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Teaching Assistants WhoInstruct PreparatoryMathematics to Academically-Challenged First-Year CollegeStudentsDanté A. L. TawfeeqPublished online: 13 Jan 2011.

To cite this article: Danté A. L. Tawfeeq (2010) Teaching Assistants Who InstructPreparatory Mathematics to Academically-Challenged First-Year College Students,PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:1,4-13, DOI: 10.1080/10511970902855746

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PRIMUS, 21(1): 4–13, 2011Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970902855746

Teaching Assistants Who Instruct PreparatoryMathematics to Academically-Challenged

First-Year College Students

Danté A. L. Tawfeeq

Abstract: Teaching preparatory mathematics to first-time college students—whocome from economically impoverished high schools that have not prepared their stu-dents to do college level mathematics—can be a daunting task for teaching assistants(TAs). The preparation of TAs to assist such students in the mastery of mathematicalcontent is a complex endeavor involving a balance between methodology and contentissues related to the training of TAs from a teacher-centered perspective versus a stu-dent-centered perspective. We discuss the training of TAs who teach students who cometo college with significant mathematical deficiencies, and describe a different approachfor the training of college TAs. A tutor-based TA training model that borrows elementsfrom 7–12 grade pre-service teacher preparation programs is suggested.

Keywords: Developmental, preparatory, mathematics, teaching, college.

1. INTRODUCTION

The ability of students to learn college-level mathematics is important to theirall-around success as college students. Furthermore, cultivating a quantitativelyliterate university student population is an important part of the work of depart-ments of mathematics [12]. Though most students at the university level are notmathematics majors, many students may take two to three lower-division math-ematics courses during their collegiate tenure [22]. In a 20-year period, from1970 to 1990, demand for collegiate mathematics courses increased more thantwice that of other courses [18]. It is possible that this trend has continued, con-sidering the fact that the number of college-bound students has increased overthe past 3 years [16].

Address correspondence to Danté A. L. Tawfeeq, School of Education, AdelphiUniversity, 1 South Ave, Long Island, NY 11530, USA. E-mail: tawfeeq@adelphi.edu

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Traditionally, university mathematics courses that are at the level of calcu-lus and above account for less than 30% of the 3.3 million total enrollmentof mathematical science courses [9]. Thus, with an increase in the numberof college-bound students, it is reasonable to assume that there will be anincrease in the enrollment of lower-division mathematics courses, which con-stitute courses lower than calculus. These lower-division courses mirror thegeneral makeup of mathematics courses found at the high school level such asalgebra I, geometry, algebra II with trigonometry, pre-calculus, and AP calculus[22]. An increasing concentration of post-secondary mathematics enrollmentin the United States is in developmental mathematics, in courses that have analgebra emphasis [10].

While two-year colleges are increasingly becoming the venue of lower-division work in mathematics [3], four-year colleges and universities still havea great number of students who are required to take lower-division courses,particularly preparatory mathematics courses which are non-credit at most uni-versities. In this article, the phrase “lower division” will be used as a descriptionof those courses that are pre-calculus and below, credit-earning developmentalcourses, or non-credit-earning preparatory courses. The TAs ability to facili-tate the learning of students in these developmental courses is the focus of thisarticle.

Many students in lower-division courses come from urban high schoolslocated in low-income areas. Such high schools may not have equipped theirstudents with the mathematical capacity to rigorously study college-level math-ematics [1, 7]. These students may not have taken enough mathematics classesor mastered the material at the high-school level, leaving them unpreparedto take mathematics courses that are at the calculus I level or above duringtheir first year of college, as suggested by the High School Longitudinal Survey[13] and High School Coursetaking [14]. Both studies were conducted by theNational Center for Educational Statistics [13, 14]. Oakes, Joseph, and Muir’s[19] study suggests that high schools in low-income areas, where a number ofunder-prepared students graduate, tend not to have qualified teachers of math-ematics who can help these students bridge the intellectual span between highschool and college mathematics curricula.

