Most of the studies over the three decades , directly or indirectly, focused on the difficulties or deficiencies teachers exhibited for particular mathematics concepts or processes. For example, addressing knowledge about numbers and operations, Linchevsky and Vinner (1989) investigated the extent to which inservice and preservice elementary teachers were flexible when the canonical whole was replaced by another whole for fractions of continuous quantities. They found all of the expected misconceptions and confusions associated with canonical representations of fractions and the teachers’ visual representations of fractions incomplete, unsatisfactory and not sufficient to form a complete concept of fractions. In a later paper, Llinares and Sánchez (1991) studied preservice elementary teachers’ pedagogical content knowledge about fractions and found that many of the participants displayed incapacity to identify the unity, to represent some fractions with chips and to work with fractions bigger than one.
A few of the studies addressed teachers’ knowledge in arithmetic in relation to a particular theoretical model. For example, Tirosh, Graeber and Glover (1986) explored pre service elementary teachers’ choice of operations for solving multiplication and division word problems based on the notion of primitive models in which multiplication is seen as repeated addition (with a whole number operator) and division as partitive (with the divisor smaller than the dividend). The findings indicated that the teachers were influenced by the primitive, behavioral models for multiplication and division. The teachers’ errors increased when faced with problems that did not satisfy these models. Greer and Mangan (1986) used a similar notion of primitive models. Their study included pre service elementary teachers and focused on the results on single-operation verbal problems involving multiplication and division. They also found that primitive operations affected the participants’ interpretation of multiplicative situations.
Another theoretical framework used in some of these studies was the distinction between “concept image”, the total cognitive structure that is associated with a concept, and “concept definition”, the form of words used to specify that concept (Vinner and Hershkowitz, 1980). For example, Pinto and Tall (1996) investigated seven secondary and primary mathematics teachers’ conceptions of rational numbers. Findings indicated that three of the teachers gave formal definitions containing implicit distortions, three gave explicit distorted definitions and one was unable to recall a definition. None consistently used the definition as the source of meaning of the concept of rational number; instead they used their concept imagery developed over the years to produce conclusions, which were sometimes in agreement with deductions from the formal definition, but often were not. Whole numbers and fractions were often seen as “real world” concepts, while rationals, if not identified with fractions, were regarded as more technical concepts. In geometry, Hershkowitz and Vinner (1984) reported on a study that included comparing elementary children’s knowledge with that of preservice and inservice elementary teachers. They found that the teachers lacked basic geometrical knowledge, skills and analytical thinking ability. Using the van Hiele levels as theoretical framework, Braconne and Dionne (1987) investigated secondary school students and their teachers’ understanding of proof and demonstration in geometry, and what kind of relationship could exist between the understanding of a demonstration and the van Hiele levels. Findings indicated that proof and demonstration were not synonymous for the teachers or for the students. Proofs belong to different modes of understanding but demonstration always pertains to the formal one, teachers emphasizing representation and wording. Furthermore, there was no obvious relationship between the understanding of a demonstration and the van Hiele levels.
Another topic studied extensively was teachers’ knowledge about functions. Ponte (1985) investigated preservice elementary and secondary teachers’ reasoning processes in handling numerical functions and interpreting Cartesian graphs. Findings indicated that many of the participants did not feel at ease processing
geometrical information and had trouble making the connection between graphical and numerical data. Later, Even (1990) studied preservice secondary teachers’ knowledge and understanding of inverse function. She found that many of the preservice teachers, when solving problems, ignored or overlooked the meaning of the inverse function. Their “naïve conception” resulted in mathematics difficulties, such as not being able to distinguish between an exponential function and a power function, and claiming that log and root are the same things. Besides, most of teachers did not seem to have a good understanding of the concepts (exponential, logarithmic, power, root functions). In another function study, Harel and Dubinsky (1991) investigated preservice secondary teachers in a discrete mathematics course for how far beyond an action conception and how much into process conceptions of function they were at the end of the instructional treatment framed in constructivism. Findings indicated that the participants had starting points varying from primitive conceptions to action conceptions of function. How far they progressed depended on several factors such as (i) manipulation, quantity and continuity of a graph restrictions; (ii) severity of the restriction; (iii) ability to construct a process; and (iv) uniqueness to the right condition. Thomas (2003) investigated preservice secondary teachers’ thinking about functions and its relationship to function representations and the formal concept. Findings indicated a wide range of differing perspectives on what constitutes a function, and that these perspectives were often representation dependant, with a strong emphasis in graphs. Similarly, Hansson (2005), in his study of middle school pre service teachers’ conceptual understanding of function, found that their views of it contrast with a view where the function concept is a unifying concept in mathematics with a larger network of relations to other concepts.
Other topics and mathematics processes and understandings addressed include the following examples. Van Dooren, Verschaffel and Onghena (2001) investigated the arithmetic and algebraic word–problem-solving skills and strategies of preservice elementary and secondary school teachers both at the beginning and at the end of their teacher training. Results showed that the secondary teachers clearly preferred algebra, even for solving very easy problems for which arithmetic is appropriate. About half of the elementary teachers adaptively switched between arithmetic and algebra, while the other half experienced serious difficulties with algebra. Barkai et al. (2002) examined in service elementary teachers’ justification in number-theoretical propositions and existence propositions, some of which are true while others are false. Findings indicated that a substantial number of the teachers applied inadequate methods to validate or refute the proposition and many of them were uncertain about the status of the justification they gave. Shriki and David (2001) examined the ability of inservice and preservice high school mathematics teachers to deal with various definitions connected with the concept of parabola. Findings indicated that both groups shared similar difficulties and misconceptions. Only a few participants possessed a full concept image concerning the parabola and were capable of perceiving the parabola in its algebraic as well as in its geometrical contexts or to identify links between them. Finally, Mastorides and Zachariades (2004), in their study of pre service secondary school mathematics teachers aimed to explore their understanding and reasoning about the concepts of limit and continuity, found that the teachers exhibited disturbing gaps in their conceptualization of these concepts. Most had difficulties in understanding multi quantified statements or failed to comprehend the modification of such statements brought about by changes in the order of the quantifiers.
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