The International Online Journal of Science and Mathematics Education

A questionnaire was designed to identify teachers’ mathematical beliefs and practices in the teaching and learning of mathematics thematically. The questionnaire used in this study consisted of a Belief and a Practice Scale with a total number of 38 items. These items were written reflecting a constructivist and a behaviorist perspective to the teaching and learning of mathematics thematically (Handal, Bovis, & Grimison, 2001). Following the discussion above, the behaviorist and constructivist constructs were adopted in the light of some literature that present them as contrasting views on mathematics education. As teaching thematically is embedded in a constructivist perspective of teaching and learning (Freeman & Sokoloff, 1995; Seely, 1995), the constructivist items on the Belief and Practice Scales are those supporting the thematic philosophy goals of the Standard course. This is in line with Freeman and Sokoloff (1995) and Seely (1995) who support a constructivist perspective that advocates more thematic and investigational forms of instruction as opposed to Stein, Silbert, & Carnine’s (1997) perspective who operating within a behaviorist pedagogical perspective recommend more direct and explicit-oriented teaching approaches for low achievers in mathematics.

The listing and arrangement of the Belief and Practice Scales items by constructivist and behaviorist orientations are presented in Table 2. As the literature ascribes a variety of meanings to the terms constructivism and behaviorism (Biggs & Moore, 1993; Clements & Battista, 1990; Leder, 1994; Murphy, 1997; Phillip, 1995; Seely, 1995; Wood, Cobb, & Yackel, 1991), the arrangement in Table 2 represented the operational conceptualization of these two constructs for the purpose of this study. This conceptualization is therefore epistemologically restrictive to this study in that it specifically applies to teachers’ mathematical beliefs in the teaching of mathematics thematically in the context of the Standard course. The same limitation applies to the operational conceptualization described below for the items in the Practice Scale.

Item No.	Item	Behaviorist Item 1(d)	Constructivist Item 1(r)	Curricular Orientation
BELIEF SCALE ITEMS
1(a)	Teaching in THEMES should only be taught in the Standard Course	0.254**	– 0.008	Behaviorist
1(b)	There is a “best way” to do a maths problem	0.381**	– 0.235**	Behaviorist
1(c)	Teaching in THEMES should be taught also in the Advanced and Intermediate courses	– 0.085	0.078	Behaviorist
1(d)	Maths is a set of rules and formulas	–	– 0.313**	Behaviorist
1(e)	Word problems can be solved by common sense and without using rules	0.005	0.237**	Constructivist
1(f)	Trial and error should not be allowed in solving a maths problem	0.312**	– 0.128	Behaviorist
1(g)	There are many ways to solve a maths problems	– 0.155	0.234**	Constructivist
1(h)	Teaching in THEMES is also for high achievers	– 0.236**	0.194*	Constructivist
1(i)	The study of maths should be integrated with other subjects (English, Arts, History)	– 0.150	0.190*	Constructivist.
1(j)	Teaching in THEMES should be made optional and not compulsory	0.154	– 0.078	Behaviorist
1(k)	Estimation is an important maths skill	– 0.143	0.255**	Constructivist
1(l)	The applications of a mathematical result are more important than its proof	0.074	0.071	Constructivist
1(m)	Teachers should teach exact procedures for solving word problems	0.124	0.054	Behaviorist
1(n)	Maths consist of unrelated topics (e.g., algebra, arithmetic, and geometry)	0.196*	– 0.008	Behaviorist
1(o)	Teaching in THEMES is worth the time and effort	– 0.040	0.127	Constructivist
1(p)	Applications problems are best left to the end of the topic in maths	0.320**	– 0.211*	Behaviorist
1(q)	Teaching in THEMES is only for low achievers	0.333**	– 0.088	Behaviorist
1(r)	Students can create and invent maths by themselves	– 0.313**	–	Constructivist
PRACTICE SCALE ITEMS
2(a)	Individual and independent work	– 0.271**	0.239**	Behaviorist
2(b)	Pair and small groups	– 0.217*	0.269**	Constructivist
2(c)	Teacher’s exposition (lecture)	– 0.005	0.156	Behaviorist
2(d)	Whole-class discussion of the real-life situation of the problem	– 0.089	0.158	Constructivist
2(e)	Drill-and-practice exercises	0.164	– 0.137	Behaviorist
2(f)	Language-based discussions of word problems	– 0.120	0.144	Constructivist
2(g)	Hands-on experiences (practical exploration)	– 0.107	0.246**	Constructivist
2(h)	Out-of-class activities (field work)	– 0.111	0.172	Constructivist
2(i)	Open-ended students’ investigational projects of real-life situations	– 0.036	0.182**	Constructivist
2(j)	Use of computer programs	– 0.016	0.070	Constructivist
2(k)	Teaching of problem-solving strategies (e.g., looking for a pattern)	– 0.074	0.093	Constructivist
2(m)	Give paper-and-pencil tests to students	0.106	– 0.088	Behaviorist
2(n)	Keep students’ portfolios for assessment	– 0.004	– 0.061	Constructivist
2(o)	Assess students’ oral presentations	– 0.048	0.127	Constructivist
2(p)	Require students to produce written reports	– 0.216*	0.221*	Constructivist
2(r)	Real-life or word problems are from newspapers/magazines	– 0.112	0.051	Constructivist
2(s)	Real-life or word problems are from the textbook	0.118	– 0.020	Behaviorist
2(t)	I make up my own problems	– 0.114	0.187*	Constructivist
2(u)	I avoid teaching through real-life problems	0.038	0.002	Behaviorist
2(v)	Real-life or word problems are shared from other colleagues	– 0.115	0.000	Constructivist

