The Community of MathTeachers, fromElementary School toGraduate SchoolSybilla Beckmann

Why should mathematicians be in-terested and involved in pre-K–12mathematics education? What arethe benefits of mathematiciansworking with school teachers and

mathematics educators?1 I will answer thesequestions from my perspective of research math-ematician who became interested in mathematicseducation, wrote a book for prospective elemen-tary teachers, and taught sixth-grade math a fewyears ago. I think my answers may surprise youbecause they would have surprised me not longago.

It’s Interesting!If you had told me twenty-five years ago, when I wasin graduate school studying arithmetic geometry,that my work would shift toward improving pre-K–12 mathematics education, I would have told youthat you were crazy. Sure, I would have said, that is

Sybilla Beckmann is professor of mathematics at the Uni-

versity of Georgia. Her email address is sybilla@math.

uga.edu.1A note on terminology: By “mathematician” I mean in-

dividuals in mathematics departments at colleges and

universities who teach mathematics courses and who

have done research in math. By “teacher” I mean individ-

uals who teach within pre-kindergarten through grade

12. By “mathematics educator” I mean individuals who

teach mathematics methods courses, professional devel-

opment seminars, or workshops or who supervise or

coordinate math teaching or curricula in schools and

who have done research in mathematics education. I ac-

knowledge that these categories are neither exhaustive

among mathematics professionals nor mutually exclu-

sive, that the descriptions of these categories should be

viewed as somewhat fuzzy and approximate, and that

the names of these categories are not fully descriptive.

important work, it’s probably hard, and somebodyneeds to do it, but it doesn’t sound very interesting.Much to my surprise, this is the work I am now fullyengaged in. It’s hard, and I believe what I’m doing isuseful to improving education, but most surprisingof all is how interesting the work is.

Yes, I find it interesting to work on improvingpre-K–12 math! And in retrospect, it’s easy to seehow it could be interesting. Math at every levelis beautiful and has a wonderful mixture of intri-cacy, big truths, and surprising connections. Evenpreschool math is no exception.

Consider this connection between preschoolmath and number theory. Young children playwith pattern tile sets that consist of the shapesshown in Figure 1. Playing with these shapes,

these shapes“mix” and are related

these two shapes“mix” and are related

Figure 1. Pattern tiles that young children playwith.

children discover that some of them can be puttogether to make others (e.g., three triangles fit to-gether to make the trapezoid) but that the squaresand thin rhombuses are different. In fact, shapesthat are made without the squares and thin rhom-buses, such as the shape in Figure 2, can never bemade in a different way using the squares or thinrhombuses. Why not? Because the square root ofthree is irrational! The square and thin rhombushave rational area (in terms of square inches), butthe other shapes’ areas are rational multiples ofthe square root of three.

368 Notices of the AMS Volume 58, Number 3

There are manyways to makethis big rhombus usingpattern tilesbut never

using squaresor thin rhombuses

Figure 2. A shape made from pattern tileswithout using the square or thin rhombus.

Of course it is interesting to find connectionsbetween elementary math and more advancedmath (such as my example with the pattern tiles,which delighted me to discover). We can discoverthese connections without ever interacting withchildren, their teachers, or with mathematics edu-cators. But what I have learned from mathematicseducators is how interesting it is to find out howstudents—our own students in college classes aswell as younger students in school—think aboutmathematical ideas. I’ve always enjoyed teachingbut before I interacted with mathematics educa-tors, I didn’t realize it would be both a usefulteaching tool and also interesting to find out howmy students were thinking about the math I wastrying to teach them. In retrospect, this lack ofawareness is surprising. Most (all?) mathemati-cians enjoy talking to each other about math tofind out how others are approaching problems andthinking about ideas. But all humans are capableof mathematical thought. Why not delight in it atevery level? From the four-year-old who realizesthat 8 + 9 is 17 because she knows 8 + 8 is 16and so 8+ 9 must be one more, to the prospectivemiddle-grades teachers in my geometry class thissemester who devised the argument for why thesum of the angles in a triangle is 180◦ that issketched in Figure 3, students can come up withways to solve problems that we might not havethought of ourselves.

start here

turn

turn

turn

go backwards

go f

orw

ard

Going all the way around the triangle, the pencil turned a half-turn, which was the sum of the angles in the triangle.

end here

Figure 3. An explanation for why the sum ofthe angles in a triangle is 180◦180◦180◦.