2. TAs AND THE TRAINING OF SECONDARY TEACHERSOF MATHEMATICS

Teaching mathematics to college students who have an inadequate high schoolmathematics background can be difficult for even an experienced professorof mathematics, and certainly for a new teaching assistant (TA) who mayhave no formal or informal training in the teaching and learning of mathemat-ics. This raises the question, how can a TA assist students with mathematicaldeficiencies if the TAs training is void of the cognitive issues related to the

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learning of mathematics? To know mathematics “for self” is entirely differentfrom knowing mathematics for the “sake of others.” Since incoming freshmenwith mathematical deficiencies often score low on college placement exams,they are required to take preparatory mathematics classes. Departments ofmathematics should prepare their TAs to deal with the learning issues facingsuch students. While TAs are expected to have a strong mathematical back-ground, they should also be aware of the psychological and social aspects ofthe learning of mathematics by those they will encounter in the classroom.

By comparison, the preparation of 7–12 grade mathematics teachers is adynamic endeavor involving issues related to content and to the pedagogicalaspects of teaching [6]. Similarly, the preparation of TAs who instruct lower-division and preparatory mathematics at the college level should also includeopportunities for new college teaching assistants to learn about basic theoriesof pedagogy and learning. The premise of this article is that new TAs, whoinstruct mathematics to first-year college students enrolled in developmentalor preparatory mathematics courses, share similarities with their professionalcounterparts at the high school level, new teachers of mathematics for 11th and12th grade students. Consider the following:

1. Course Load – A graduate mathematics major (MA- or MS-seeking) whois a new TA may teach classes at the pre-calculus level or lower, as will afirst-year teacher of mathematics who instructs 11th and 12th grade.

2. Mathematics Background – Depending on the institution where the pre-service teacher of mathematics is trained (there is a push for mathematicseducation majors to have the same number of mathematics courses in theirprogram as mathematics majors have), a first-semester graduate studentwho is a TA may have taken only a few more mathematics classes, as anundergraduate, then his or her counterpart at the high school who is a newfirst-semester in-service teacher of mathematics.

3. Students’ Abilities – Because they are recent graduates of high school, themathematical maturity of college underclassmen might well be similar tothat of a high school junior or senior.

One significant difference between a new TA, who has no background in K–12teaching, and a new teacher of 7–12 mathematics is that the educational back-ground of a 7–12 mathematics teacher includes a semester of student teachingwith additional hours of field-based clinical experiences. For example, in thestate of New York, pre-service teachers of mathematics are required to have100 hours of field observation that includes participating in a middle and highschool classroom prior to their student teaching.

Furthermore, a total of 480 hours of student teaching are also requiredprior to graduation. This is done so that pre-service teachers of mathematicscan attend to the variety of classroom issues that arise in the daily teach-ing of mathematics. During a pre-service teacher’s mathematics course work,

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theories of teaching and learning are studied, explored, and experienced inthe classroom. Student teaching allows the pre-service teacher of mathemat-ics to practice pedagogical techniques investigated during their course andfield work. This same training structure, truncated perhaps, would be usefulin developing the instructional habits of new TAs who work with studentsin lower-division courses. The remainder of this article will explore whatthis truncated framework could look like, and offers practical suggestions forimplementation.

3. POORLY TAUGHT STUDENTS AND POORLY TAUGHT TAs

Poor teaching preparation for TAs can compound instructional issues whenattempting to support freshmen students who come from school systems thathave not provided them with the mathematical basics to successfully engagecollege-level mathematics. Though most TAs are likely skilled in acquiringmathematical knowledge for themselves, many TAs may have little or no ideaof how to facilitate acquisition of mathematical knowledge by others—that is,TAs have no pedagogical content knowledge, as established by Shulman [21],for the instruction of students at the university level who lack the secondarymathematics content skills needed to navigate through the tertiary curriculum.

Pedagogical content knowledge is:

Examples of this “work of teaching” include explaining terms and con-cepts to students, interpreting students’ statements and solutions, judgingand correcting textbook treatments of particular topics, using representa-tions accurately in the classroom, and providing students with examplesof mathematical concepts, algorithms, or proof [5, p. 373]

The Program for International Students Assessments (PISA) has pro-vided an indication of how well a cross-section of America’s high schoolstudents maneuver though mathematical tasks. The PISA, which comparesthe capabilities of 15-year-old students internationally relative to mathemati-cal literacy and problem solving, has revealed that America’s 15-year-olds in2006 scored in the bottom 25 percentile when compared to their internationalcounterparts [15]. Much worse, when statistics of the PISA are disaggregated,America’s population of 15-year-old African-American and Latino students’mathematical performance is lower than their Caucasian, Asian-American,and international counterparts. These under-achieving students, many of whomcome from lower socioeconomic backgrounds are often first-generation collegestudents, and would benefit from well-trained TAs to support their learningefforts.

