Strengthening theMathematical Content

Knowledge of Middle andSecondary Mathematics

TeachersIra J. Papick

As a research mathematician and ateacher of mathematics, I have continu-ally enjoyed the thrill of discovering (orre-discovering) mathematics and theexcitement of discussing the beauty

and utility of mathematics with colleagues andstudents. Stimulating mathematical conversationshave made each day interesting and unique. Al-though I have taught several different mathematicsundergraduate and graduate courses involving avariety of majors, much of my career has beenspent working with prospective/practicing mid-dle and secondary mathematics teachers. Thisteaching trajectory was not accidental, since myoriginal occupational plan was to become a highschool mathematics teacher. A desire to continuemy studies of advanced mathematics altered thisplan, and although my enthusiasm and passion forthe subject has not directly impacted middle andsecondary students, the mathematical preparationof their teachers has been central to my collegiatelife.

Since my primary research area is in com-mutative algebra, I have taught and developedcourses in linear and abstract algebra for prospec-tive/practicing middle and high school teachers.These courses were rich in mathematical ideas, butthe connections to important concepts in schoolmathematics were not always explicitly detailed.Several factors contributed to this shortcoming,but a most significant one was the lack of excel-lent curricular materials (both at the school and

Ira J. Papick is professor of mathematics at the Univer-

sity of Nebraska-Lincoln. His email address is ipapick2

@math.unl.edu.

university levels) to help illustrate and demon-strate the critical links. Without such materials,

it is challenging and time-consuming for mathe-

maticians, who primarily teach content courses for

prospective teachers and who are typically unfa-miliar with school mathematics curricula, to make

these critical connections. A natural consequence

of this predicament was frustrated students (What

does this have to do with teaching mathematics inmiddle or secondary school?) and my own bewil-

derment (Why can’t these students appreciate the

beautiful theorems we are proving?).

To help address the need for specialized coursesand materials for mathematics teachers, the Con-

ference Board of Mathematical Sciences, in concert

with the Mathematical Association of America (with

funding provided by the United States Departmentof Education), developed the Mathematical Ed-

ucation of Teachers Report (MET Report ), 2001.

This document carefully articulates a framework

for mathematics content courses for prospec-tive teachers that is built upon the premise that

“the mathematical knowledge needed for teaching

is quite different from that required by collegestudents pursuing other mathematics-related

professions.”

Mathematics teachers should deeply understand

the mathematical ideas (concepts, procedures, rea-soning skills) that are central to the grade levels

they will be teaching and be able to communicate

these ideas in a developmentally appropriate man-

ner. They should know how to represent and con-nect mathematical ideas so that students may com-

prehend them and appreciate the power, utility, and

diversity of these ideas, and they should be able to

March 2011 Notices of the AMS 389

understand student thinking (questions, solution

strategies, misconceptions, etc.) and address it ina manner that supports student learning.

To further clarify the notion of “mathematicalknowledge for teaching”, consider some authentic

mathematics students’ questions that school alge-bra teachers regularly encounter in their teaching

practice and should be prepared to address in amathematically meaningful way.

1. My teacher from last year told me thatwhatever I do to one side of an equation, I

must do the same thing to the other sideto keep the equality true. I can’t figure out

what I’m doing wrong by adding 1 to thenumerator of both fractions in the equality12= 2

4and getting 2

2= 3

4.

2. Why does the book say that a polynomial

anxn +an−1x

n−1+ · · ·+a1x+ a0 = 0 if andonly if each ai = 0, and then later says that

2x2 + 5x+ 3 = 0?3. You always ask us to explain our thinking.

I know that two fractions can be equal, buttheir numerators and denominators don’t

have to be equal. What about ifa

b= c

d, and

they are both reduced to simplest form.

Does a = c and b = d, and how should weexplain this?

4. I don’t understand why (−3) × (−5) = 15.Can you please explain it to me?

5. The homework assignment asked us tofind the next term in the list of numbers

3,5,7, . . . ? John said the answer is 9 (hewas thinking of odd numbers), I said the

answer is 11 (I was thinking odd primenumbers), and Mary said the answer is 3

(she was thinking of a periodic pattern).Who is right?