One surprise in listening to how students thinkabout math is to find that insightful ideas can

come even from students who have big gaps intheir mathematical knowledge. I have found thisnot only with college students, but also withschoolchildren. A few years ago I taught sixth-grade math, every morning for a whole year, toa group of students who were acknowledged byother teachers at the school to be functioningbelow grade level in math. Many of the studentswere still struggling with basic arithmetic facts.Near the beginning of the year, when I asked mystudents to write a word problem for whole num-ber division, most of the students couldn’t writeany problems at all. But, despite the deficits, stu-dents still came up with valuable comments andinsights throughout the year, and their interestin abstract mathematical ideas surprised me attimes. When we discussed the circumference andarea of a circle, I showed the students a printoutof several thousand digits of pi. I told them thatthe digits go on forever without stopping andwithout a repeating pattern. Their eyes grew big.“For real!?” they said. When I asked the studentswhere pi would be on a number line, Santiagodescribed how he thought about the location ofpi, explaining that we’d have to keep zooming inforever on the number line to see exactly where piis located.

In all my teaching, whether sixth grade or atthe college and graduate levels, I’ve found thatgaps and difficulties can coexist with insightfulthoughts and interest in mathematical ideas andwith enthusiasm for math. It’s easy to get frus-trated with our students’ knowledge gaps andmisconceptions, but by recognizing that all ofour students have mathematical potential and byseeking out our students’ ideas, we can make ourteaching more satisfying and more interesting.

What Can We Contribute to Pre-K–12Education and What Can We Learn?It’s not surprising when I say that mathematicianshave much to offer teachers and mathematicseducators because of their broader, deeper viewof mathematics. Mathematicians can help teachersand mathematics educators learn more math andlearn connections between school math and moreadvanced math. But, perhaps surprisingly, thereis plenty of mathematics that teachers and math-ematics educators know but that mathematiciansmay not know explicitly or may not know in a waythat applies to school mathematics.

For example, imagine that you are teachingthird graders about division and that you wantthem to solve a variety of division word problems.What kinds of problems will you give them for15 ÷ 3? You will surely have the students solveproblems about dividing 15 objects equally among3 groups, such as dividing 15 cookies equallyamong 3 people, or dividing 15 blocks equallyamong 3 containers. But you might not think tohave students solve problems that involve dividing

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15 objects into groups of 3 each, such as dividing15 cookies into packages of 3 each, or dividing15 blocks into containers that each hold 3 blocks.These two different perspectives on what divisionmeans correspond to two different equations,which are related by commutativity.

3×? = 15

?× 3 = 15

We know not to take commutativity for grantedbecause of the existence of nonabelian groups andnoncommutative rings. The commutative propertyof multiplication is important to third graders toobecause it helps them lighten the load of learningthe single-digit multiplication facts. But will thirdgraders understand that whole number multiplica-tion is commutative? Is it obvious? In fact, no; evenfrom a third-grade perspective, the commutativityof multiplication of whole numbers is not obvious,as shown in Figure 4.

3 × 5

5 × 3

is the total in 3 groups of 5

is the total in 5 groups of 3

Figure 4. A third-grade perspective on whycommutativity of multiplication is not obvious.

After seeing many examples, third graders maycome to expect that multiplication really is com-mutative, but what is a third-grade way to see whywhole number multiplication is commutative?(Note: no Peano axioms!) The existence of two-dimensional arrays, which can be decomposedeither into equal rows or into equal columns, asin Figure 4, shows why whole number multiplica-tion is commutative. As simple as arrays are, theexistence of these structures now strikes me assaying something much deeper and more surpris-ing about two-dimensional Euclidean space than Ihad previously appreciated.

3 × 5 5 × 3=

Figure 5. A third-grade perspective on thecommutativity of multiplication.

My examples so far have concerned only wholenumber multiplication and division. But examplesof surprisingly intricate details that are involvedin understanding elementary math are everywhere.Did you know, for example, that there are manyways to explain why it makes sense to divide frac-tions according to the “invert and multiply” rule,including ways that involve analyzing word prob-lems and drawing simple pictures? Who knew!

Even if you aren’t interested in learning coolways of explaining why “invert and multiply” isvalid, what can mathematicians learn from thework of mathematics educators and teachers? Ican summarize the most important thing I havelearned: to improve teaching and learning in math-ematics, we must take into account not only themathematics itself—how to organize it, how toexplain lines of reasoning clearly and logically,how the mathematical ideas are connected toother ideas both in and outside of math—butalso what students think—what paths they tend totake as they develop understanding of mathemat-ical ideas, where the difficulties lie, what errorsand misconceptions tend to occur, what capturesstudents’ interest. We must attend to where ourstudents are in their understanding of the materialwe are trying to teach them, not just by markingtheir answers right or wrong (which of course isimportant), but also by looking into the source ofour students’ errors. What ideas have our studentsnot yet grasped and how can we help them learnthose ideas? What misconceptions do they haveand how can we help them see why these aremisconceptions? What gets students excited aboutmath and interested in learning it?

