A Science-of-LearningApproach to MathematicsEducationFrank Quinn

The focus in this approach is directly onindividual, mathematics-specific learn-ing. It is “science” in the sense that itdeveloped bottom-up from micro-scaleobservations. It addresses long-term

needs for abilities and skills because these imposestrong constraints on elementary education. Oureventual goal is to produce competent citizens andcapable scientists and engineers. To do this, collegefaculty need certain skills from high school; todevelop these skills, high-school teachers need cer-tain outcomes from middle school; and so on down.Finally the account draws on, and is consistentwith, modern professional practice. These featuresare described a bit more in the next section and indetail in [4], from which this material was drawn.

Much of the output from this approach is alsomicro-scale: highly effective ways to treat fractions,polynomial multiplication, word problems, and soforth, but with little indication as to how thesemight be put together in a course or curriculum–inother words, a toolkit without much assemblyinstruction, in contrast to educational-philosophyapproaches that provide assembly instructions(pedagogy and content-independent methodology)without much toolkit. There are, however, strategiesto make the tools work, and we invert the logicalprogression and begin with these in the sectionon Mathematics and Teaching. The section onExamples provides examples, mostly from [4].Topics in the first four subsections are fractions,multiplication of polynomials, and partial fractionsin both integers and polynomials. The goal is toindicate how topics might develop over time and

Frank Quinn is professor of mathematics at Virginia Poly-technic Institute and State University. His email address isquinn@math.vt.edu.

to suggest that a cognitively adapted and logicallyclear development might be more ambitious. Wordproblems are the last topic in the section. The finalsection concerns teacher preparation relevant tothis approach. Among other things, it throws lighton why study of advanced topics does not seem toimprove K–12 teaching.

This article was developed as a response to arti-cles in the AMS Notices special issue on educationin March 2011 [1]. The supplement [5] includescomparisons to these articles.

BackgroundThe most significant background for this materialis over a thousand hours of one-on-one diagnosticwork with students.1 The procedure is that I go tostudents when they need help. I review their work,diagnose the specific problem, and show them howto fix it to avoid a recurrence. This is not tutoring.The students do most of the talking and most ofthe work, and I leave them to finish on their ownas soon as they are past the specific difficulty. Thefocus on diagnosis also helps students see how todiagnose and correct their own errors.

The context for the diagnostic work is acomputer-tested engineering calculus course Ideveloped. The courseware is designed to empha-size structure and encourage work habits thatavoid problems. The material is also designed sothat pre-existing learning errors will cause seriousdifficulty. The goal is to expose these errors sothey can be diagnosed and repaired.

A final ingredient is an extensive bottom-upanalysis of contemporary mathematical practice[3]. This draws on my own research experience,

1In the Math Emporium at Virginia Tech.

1264 Notices of the AMS Volume 58, Number 9

editorial work, work on publication policy, cog-nitive neuroscience, history of mathematics, thework with students mentioned above, and manyother sources. It is well known that modern pro-fessional practice is quite different from that inthe nineteenth century and from models used ineducation and most philosophical accounts. It wasno surprise to find that contemporary practice isbetter adapted to the subject. It was a surpriseto find that, within the subject constraints, it ismuch better adapted to human cognitive facilities.It seems that contemporary practice is not just forfreaks: carefully understood, it can be a powerfulresource for education.

These experiences have given a fine-grainedperspective on mathematical learning and itsdifficulties. Moreover, the perspective directlyconcerns individual learning: not teaching, andnot learning mediated by teachers. It has becomeuncomfortably clear that teaching is not nearly asclosely connected with learning as educators liketo think.

Mathematics and TeachingMathematics offers insights into the world and itsown realms to explore, but it is not a spectatorsport. Current educational approaches do notprovide enough functionality to bring it to life.Current introductions to numbers and fractions,for instance, have rather passive “understanding”as goals rather than doing things with them.The difference for fractions is illustrated in theexamples.

The goal here is to suggest that a more activeapproach could be successful in real-life settings,but there are very strong constraints. First, weakgoals permit relaxed attitudes toward educationalphilosophy and individual practice. Achievingstrong goals is delicately dependent on approachand provides a criterion that identifies manyapproaches as “wrong”. To achieve success onemust first avoid failure. Second, there are guidelinesfor “right” approaches, but no easy formula.Successful practice is a delicate negotiation betweenmathematical structure and the complex odditiesof human cognition. This has to be done on acase-by-case (micro-scale) basis, but in ways thatfit into the long-term development.

This section provides an overview of structureand practice of mathematics, extracted from adetailed account in [3] and formulated for use inelementary education. However, it is an after-the-fact description of the methodological constraintsfound in the micro-scale material, not a philosoph-ical foundation from which methodology can bededuced. Again these are—at best—guidelines thathelp avoid failure, not formulas that guaranteesuccess.

Precision and Functionality

Mathematics has evolved a set of explicit rules ofreasoning with the following astounding property:arguments without rule violations have completelyreliable conclusions!2 People often have troublelearning to use these rules effectively because rulesin most other systems are sufficiently vague andineffective that precision is a waste of time, whilein mathematics anything less than full precision(no rule violations) is a waste of time. However,mathematical rules are simple compared to thoseof the tax code, religion, politics, physical science,and so on. Most people can master the basics.

Precision. Precision is particularly important forfunctionality in elementary mathematics: sloppythinking that is harmless to experienced userscan seriously confuse beginners. Confusion aboutfractions, for instance, results from failing todistinguish between things and names for them;between representatives of an equivalence classand the equivalence class; and from commonambiguity in the use of “=”. This is acceptable ifthe goal is passive “understanding”, but ambitiousgoals require much more clarity. The followingsubsections suggest ways to avoid these and otherproblems.