6. We know how to find 22, but how do we find

22.5 or 2√

2?7. My algebra teacher said

x2+x−6x−2

= [(x+3)(x−2)](x−2)

= x + 3, but my sister’s boyfriend (who is

in college) says that they are not equal, be-cause the original expression is not defined

at 2, but the other expression equals 5 whenevaluated at 2.

8. My father was helping me with my home-work last night and he said the book iswrong. He said that

√4 = 2 and

√4 = −2,

because 22 = 4 and (−2)2 = 4, but the book

says that√

4 6= −2. He wants to know whywe are using a book that has mistakes.

9. Why should we learn the quadratic formulawhen our calculators can find the roots to

8 decimal places?10. The carpenter who is remodeling our

kitchen told me that geometry is impor-tant. He said he uses his tape measure and

the Pythagorean theorem to tell if a corneris square. He marks off 3 inches on one

edge of the corner, 4 inches on the other

edge, and then connects the marks. If theline connecting them is 5 inches long, heknows by the Pythagorean theorem thatthe corner is square. This seems differentfrom the way we learned the Pythagoreantheorem.

Remark. The Situations Project, a collaborativeproject of the Mid-Atlantic Center for MathematicsTeaching and Learning and the Center for Profi-ciency in Teaching Mathematics, is developing apractice-based framework for mathematical knowl-edge for teaching at the secondary school level.The framework creates a structure for identifyingand describing important mathematics that under-lies authentic classroom questions (“situations”)arising in teaching practice (e.g., identifying anddescribing various mathematical ideas connectedto questions such as those previously listed).

In addition to knowing and communicatingmathematics, teachers of mathematics must beprepared to:

• Assess student learning through a varietyof methods.

• Make mathematical curricular decisions(choosing and implementing curriculum),understand the mathematical content ofstate standards and grade-level expecta-tions, communicate mathematics learninggoals to parents, principals, etc.

This kind of mathematical knowledge is beyondwhat most teachers experience in standard mathe-matics courses in the United States (Principles andStandards for School Mathematics, NCTM, 2000),but there are a growing number of institutions,mathematicians, and mathematics educators whoare determined to improve their teacher educationprograms along the lines recommended in the METReport.

Two Collaborative Projects: Mathemati-cians and Mathematics Educators WorkingTogether to Improve Mathematics TeacherEducationCollaborative efforts between mathematicians,mathematics teacher educators, classroom teach-ers, statisticians, and cognitive scientists haveyielded (and continue to yield) innovative foun-dational mathematics and mathematics educationcourses and materials for prospective and prac-ticing teachers that fundamentally address theneed to improve the mathematical and pedagogicalcontent knowledge of teachers. These collabora-tions have provided a greater understanding of thevarying perspectives on important issues regard-ing the teaching and learning of mathematics andhave significantly contributed (and continue to doso) to the improvement of mathematics teachereducation in the United States. What follows aretwo examples of such fruitful collaboration.

390 Notices of the AMS Volume 58, Number 3

I. Connecting Middle School and College

Mathematics

Using the MET Report as a basic framework, a groupof research mathematicians and mathematics ed-ucators at the University of Missouri-Columbia, incombination with a group of classroom teachersfrom Missouri, jointly developed four foundationalcollege-level mathematics courses for prospec-tive and practicing middle-grade teachers andaccompanying textbooks as part of the NSF-fundedproject, Connecting Middle School and CollegeMathematics [(CM)2], ESI 0101822, 2001–2006.These courses and materials were designed toprovide middle-grade mathematics teachers witha strong mathematical foundation and to connectthe mathematics they are learning with the mathe-matics they will be teaching. The four mathematicscourses focus on algebra and number theory, geo-metric structures, data analysis and probability,and mathematics of change and serve as the core ofthe twenty-nine-credit-hour mathematics contentarea of the College of Education’s middle-schoolmathematics certificate program at the Universityof Missouri-Columbia. For those practicing elemen-tary or middle-grade teachers seeking graduatemathematics experiences to improve their mathe-matics content knowledge, cross listed, extendedversions of the four core courses developed underthe (CM)2 Project are offered for graduate credit.

In an effort to help students explore and learnmathematics in greater depth, the four companiontextbooks that were developed as part of the (CM)2

Project (Algebra Connections, Geometry Connec-tions, Calculus Connections, Data and ProbabilityConnections, Prentice Hall, 2005, 2006), employ aunique design feature that utilizes current middle-grade mathematics curricular materials in thefollowing multiple ways:

• As a springboard to college-level mathe-matics.

