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 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). Despite
professed value of mathematics in socio-economic and technological development most secondary
classrooms are characterized by lack of enthusiasm to learn mathematics. That is, many ignore or
are unaware of applicability of mathematics– a phenomena believed to be a weakness of school
mathematics (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 use
of hands-on activities. Moreover, envisioned changes favour students’ active participation and a
shift in emphasis from teaching to learning. Furthermore, computers and calculators are believed
to enrich learning experiences and align schools mathematics with technological developments
(NCTM, 1989). Nevertheless, recent developments (e.g., AAAS, 2006; NCTM, 2006) indicate
apparent 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 has
ignored 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. of teachers to effectively manage reforms has been a bane to mathematics education (e.g., New
Math). 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 secondary
level, 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, and
not as it appears, lack of innovative practices.
Moreover envisaged interventions are fraught with some dilemma, and imply considerable
sensitivity in improving learning in typical schools. First, the visions of successful practices
openly advocate for student-centred instruction, but the teachers’ role towards this centredness
is never peripheral. Secondly, although poor performance in examinations has motivated calls for
innovative instructional practices, success of, especially constructivist, interventions depend on
reformed assessment programs (Frykholm, 1999). Whereas this dilemma may not be resolvable,
a substantial inclusion of classroom practices in teacher preparation and professional development
would probably minimize inefficiency in mathematics education. In particular, it is necessary to
shift from generic pedagogies and account for contextual aspects of education (by region and
educational levels). In the rest of the paper, we illustrate the significance of epistemological
knowledge 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 should blend content knowledge with an
understanding of students’ reasoning and develop pedagogical content knowledge (PCK). Conse-
quently, they ought to experience the process of learning school mathematics through tasks with
pedagogical 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 in
classroom settings– a rarity in many professional development programs.
In order to enhance learning of mathematics, there is need for epistemological knowledge of
mathematics, a professional knowledge for mathematics teaching (Steinbring, 1998). An under-
lying assumption is the developmental nature of mathematical knowledge subject to social and
theoretical constraints. That is, teaching and learning of mathematics are autonomous systems in
which 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 learning
tasks; analyze interactively constructed mathematical knowledge; and reflect on this knowledge
on the basis of scientific mathematics (Steinbring, 1998, 2005). This implies that teachers must
monitor theoretical consistency in learners’ idiosyncratic strategies from a variety of case studies.
That is, epistemological knowledge does not merely develop through reading books but through
theoretically grounded analyses of classroom episodes, e.g., the Epistemological Triangle (Stein-
bring, 2005). Moreover, it incorporates “historical, philosophical, and epistemological conceptual
ideas” (Steinbring, 1998, p. 160).
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3. 2.2 Case I: Linear Equation
A student’s strategy in generalizing a linear relationship (Figure 1) from values obtained from a
computer 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 baffled
both 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-conventional
but consistent strategy (Confrey, 1993). The epistemological validity of this strategy is confirmed
from the two-point form of linear equations (Eq. 1), where for points (a,b) & (c,d), the linear
equation 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’ solutions
2.3 Case II: Arithmetic Sequence
Teachers’ 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: http://www.jica-net.com/CD/05PRDM011_2/n1/n01.html). In
addition, analysis of students’ thinking is a complex endeavour that requires continuous and varied
experiences. 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 a
relation in the terms of the sequence to obtain the addend ‘1’.
Both solutions demonstrate the inadequacy of memorizing formula without understanding
mathematical 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, although
time might also be a factor. The solution which typifies the need for epistemological knowledge of
mathematics is equivalent to evaluating the 21st term when the ‘first’ term is shifted by a common
difference (Eq. 2)
T n = a + (n − 1)d ⇔ T n+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 classroom
in Japan (Ogwel, 2007). The purpose of the study was to understand the process of learning
mathematics in regular secondary classrooms. The following discussion is based on the solution
of Part (3) a problem given in the 10th lesson (see the transcript below):
3

4. A F D
E
I
H
B C
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 = 3:4
and 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
H
B 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’s
attention (772 & 780). In addition, he justifies use of auxiliary segment BD (EF//BD) and a
shear transformation of △BEF to △BDE (see Figure 5). He then arrives at equivalence of △BED
and △BEC (Figure 6), before finally asserting the equivalence of △BEC and △BDF (Figure 7).
4

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, and
without 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 one
uses graphic software).
That the teacher understood the plausibility of student’s reasoning, despite his expectation
that ratios were to be used is due to epistemological knowledge of mathematics. This professional
knowledge is valuable for mathematics teachers to make real-time decisions within the complexity
of classroom interactions. Similarly, the compact communication (771–785) reveals a need for
this knowledge and patience in monitoring consistency of students reasoning. Moreover, that
the scanty episode translates into coherent argument (Figures 4, 5, 6 & 7) is due to teacher’s
elaboration 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 students
access the student’s reasoning.
This episode further demonstrates the significance of problem-solving, where original task
is transformed without altering its structural properties. Similarly, it shows the value of seeing
mathematics as connections (NCTM, 1989), where the student uses equivalent areas in the unit
of 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 secondary
mathematics 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 the
lengths 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 evident
in 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 simulated
innovations. The validity of such qualitative interpretations of classroom episodes require, be-
sides well defined theoretical lenses, an understanding of the classroom culture through extended
observations (Ogwel, 2007). Video records or audio-tapes and transcripts are also invaluable in
recollecting classroom episodes. For observers and researchers, the process of transcription and
the desire to construct a coherent and convincing discourse implies immense learning (Mason,
1998) which is often not acknowledged in objective research formulations. Finally, the compact
and 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.
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6. 3 Conclusions
If learning is a cyclic process which involves planning, implementation and feedback, then teacher
education 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 potential
gains if complementarity of the two systems is harnessed. That is, initial teacher preparation must
substantially incorporate classroom experiences while professional development should enhance
theory-laden reflections. In addition, classroom-based research potentially challenges teachers and
educators’ 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 theoretical
principles automatically result into students’ learning. On the contrary, it provides opportunity
for collaboration (cf Scherer and Steinbring, 2006) and testing and revision of educational inter-
ventions, for instance Project Mathe 2000’s ‘Substantial Learning Environments’ developed by
Wittmann 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 requires
substantial internship experiences. Inevitably, the future of mathematics education lies in appro-
priate utilization of technology, thus, the benefits of mathematical software in instruction cannot
be gainsaid. Moreover, despite the observed deficiency in school mathematics, we agree with
Zbiek and Conner (2006) that the challenge is how modeling and problem-solving can be used
to enhance understanding of school mathematics. We are, however, not oblivious to the logistical
and 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 join
in enhancing quality of education, hence quality of life.
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