Potential Proofs. For users the key enabling concept([3] §4, [4](c)) is potential proof : a record ofreasoning, using methods of established reliability,and detailed enough to check for errors. Potentialproofs in this sense are very common. The scratchwork done by a child multiplying multidigit integersby hand provides a record of reasoning that can bechecked for errors. When teachers tell students to“show your work”, they mean “provide a record ofreasoning…”. The basic method is already in wideuse, and the potential-proof description mainlyclarifies the features and activities needed to makeit fully effective.

In these terms a “real” proof is a potentialproof that has been checked (and repaired ifnecessary) and found to be error-free. However, thebenefits come from reasoning and checking. Errorshappen. A flawed first attempt with sufficientdetail can provide a framework for diagnosisand improvement. A flawed first attempt withinsufficient detail is useless, and the user muststart again from scratch. Focusing on the formaland error-free aspects of completed proofs actuallydistracts from the features that provide power tousers.

2For all practical purposes; see [3], §2 for nuances.

October 2011 Notices of the AMS 1265

Work Templates. To exploit the power of potentialproofs, students should be taught to record theirreasoning in a way that can be checked for errors.Criteria for good work formats are:

(1) record enough detail so reasoning can bereconstructed and checked for errors;

(2) be compact and straightforward; and(3) help organize the work in ways consistent

with human cognitive constraints.

Designing such a format is hard for professionalsand cannot be expected of students, so theymust be provided with templates. These can rangefrom literal fill-in-the-blank forms (cf. the Solutionsubsection of Multiplying Polynomials) to routinelyrecording crucial intermediate steps (cf. the sectionWord Problems).

The standard formats for multidigit multiplica-tion and long division are well-known examples oftemplates, though they are not fully satisfactory.They satisfy the first two conditions but are lesssuccessful with (3) because they have been opti-mized for production arithmetic by experiencedusers rather than to provide support for beginners.A less efficient but potentially more supportivetemplate for multiplication is explored in [4](a).

Accuracy and Diagnosis. Mathematical methodsenable near 100 prcent accuracy, and extendedarguments depend on this. If a single operationcan be done with 70 percent accuracy, then thelikelihood of getting the right answer in problemswith ten operations is under 3 percent. If one wants70 percent accuracy in the ten-operation problem,then individual operations must be done with 97percent accuracy. Elementary teachers who accept70 percent success rates are not only wasting theunique potential of mathematics but also settingtheir students up for failure in later courses. Theproposal, then, is that the goal should be quality,not quantity, and not speed; fewer problems, butexpected to be 100 percent right. Is this reasonablein education? Maybe not, but I sketch strategiesthat should make it possible to get close. One isdescribed above: carefully designed work templatesfor students to emulate.

The most important strategy is teacher diagnosisof errors that students can’t find themselves.Ideally, every wrong answer should be corrected.For diagnosis, the student should explain his orher reasoning following his or her written workrecord. If the record is illegible, steps were skipped,or appropriate templates were not used, then thework should be redone before being diagnosedfor specific errors; it is remarkable how often thisresolves the problem, and it is valuable for studentsto see this. If the work record is appropriate, thenit can be reviewed efficiently and mistakes quicklypinpointed. Premature guesses about the difficultyare often wrong and will make confusion worse,so the teacher should wait until the student has

come to the error before doing anything. Further,the student learns more if he or she spots theerror him- or herself; wait and see if he or she says,“oh, now I see what went wrong.” In such casesthe teacher can ask what happened, to see if theinsight is right, but if so then the teacher shouldleave well enough alone and not explain further,even if a clearer description can be given. If thisapproach is used systematically, then errors willbecome remarkably uncommon. Confusions don’taccumulate and students learn to diagnose theirown work.

Definitions and Key Statements

Precise reasoning requires precise descriptions:vague or intuitive things are inaccessible to math-ematics. I discuss four aspects: when explicitdescriptions are necessary; how things are de-scribed precisely; where good descriptions comefrom; and how they support the intuitive processesthat usually precede precise reasoning.

When Statements Are Needed. There are no hard-and-fast rules about when things should beformalized because it depends more on howour brains work than on mathematical structure.Professional practice gives clues: mathematicianshave essentially the same neural equipment as ev-eryone else and are very ambitious about exploitingit to the fullest.

For instance, addition seems to be internalizedas a kind of behavior, and work habits learnedin the integers transfer relatively easily to muchmore general contexts that behave in the same way.This is formalized as: any binary operation thatis commutative, associative, and has a neutral ele-ment and inverses is entitled to be called “addition”and denoted by “+”. Similarly, a second operationthat distributes over + is entitled to be denoted as“multiplication”. This enables transparent transferof work habits. That this is a cognitive strategyrather than just mathematical structure becomesclear when people move out of their comfort zone.Technical examples: homology is written additively,and the bar construction multiplicatively, and it israther awkward to translate these to the other no-tation. It can be amusing to watch mathematiciansencountering tropical arithmetic in the integers forthe first time. This has two operations, and onedistributes over the other just as multiplicationdoes over addition. However, the operation thatdistributes like multiplication is ordinary addition.This means it cannot be rewritten as multiplication,so distributivity looks backward. The operationanalogous to addition is the max function, whichdoes not have inverses and also does not followfamiliar patterns.

The point is that while it is a good idea tohave the associative and commutative axiomsavailable for reference, well-internalized work

1266 Notices of the AMS Volume 58, Number 9

habits make it unnecessary and distracting torequire students to work with them explicitly.See [4](b) for an elaboration. In contrast, makingexplicit the definition of fractions and the quadraticformula has significant cognitive benefits.

Formats. Contemporary practice uses axiomaticdefinitions to provide precision, but the formatis not forced by the job to be done. In fact, thedefinition format seems to have won out over thealternatives because it is more effective for humanuse; see [3], §5. In other words, precise definitionscan be an enabling technology for people. Similarlyfor concise theorems.

