = than half=20 of U.S. states now require Algebra II for almost = all=20 high school graduates (Zinth, 2006).

Advocates of algebra advance = several=20 arguments for this dramatic change in education = policy:=20

Workforce projections suggest a growing = shortage=20 of U.S. citizens having the kinds of technical = skills=20 that build on such courses as Algebra II = (Committee on=20 Science, Engineering, and Public Policy, = 2007).=20
Employment and education data show that = Algebra II=20 is a =E2=80=9Cthreshold course=E2=80=9D for = high-paying jobs. In=20 particular, five in six young people in the = top=20 quarter of the income distribution have = completed=20 Algebra II (Carnevale & Desrochers, 2003). =
Algebra II is a prerequisite for College = Algebra,=20 the mathematics course most commonly required = for=20 postsecondary degrees. Virtually all college = students=20 who have not taken Algebra II will need to = take=20 remedial mathematics.=20
Students most likely to opt out of algebra = when it=20 is not required are those whose parents are = least=20 engaged in their children's education. The = result is=20 an education system that magnifies inequities = and=20 perpetuates socioeconomic differences from one = generation to the next (Haycock, 2007). =

Skeptics of Algebra II = requirements=20 note that other areas of mathematics, such as = data=20 analysis, statistics, and probability, are in = equally=20 short supply among high school graduates and are = generally more useful for employment and daily = life.=20 They point out that the historic association of = Algebra=20 II with economic success may say more about = common=20 causes (for example, family background and peer = support)=20 than about the usefulness of Algebra II skills. = And they=20 note that many students who complete Algebra II = also=20 wind up taking remedial mathematics in = college.

Indeed, difficulties quickly = surfaced=20 as soon as schools tried to implement this new = agenda=20 for mathematics education. Shortly after = standards,=20 courses, and tests were developed to enforce a = protocol=20 of =E2=80=9CAlgebra II for all,=E2=80=9D it = became clear that many=20 schools were unable to achieve this goal. The = reasons=20 included, in varying degrees, inadequacies in=20 preparation, funding, motivation, ability, and=20 instructional quality. The result has been a=20 proliferation of =E2=80=9Cfake=E2=80=9D = mathematics courses and lowered=20 proficiency standards that enable districts and = states=20 to pay lip service to this goal without making = the=20 extraordinary investment of resources required = to=20 actually accomplish it (Noddings, 2007).

Several strands of evidence = question=20 the unarticulated assumption that additional = instruction=20 in algebra would necessarily yield increased = learning.=20 Although this may be true in some subjects, it = is far=20 less clear for subjects such as Algebra II that = are=20 beset by student indifference, teacher = shortages, and=20 unclear purpose. For many of the reasons given,=20 enrollments in Algebra II have approximately = doubled=20 during the last two decades (National Center for = Education Statistics [NCES], 2005a). Yet during = that=20 same period, college enrollments in remedial = mathematics=20 and mathematics scores on the 12th grade = National=20 Assessment of Educational Progress (NAEP) have = hardly=20 changed at all (NCES, 2005b; Lutzer, Maxwell, = &=20 Rodi, 2007). Something is clearly wrong.

Although we cannot conduct a = randomized controlled study of school = mathematics, with=20 some students receiving a treatment and others a = placebo, we can examine the effects of the = current=20 curriculum on those who go through it. Here we = find more=20 disturbing evidence:=20

One in three students who enter 9th grade = fails to=20 graduate with his or her class, leaving the = United=20 States with the highest secondary school = dropout rate=20 among industrialized nations (Barton, 2005). = Moreover,=20 approximately half of all blacks, Hispanics, = and=20 American Indians fail to graduate with their = class=20 (Swanson, 2004). Although mathematics is not = uniquely=20 to blame for this shameful record, it is the = academic=20 subject that students most often fail.=20
One in three students who enter college = must=20 remediate major parts of high school = mathematics as a=20 prerequisite to taking such courses as College = Algebra=20 or Elementary Statistics (Greene & = Winters, 2005).=20
In one study of student writing, one in = three=20 students at a highly selective college failed = to use=20 any quantitative reasoning when writing about = subjects=20 in which quantitative evidence should have = played a=20 central role (Lutsky, 2006).=20
College students in the natural and social = sciences consistently have trouble expressing = in=20 precise English the meaning of data presented = in=20 tables or graphs (Schield, 2006).

One explanation for these = discouraging=20 results is that the trajectory of school = mathematics=20 moves from the concrete and functional (for = example,=20 measuring and counting) in lower grades to the = abstract=20 and apparently nonfunctional (for example, = factoring and=20 simplifying) in high school. As many observers = have=20 noted ruefully, high school mathematics is the = ultimate=20 exercise in deferred gratification. Its payoff = comes=20 years later, and then only for the minority who = struggle=20 through it.