• To expose future (or present) teachers tocurrent middle-grade curricular materials.

• To provide strong motivation to learn moreand deeper mathematics.

• To support curriculum dissection—criticallyanalyzing middle-school curriculum con-tent—developing improved middle-gradelessons through lesson study approach.

• To use college content to gain new per-spectives on middle-grade content and viceversa.

• To apply middle-grade instructional strate-gies and multiple forms of assessment tothe college classroom.

Throughout each book, the reader finds a numberof classroom connections, classroom discussions,and classroom problems. These instructional com-ponents are designed to deepen the connectionsbetween the mathematics that students are study-ing and the mathematics that they will be teaching.

The classroom connections are middle-grade in-vestigations that serve as launching pads to thecollege-level classroom discussions, classroomproblems, and other related collegiate mathemat-ics. The classroom discussions are intended to bedetailed mathematical conversations between col-lege teachers and preservice middle-grade teachersand are used to introduce and explore a variety ofimportant concepts during class periods. The class-room problems are a collection of problems withcomplete or partially complete solutions and aremeant to illustrate and engage preservice teachersin various problem-solving techniques and strate-gies. The continual process of connecting what theyare learning in the college classroom to what theywill be teaching in their own classrooms providesteachers with real motivation to strengthen theirmathematical content knowledge.

II. Nebraska Algebra

(Part of the NSF project, NebraskaMATH, DUE-0831835, 2009–2014). For several decades, schoolalgebra has occupied a unique position in the mid-dle and secondary curricula and even more so inrecent times with the expectations of “algebra forall” (Kilpatrick et al., 2001). Not only is algebra acritical prerequisite for higher-level mathematicsand science courses, but also it is essential for suc-cess in the work force (ACT, 2005). Most recently,several national reports have called for an intensi-fied focus on the learning and teaching of schoolalgebra (National Mathematics Advisory Panel;NCTM Focal Points; MET; MAA report, Algebra:Gateway to a Technological Future). Although thespecific recommendations of these reports havesome differences, all of them agree that “strategiesfor improving the algebra achievement of middleand high school students depend in fundamentalways on improving the content and pedagogicalknowledge of their teachers” (Katz, 2007).

Employing the teacher-education recommen-dations of the aforementioned reports, withthe ultimate goal of extending success in alge-bra to all students in Nebraska, a collaborativegroup of mathematicians, mathematics educators,classroom teachers, statisticians, and cognitivepsychologists recently developed an integratednine-graduate-credit-hour sequence designed tohelp practicing Nebraska Algebra I teachers tobecome master Algebra I teachers with specialstrengths in algebraic thinking and knowledge forteaching algebra to middle and high school stu-dents. Two of the courses in the program (Algebrafor Algebra Teachers and Seminar in EducationalPsychology: Cognition, Motivation, and Instructionfor Algebra Teachers) are taught in a two-week sum-mer institute (the first two cohorts completed thesecourses in the summers of 2009 and 2010) and,during the academic year following their partici-pation in the Nebraska algebra summer institute,teachers return to the classroom and work with

March 2011 Notices of the AMS 391

an instructional coach or teaching mentor as theystrive to transfer knowledge gained in the summerinstitute into improved classroom practice. In ad-dition, teachers take a three-graduate-credit-houryearlong pedagogy class focused on enhancingtheir ability to teach algebra to all students and tobecome reflective practitioners.

The Algebra for Algebra Teachers course was de-signed to help teachers better understand the con-ceptual underpinnings of school algebra and howto leverage that understanding into improved class-room practice. Course content and pedagogy devel-opment was strongly influenced by national reportsand research findings, as well as by the collabora-tive expertise of mathematicians, mathematics ed-ucators, and classroom teachers.