Note: *p < 0.05 (two-tailed), **p < 0.01 (two-tailed) Items 2(l), 2(p), and 2(w)
were open-ended questions

Nine of the 18 items in the Belief Scale were meant to represent behaviorist-oriented beliefs while the other nine items replicated the first half, but were written to mirror beliefs representing a constructivist perspective. The Belief Scale intended to measure teachers’ beliefs on nine major theoretical issues such as: (a) the nature of mathematics as a school subject; (b) the nature of mathematics as a discipline; (c) alternative methods in reaching a solution; (d) application versus pure mathematics; (e) discovery versus explicit teaching; and (e) approaches in solving a mathematics problem. The remaining three theoretical issues focused on characterizing teachers’ beliefs on the thematic approach. In turn, the Practice Scale was designed to characterize teachers’ espoused instructional practices in the teaching and learning of mathematics thematically in the areas of teaching and assessment techniques and use of resources. Seven choices of response for each item in both scales were presented on a Likert-type scale, which vary in their degree and agreement to the item.

In general, a constructivist-oriented teacher, as opposed to a behaviorist-oriented counterpart, would believe that teaching mathematics in themes is necessary for both low and high achievers. Such a teacher would also believe that thematic teaching should be made compulsory not only in the Standard course but also in the Intermediate and the Advanced levels of years 9 and 10. A constructivist-oriented teacher would also be of the opinion that mathematics is more than rules and formulas; that is, he or she would believe that students can create or do mathematics and that trial and error should be allowed in mathematics problem solving. Likewise, a constructivist-oriented teacher would believe that applications of mathematics are an important learning outcome and should not be left at the end of a unit. In terms of discovery versus explicit teaching, a constructivist-oriented teacher would believe that teachers should not teach exact procedures for solving word problems and that there is not a “best way” to do a mathematical problem. As with instructional practice, a constructivist-oriented teacher would focus on whole-class discussion of real-life problems, avoid lecturing, and be more engaged in the teaching of problem-solving strategies. Similarly, such a teacher will favor cooperative learning in small groups, out-of-class activities, open-ended students’ investigation of real-life situations, and use of technology in learning mathematics. In terms of assessment techniques, a constructivist-oriented teacher would use a diversity of assessment techniques such as students’ portfolios, oral presentations, and written reports rather than relying only on paper-and-pencil tests. Furthermore, such a teacher will rely less on textbook instruction and will make his or her own resources.