Definitions and theorems provide precision, but,as with any format, they do not turn junk intogold. Effective definitions usually result from longand painful searches. Mathematicians will havesome intuitive goal, make a precise descriptionthat they hope will realize their intuition, workwith it for a while, and most of the time it doesn’tdo what they hope for. They junk the attempt andtry again. In some areas these false starts probablyaccount for 50 percent of all mathematical effort.A really good definition is therefore a treasure, andbecause it can efficiently guide newcomers aroundyears of confusion, a powerful legacy. Really sharpand effective theorem statements are similarlyprecious.

Suggestion. Important objects, terminology, andproperties (fraction, prime, relatively prime, properfraction, quadratic formula, …) should be givenbrief and genuinely precise formulations.

• Formulations should be constructed pri-marily by professional mathematicians,with education feedback. Professionals ha-bitually hone and fine-tune them to be briefand effective and can ensure that they arecompatible with later material.

• Ideally, students should memorize themso they can be reproduced exactly. Defini-tions provide anchor points, and genuinelyfunctional understanding nucleates anddeepens around the definition. This isparticularly true for weaker students.

• Explanation of what a definition “means” isbest given after the definition, not before.Putting too much explanation first almostguarantees confusion.

Compared with current educational practice, thisis very rigid, but it is a baby version of the waymathematicians approach unfamiliar material, andit works. More precisely, less rigid approaches takelonger, are less effective, and were abandoned whenthis approach became available about a centuryago.

Is this approach reasonable in education? Well-crafted definitions evoke mathematical objectswith the economy and grace of poetry. Asking

students to memorize them is like asking them tomemorize unusually powerful poems. There arenot so many that this is burdensome, and if it isdone consistently it will become routine. Studentswill also see quick payoffs because good definitionsare immediately functional.

Intuition. The final point concerns the role ofintuition. A great deal of mathematical activity isat least guided by intuition, but this makes precisedefinitions more important. Intuitions developedby working with a good definition are often effective.Intuitions based on sloppy “definitions”, or naiveintuitions “refined” by exposure to definitions, tendto be unreliable. Intuitions based on “meaning” orphysical analogues are almost never satisfactory.This is acceptable if passive understanding is thegoal, but it strongly limits long-term outcomes.

Limits of Mathematics

The need for complete precision provides power butalso limits the scope of mathematics. In particular,nothing in the physical world can be described withmathematical precision, so mathematical methodsdo not apply directly. People applying mathematicsaccommodate this with an intermediate step: asymbolic mathematical model is developed torepresent the physical situation, and it is the modelthat is analyzed mathematically. Mathematicalconclusions about the model are reliable in a waythat the connection between the physical situationand the model cannot be (see Example 1: BuyingPencils and [5] for examples), and working withouta model confuses this structural difference.

Historically, systematic use of modeling devel-oped in the seventeenth century as an essentialpart of the “scientific revolution”.

There are additional reasons that the mixingof modeling and analysis has been obsolete forfour centuries: they are quite different cognitively,and mixing causes cognitive interference. In fact,cognitive interference in word problems may bethe most serious single difficulty I have seen instudents. For weaker students “crippling” may notbe too strong a word. Examples are given here inthe section Word Problems, and there are manymore in [4].

Most contemporary educators are philosophi-cally committed to word problems as a differentformat rather than a different activity. They encour-age a holistic approach with reasoning “in context”and discourage modeling. In other words, theyfollow sixteenth-century practice and, naturally,get sixteenth-century outcomes.

Technology

A great deal of [4] is concerned with educationaluse (or misuse) of technology. The issue is too

October 2011 Notices of the AMS 1267

complex to be addressed here, but I make onecomment.

The traditional classroom was a tightly boundpackage, and technology is making it come apart.In particular, it is loosening the link betweenteaching and learning. For instance, calculatorsappear to make teaching easier and improvestudent performance on traditional test problemtypes. However, these problem types no longerhave the learning correlates they once did. Easierteaching and better test performance are, in somecases, masking learning declines. A genuinelylearning-oriented approach reveals this.

ExamplesExplicit problem-oriented methods for student useare the main products of the science-of-learning ap-proach and are the bases for generalities describedin the previous section. Space constraints do notpermit really illuminating the generalities, but wegive enough to illustrate how they fit together. Thefirst section gives a precise but unconventionalapproach to fractions. It does not immediatelygive the usual passive “parts-of-a-whole” picture,though that could be given as an application ofinteger fractions. However, it does immediatelygive effective contexts for partial fractions in boththe integers and the polynomials (both in thesubsection Integer Partial Fractions). An unconven-tional procedure for multiplying polynomials inthe subsection Multiplying Polynomials carefullyexploits structure to give access to much morecomplex problems than usual and is adapted tofind coefficients in the Integer Partial Fractions sub-section. Many more examples and interconnectionsare given in [4]. The polynomial multiplicationprocedure, for instance, is adapted to give analgorithm for multiplying multidigit integers in[4](a, c).

The subsection Word Problems illustrates theearlier discussion of word problems in Limits ofMathematics.

Fractions

These are a perennial source of trouble, and thereseem to be two reasons. First, a fraction is a namefor, or description of, something, not the thing itself.The expression 9

4 = 2.25 really means “2.25 can be

written as (is a name for) 94 ”. The different names

encode different properties, and we want to workwith both because we want to exploit the differentinformation they encode. Of course all our symbolsare names, not things. This point is too subtle forexplicit use in elementary education, but studentslearn a great deal subliminally from the way thingsare presented, so confusion can be avoided ifteachers are aware of it and accommodate it intheir phrasing (see the later subsection Caution).

The second point about fractions is that theyspecify things implicitly. The name 2.25 encodes anexplicit procedure for assembling a number fromsingle-digit integers and powers of 10. Fractionsand names such as

√2 encode properties that

determine the thing but do not encode a procedure.See [5] for a completely analogous description ofsquare roots.

Definition. “� can be expressed as ab ” means

b� = a.

Notes.

• Examples in the rest of this section demon-strate that this is an effective approachfor high and middle school. It is not yetclear whether it can be used directly inelementary grades, but it should be clearthat whatever approach is used should becarefully upward-compatible with this.