In the past, schools offered = this=20 abstract and ultimately powerful mainstream = mathematics=20 curriculum to approximately half their = students=E2=80=94those=20 headed for college=E2=80=94and little if = anything worthwhile to=20 the other half. The conviction that has emerged = in the=20 last two decades that all students should be = offered=20 useful and powerful mathematics is long overdue. = However, it is not yet clear whether the best = option for=20 all is the historic algebra-based mainstream = that is=20 animated primarily by the power of increasing=20 abstraction.

Mastering Mathematics

Fractions and algebra may be = among the=20 most difficult parts of school mathematics, but = they are=20 not the only areas to cause students trouble. = Experience=20 shows that many students fail to master = important=20 mathematical topics. What's missing from = traditional=20 instruction is sufficient emphasis on three = important=20 ingredients: communication, connections, and=20 contexts.

Communication

Colleges expect students to=20 communicate effectively with people from = different=20 backgrounds and with different expertise and to=20 synthesize skills from multiple areas. Employers = seek=20 the same things. They emphasize that formal = knowledge is=20 not, by itself, sufficient to deal with today's=20 challenges. Instead of looking primarily for = technical=20 skills, today's business leaders talk more about = teamwork and adaptability. Interviewers examine=20 candidates' ability to synthesize information, = make=20 sound assumptions, capitalize on ambiguity, and = explain=20 their reasoning. They seek graduates who can = interpret=20 data as well as calculate with it and who can=20 communicate effectively about quantitative = topics=20 (Taylor, 2007).

To meet these demands of = college and=20 work, K=E2=80=9312 students need extensive = practice expressing=20 verbally the quantitative meanings of both = problems and=20 solutions. They need to be able to write = fluently in=20 complete sentences and coherent paragraphs; to = explain=20 the meaning of data, tables, graphs, and = formulas; and=20 to express the relationships among these = different=20 representations. For example, science students = could use=20 data on global warming to write a letter to the = editor=20 about carbon taxes; civics students could use = data from=20 a recent election to write op-ed columns = advocating for=20 or against an alternative voting system; = economics=20 students could examine tables of data concerning = the=20 national debt and write letters to their = representatives=20 about limiting the debt being transferred to the = next=20 generation.

We used to believe that if = mathematics=20 teachers taught students how to calculate and = English=20 teachers taught students how to write, then = students=20 would naturally blend these skills to write = clearly=20 about quantitative ideas. Data and years of = frustrating=20 experience show just how na=C3=AFve this belief = is. If we=20 want students to be able to communicate = mathematically,=20 we need to ensure that they both practice this = skill in=20 mathematics class and regularly use quantitative = arguments in subjects where writing is taught = and=20 critiqued.

Connections

One reason that students = think=20 mathematics is useless is that the only people = they see=20 who use it are mathematics teachers. Unless = teachers of=20 all subjects=E2=80=94both academic and = vocational=E2=80=94use=20 mathematics regularly and significantly in their = courses, students will treat mathematics = teachers'=20 exhortations about its usefulness as = self-serving=20 rhetoric.

To make mathematics count in = the eyes=20 of students, schools need to make mathematics = pervasive,=20 as writing now is. This can best be done by=20 cross-disciplinary planning built on a = commitment from=20 teachers and administrators to make the goal of = numeracy=20 as important as literacy. Virtually every = subject taught=20 in school is amenable to some use of = quantitative or=20 logical arguments that tie evidence to = conclusions.=20 Measurement and calculation are part of all = vocational=20 subjects; tables, data, and graphs abound in the = social=20 and natural sciences; business requires = financial=20 mathematics; equations are common in economics = and=20 chemistry; logical inference is fundamental to = history=20 and civics. If each content-area teacher = identifies just=20 a few units where quantitative thinking can = enhance=20 understanding, students will get the = message.

The example of many = otherwise=20 well-prepared college students refraining from = using=20 even simple quantitative reasoning to buttress = their=20 arguments shows that students in high school = need much=20 more practice using the mathematical resources=20 introduced in the elementary and middle grades. = Much of=20 this practice should take place across the = curriculum.=20 Mathematics is too important to leave to = mathematics=20 teachers alone.

Contexts

One of the common criticisms = of school=20 mathematics is that it focuses too narrowly on=20 procedures (algorithms) at the expense of = understanding.=20 This is a special problem in relation to = fractions and=20 algebra because both represent a level of = abstraction=20 that is significantly higher than simple integer = arithmetic. Without reliable contexts to anchor = meaning,=20 many students see only a meaningless cloud of = abstract=20 symbols.

