Educating tomorrow's mathematics_teachers


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Educating tomorrow's mathematics_teachers

  1. 1. Educating Tomorrow’s Mathematics Teachers: The Role of Classroom-Based Evidence∗ Ateng’ Ogwel . Mathematics education in many countries is characterized by low level of motivation especially among secondary school students. Besides, specificity of school mathematics mo- tivates procedural learning and little appreciation of mathematics in real life situations. Thus, reforms in mathematics education require teachers to provide opportunities for students to take responsibility for their learning and engage in worthwhile learning activities. However, there is paucity of corresponding provisions for teachers to learn and develop necessary skills and knowledge for the desired instructional practices. Consequently, classroom-based quali- tative research, including analysis of students’ thinking processes is necessary for teachers to develop pedagogical content knowledge. This paper illustrates the necessity for epistemolog- ical knowledge of mathematics and call for enhanced collaboration amongst teachers, teacher educators, curriculum developers and other potential players in mathematics education.1 IntroductionMathematics classrooms across many countries mirror the images of past decades despite signif-icant efforts and research targeting improvement (e.g., Frykholm, 1999; Wiliam, 2003). Despiteprofessed value of mathematics in socio-economic and technological development most secondaryclassrooms are characterized by lack of enthusiasm to learn mathematics. That is, many ignore orare unaware of applicability of mathematics– a phenomena believed to be a weakness of schoolmathematics (Onion, 2004), rather exam-oriented mathematics. Similarly, unsatisfactory perfor-mance in this mathematics in national and international assessments continues to trouble a parents,educators and policy makers. Efforts to address these problems have in the past focused on cur-ricula reviews, provision of real life experiences, incorporation of non-routine problems, and useof hands-on activities. Moreover, envisioned changes favour students’ active participation and ashift in emphasis from teaching to learning. Furthermore, computers and calculators are believedto enrich learning experiences and align schools mathematics with technological developments(NCTM, 1989). Nevertheless, recent developments (e.g., AAAS, 2006; NCTM, 2006) indicateapparent review of some visions (e.g., hands-on activities and use of calculators). Major reforms in mathematics education have been reactionary (e.g., against Sputnik, Na-tion at Risk and TIMSS) (Klein, 2003; NCTM, 1989), and urgency to showcase their success hasignored appropriate research (Good, Clark and Clark, 1997). Similarly, inadequate preparation ∗ A paper for the Workshop on ‘Modeling in Mathematics Learning: Approaches for Classrooms of the Future’,Makerere University July, 23–25 2007
  2. 2. of teachers to effectively manage reforms has been a bane to mathematics education (e.g., NewMath). For instance, disparity between teachers’ espoused beliefs and their classroom practices(Frykholm, 1999) is probably due to inherent fashion and policy advocacy for educational reforms.Also, the tendency to adopt educational interventions– from the developed to developing coun-tries; and transfer of discourse patterns from elementary schools or universities to the secondarylevel, has impeded meaningful and sustainable improvements in mathematics education. We sub-mit that a major problem in mathematics education is the prevalence of traditional practices, andnot as it appears, lack of innovative practices. Moreover envisaged interventions are fraught with some dilemma, and imply considerablesensitivity in improving learning in typical schools. First, the visions of successful practicesopenly advocate for student-centred instruction, but the teachers’ role towards this centrednessis never peripheral. Secondly, although poor performance in examinations has motivated calls forinnovative instructional practices, success of, especially constructivist, interventions depend onreformed assessment programs (Frykholm, 1999). Whereas this dilemma may not be resolvable,a substantial inclusion of classroom practices in teacher preparation and professional developmentwould probably minimize inefficiency in mathematics education. In particular, it is necessary toshift from generic pedagogies and account for contextual aspects of education (by region andeducational levels). In the rest of the paper, we illustrate the significance of epistemologicalknowledge of mathematics.2 Epistemological Knowledge of Mathematics2.1 Development Epistemological Knowledge of MathematicsAlthough teachers’ advanced knowledge of mathematics is necessary, it is insufficient for im-proving students learning of mathematics. Teachers should blend content knowledge with anunderstanding of students’ reasoning and develop pedagogical content