• The structure of the definition implies thatthe default way to work with fractions isto clear denominators. The usual rules areshortcuts that sometimes avoid this, butthese shortcuts are verified by clearingdenominators, and if you cannot see howto use shortcuts, then clear denominators.

• Note that 0� = 0, so if a ≠ 0, then nothingcan be expressed as a

0 . It is a uselessnotation, and we avoid it by requiringdenominators to be nonzero.

Examples.

(1) The decimal 2.25 can be expressed as 94

because it satisfies 4× 2.25 = 9.(2) Show that if c ≠ 0, then ca

cb can be expressedas a

b . To check, set cacb = � and clear

denominators. This gives ca = cb�. Ifc ≠ 0, then c can be canceled to givea = b�, and this means � = a

b .

(3) Is a− 2?= a2−4

a+2 ? To find out, clear denomi-nators:

(a+ 2)× (a− 2) = a2 − a2+ 2a− 4 = a2 − 4;

this is true, so the answer is “yes” (but seethe subsection Caution).

(4) Is ax +

by

?= a+bx+y ? Clear denominators (mul-

tiply by xy(x + y)) to get a(y(x + y)) +b(x(x+ y)) ?= (a+ b)(xy). Expanding and

canceling gives ay2 + bx2 ?= 0. This hardlyever works, so the answer in general is“no”.

(5) Express ax +

by as a fraction. First give

it a neutral name, � = ax +

by , and clear

denominators by multiplying by x and y :xy � = ya + xb. But this means � can bewritten as � = ya+xb

xy .

1268 Notices of the AMS Volume 58, Number 9

Comments and Cautions. First, I explain the nota-

tions � and?=.

�. If we are trying to express something as afraction, then we need another name for it to useduring the process. The thing is usually given witha compound name (e.g., 2

a + 7× 2b ), but it is hard

to work with this and still think of it as a singleobject. It helps to use a temporary name, e.g., “letA = 2

a + 7 × 2b ”. But using a symbol means we

have three symbolic names in play, and recycling asymbol (for instance using “A” in both Examples (2)and (5)) invites confusion. The neutral placeholder,�, encodes no structure and has no hidden orlasting significance.

?=. This addresses problems with sloppy use of“=”. In common use this can mean “define x by…”;“it is always true that…”; “is it always true that…?”,or “determine when it is true that…”. The equationalone is incomplete, and the intended meaning is tobe inferred from context. This ambiguity becomesinvisible to experienced users, but students—andmany teachers—do not understand it and it is aconstant source of confusion (see question 2, p. 390,in [2]). Computers do not understand it either,and the programming language of Mathematicaprovides at least six different symbols for differentmeanings of “=” . In the example here we wantto begin with an equality posed as a questionand end with the same expression as an assertion.Confusion is inevitable if the notation does notdisplay the difference between the two.

Caution. There is an additional ambiguity in theuse of � = a

b as short for “� can be written as ab ”.

The difficulty can be seen by rephrasing Example(2) as “show that ca

cb =ab ”. The typical response is

“clear denominators by multiplying by cb and bto get b(ca) ?= cb(a). This is true so the originalexpression is true.” But this logic must be flawedbecause it does not require c to be nonzero.

The source of the flaw is that fractions (asnames) encode a property as well as an underlyingobject. The symbol “=” only concerns underlyingobjects, while “can be expressed as” includes theadditional information. Technically this leads peo-ple to overlook one of the two implications in the“if and only if” aspect of the definition. This error isessentially harmless, especially after “don’t divideby 0” is firmly internalized, and I think trying tofix or explain it would cause more problems thanit would solve. I suggest teachers should be awareof it and use the more precise phrasing to evokecorrect reasoning when appropriate—especiallywhen verifying rules—and to answer questionsabout meaning. In ordinary practice the flawedversion should be accepted.

More about Names. Using care about the differencebetween things and names resolves a lot ofconfusion. Again, this is too subtle for explicit usein elementary education, but teachers should beaware of it and terminology should be consistentwith it. Examples:

• “Express 1.23+ 45.6 as a decimal” ratherthan “find 1.23 + 45.6”. The expressionspecifies a number, so it is already “found”.The decimal name encodes a particularconstruction of the number in addition tothe number itself, and the objective is tofind the construction-describing name.

• “Express 12 +

23 as a fraction” rather than

either “find” or “simplify”. As above, theexpression already specifies a number, so itis not “lost”. We could convert the fractionsto decimals and “find” it by adding these,but this does not give the desired form.Finally, “simplify” is ambiguous: the sumis the partial-fraction form of the rationalnumber, so is already “simplified” in thissense.

Integer Partial Fractions

One objective is to foreshadow a standard rational-function topic in the Polynomial Partial Fractionssubsection. This might also illustrate that there arelots of interesting problems within mathematics.

First, a theorem. Theorems are great labor-savers: they explain what you can do with completeconfidence, and what the limits are. In this case,for instance, you should expect that attempts toseparate nonrelatively prime factors will fail insome way.

Theorem 1. If b and c are relatively prime, thenfractions with denominator bc can be expanded asabc =

xb +

yc , for some x, y .

This is true essentially whenever the terms makesense (any commutative ring) and is an immediateconsequence of the definition of relatively prime:a, b are relatively prime if there are m,n so thatam+ bn = 1. This is the correct general definition,but it is cumbersome to use, and in practice (integershere, and real polynomials in the Polynomial PartialFractions subsection) one uses refinements of both“relatively prime” and the expansion.

Theorem 2 (Refinements for integers).

(1) For integers, “relatively prime” is the sameas “have no common factor” (except 1, ofcourse).

(2) if b1, b2, . . . , bn are pairwise relativelyprime, then there is a unique expansion

ab1 × · · · × bn

= r1b1+ r2b2+ · · · + rn

bn+ q

with all the ribi

proper fractions and q aninteger.

