Algebra Teachers' Utilization of Problems Requiring Transfer Between Algebraic, Numeric, and Graphic Representations

Robert F. CunninghamThe College of New JerseyFor students to develop an understanding of functions, they must have opportunities to solveproblems that require them to transfer between algebraic, numeric, and graphic representations(transfer problems). Research has confirmed student difficulties with certain types of transferproblems and has suggested instructional factors as a possible cause. Algebra teachers (n = 28)were surveyed to determine the amount of class time they devote to different types of transferproblems and how many times these problems appear on their teacher-made assessments. Resultssuggest that teachers dedicate less class time to graphic to numeric transfer problems than to anyother type of transfer problem and that these problems appear less frequently on assessments.These are exactly the types of transfer problems that pose the most difficulty for students. It isconjectured that teachers' familiarity with these problems, combined with assumed studentmastery, contribute to this mismatch.There is much debate about the central role func-tions play in the study of mathematics (Beckmann,Senk, & Thompson, 1999; Fererrini-Mundy & Lauten,1993; Schwartz, 1992), and there is increasing recog-nition thatbeing able to move or transferbetween theirrepresentations plays a key role in student understand-ing of them (Knuth, 2000; Porzio, 1999; Wilson K Dunham & Osborne, 1991; Eisenberg& Dreyfus, 1991; Goldenberg, 1988; Knuth, 2000;Moschkovich, Schoenfeld, & Arcavi, 1993; Porzio,1999) have documented student difficulty with suchtransfers in courses from algebra through calculus.Eisenberg and Dreyfus (1991) indicated that somestudents are reluctant to use visual or graphic repre-sentation, and they have not had the opportunity tolearn their complexity. Knuth (2000) reported that amajority of students fail to recognize or create transferbetween graphic and algebraic representations. Knuth,as well as Dunham and Osborne (1991), have pointedout that symbolic manipulation and reliance on alge-braic representation are overemphasized in schools,suggesting that student difficulties stem more frominstructional factors than from cognitive factors.Multiple representations of functions is a recurringtheme throughout mathematics, but its position in alge-bra is unique and worthy of examination. Yerushalmyand Gafni (1992) stated that, "Function in its multiplerepresentations is the fundamental object of algebrawhich ought to be presented through the learning andteaching of any topic in algebra" (p. 318). Althoughstudent and teacher understandings of the concept offunction have been extensively studied, fewer studieshave been done involving transfer between represen-tations (Knuth, 2000). It remains true that the numberof studies involving algebra teachers has been sparse(Kieran, 1992), as has been research exploring teach-ers' instructional use of transfer problems (ones thatrequire transfers between representations).Accordingly, student difficulties with certain typesof transfer problems have been documented, andinstructional factors have been indicated as a possiblesource of these difficulties. Teachers, regardless ofcurriculum and textbook, make the ultimate instruc-tional and assessment decisions on how to integratetransfer problems into their courses. Further, algebrateachers and the opportunities theyprovide for studentsVolume 105(2), February 2005