While in graduate school, I noticed that the vast majority of courses taughtby TAs were lower-division classes such as preparatory mathematics, liberal

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arts mathematics, college algebra, trigonometry, and pre-calculus. These TAswere required to take part in a single, two-week seminar on issues related tothe teaching of mathematics. As a doctoral student of mathematics educationwith 6 years of teaching experience at the middle and high school levels and3 years of teaching as an adjunct at a neighboring university, I found the TAs’training to be generally myopic, in the sense that their training was teacher-centered and lecture-intensive. It was void of student-centered instruction and aproper understanding of theories of learning. This article is not suggesting thata student-centered philosophy is more or less effective than a teacher-centeredphilosophy with regard to the training of TAs.

However, equal attention should be given to both philosophies so as tocreate a holistic training program for TAs. Joel David Hamkins [4] presents avideo and peer feedback model for the training of Graduate Student Instructors.Though this is a superb approach to training TAs—and an approach that issimilar to one I am currently taking with several teachers of mathematics ina high-needs high school—I found the results can be insignificant when thiswork is done in the absence of conversations regarding students’ learning.

4. TRAINING TAs THROUGH A TUTORING MODEL

Structuring training for TAs that focuses on discourse between the students theyteach and themselves is important for their instructional development. Differentmodes of verbal instruction facilitate different modes of mathematical learningfor students. For example, the didactic approach is linear and teacher-/content-centered and promotes monologue. In contrast, discourse that is probing andopen-ended is circular and promotes dialogue. Though the teaching of one,two, or three students differs from what TAs do in a classroom with a largenumber of students, experiences with tutors’ “local discourse” about mathe-matical concepts, with a small number of students, may promote intellectualintimacy between TA and student, and prepares TAs to be better facilitatorsof mathematical learning through applyng mini-lessons during each tutoringencounter.

By comparison, pre-service teachers of mathematics do something sim-ilar a semester prior to student teaching in what is called a field clinical. Inthe field clinical pre-service teachers are required to spend time in the class-room observing lessons, supporting classroom teachers, and working withstudents in small learning groups. As with pre-service teachers of mathematicsin their field clinical, tutors are exposed to small episodes of teaching whiletutoring small groups or individuals. The phrase “local discourse” is a substi-tute for the sociolinguistic learning proximity that one, two, or three studentshave with a tutor in a tutoring session. I see “local discourse” as dialoguebetween the learning facilitator (TA) and students where there is a synthe-sis of ideas, ideas that at worst are parallel but headed for convergence after

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negotiation. These ideas emerge after the group investigation of mathemat-ical concepts as ideas, first, and then mathematical concepts as facts, later.Furthermore, the dialogue between learning facilitator and student need notbe contentious or inflexible. This local dialogue is comprised of the followingdomains:

1. An open discussion about each student’s and TA’s mathematical knowl-edge. A comfort zone must be established and extended.

2. A free exchange of ideas, which provides an arena to explore the mathe-matical talk as described by Pimm [20].

3. A debate between students, mixed with the learning facilitator’s open-ended questioning. This promotes mastery of the language of mathematicsthrough conversation, as presented by Lakotas [8].

4. A considerable amount of verbal feedback from the students regarding theirfeelings about the learning session.

5. A journal kept by the TA that is comprised of his or her feelings aboutaccomplishments and failures in the classroom.

A synthesis of these domains is most important in this model for the train-ing of TAs. This model allows for a maximum amount of input from the vantagepoint of the TA and the students they tutor while seeking a fit between theirideas resulting in an integration and uniformity that each value as a result oftheir dialogue. Usually, TAs are unable to participate in an entire semester ofstudent teaching like pre-service teachers of mathematics. However, they cando an entire semester of tutoring with a small number of students, relative tothe above model, in their institution’s mathematics study lab prior to teaching aclass. This can help prepare TAs to facilitate the learning of mathematics withlarger groups of students.

This model for the training of TAs is an aggregate of concepts dis-cussed in the National Council of Teachers of Mathematics’s 2000 Principlesand Standard [17], the College Board for the Mathematical Science’s TheMathematical Education of Teachers [2], and observations I have made as aresearcher of mathematics education and as an instructor of mathematics at themiddle school, high school, and college levels. It is unlikely that this modelcan be “all things to all people.” However, this model is a guide that can beutilized by departments of mathematics. For further information about sim-ilar ideas, see Nardi, Jaworski, and Hegedus [11] who describe a four-levelspectrum of pedagogical awareness that emerged during an investigation ofsix tutors of undergraduate mathematics at Oxford University over an eight-week period. Their research richly details how tutors moved though a learningcontinuum with regard to their understanding of teaching of mathematics tosmaller populations of students.