October 2011 Notices of the AMS 1269

For integer fractions, ab proper means that

b > a ≥ 0; see the section Polynomial PartialFractions for the polynomial version. Note that theversion for a single factor, ab =

rb +q with r

b proper,is the usual quotient-with-remainder expressionfor a÷ b.

This refinement is true because the Euclideanalgorithm works in the integers. In fact, there isan algorithm for the numerators in the expansion,see the Wikipedia entry on the extended Euclideanalgorithm. The algorithm is cumbersome andobscure, so modular arithmetic is used here.

Problem. Express 47/180 as a sum of an integer andproper fractions with prime-power denominators.

Solution. 180 = 22325 so the maximal partialfraction expansion is of the form a

4 +b9 +

c5 + d.

The first step must be to clear denominators:fractions are defined implicitly, and we mustconvert to an explicit form to work with them (seethe earlier subsection Fractions). This gives

47 = (4 · 9 · 5)(a4+ b

9+ c

5+ d)

= 9 · 5a+ 4 · 5b + 4 · 9c + 4 · 9 · 5d.

Reduce mod 4 to get 474≡ 3

4≡ 45a4≡ a. This

means a = 3 + 4r , and the numerator that givesa proper fraction is 3. Next reduce mod 9 to get

29≡ 2b, so b

9≡ 1. Finally reduce mod 5 to get 25≡ c.

We now know that 47/180 = 34 +

19 +

25 + d, with

d an integer. We can find d by converting thepartial fractions back to a common denominator,but we could also use a calculator. The left side isapproximately 0.2611, while the fractions on theright add up to about 1.2611. Since d is an integerit must be −1, and

47/180 = 34+ 1

9+ 2

5− 1.

Multiplying Polynomials

This section illustrates how careful attention tocognitive issues and mathematical structure cansignificantly extend the problem types studentscan do. The main cognitive concern is that dif-ferent activities (here organization, addition, andmultiplication) should be separated as much aspossible. Structure is used to make the procedureflexible; see the next section for a variation. See[4](a) for detailed discussion.

Problem. Write the product (2y3 + y2 − 9)(−y2 +5y − 1) as a polynomial in y , in standard form.

“Standard form” here means coefficients timespowers of the variable, with exponents in descend-ing order. The factors are given in this standardform. Sometimes ascending order is used.

Solution. The first step is purely organizational.We see that the outcome will be a polynomial ofdegree 5, so set up a template for this:

y5[ ]+ y4[ ]+ y3[ ]+ · · · .(with enough space between brackets for the next step).

Next scan through the factors and put coefficientproducts in the right places. To get they3 coefficient,for example, begin with the leftmost term in the firstfactor. Record the coefficient, (2). The exponent is3, so the complementary exponent is 0. Beginningat the right in the second factor, we see that thecoefficient on y0 is −1, so we record this as aproduct (2)(−1). Now move one place to the rightin the first factor and record coefficient (1). Theexponent is 2, so look for complementary exponent1 in the second factor. The coefficient on thisis 5, so record this as (1)(5). Again move oneplace to the right in the first factor and recordcoefficient (−9). The exponent is 0, so we look forcomplementary exponent 3 in the second factor.There is none, so we record coefficient 0 as (−9)(0).The template will now look like

· · · + y3[(2)(−1)+ (1)(5)+ (−9)(0)] · · · .It is quite important that no arithmetic be donein the organizational phase, not even (1)(5) = 5.Also, we usually quit scanning the first factor whencomplementary exponents are above the degree ofthe second, but it is better to record overruns suchas (−9)(0) than to worry about this. The reasonis that even trivial on-the-fly arithmetic requiresa momentary break in focus that significantlyincreases error rates in both organization andarithmetic. For the same reason, every coefficientis automatically enclosed in parentheses whetherthey are needed or not.

The first arithmetic step is to do the multiplica-tions:

y5[(2)(−1)︸ ︷︷ ︸−2

]+ y4[(2)(5)︸ ︷︷ ︸10

+ (1)(−1)︸ ︷︷ ︸−1

]

+ y3[(2)(−1)︸ ︷︷ ︸−2

+ (1)(5)︸ ︷︷ ︸5

+ (−9)(0)︸ ︷︷ ︸0

]+ · · · .

The second arithmetic step is to do additions:

y5((2)(−1)︸ ︷︷ ︸−2

)+ y4((2)(5)︸ ︷︷ ︸10

+ (1)(−1)︸ ︷︷ ︸−1︸ ︷︷ ︸

9

)

+ y3((2)(−1)︸ ︷︷ ︸−2

+ (1)(5)︸ ︷︷ ︸5

+ (−9)(0)︸ ︷︷ ︸0︸ ︷︷ ︸

3

)+ · · · .

Additions and multiplications are separated be-cause, again, switching back and forth invites errors.Note also that this pattern requires essentially noorganizational activity during the arithmetic steps:everything is in standard places, and outcomes arewritten just below inputs.

1270 Notices of the AMS Volume 58, Number 9

The final result is

y5(−2)+ y4(9)+ y3(3)+ y2(8)+ y1(−45)+ 9.

Discussion. Standard high-school practice is torestrict to product of binomials and use theintelligence-free algorithm with acronym “FOIL”.This is so ingrained that many students say “FOILit” rather than “expand”, even when the factorsare not binomials. However, this algorithm mixesorganization and arithmetic enough that somestudents have trouble even with this simple case,and since the expansion step is not adapted tothe eventual goal, a second sorting step is needed.Finally, the near-exclusive focus on binomialsmakes larger expressions foreign territory. Somestudents are at a loss about how to deal with them,others generalize incorrectly from the pattern,and almost all are anxious and hesitant. A betteralgorithm fixes all this and opens up a much widerworld.

Polynomial Partial Fractions

The final example brings together fractions, par-tial fractions, and polynomial multiplication. Theanalogue of modular arithmetic is included inthe supplement [5]. Theorem (1) in the earliersubsection Integer Partial Fractions applies, but, asin that section, we need a refinement.