knowledge (PCK). Conse-quently, they ought to experience the process of learning school mathematics through tasks withpedagogical and mathematical challenges (Cooney, 1999), and shift their conception of mathe-matics as a static body of knowledge to a dynamic subject which allows multiple representations(Confrey, 1993; Steinbring, 1998). Whereas content knowledge develops through school and col-lege learning, pedagogical content knowledge is enhanced through encounters with learners inclassroom settings– a rarity in many professional development programs. In order to enhance learning of mathematics, there is need for epistemological knowledge ofmathematics, a professional knowledge for mathematics teaching (Steinbring, 1998). An under-lying assumption is the developmental nature of mathematical knowledge subject to social andtheoretical constraints. That is, teaching and learning of mathematics are autonomous systems inwhich the role of the teacher is not to simply transmit scientific mathematics. On the contrary,it is to provide learning tasks for learners to subjectively interpret and reflect on; revise learningtasks; analyze interactively constructed mathematical knowledge; and reflect on this knowledgeon the basis of scientific mathematics (Steinbring, 1998, 2005). This implies that teachers mustmonitor theoretical consistency in learners’ idiosyncratic strategies from a variety of case studies.That is, epistemological knowledge does not merely develop through reading books but throughtheoretically grounded analyses of classroom episodes, e.g., the Epistemological Triangle (Stein-bring, 2005). Moreover, it incorporates “historical, philosophical, and epistemological conceptualideas” (Steinbring, 1998, p. 160). 2
  3. 3. 2.2 Case I: Linear EquationA student’s strategy in generalizing a linear relationship (Figure 1) from values obtained from acomputer program illustrates the need for teachers to closely listen to students and monitor the-oretical consistency of non-conventional constructions (Confrey, 1993). The strategy that baffledboth the teacher and researcher involved computing the products yi x j & xi y j , evaluating their dif-ference to obtain the y-intercept, c (i.e., yi x j - xi y j ⇒ 15-14 = 1). To obtain the gradient, m, the student evaluated the difference between successive y values(i.e., y j -yi ⇒ 2). The dilemma was that the student obtained y = 2x + 1 using a non-conventionalbut consistent strategy (Confrey, 1993). The epistemological validity of this strategy is confirmedfrom the two-point form of linear equations (Eq. 1), where for points (a,b) & (c,d), the linearequation is: (d − b)x + (bc − ad) y= (1) c−a x 1 2 3 4 5 xi xj 7-4 = 3 3 x 20 = 60 20 x 3 = 60 y 3 5 7 9 11 yi yj 60 + 4 = 64 60 + 1 = 61 2 2 yj - yi Student A Student B Figure 1: A student’s solution Figure 2: Two students’ solutions2.3 Case II: Arithmetic SequenceTeachers’ decision to probe students’ thinking may be due to correct answers from doubtful/unfamiliar processes, as exemplified in the solution of the 20th term of the sequence 4, 7, 10,13, . . (Source: Inaddition, analysis of students’ thinking is a complex endeavour that requires continuous and variedexperiences. For instance, the two students have “20 × 3” although their final values are different(Figure 2). While the first one (A) appears to use a conventional approach, student (B) used arelation in the terms of the sequence to obtain the addend ‘1’. Both solutions demonstrate the inadequacy of memorizing formula without understandingmathematical properties. Besides, the failure to link the second solution to conventional one is pri-marily due to inadequate conception of sequences and over-emphasis on manipulatives, althoughtime might also be a factor. The solution which typifies the need for epistemological knowledge ofmathematics is equivalent to evaluating the 21st term when the ‘first’ term is shifted by a commondifference (Eq. 2) T n = a + (n − 1)d ⇔ T n+1 = a − d + nd (2)2.4 Case III: Similarity of FiguresThis case is drawn from a study in which eleven lessons were observed in a Grade 9 classroomin Japan (Ogwel, 2007). The purpose of the study was to understand the process of learningmathematics in regular secondary classrooms. The following discussion is based on the solutionof Part (3) a problem given in the 10th lesson (see the transcript below): 3