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5. A MODEL IN ACTION

Reflecting on one’s own teaching experience plays a significant role in themodification of pedagogical techniques and ideas regarding learning. Hence,the bases of my views regarding the training of TAs who will instruct prepara-tory mathematics to students comes from 15 years of teaching mathematics to alarge number of at-risk students at the middle, secondary, and university levels.While teaching mathematics at Florida A&M University, the largest histori-cally black university in the country, I had the opportunity to supervise tutorsin the School of General Studies’ Academic Skills Center and to work withadjuncts (TAs) who were first-time instructors of preparatory mathematics. Itwas vital that I, along with other faculty, create a training program that pro-vided pedagogical techniques which addressed the needs of the at-risk studentpopulation TAs were to instruct.

A large number of freshmen at Florida A&M University, over a quarterof the incoming population on average, are required to take preparatory math-ematics. Registering for preparatory mathematics is compulsory for studentswho have not taken enough high school mathematics classes, or have scoredextremely low on entrance exams, or received a General Education Diploma.Such academic deficiencies can be found among students from communitiesthat have historically been denied access to equal education.

Although great progress has been made for all students in terms ofeducation in this country, social and educational inequities of the past arestill overwhelmingly apparent in the student populations at Florida A&MUniversity and other post-secondary schools that enroll such students. By pro-viding training and modeling effective teaching strategies, similar to what Ipresented earlier in this article, the tutors and adjuncts (TAs) at Florida A&MUniversity were better able to serve the needs of their students. The effective-ness of the model used to train these instructors was determined by the numberof students who passed a state-mandated exit exam. During my time at theuniversity, we raised the passing rate on the state exit exam from about 43% toabout 86%. We attributed the success of our students to the training program wecreated for our tutors and adjuncts (TAs) in the School of General Education.

6. CONCLUSION

Mathematics departments at the university level cannot ignore the impactthat the teaching of mathematics in middle and high school has on studentsthat TAs will teach. Issues related to the learning of mathematics at thecollege/university level, particularly with regards to preparatory mathemat-ics, are analogous to the learning of mathematics at the high school level.Furthermore, just as instructional practices are evolving to meet the needsof academically challenged students at the high school level, so too must

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the instructional practice of TAs who will teach many of the academicallychallenged at the university level evolve. These changes must be accom-plished in order to meet the needs of students who will, because of insufficientmathematical instruction, have limited opportunities in their choice of a major.

The thesis of this article is a result of my experiences gained as an under-graduate student and instructor of mathematics/mathematics education at aHistorically Black College and University (HBCU). HBCUs often provide edu-cational opportunities for students who come from school systems that have notadequately prepared them to effectively engage in college-level mathematics.To teach such students who are only prepared to do preparatory mathematicsrequires a great deal of skill and planning. Furthermore, the problems of teach-ing and learning in lower-division mathematics occurs at all public and privatecolleges and universities throughout this nation, and thus the problem meritsattention. I have offered and outlined a strategy to address some of the trainingof TAs who will instruct preparatory mathematics.

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1. Burton, N. W., N. B. Whitman, M. Yepes-Baraya, F. Cline, andR. Myun-in Kim. 2002. Minority Student Success: The Role of Teachersin Advanced Placement Program (AP) Courses: College Board Report2002–2008. New York. http://professionals.collegeboard.com/ profdown-load/pdf/researchreport 20028_18660.pdf. Accessed 26 November 2010.

2. Conference Board of the Mathematical Sciences. 2001. The MathematicalEducation of Teachers. American Mathematical Society and AmericanMathematical Association, Providence, RI, Washington, DC.

3. FitzSimons, G., and G. Godden. 2000. Perspective of Adults In LearningMathematics (pp. 13–33). Kluwer Academic Publishers, Boston.

4. Hamkins, J. D. 1999. Using video and peer feedback to improve teach-ing. In Bonnie, G., S. Z. Keith, and W. A. Marion (Eds.), MathematicalAssociation of America’s Notes # 49 (pp. 267–277). MathematicalAssociation of America, Washington, DC.