Theorem 3 (Refinement for polynomials).

(1) For polynomials “relatively prime” is thesame as “have no common factor” and thesame as “have no common roots” (includingcomplex roots).

(2) If b1, b2, . . . , bn are pairwise relatively primereal-coefficient polynomials, then there is aunique expansion

ab1 × · · · × bn

= r1b1+ r2b2+ · · · + rn

bn+ q

with ri , q polynomials and each ribi

a properfraction.

A polynomial fraction is called proper if thedegree of the numerator is strictly smaller than thedegree of the denominator. The term has the samesignificance as the integer version even though thedefinitions are different. As with integers, q is theresult of division (the “quotient”) and the ri areremainders.

This statement is almost identical to the integerstatement. This emphasizes commonality of theunderlying structure, but the main reason isefficiency: since the underlying structures are thesame, a maximally efficient statement for onecontext will also be maximally efficient for theother.

Problem. Find the real partial-fraction expansionof the polynomial fraction

3x3 + 2x+ 9(4x2 − 4x+ 1)(x2 − 2x+ 3)

.

Solution, Setup. The first quadratic in the denomi-nator factors as (2x− 1)2, but these factors cannotbe separated because they are not relatively prime.The second quadratic has complex roots, so itcan be factored over the complexes as a productof linear terms. These could be separated in apartial-fraction expansion over C, but not overR. The denominators in the partial fractions aretherefore the two quadratics. Finally, the inputfraction is proper, so the expansion cannot havea (nonfraction) polynomial term (note this conclu-sion is special to polynomials: it does not work forintegers).

Both pieces in the expansion have denominatorsof degree 2, and we know that the numerators havesmaller degree. The expansion is therefore of theform

3x3 + 2x+ 9(4x2 − 4x+ 1)(x2 − 2x+ 3)

= ax+ b4x2 − 4x+ 1

+ cx+ dx2 − 2x+ 3

.

As usual, to work with fractions we have to cleardenominators. This gives

3x3 + 2x+ 9 = (ax+ b)(x2 − 2x+ 3)

+ (cx+ d)(4x2 − 4x+ 1).(1)

Solution, Linear-Algebra Approach. Here we illus-trate the use of linear algebra to find the coefficientsin the expansion above. The supplement [5] alsoincludes a modular-arithmetic approach analogousto the integer procedure. Equation (1) is an equalityof polynomials, so the coefficients must be equal.The coefficient equations give the linear systemthat determines a–d.

The plan is as follows: for each exponent nscan through the products above and pick outcoefficients on xn, exactly as in the section onpolynomial products. Recording coefficients gives

x3 : 3 = a(1)+ c(4)x2 : 0 = a(−2)+ b(1)+ c(−4)+ d(4)x1 : 2 = a(3)+ b(−2)+ c(1)+ d(−4)x0 : 9 = b(3)+ d(1).

Writing this linear system in matrix form gives1 0 4 0−2 1 −4 4

3 −2 1 −40 3 0 1

abcd

=

3029

.Problems this involved are necessary to illustratesome of the phenomena, but full solutions are notalways necessary. A possibility is to ask “find alinear system that determines the coefficients…”to avoid the linear algebra. Setting up the system is

October 2011 Notices of the AMS 1271

an important goal in any case because in the longrun they would enter

LinearSolve[{{1,0,4,0},{-2,1,-4,4},{3,-2,1,-4},{0,3,0,1}},{3,0,2,9}]

in a computer-algebra system,3 and obtain( 319

81 ,21481 ,−

1981 ,

2927). Even if a full solution is required,

then the problem statement should be “find alinear system that determines the coefficients,and then solve to find the expansion.” Requiringexplicit display of the intermediate step hastwo functions: first, if the final answer is wrong,then the intermediate step can be used to locateerrors. The second reason is to ensure thatstudents carefully formulate the intermediate step,especially if the system is to be solved by hand.Many will try to save writing (and thinking) bysolving on the fly as coefficients are found (e.g., forthe x3 coefficient writing a = 3− 4c and using thisto eliminate a). They could use this strategy tosolve the system after it is set up, but mixing thesteps increases the error rate and in the long termwill limit the problems they can handle.

As a final note, if the input fraction were notproper, then the q term in Theorem 3 wouldbe nonzero. It can be included in the cleared-denominator form (1) and handled the sameway. Most approaches recommend first using longdivision to get q (the quotient) and a proper fraction(with remainder as numerator), and then expandingthe proper fraction. The division algorithm givesthe coefficients in the quotient, so the remainingcoefficients give a smaller linear system. Butdivision takes a lot more work. The extra partof the larger linear system has lower triangularcoefficient matrix so does not add much to thedifficulty.

Word Problems

This is a crucial topic because contemporaryeducators put great emphasis on word problems,but they use a problematic approach. There isan abstract discussion in the earlier subsectionLimits of Mathematics; here I illustrate the use ofmodeling to avoid the problem.

A mathematical model is a translation of a real-world or word problem to a symbolic form suitablefor mathematical analysis. The key feature is thatlittle or no analysis is done during modeling, andno modeling is done during analysis of the model.Three examples are given. The first and second aretypical number-and-word problems with little orno mathematical content. The third illustrates theuse of algebra to explore a problem about statistics.The supplement [5] has additional examples andillustrates difficulties with “trick problems”.

3This is Mathematica syntax.

Example 1: Buying Pencils. This was suggested tome as representative of a common and importantproblem type. Educational goals are discussedbelow.

Frank bought 8 pencils at 32 cents apiece.How much did he pay?

Mathematical Model. This is a rate problem, withmodel:

(number of pencils, items)×(cost rate, cents/item)

= (total cost, cents).

Plugging in values gives 8 × 32 = 256 cents, or$2.56.

Rate equations and the format that includesunits (here items and cents) are discussed in thenext problem.

Problems with the Model. As a reality check, Frankactually did buy some pencils.

