The International Online Journal of Science and Mathematics Education

Establishing Conceptual Bases for the Measurement of Volume

Profiling Teachers:
Constructivist- and Behaviorist-Oriented Mathematics

Boris Handal
The University of Sydney

borishandal@optusnet.com.au

AbstractIntroductionThe StudyMethodology
ConclusionReferences

Results and Discussion

Correlation Analysis

A Cronbach’s Alpha analysis resulted in a satisfactory coefficient of rational equivalence reliability of 0.814 for the Belief and Practice Scales. A Pearson’s product-moment correlation coefficient analysis was applied to the scores on the Belief and Practice Scales to determine the degree of association between teachers’ beliefs and their instructional practices. The Pearson’s r product-moment correlation coefficient (r = 0.05) was found to be significant and positive for the total sample (r = 0.432, N = 120). This r coefficient indicates a low-moderate correlation between scores on the Belief and Practice Scales for the total sample. Consequently teachers’ beliefs can be considered a predictor of instructional practice.

To examine the possibility of categorizing teachers’ beliefs in two distinct sets, a correlation analysis was carried out as shown in Table 2. Two items on the Belief and Practice Scales were selected by the researcher as organizers of the constructivist and behaviorist belief systems; namely, “Students can create and do maths by themselves” (Item 1[r]) and “Maths is a set of rules and formulas” (Item 1[d]). The former represented the constructivist construct while the latter represented the behaviorist one. These two items were chosen because they represented two opposite philosophical notions of mathematics as a discipline. Mathematics in the constructivist perspective is associated with a creative activity in which “knowing mathematics is doing mathematics” (National Council of Teachers of Mathematics, 1989, p. 7). In turn, in the behaviorist perspective, learning mathematics is associated with activity that predominantly emphasizes memorizing facts and procedures (Leder, 1994). It is also noteworthy that both items are negatively correlated (r = –0.313, p < 0.01). In addition, in the factor analysis shown in Table 3, item 1(d) “Maths is a set of rules and formulas” obtained a loading of 0.369 on the factor associated with the behaviorist construct while the constructivist-oriented item “Students can create and do maths by themselves” (Item 1[r]) obtained a loading of 0.312 on the factor associated with the constructivist construct in the factor analysis (see Table 3). This methodology of choosing two items as organizers of a correlation analysis has been previously used in educational research (Howard, Perry, & Lindsay, 1997; Perry, Howard, & Tracey, 1999).

It was assumed that behaviorist-oriented items were correlated positively and significantly among themselves. Similarly, it was expected that constructivist-oriented items were correlated positively and significantly with each other. With the exception of the behaviorist practice item 2(a) “Individual and independent work,” the constructivist item 1(r) correlates positively and significantly with the other constructivist items and correlates negatively and significantly with the behaviorist items. Likewise, with the exception of the same item behaviorist-practice item 2(a), the behaviorist item “Maths is a set of rules and formulas” (Item 1[d]) correlates positively and significantly with the other behaviorist items, and negatively and significantly with the other constructivist items. This apparent contradiction of the correlation of the behaviorist item 2(a) with the other behaviorist and constructivist items may mean that teachers from both profiles did not fully embrace the teaching technique “Individual and independent work” as part of their belief systems.

Table 3
Factor Loading of Items on the Belief and Practice Scales

Items	Intended Factor^a	Factor 1	Factor 2
1(c)	C	7.181E–02	– 0.568
1(e)	C	0.194	3.466E–02
1(g)	C	0.323	– 6.579E–02
1(h)	C	0.128	– 0.605
1(i)	C	0.322	– 0.273
1(k)	C	0.270	0.250
1(l)	C	0.192	0.250
1(o)	C	0.290	– 0.304
1(r)	C	0.312	– 0.103
2(b)	C	0.533	– 5.361E–02
2(d)	C	0.525	0.164
2(f)	C	0.424	2.655E–02

2(g)	C	0.769	– 1.020E–02
2(h)	C	0.700	4.441E–02
2(i)	C	0.695	– 1.970E–02
2(j)	C	0.370	– 0.109
2(k)	C	0.391	– 1.694E–03
2(n)	C	0.235	1.336E–02
2(o)	C	0.514	4.191E–02
2(p)	C	0.621	– 0.126
2(r)	C	0.492	– 6.078E–02
2(t)	C	0.413	– 3.708E–02
2(v)	C	0.425	– 0.119
1(a)	B	9.934E–02	0.452
1(b)	B	– 4.046E–02	0.396
1(d)	B	– 0.124	0.396
1(f)	B	– 0.162	0.193
1(j)	B	– 0.132	0.411
1(m)	B	– 9.736E–02	0.140
1(n)	B	– 9.6141E–02	7.674E–02
1(p)	B	– 1.261E–02	0.391
1(q)	B	5.702E–02	0.618
2(a)	B	0.313	– 0.122
2(c)	B	4.622E–02	0.163
2(e)	B	– 1.907E–02	0.288
2(m)	B	– 7.852E–02	0.277
2(s)	B	– 0.114	0.181
2(u)	B	– 0.325	– 4.857E–02

Note:     Extraction method: Principal Axis Factoring
           Rotation method: Oblique with Kaiser Normalization
             Rotation converged in 8 interations.

^a C stands for constructivist-oriented items, B stands for
behaviourist-oriented items.

In general, the correlation pattern between behaviorist and constructivist items suggests the existence of two distinctive constructs underlying teachers’ beliefs and practices. Although some of the correlation coefficients are small, their statistical significance confirms the existence of these two contrasting curricular orientations in teachers’ instructional beliefs and espoused practices.