  4. 4. A F D E I H B C Figure 3: Problem on Similarity of FiguresIn Figure 3, ABCD is a parallelogram, with E and F on AB and AD respectively; AF:FD = 3:4and EF//BD; BD intersects EC and FC at H and I respectively. Find (1) EB:DC, (2) (a) EH:HC,(b) EF:HI;(3) If the area of △BCE is 10 sq. units, find the area of △BDF. 770.TR: (10:33:00;. . . S36 solving Problem 6). Now finally, at long last. I feel that whenever I prepare, it comes to pass. Now, err, where is △BCE? △BCE now, is it this? There are several colours here (inaudible). We know that this area is 10 sq. units. The area is 10 sq. units (in low tone). In which case, now △BDF, BD, err oh! Is it this? Oh! Can this be known? (S36 continues with problem 6). How do we do it? Let me ask? S14, (S30 turns to S37’s desk), How do we do this? (S37 explains something to S30 on problem 1; S30 nods, turns back to her desk). 771.S14: (10:34:00; points at the board) in BDF 772.TR: In BDF? 773.S14: B is the apex (briefly looks at own worksheet) 774.TR: Yea 775.S14: Since BD and 776.TR: Yes (S26, S15 and S1 appear to be listening attentively) 777.S14: EF are parallel 778.TR: Yea. 779.S14: F moves to D 780.TR: F? 781.S14: Is the apex above E 782.TR: Here? Oh, yeah 783.S14: Then the base is the same 784.TR: I see 785.S14: Inaudible 786.TR: (takes about 10 seconds looking at the figure on the board) That is great. (To the other students) Oh! Please do you understand why S14 has just said that? So it can also be done that way? (Erases the board) . . .. A F D A F D A D A F D E E E E I HB B C B B C C C Figure 4: Desired Figure 5: In BDFE Figure 6: In BCDE Figure 7: Final The student begins by focusing on △BDF and △BDE (Figures 4 & 5), prompting the teacher’sattention (772 & 780). In addition, he justifies use of auxiliary segment BD (EF//BD) and ashear transformation of △BEF to △BDE (see Figure 5). He then arrives at equivalence of △BEDand △BEC (Figure 6), before finally asserting the equivalence of △BEC and △BDF (Figure 7). 4
  5. 5. The auxiliary line is used to demonstrate equivalence through reflexive (Figure 4), symmetric(Figures 5 & 6) and transitive (Figure 7) relations of triangles in an unusual orientation, andwithout reference to ‘10’. This solution involves structural reasoning beyond empirical quantities,which would involve drawing a parallelogram in which the ratios (3:4) represent actual lengths.However, it may be difficult to ensure that the area of △BCE is exactly 10 sq. units (unless oneuses graphic software). That the teacher understood the plausibility of student’s reasoning, despite his expectationthat ratios were to be used is due to epistemological knowledge of mathematics. This professionalknowledge is valuable for mathematics teachers to make real-time decisions within the complexityof classroom interactions. Similarly, the compact communication (771–785) reveals a need forthis knowledge and patience in monitoring consistency of students reasoning. Moreover, thatthe scanty episode translates into coherent argument (Figures 4, 5, 6 & 7) is due to teacher’selaboration of inaudible students’ responses and interpretation of their non verbal communication.That is, communication pattern was not only cultural, but enabled the observer and other studentsaccess the student’s reasoning. This episode further demonstrates the significance of problem-solving, where original taskis transformed without altering its structural properties. Similarly, it shows the value of seeingmathematics as connections (NCTM, 1989), where the student uses equivalent areas in the unitof similarity of figures. The study also revealed a sharp contrast with prevalent classroom dis-course in elementary schools; and that mathematical training was an explicit aim in the class(Ogwel, 2007). That is, the lessons showed attempts to address transitional demands of secondarymathematics education. Besides, classroom interactions depended on the nature of problems–for example Figure 8, where a student’s insinuation that AD//BC prompted prolonged discussion.This problem involves surds and angle properties of a circle, an element of coherence in curricu-lum.Figure 8 shows a quadrilateral ABCD inscribed in a circle. If BC = 3, CD = 6, CP= 2. Find thelengths of (1) AP, and (2) BD. A P D B C Figure 8: Conditions for Similarity Furthermore, collaboration among teachers, educators and university professors was evidentin publication of textbooks. More significantly, the teacher’s willingness to be observed in a typ-ical class demonstrates that potential progress in mathematics education lies beyond simulatedinnovations. The validity of such qualitative interpretations of classroom episodes require, be-sides well defined theoretical lenses, an understanding of the classroom culture through extendedobservations (Ogwel, 2007). Video records or audio-tapes and transcripts are also invaluable inrecollecting classroom episodes. For observers and researchers, the process of transcription andthe desire to construct a coherent and convincing discourse implies immense learning (Mason,1998) which is often not acknowledged in objective research formulations. Finally, the compactand mostly inaudible communication by students and the inadequacy of the epistemological tri-angle in analyzing communicative aspects of interactions is not a failure in the design of research,but an indicator of the need for further research in regular secondary mathematics classrooms. 5