• Pencils are now sold in blister packs. Igot one pack of 8 for $2.57. Math wasunnecessary.

• I paid $2.70, including $0.13 Virginia salestax.

No one really expects problems like this to havemuch relation to reality, but this illustrates a pointmade in the subsection Limits of Mathematics. Ifthis is a math problem, then, evidently, math doesnot connect well with reality. Separating modelingand analysis gives a better picture: it is the modelthat is faulty, and the mathematical analysis of themodel is perfectly correct.

Problems with Educational Goals. There are twopoints of view on such problems. One is that theybring math to life. Teachers reading them canalmost smell the pencils. The other is that theyare a “thinly disguised invitation to multiply”, withsome unpackaging activity that does not qualify asmodeling.

The students I have asked about this almostunanimously despise these problems and ig-nore the charming context. They parse it as“…8…32…apiece”: two numbers, multiply. Someeducators even teach a “keyword” approach todetermine the operation. So the bring-to-life ideafails, but pencils no longer smell good anyway.

I have serious concerns with the second view.First, why are large numbers of thinly disguisedtrivial arithmetic problems a good thing? Theyimprove grades and test scores, but do theycontribute in any real way to learning? Second,it is not a good idea to encourage “unpackaging”without modeling. The carefully formulated modeluses a rate equation, and making it explicitconnects with more complicated work (see the nextproblem). Conversely, hiding rate equations willcause confusion later. More generally, “reasoning

1272 Notices of the AMS Volume 58, Number 9

in context” with intuitive “unpackaging” is a hard-to-break habit that seriously limits outcomes inlater courses.

Example 2: Leaky Fuel Tank.

A car begins a trip with 40 liters of fuel anddrives at a constant speed 70 km/hr. At thisspeed the car gets 12 km/liter, but it runsout of fuel after 360 km. Apparently the fueltank is leaking. If it leaks at a constant rate,what is the rate (liter/hr)?4

Model. The rate of loss is (fuel Lost)/(Time); denotethis by L/T . There are two rate equations, and therelation of (fuel Lost) to (fuel Used):

(1) (speed, km/hr)×(Time, hr) = (distance, km)(2) (distance rate, km/liter)×(Used, liter) =

(distance, km)(3) (Lost, liter) = (total, liter) - (Used, liter).

Now put in the numbers from the problemstatement:

(1) 70T = 360(2) 12U = 360(3) L = 40−U .

Solve to getT = 36/7,U = 30,L = 10, and thereforeL/T = 70/36 = 35/18 ' 1.94 liters/hour.

Discussion. The first step was to write the rateequations in a symbolic form designed to be easilyand reliably remembered. Neither numerical valuesnor any analytic reasoning are done in this step.Trying to look ahead, for instance “I need to knowthe time, so write the speed rate equation as(time)=(distance)/(speed)…” invites errors. Thisproblem is significantly more difficult withoutmodeling because there are so many opportunitiesfor such errors. Note also that the rate equationsare written with units, so dimensional analysiscan be used as a check that they are correct. Inparticular, the rate in the fuel use equation is givenas a distance rate (km/liter) rather than a fuel rate(liter/km). The latter seems more logical (see thenext problem), and confusion on this point is likelyto be a source of error.

The last step is to substitute numbers into thesymbolic forms, and then solve. This is routine,and easy because the analysis is protected fromcognitive interference with modeling.

As an aside, it is a common criticism of suchproblems that “constant rate” cannot be madeprecise without derivatives (cf. [6]). This is notcorrect: “constant rate” means a rate equationapplies, and this makes perfect sense if it is madeexplicit. Variable rates need derivatives.

4Adapted from a Massachusetts high school problem viaa “puzzler” from the National Public Radio program CarTalk.

Example 3: Fuel Efficiency. This illustrates a com-mon way to misuse statistics. Comparison withthe degraded version illustrates the benefits ofsymbols rather than the usual inert numbers.

A regulatory agency wants to promote devel-opment of fuel-efficient cars. The regulationsrequire a certain average efficiency to provideautomakers flexibility and incentive: they canoffer inefficient models if these are balancedby super-efficient ones. However, an environ-mental group claims that if the regulatorsuse the traditional km/liter measure of auto-motive efficiency, then more efficient modelswill actually lead to increased fuel use. Arethey right? Which measure should be used toavoid this?

Symbolic Specific Problem. Suppose there is a targetefficiency T km/liter. A manufacturer sells twomodels: an efficient one that gets rT km/liter forsome efficiency multiplier r > 1, and an inefficientone that gets b km/liter so that the average of rTand b is T .

(1) Find the average of the fuel required by thetwo models to go distance d. Express thisas the product of the fuel required by a carwith average efficiency T and a functionh(r).

(2) Evaluate h(r) for r = 4/5 and 5/3. Plot h(r)(by calculator or computer) on the interval[1,1.8]. For which efficiency multipliers rdoes the two-model fleet use more fuelthan average-efficiency cars?

(3) Find r so that the two-model fleet uses twiceas much fuel as a fleet of average-efficiencycars.

(4) Explain what could theoretically happenif the automakers developed a car withefficiency twice the target (i.e., r = 2).

Discussion. The previous problem provides anappropriate template for the modeling step, so anexplicit solution is not given here.

Is there a better measure? Here we are interestedin total fuel use. The km/liter measure has rateequation

(fuel, liter) = (distance, km)(distance rate, km/liter)

so it is inversely related to fuel use. But standardstatistical methods are additive: they work well ifthe data have a linear structure, poorly otherwise.This explains why the km/liter measure workspoorly. The linearly related efficiency measure isthe inverse, the fuel rate in liter/km with rateequation

(fuel, liter) = (distance, km)(fuel rate, liter/km).

It would therefore be more honest to use the fuelrate in the statistical analysis.

October 2011 Notices of the AMS 1273

Degraded Specific Problem. The regulatory agencysets 12 km/liter as the target efficiency. A man-ufacturer sells two models: an efficient one thatgets 18 km/liter and an inefficient one that gets bkm/liter so that the average of 18 and b is 12.