Factor Analysis

An exploratory factor analysis using principal axis factoring with oblique rotation was carried out. The purpose of the factor analysis was to further explore the constructivist and behaviorist dimensions underlying the construction of the Belief and Practice Scales (Cooksey, 1984; Perry, Howards, & Tracey, 1999). The two-factor solution shown in Table 3 accounted for 23.78% of the variance.

A positive loading reveals that the item is associated with a certain factor. All the 38 items in the Belief and Practice Scales displayed a loading on one factor and a smaller loading on the other. These included 23 constructivist-oriented and 15 behaviorist-oriented items. Twenty-five of these 38 items displayed a positive loading on one factor and a negative loading on the other.

The entire 23 constructivist items loaded positively on Factor 1. Of these constructivist items, 15 items loaded positively on Factor 1 and negatively on Factor 2 at the same time. With the exception of items 2(a) and 2(u), all the behaviorist-oriented items loaded positively on Factor 2. Ten of the 15 behaviorist-oriented items loaded negatively on Factor 1 and positively on Factor 2. Moreover, factors 1 and 2 were basically independent of each other (r = –0.232) for the Belief and Practice Scales.

The two behaviorist-oriented items that loaded negatively on Factor 2 were items 2(a) and 2(u). That the behaviorist-oriented items 2(a) “Individual and independent work” and 2(u) “I avoid teaching through real-life problems” did not fit on either of the two profiles further confirmed the suggestion that some beliefs can be contradictory within a belief system (Andrews & Hatch, 1999).

Although some of the loadings were small, this two-factor solution further suggests the existence of the constructivist and behaviorist constructs underlying the design of the Belief and Practice Scales. In addition, this two-factor solution suggests the existence of a link between beliefs and practices within the same construct and, in conjunction with the correlation analysis in the previous section, provided construct validity of the Belief and Practice Scales in the questionnaire component of the study.

Analysis of Teachers’ Open-ended Responses

Teachers’ responses to the open-ended questions of the questionnaire also suggested the existence of constructivist and behaviorist curricular orientations as evidenced in the above correlation and factor analyses. Generally speaking, constructivist-oriented teachers declared their adherence to the goals of teaching mathematics thematically while the behaviorist group expressed their dissatisfaction. On one side, teachers who were classified as constructivist oriented according to their scores on the Belief Scale tended to regard the teaching of mathematics thematically as a way to promote the usefulness of school mathematics to students usually perceived by the educational community as unmotivated to do mathematics. For example, a constructivist-oriented teacher remarked: “The themes approach works well with Standard students because it provides students and teachers with immediate applications of skills ... I feel it is better to develop mathematical appreciation like artistic appreciation …” On the other side, behaviorist-oriented teachers confirmed their dissatisfaction with the thematic approach and constructivist ideas. For example, one behaviorist-oriented teacher commented: “[I] totally disagree with the themes concept.” Another behaviorist-oriented teacher commented: “Most of us would rather teach a level where there is some deductive thought and reasoning and not the simplicity of content at the Standard level.” In addition, another behaviorist teacher stated, “Group discussions and activities only help them [Standard students] to waste time.”

Interview Analysis

Subsequent interviews assisted in validating the behaviorist and constructivist constructs found through correlation analysis. For this purpose, two interviewees that scored the highest scores on the Belief Scale were selected as being constructivist teachers (Kim and Sean) while the last two teachers in the ranking were considered as being representative of behaviorist teachers (Peter and Jenny). Indeed, Sean and Kim stand in the constructivist group because of their views of teaching and learning mathematics thematically. Sean stated that the “thematic approach is a fantastic idea.” For example, he provided his students with research projects or miniassignments through which they can work at their own pace. Sean begins his lessons from a particular theme and uses the theme to initiate a brainstorming session. Later on, he manages to connect this central idea to the development of the basic skills. Sean considers that students are better off learning mathematics thematically while conceding that teaching via topics is easier. In turn, Kim was of the opinion that themes are a good idea for low-ability students because of the practical nature of themes. He also believes that students should learn mathematics in a meaningful context. As a head teacher, Kim encourages alternative methods for assessments such as projects or miniassignments within his faculty. In his description of the sequencing of a typical thematic unit, Kim spoke of commencing his lessons with a theme as a generic idea and later posing a problem that led the class into a discussion and eventually to the learning of mathematical concepts. According to Kim: “When teaching in themes a lot of interaction needs to go on to gain an understanding of the problem or investigation.” Kim’s faculty had embarked on the production of thematic units that were adapted to the characteristics of their students thus replacing textbook-aligned teaching.

Behaviorist-oriented teachers like Jenny and Peter held opposite views in some regards. Jenny considered that themes should not be taught in the Advanced classes and emphasized that themes were good only for the lower classes because low-achieving students in mathematics needed to see the relevance of mathematics. In her words, “the higher kids . . . enjoy the [abstract] mathematics and don’t see where it is leading to.” As stated earlier, she taught via topics using a textbook launched previous to the release of the Standard course; her teaching was more concerned with the teaching of basic skills: “I like to teach one topic and get it over and done, then teach the next topic. But, with themes it’s bits and pieces.” According to Peter, another head teacher, “Teaching through topics is more effective than themes because in topics the teacher covers all the content at once before he or she moves to the next unit.” To Peter, the thematic approach is not good for low-ability students who usually lacked good recollection of mathematical concepts. Peter has also “a concern about the effectiveness of group work.” According to him, the books are not good because there are “very few drill lessons on what they should be doing. Not enough drill.” His teaching is centered on basic skills; applications are left at the end of the unit.

AbstractIntroductionThe StudyMethodology
ConclusionReferences