  6. 6. 3 ConclusionsIf learning is a cyclic process which involves planning, implementation and feedback, then teachereducation must also reflect this process. Typical weaknesses in pre-service teacher education (the-oretically oriented) and in-service teacher education (practical-based) may be turned into potentialgains if complementarity of the two systems is harnessed. That is, initial teacher preparation mustsubstantially incorporate classroom experiences while professional development should enhancetheory-laden reflections. In addition, classroom-based research potentially challenges teachers andeducators’ beliefs and conceptions; produces data that can be interpreted from multiple theoreti-cal perspectives; and offers authentic learning opportunities for teachers and researchers. Besides,classroom-based evidence demystifies notions that curricula, instructional materials or theoreticalprinciples automatically result into students’ learning. On the contrary, it provides opportunityfor collaboration (cf Scherer and Steinbring, 2006) and testing and revision of educational inter-ventions, for instance Project Mathe 2000’s ‘Substantial Learning Environments’ developed byWittmann and Muller, and analysis of their use done by Steinbring (cf Steinbring, 2005). Consequently, there is need to review generic approaches in mathematics teacher education;promote collaboration among curriculum developers, policy makers, mathematicians, mathemat-ics educators and teachers; and re-conceptualize that, like other professions, teaching requiressubstantial internship experiences. Inevitably, the future of mathematics education lies in appro-priate utilization of technology, thus, the benefits of mathematical software in instruction cannotbe gainsaid. Moreover, despite the observed deficiency in school mathematics, we agree withZbiek and Conner (2006) that the challenge is how modeling and problem-solving can be usedto enhance understanding of school mathematics. We are, however, not oblivious to the logisticaland immense resource implications for the proposed approach. However, the potential gains out-weigh the costs, and players in corporate and private sectors in African countries could also joinin enhancing quality of education, hence quality of life.ReferencesAmerican Association for the Advancement of Science (2006). Finding common ground in the US math wars. Science, 312(5776), 969–1069.Confrey, J. (1993). Learning to see children’s mathematics: Crucial challenges in constructivist reform. In K. Tobin (Ed.), The practice of constructivism in science education (pp. 299–321). Hillsdale, NJ: Lawrence Erlbaum.Cooney, T. J. (1999). Conceptualizing teachers’ ways of knowing. Educational Studies in Mathematics, 38, 163–187.Frykholm, J. A. (1999). The impact of reform: The challenges of teacher preparation. Journal of Mathematics Teacher Education, 2, 79–105.Good, T. L., Clark, S. N. & Clark, D. C. (1997). Reform efforts in American schools: Will faddism continue to impede meaningful change? In B. J. Biddle, T. L. Good & I. F. Goodson (Eds.), International handbook of teachers and teaching (pp. 1387–1427). Dordrecht: Kluwer Academic Publishers.Klein, D. (2003). A brief history of American K–12 mathematics education in the 20th century. In J. M. Royer (Ed.), Mathematical Cognition (pp. 175–225). Connecticut: Information Age Publishing.Mason, J. (1998). Researching from the inside in mathematics education. In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics education as a research domain: A search for identity. An ICMI Study (pp. 357–377). Dordrecht: Kluwer Academic Publishers.National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through Grade 8 mathematics: A quest for coherence [Electronic Version]. Reston, VA: cfp/full_document.pdf [November 11,2006]: NCTM.Ogwel, J. C. A. (2006). Interactive learning of mathematics in secondary schools: Three core elements of regular classrooms. Journal of JASME: Research in Mathematics Education, 12, 189–200.Ogwel, J. C. A. (2007). Interactive learning of mathematics in secondary schools. Unpublished master’s thesis, Graduate School of Education, Hiroshima University. 6
  7. 7. Onion, A. J. (2004). What use is maths to me? A report on the outcomes from student focus groups. Teaching Mathematics and Its Applications, 23(4), 189–194.Scherer, P. & Steinbring, H. (2006). Noticing children’s learning processes–teachers jointly reflect on their own classroom interaction for improving mathematics teaching. Journal of Mathematics Teacher Education, 9(2), 157–185.Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers. Journal of Mathematics Teacher Education, 1(2), 157–189.Steinbring, H. (2005). The Construction of new mathematical knowledge in classroom interactions – An epistemo- logical perspective. New York: Springer.Wiliam, D. (2003). The impact of educational research on mathematics education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick and F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 471–490). Dordrecht: Kluwer Academic Publishers.Zbiek, R. M. & Conner, A. (2006). Beyond motivation: Exploring mathematical modeling as a context for deepening students’ understanding of curricular mathematics. Educational Studies in Mathematics, 63(1), 89–112. 7