We might think that studying student think-ing is only the job of mathematics educationresearchers and that the rest of us who teachmath could safely dispense with it. Top-notchteaching might seem to be just a matter of havinga well-structured course and a good book andthen presenting the material clearly and enthusi-astically in class, assigning good homework, andholding students accountable by giving tests. Allthese things are components of good teachingand can contribute to student learning, but theyare not enough for excellent teaching. Most of uswho teach have had the experience of deliveringsome beautifully polished lessons and carefullydesigned homework sets only to find out fromstudents’ performance on the test that they didn’tactually grasp the ideas. What was missing? Mostlikely, our lectures didn’t connect with students’existing knowledge and didn’t help students en-gage with the material at a level where they couldmake sense of it. In our enthusiasm to share ex-citing mathematical ideas, we might have failed tosee that our students weren’t ready to appreciatethe ideas. We probably gave answers to questionsbefore students even grasped what the questionswere and why the questions were significant. We

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showed students mathematical tools for solvingproblems before the students saw the need forthose tools. We didn’t learn how our students werethinking and therefore we weren’t able to helpthem build the ideas up in their own minds.

So I have learned from mathematics educatorsthat there is no “royal road” to mathematics teach-ing and learning. It will never be just a matterof getting students who are adequately preparedwhen they enter our classes, and it will neverbe just a matter of delivering polished material.Teaching is a deeply human activity because, likeconversation, it requires a give and take betweenthe teacher and the students. Good teaching willalways be hard work because it requires a teacherto know the mathematics and to take his or herstudents’ thinking into account when makinginstructional decisions. Good teaching requiresknowing the mathematical ideas and how to con-nect and scaffold them to make them accessible tostudents, and it requires finding out how studentsare thinking and then using this information inlectures, problems, and activities. Good learningwill always be hard work for students because itrequires them to engage actively with the material,to think about what they do and don’t understand,and to persevere in making sense of the ideas.

Even if you are not interested in learning moreabout pre-K–12 math or in learning about thework of mathematics educators and about re-sults from mathematics education research, whyshould mathematicians, mathematics educators,and teachers work together?

We Are All in This Together: CollectiveResponsibility for Improving Pre-K–College Math EducationIf we care about the pipeline of students goinginto math and about the strength of our pro-fession in the future, then we simply must takethe whole system of mathematics education intoaccount. Students arrive at college with a longhistory of learning math, and that history affectstheir initial choices of math in college and theirattitudes toward math as they enter their initialcollege math classes. These initial classes, to-gether with a student’s mathematical background,affect a student’s decision to take further mathclasses or not, and they affect whether the studentdecides to become a math teacher. This meansthat all of us who teach math, pre-K teachers,elementary school teachers, middle school teach-ers, high school teachers, college teachers—all ofus—must think collectively and systemically aboutimproving our system.

Think about this: if you teach a calculus course,some of your students may go on to become teach-ers who will teach high school, middle school, orelementary school students. These students’ ex-periences in your math class inform them aboutwhat math is and how it’s done. Do your students

view explaining ideas and making sense of linesof reasoning as an important part of math? Or dothey see math as plowing through a large volumeof stuff that doesn’t make sense? Students’ expe-riences and views—not just your intentions—willinform their future teaching if they become teach-ers. So whether you want to be involved in pre-K–12mathematics education or not, if you teach mathto college students, you are involved in pre-K–12mathematics education because some of your stu-dents might someday become teachers.

If we—mathematicians, mathematics educators,and teachers—are the community that is respon-sible for improving the mathematics education ofall students, then we all bear collective as well asindividual responsibility for improvement of themathematics education system as a whole. Indi-vidually, we are responsible for constantly seekingto improve our own teaching. Collectively, enoughof us must work together to cause the commu-nity as a whole to move along a path of constantimprovement.

But here is something puzzling: why is it thatour system of doing research promotes vigorousactivity and striving for excellence, whereas at nolevel of teaching, from pre-K through the graduatelevel, do we have such a system? In research, wehave a system of publication, presentation, andpeer review in which we build on each other’s ideasand constantly strive to move the field forward.The acts of publishing and presenting researchfindings are public activities, and because theseactivities are filtered by a peer review system, theyallow us to compete for each other’s admiration,and thus they provide us with an incentive to thinkhard about our work and to keep trying to improveit.

Wouldn’t it be wonderful if teaching were a pub-lic activity, in the way that research is, in which webuild on other people’s good ideas and competefor each other’s admiration? Wouldn’t it be greatif all of us who teach math were to take pride inthe things we know well and yet at the same timebe humble, expect to learn more, and recognizethat in each one of us, knowledge, skill, and in-sightfulness coexist with gaps and areas that needimprovement? I think it would be truly exciting tohave a vibrant community of math teachers at alllevels—the community of math teachers from pre-kindergarten through graduate school—thinkingtogether about mathematics teaching and spurringeach other on to do better and better work for thesake of all of our students.

AcknowledgmentsI would like to thank Michael Ching, Pete Clark, andMark Saul for commenting on earlier drafts.

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