5. Hill, H., B. Rowan, and D. Ball. 2005. Effects of teachers’ mathemati-cal knowledge for teaching on student achievement. American EducationResearch Journal. 42(2): 371–406.

6. Imig, D., and S. Imig. 2006. What do beginning teachers need to know?An essay. Journal of Teacher Education. 57(3): 286–291.

7. King, J. E. 1996. Improving the Odds: Factors that Increase the Likelihoodof Four-Year College Attendance Among High School Seniors: CollegeBoard Report 96-2. New York http://professionals.collegeboard.com/data-reports-research/cb/improving-the-odds. Accessed 5 September 2007.

8. Lakatos, I. 1976. Proofs and Refutations. Cambridge University Press,Cambridge.

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9. Loftsgarden, D. O., D. C. Rung, and A. E. Watkins. 1997. StatisticalAbstracts of Undergraduate Programs in the Mathematical Sciences inthe United States: Fall 1995 CBMS Survey. Mathematical Association ofAmerica, Washington, DC.

10. Lutzer, D. J., J. W. Maxwell, and S. B. Rodi. 2002. StatisticalAbstract of Undergraduate Programs in the Mathematical Sciences in theUnited States: Fall 2000 CBMS Survey. American Mathematical Society,Providence, RI.

11. Nardi, E., B. Jaworski, and S. Hegedus. 2005. A spectrum of pedagogicalawareness for undergraduate mathematics: From “tricks” to “techniques.”Journal for Research in Mathematics Education. 36(4): 284–316.

12. Mathematical Association of America. 2001. Mathematics andthe Mathematical Sciences in 2010: What Should Students Know?Mathematical Association of America, Washington, DC.

13. National Center for Educational Statistics. 2007. High SchoolCoursetaking. http://nces.ed.gov/programs/coe/2007/analysis/sa02a.asp.Accessed 1 September 2007.

14. National Center for Educational Statistics. 2009. High SchoolLongitudinal Survey. http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=201001. Accessed 24 November 2010.

15. National Center for Educational Statistics. 2007. Program for InternationalStudent Program Assessment. http://nces.ed.gov/surveys/PISA/PISA2003highlights.asp. Accessed 20 November 2010.

16. National Center for Educational Statistics. 2009. Enrollment inPostsecondary Institutions, Fall 2007; Graduation Rates, 2001 and2004 Coharts; Financial Statistics, Fiscal Year 2007. http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2009155. Accessed 20 November 2010.

17. National Council of Teachers of Mathematics. 2000. Principles andStandards for School Mathematics. National Council of Teachers ofMathematics, Reston, VA.

18. National Research Council. 1991. Moving Beyond Myths: RevitalizingUndergraduate Mathematics. National Academy Press, Washington, D.C.

19. Oakes, J., R. Joseph, and K. Muir. 2001. Access and achievement inmathematics and science. In J. A. Banks, and C. A. McGee Banks(Eds.), Handbook of Research on Multicultural Education (pp. 69–90).Jossey-Bass, San Francisco.

20. Pimm, D. 1987. Speaking Mathematically: Communications in Mathe-matics Classrooms. Routledge & Kegan Paul Inc., New York.

21. Shulman, L. S. 1986. Those who understand: Knowledge growth inteaching. Educational Researcher. 15(2): 4–14.

22. Steen, L. A. 1999. Assessing assessment. In Bonnie, G., S. Z. Keith, andW. A. Marion (Eds.), Assessment Practice in Undergraduate Mathematics:MAA Notes #49 (pp. 1–6). Mathematical Association of America,Washington, D.C.

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BIOGRAPHICAL SKETCH

Danté A. L. Tawfeeq is an Assistant Professor of Mathematics Education in theRuth S. Ammon School of Education, located in Long Island, NY. He was a2004–2005 Dolciani-Holloran Foundation Project NExT Fellow, selected as aFulbright Senior Specialist candidate, and received his PhD from Florida StateUniversity. His research interests include secondary teachers’ knowledge ofmathematical content and pedagogy, the learning of mathematics at the col-lege level, and the learning of mathematics by African American and Latino(a)students. In addition to teaching mathematics and methods of teaching mathe-matics courses, he enjoys jazz music, martial arts (23 years!), attending profes-sional boxing matches, playing chess, and watching SpongeBob SquarePantswith his children, Jibril, Bashir, Sage, and Capri.

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