The course content begins with a review of keyfacts about the integers, including the Euclideanalgorithm and the fundamental theorem of arith-metic. The integers modulo n are studied as a toolto broaden and deepen students’ knowledge ofthe integers, since questions concerning integerscan often be settled by translating and analyzingthem within the framework of this allied system.From this foundation, the course of study involvespolynomials, roots, polynomial functions, poly-nomial interpolation, and polynomial rings k[x],where k is the field of rationals, reals, or complexnumbers. Special attention is paid to linear andquadratic polynomials/functions in connection totheir importance in school algebra. Fundamentaltheorems established in the context of the ring ofintegers are studied in the context of k[x], e.g.,the division algorithm, Euclidean algorithm, irre-ducibility, and unique factorization. Additionally,applications (such as the remainder theorem, factortheorem, etc.) are considered, and other results inpolynomial algebra (such as the rational root test,multiple roots and formal derivatives, Newton’smethod, etc.) are studied in depth.

The course pedagogy combines collaborativelearning with direct instruction and was designedto provide teachers with dynamic learning andteaching models that can be employed in theschool classroom. Course assessments includeindividual and collective presentations, writtenassignments, historical assignments, mathematicalanalyses of school curricula, extended mathematicsprojects, and a final course assessment.

ConclusionA fundamental tenet of our courses and materialdevelopment is that mathematics teachers shouldnot only learn important mathematics, but theyshould also explicitly see the fundamental con-nections between what they are learning and whatthey teach (or will teach) in their own classrooms.Moreover, while learning this mathematics, theyshould directly experience exemplary classroompractice, creative applications to a wide variety ofstate-of-the-art technology, and multiple forms of

authentic assessment. The work we have accom-plished, and that which we hope to accomplish,has occurred through the collaboration of math-ematicians, mathematics educators, classroomteachers, statisticians, and cognitive psychologists.The combined expertise and perspective of theseprofessionals have significantly strengthened ourefforts, and we are hopeful that these and futurecollaborations will help contribute to the improve-ment of mathematics teacher education in theUnited States.

References1. ACT, Crisis at the Core: Preparing All Students for

College and Work, ACT, 2005.

2. John Beem, Geometry Connections, Prentice Hall,

Upper Saddle River, NJ, 2005, 320 pp.

3. Conference Board of Mathematical Sciences,

The Mathematical Education of Teachers Report,

Mathematical Association of America, Washington,

DC, 2001.

4. Asma Harcharras and Dorina Mitrea, Calculus

Connections, Prentice Hall, Upper Saddle River, NJ,

2006, 496 pp.

5. V. Katz (editor), Algebra, Gateway to a Technolog-

ical Future, MAA Reports, Washington DC, 2007, p.

27.

6. J. Kilpatrick, J. Swafford, and B. Findell (editors),

Adding It Up: Helping Children Learn Mathematics,

Mathematics Learning Study Committee, National

Research Council, Washington, DC, 2001.

7. National Council of Teachers of Mathematics,

Principles and Standards for School Mathematics, Na-

tional Council of Teachers of Mathematics, Reston,

Virginia, 2000.

8. , Curriculum Focal Points for Prekindergarten

through Grade 8 Mathematics, National Council of

Teachers of Mathematics, Reston, Virginia, 2000.

9. Ira Papick, John Beem, Barbara Reys, and Robert

Reys, The Missouri Middle Mathematics (M3) Project:

Stimulating standards-based reform, Journal of

Mathematics Teacher Education 2 (1999), 215-222.

10. Ira Papick and James Tarr, Collaborative Efforts

to Improve the Mathematical Preparation of Mid-

dle Grades Mathematics Teachers: Connecting Middle

School and College Mathematics, AMTE monograph,

Volume 1, 2004, pp. 19–34.

11. Ira Papick, Algebra Connections, Prentice Hall,

Upper Saddle River, NJ, 2006, 368 pp.

12. , Research Mathematician and Mathematics

Educator: A Foot in Both Worlds, A Decade of

Middle School Mathematics Curriculum Implemen-

tation: Lessons Learned from the Show-Me Project,

Information Age Publishers, Charlotte, NC, 2008.

13. Debra Perkowski and Michael Perkowski, Data

Analysis and Probability Connections, Prentice Hall,

Upper Saddle River, NJ, 2006, 416 pp.

14. Situations Project of the Mid-Atlantic Cen-

ter for Mathematics Teaching and Learning

(Universities of Delaware and Maryland and Penn

State University) and the Center for Proficiency

in Teaching Mathematics (Universities of Georgia

and Michigan) (2009), Framework for Mathematical

Proficiency for Teaching (draft document).

392 Notices of the AMS Volume 58, Number 3