(1) Find the average of the fuel required bythe two models to go 100 km. Comparewith the fuel required by a car with averageefficiency 12.

(2) Repeat for high-efficiency rates 20 and 22.Describe the pattern you see as the betterefficiency increases.

The use of specific numbers is typical of con-temporary practice. This makes it easier to dowithout modeling and more accessible to directevaluation on a calculator. On the other hand, agreat deal of mathematical structure has been lost.The average fuel use as a function of the upperefficiency is now represented by a few numericalvalues. These suggest part of the pattern but do notmake obvious the infinite limit at upper efficiency24. Further, the use of a specific numerical targethides the fact that the controlling parameter is thequotient (max efficiency)/(target efficiency), calledr in the first version. All this structure emergesfrom essentially the same work done symbolically.

The use of numerical values is also a favoritetactic of test designers. The first version is a singleproblem. Plugging in numbers gives a large numberof seemingly different problems. But don’t we wantstudents to see structure and patterns? Destroyingit for the convenience of test design and calculatorevaluation seems inappropriate.

Teacher PreparationOutcomes from K–12 education are unsatisfactoryand not improving. Angst about this has mainly fo-cused on teacher preparation, implicitly assumingthat educational methodology is effective and theproblem is incompetent teachers. The discussionhere suggests an alternative: it is the methodologythat is incompetent, and worse outcomes mightactually reflect better teaching of the methodology.If this is the case, then the methodology has tobe straightened out before there is much point indiscussing teacher preparation.

The issue is approached through analysis oftwo common but conflicting viewpoints aboutadvanced study.

Advanced Study Not Necessary?

The school-of-education argument is essentially“our students have trouble with advanced coursesand don’t get much from them. Pedagogy is moreimportant than content anyway.”

Wu [6] tries to give this point of view moresubstance by citing an article of Begle that found nocorrelation between advanced coursework and bet-ter student outcomes. Wu concludes that advanced

viewpoints are irrelevant to elementary educationand uses fractions to illustrate the reasons. How-ever, his description of fractions is inaccurate(see the supplement [5] for discussion), and whatthis actually shows is that imprecision and sloppynotation that are harmless at advanced levels canrender the presentation irrelevant to elementaryteaching. More care can give more relevance: see[4](b) for a relatively painless extension of thefraction material in the subsection Fractions tocommutative rings. This includes dealing withzero-divisors; not K–12 classroom material, but itcertainly clarifies why having 0 in the denominatoris a bad idea. It also connects with modern themesby observing that Grothendieck groups are just afancy name for fractions when they are approachedthis way. On the other hand, I agree that coursessuch as real and complex analysis are unnecessary:they have little to do with K–12 topics and benefitsto survivors are unlikely to justify the severeattrition among potential teachers.

Examples such as the above suggest that modernadvanced courses with appropriate topics andprecision could connect nicely with K–12 topics,but the next discussion reveals a problem withmethodology.

Advanced Study Is Necessary?

Most people with technical backgrounds feel teach-ers need more mathematics but cannot clearlyarticulate why. It seems to me the essence is “mybackground leads me to feel that educators aredoing bad things. If they had stronger backgroundsthey would feel the same way, and do better.”Roughly, the need is for more sophistication andmathematically disciplined thinking. The standardway to get these is subliminally from the materialduring extensive study, so more study seems tobe the solution. According to Begle, however, theteachers who did have enough advanced study tointernalize the mindset do not do any better asteachers.

There is a disturbing explanation for this. Ad-vanced study leads to internalization of modernmethodology. Education is modeled on a philosoph-ical description of mathematical practice in thenineteenth century and before,5 centered aroundromanticized ideas about the power of intuition.In professional practice this was abandoned asineffective in the early twentieth century. However,educators reject contemporary professional meth-ods as inappropriate for normal humans. Thisbias makes internalization from advanced coursesuseless. In fact, the conviction that the nineteenthcentury was a better time for children is so strongthat mathematicians working in education tend to

5For extensive historical and technical background on thissee [3].

1274 Notices of the AMS Volume 58, Number 9

buy into it and suspend their professional skillsand sensibilities.

Conclusions

Genuinely productive discussion of teacher prepa-ration must wait for resolution of deeper problems:

(1) Core methodology for mathematics educa-tion is stuck in the nineteenth century.

(2) The approach to applications (word prob-lems) is stuck in the sixteenth century, andmuch of geometry is stuck in the thirdcentury BCE.

(3) Until this changes, we should not expectoutcomes better than those of antiquity.

(4) Until this changes, advanced courseworkwith twentieth-century methodology willbe irrelevant.

On a brighter note, the science-of-learning method-ology described here is compatible with modernmathematics, and advanced coursework wouldbe relevant to teaching based on this approach.Moreover, students trained this way would notfind modern mathematics so totally alien, and thecalculus/proof transition would not be the killerthat it is today.

References[1] Special Issue on Education, Notices of the American

Mathematical Society 58 (March 2011).[2] Ira Papick, Strengthening the mathematical content

knowledge of middle and secondary mathematicsteachers, pp. 389–393 in [1].

[3] Frank Quinn, Contributions to a science of con-temporary mathematics, preprint; current draft athttp://www.math.vt.edu/people/quinn/.

[4] , Contributions to a science of mathemati-cal learning, preprint; current draft at http://www.math.vt.edu/people/quinn/.

a: Neuroscience experiments for mathematicseducation

b: Proof projects for teachersc: Contemporary proofs for mathematics educa-

tion (to appear in the proceedings of ICMI Study19, held in Taipei May 2009).

[5] , “Supplement to ‘A science-of-learning ap-proach to mathematics education”’, at the websiteabove.

[6] H. Wu, The mis-education of mathematics teachers,pp. 372–384 in [1].

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October 2011 Notices of the AMS 1275