Educating Tomorrow’s Mathematics Teachers: The Role ofClassroom-Based Evidence∗

Ateng’ Ogwel

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Mathematics education in many countries is characterized by low level of motivationespecially 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 totake responsibility for their learning and engage in worthwhile learning activities. However,there is paucity of corresponding provisions for teachers to learn and develop necessary skillsand knowledge for the desired instructional practices. Consequently, classroom-based quali-tative research, including analysis of students’ thinkingprocesses is necessary for teachers todevelop pedagogical content knowledge. This paper illustrates the necessity for epistemolog-ical knowledge of mathematics and call for enhanced collaboration amongst teachers, teachereducators, curriculum developers and other potential players in mathematics education.

1 Introduction

Mathematics 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 learnmathematics. 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 inthismathematics 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 activitiesand use of calculators).

Major reforms in mathematics education have been reactionary (e.g., againstSputnik, Na-tion at Riskand 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

of teachers to effectively manage reforms has been a bane to mathematics education (e.g., NewMath). For instance, disparity between teachers’ espousedbeliefs 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 forstudent-centredinstruction, but the teachers’ role towards thiscentrednessis 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 illustratethe significance of epistemologicalknowledge of mathematics.

2 Epistemological Knowledge of Mathematics

2.1 Development Epistemological Knowledge of Mathematics

Although teachers’ advanced knowledge of mathematics is necessary, it is insufficient for im-proving students learning of mathematics. Teachers shouldblend 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 ofmathematics 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 subjectivelyinterpret 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 developthrough reading books but throughtheoretically grounded analyses of classroom episodes, e.g., theEpistemological Triangle(Stein-bring, 2005). Moreover, it incorporates “historical, philosophical, and epistemological conceptualideas” (Steinbring, 1998, p. 160).

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2.2 Case I: Linear Equation

A student’s strategy in generalizing a linear relationship(Figure1) 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 productsyixj & xiyj, evaluating their dif-ference to obtain the y-intercept,c (i.e.,yixj - xiyj ⇒ 15-14= 1).

To obtain the gradient,m, the student evaluated the difference between successivey values(i.e.,yj -yi ⇒ 2). The dilemma was that the student obtainedy = 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:

y =(d − b)x+ (bc− ad)

c− a(1)

x 1 2 3 4 5 xi xjy 3 5 7 9 11 yi yj

2 2 yj - yi

Figure 1: A student’s solution

7-4 = 33 x 20 = 6060 + 4 = 64

20 x 3 = 6060 + 1 = 61

Student A Student B

Figure 2: Two students’ solutions

2.3 Case II: Arithmetic Sequence

Teachers’ decision to probe students’ thinking may be due tocorrect answers from doubtful/unfamiliar processes, as exemplified in the solution of the 20th term of the sequence4, 7, 10,13, . . (Source:http://www.jica-net.com/CD/05PRDM011_2/n1/n01.html). 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(Figure2). 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)

Tn = a+ (n− 1)d⇔ Tn+1 = a− d+ nd (2)

2.4 Case III: Similarity of Figures

This 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):

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A F D

E

B C

H

I

Figure 3: Problem on Similarity of Figures

In 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 severalcolours here (inaudible). We know that this area is 10 sq. units. The area is 10sq. units (in lowtone). In which case, now△BDF, BD, err oh! Is it this? Oh! Can this be known?(S36 continueswith problem 6). How do we do it? Let me ask? S14, (S30 turns to S37’s desk), How do we dothis? (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 otherstudents) Oh! Please do you understandwhy S14 has just said that? So it can alsobe done thatway? (Erases the board) . . ..

A F D

E

B C

H

I

Figure 4: Desired

A F D

E

B C

Figure 5: In BDFE

A D

E

B C

Figure 6: In BCDE

A F D

E

B C

Figure 7: Final

The student begins by focusing on△BDF and△BDE (Figures4& 5), prompting the teacher’sattention (772 & 780). In addition, he justifies use of auxiliary segment BD (EF//BD) and ashear transformationof △BEF to △BDE (see Figure5). He then arrives at equivalence of△BEDand△BEC (Figure6), before finally asserting the equivalence of△BEC and△BDF (Figure7).

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The auxiliary line is used to demonstrateequivalencethrough reflexive (Figure4), symmetric(Figures5 & 6) and transitive (Figure7) 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 studentuses 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 Figure8, 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.

PB

A

C

D

Figure 8: Conditions for Similarity

Furthermore, collaboration among teachers, educators anduniversity 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 discourseimplies immense learning (Mason,1998) which is often not acknowledged in objective researchformulations. Finally, the compactand mostly inaudible communication by students and the inadequacy of theepistemological tri-anglein analyzing communicative aspects of interactions is not afailure in the design of research,but an indicator of the need for further research in regular secondary mathematics classrooms.

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3 Conclusions

If 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 canbe 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 inmathematics 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.

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