As the level of abstraction = increases,=20 algorithms proliferate and their links to = meaning fade.=20 Why do you invert and multiply? Why is (a = +=20 b)² =E2=89=A0 = a² +=20 b²? The reasons are obvious if = you=20 understand what the symbols mean, but they are=20 mysterious if you do not. Understandably, this = apparent=20 disjuncture of procedures from meaning leaves = many=20 students thoroughly confused. The recent = increase in=20 standardized testing has aggravated this problem = because=20 even those teachers who want to avoid this trap = find=20 that they cannot. So long as procedures = predominate on=20 high-stakes tests, procedures will preoccupy = both=20 teachers and students.

There is, however, an = alternative to=20 meaningless abstraction. Most applications of=20 mathematical reasoning in daily life and typical = jobs=20 involve sophisticated thinking with elementary = skills=20 (for example, arithmetic, percentages, ratios), = whereas=20 the mainstream of mathematics in high school = (algebra,=20 geometry, trigonometry) introduces students to=20 increasingly abstract concepts that are then = illustrated=20 with oversimplified template exercises (for = example,=20 trains meeting in the night). By enriching this = diet of=20 simple abstract problems with sophisticated = realistic=20 problems that require only simple skills, = teachers can=20 help students see that mathematics is really = helpful for=20 understanding things they care about (Steen, = 2001).=20 Global warming, college tuition, and gas prices = are=20 examples of data-rich topics that interest = students but=20 that can also challenge them with surprising=20 complications. Such a focus can also help combat = student=20 boredom, a primary cause of dropping out of = school=20 (Bridgeland, DiIulio, & Morison, 2006).

Most important, the = pedagogical=20 activity of connecting meaning to numbers needs = to take=20 place in authentic contexts, such as in history, = geography, economics, or = biology=E2=80=94wherever things are=20 counted, measured, inferred, or analyzed. = Contexts in=20 which mathematical reasoning is used are best = introduced=20 in natural situations across the curriculum. = Otherwise,=20 despite mathematics teachers' best efforts, = students=20 will see mathematics as something that is useful = only in=20 mathematics class. The best way to make = mathematics=20 count in the eyes of students is for them to see = their=20 teachers using it widely in many different = contexts.

My = =E2=80=9CAha!=E2=80=9D Moment

Douglas Hofstadter, = Distinguished Professor of Cognitive = Science,=20 Indiana University, Bloomington.

I first realized the = deep lure=20 of mathematics when, at about age 3, I = thought up=20 the =E2=80=9Cgreat idea=E2=80=9D of = generalizing the concept of 2=20 =C3=97 2 to what seemed to me to be the = inconceivably=20 fancier concept of 3 =C3=97 3 =C3=97 3. My = inspiration was=20 that since 2 =C3=97 2 uses the concept of = two-ness=20 twice, I wanted to use the concept = of=20 three-ness thrice! It wasn't = finding out=20 the actual value of this expression (27,=20 obviously) that thrilled me=E2=80=94it was = the idea of the=20 fluid conceptual structures that I could = play with=20 in my imagination that turned me on to = math at=20 that early age.

Another = =E2=80=9Caha=E2=80=9D moment came a few=20 years later, when I noticed that = 3² =C3=97=20 5² is equal to (3 =C3=97 = 5)².=20 Once again I was playing around with = structures,=20 not trying to prove anything. (I didn't = even know=20 that proofs existed!) It thrilled me to = discover=20 this pattern, which of course I verified = for other=20 values and found mystically exciting.

I believe that = teachers should=20 encourage playfulness with mathematical = concepts=20 and should encourage the discoveries of = patterns=20 of whatever sort. Any time a child = recognizes an=20 unexpected pattern, it may evoke a sense = of=20 = wonder.

References

American Diploma Project. = (2004).=20 Ready or not: Creating a high school diploma = that=20 counts. Washington, DC: Achieve. Available: = www.achieve.org/files/ADPreport_7.pdf

Barton, P. E. (2005). = One third=20 of a nation: Rising dropout rates and declining=20 opportunities. Princeton, NJ: Educational = Testing=20 Service. Available: www.ets.org/Media/Education_Topics/pdf/onethird.pdf=20

Best, J. (2001). Damned = lies and=20 statistics: Untangling numbers from the media,=20 politicians, and activists. Berkeley: = University of=20 California Press.

Best, J. (2007, June). = Beyond=20 calculation: Quantitative literacy and critical = thinking=20 about public issues. Paper presented at = Johnson=20 Foundation Conference on Quantitative Literacy = and Its=20 Implications for Teacher Education, Milwaukee, = WI.

Bridgeland, J. M., = DiIulio, J. J.,=20 & Morison, K. B. (2006). The silent = epidemic:=20 Perspectives of high school dropouts. = Washington,=20 DC: Peter D. Hart Research. Available: www.gatesfoundation.org/nr/downloads/ed/TheSilentEpidemic3-06= FINAL.pdf=20

Carnevale, A. P., & = Desrochers,=20 D. M. (2003). Standards for what? The = economic roots=20 of K=E2=80=9316 reform. Princeton, NJ: = Educational Testing=20 Service. Available: www.transitionmathproject.org/assets/docs/resources/standards= _for_what.pdf=20

Committee on Science, = Engineering,=20 and Public Policy. (2007). Rising above the = gathering=20 storm: Energizing and employing America for a = brighter=20 economic future. Washington, DC: National = Academies=20 Press. Available: www.nap.edu/catalog/11463.html =

Greene, J. P., & = Winters, M.=20 (2005). Public high school graduation and = college=20 readiness rates, 1991=E2=80=932002. New = York: Manhattan=20 Institute for Policy Research. Available: www.manhattan-institute.org/html/ewp_08.htm=20

Haycock, K. (2007). Kati = Haycock's=20 Testimony before the Subcommittee on Labor, HHR, = and=20 Education, House Appropriations Committee = [Online press=20 release]. Washington, DC: Education Trust. = Available: www2.edtrust.org/EdTrust/Press+Room/Haycock+Appropriations+Te= stimony.htm=20

Lutsky, N. (2006). Quirks = of=20 rhetoric: A quantitative analysis of = quantitative=20 reasoning in student writing. Proceedings of = the=20 section on statistical education, American = Statistical=20 Association (pp. 2319=E2=80=932322). = Available: www.statlit.org/pdf/2006lutskyASA.pdf

Lutzer, D. J., Maxwell, J. = W., &=20 Rodi, S. B. (2007). CBMS (Conference Board of = Mathematical Sciences) 2005: Statistical = abstract of=20 undergraduate programs in the mathematical = sciences in=20 the United States. Providence, RI: American=20 Mathematical Society.

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National Center for = Education=20 Statistics. (2005a). Table ED4-A. In = America's=20 children: Key national indicators of = well-being.=20 Washington, DC: U.S. Department of Education. = Available:=20 www.childstats.gov/pdf/ac2007/ac_07.pdf=20

National Center for = Education=20 Statistics. (2005b). The nation's report = card:=20 Long-term trends. Washington, DC: Institute = of=20 Education Sciences, U.S. Department of = Education.=20 Available: http://nces.ed.gov/nationsreportcard/ltt/results2004/nat-math= -perf.asp=20

Noddings, N. (2007, March = 20). The=20 new anti-intellectualism in America. = Education Week,=20 26(28), 29, 32. Available: www.edweek.org/ew/articles/2007/03/20/28noddings.h26.html?qs=3D= Noddings=20

Schield, M. (2002). = Statistical=20 literacy inventory: Reading and interpreting = tables and=20 graphs involving rates and percentages. = Minneapolis,=20 MN: Augsburg College, W. M. Keck Statistical = Literacy=20 Project. Available: http://web.augsburg.edu/~schield/MiloPapers/StatLitKnowledge2= r.pdf=20

Schield, M. (2006, July).=20 Statistical literacy survey analysis: Reading = graphs=20 and tables of rates and percentages. Paper = presented=20 at the International Conference on Teaching = Statistics,=20 Salvador Bahia, Brazil. Available: www.StatLit.org/pdf/2006SchieldICOTS.pdf=20

Steen, L. A. (Ed.). = (2001).=20 Mathematics and democracy: The case for = quantitative=20 literacy. Princeton, NJ: National Council on = Education and the Disciplines, Woodrow Wilson = National=20 Fellowship Foundation. Available: www.maa.org/ql/mathanddemocracy.html

Swanson, C. B. (2004). = Who=20 graduates? Who doesn't? A statistical portrait = of public=20 high school graduation, class of 2001. = Washington,=20 DC: Urban Institute. Available: www.urban.org/publications/410934.html=20

Taylor, C. (2007, June).=20 Preparing students for the business of the = real (and=20 highly quantitative) world. Paper presented = at=20 Johnson Foundation Conference on Quantitative = Literacy=20 and Its Implications for Teacher Education, = Milwaukee,=20 WI.

Tucker, A. (2006). = Preparing for=20 fractions [Discussion paper]. Washington, = DC:=20 Mathematical Association of America. Available: = www.maa.org/pmet/resources/PrepForFractions.pdf=20

Wu, H. (2005). Chapter 2: = Fractions=20 (Draft). Berkeley: University of = California=E2=80=93Berkeley.=20 Available: http://math.berkeley.edu/~wu/EMI2a.pdf=20

Zinth, K. (2006). = Mathematics=20 graduation requirements, classes 2006 through = 2011.=20 Denver, CO: Education Commission of the States.=20 Available: www.ecs.org/clearinghouse/67/07/6707.pdf=20