Mathematics teaching and Mathematics teachers

One of the most important purposes in mathematics teaching from pre-school to university is that students learn to understand and analyse mathematical concepts and solution procedures. However, teaching of and achievements in mathematics have been criticized in several countries during the last decade. It is generally concluded that school mathematics focus to develop algorithmic skills instead of understanding of mathematical concepts (e.g. Magne 1990; Sierpinska 1994; Soro & Pehkonen 1998) and teachers devote much less time and attention to conceptual rather than procedural knowledge (e.g. Porter 1989; Menzel & Clarke 1998, 1999). Pupils learn superficially several basic concepts in arithmetic and algebra without understanding (e.g. Hiebert & Carpenter 1992; Sierpinska 1994; Tall 1996; Silfverberg 1999).

Several attempts have been made in many countries in order to renew and develop mathematics teaching on all levels (e.g. Dunkels & Persson 1980; Dunkels 1989, 1996; Andersson 2000; Klein 2002). For instance, many universities have tested new teaching methods in mathematics during the last years (e.g. Tucker 1995; Dunkels 1996; Andersson 2000). There are many motives for this. Students’ pre-knowledge in mathematics both in domestic and some foreign universities have deteriorated (Royal Statistical Society 1995; National Agency for Higher Education 1999, 47). New groups of students as immigrants and groups who are not interested in mathematics make new demands on mathematics teaching. Some universities have also made special efforts in order to attract women to mathematics and science education (e.g. Grevholm & Hanna 1993; Wistedt 1996; Lindberg & Grevholm 1996). Advanced technology has made changes in mathematics teaching possible and opened totally new opportunities in laboratory work, computer visualization of concepts, distance learning etc.

In countries, such as USA (NCTM1 1991), Sweden (The Swedish Board of Education 1994, 2000) and Finland (National Board of Education 2004), new curricula describe the new vision for mathematics teaching. Recent documents in mathematics education have placed a great responsibility for the success of curricular reform on the teacher (Romberg & Carpenter 1986; NCTM 1991; 2000 . These responsibilities include, for example, an emphasis on problem solving and reasoning, communication and discourse around mathematical topics, connections within and across content areas, increased use of technology, manipulatives, and group work. Though development programs for teachers have been carried out teachers have, for example, been introduced to instructional materials for problem solving and to small group working mathematics teaching has not changed in a desirable way (e.g. Lundgren 1972; Fennema & Nelson 1997; Darling-Hammond 1997; Stigler & Hiebert 1999).

Problems in teacher education have a similar character. Students on teacher training programmes are not much influenced by their education (cf. Grevholm 2002). As newly graduated teachers in mathematics, they often instead turn back to acquired methods from their own school time (Raymond & Santos 1995, 58; Hill 2000, 23). This problem seems to be universal. To change the ways of teaching is a laborious task and the result is not yet satisfactory (e.g. Kupari 1999, 4). Some researchers also claim that other things than curriculum, for example, traditions, teacher knowledge, and textbooks, guide teaching (Cuban 1984; Rönnerman 1993; Tyack & Tobin 1994; cf. Magne 1990; Engström & Magne 2003). In order to get pupils interested in the subject, more research in school mathematics is needed. It is also important that research results then reach teachers, teacher educators and policy makers (Wallin 1997).

Hoyles (1992) has described and analysed, in a meta-case study, the research on mathematics teaching and mathematics teachers over a period of 10 years. Before the middle of 1970s the research was grounded in the behaviourist process—product tradition (Clark & Peterson 1986), which focused on ‘what teachers did, not what they thought’ (Cooney 1994, 624). Most of this research and later investigations were however concentrated on students’ behaviour and ability and the teacher—if he/she was mentioned at all—was described as a facilitator—someone how dispenses facts and information, identifies misunderstanding or provides materials or strategies to overcome misconceptions (Hoyles 1992, 32). A new phase in research started between 1982 and 1984 (Hoyles 1992). Research focused on clarifying how teachers’ mental structure including knowledge, conceptions and attitudes influenced their action (Ernest 1989a, 13; Romberg 1984). In other words, researchers of teaching began to alter their view of the teacher to encompass a more active, cognising agent whose thoughts and decisions influence all aspects of classroom instruction and learning (Clark & Peterson 1986; Peterson, Fennema, Carpenter & Loef 1989). This change in the conceptualization of the teacher coincided with a move from assessing the teachers’ knowledge in quantitative terms, such as the number of college courses completed or scores on standardized tests (Ball 1991), to the more recent qualitative attempts to describe teachers’ conceptions of their subject areas. By the end of 1980s and in the beginning of 1990s the character of teacher’s beliefs and conceptions in mathematics were understood better and the awareness increased that teacher’s beliefs, conceptions, knowledge, thoughts and decisions have an influence on teaching and pupils’ learning (Pajares 1992; Thompson 1992). A constructivist view of knowledge has also partly influenced investigations in teachers’ knowledge structures and has received a large attention in recent research (Davis, Maher & Nodding 1990; Cooney 1994, 612).

Teachers’ mathematics-related beliefs and conceptions have been investigated in numerous research reports in the last decade (e.g. Sandqvist 1995; Hannula 1997; Adams & Hsu 1998; Pehkonen 1998a; Kupari 1999; Perkkilä 2002). Also student teachers’ conceptions of mathematics teaching have been studied (e.g. Trujillo & Hadfield 1999; Kaasila 2000; Pietilä 2002). The largest part of the research on teachers’ mathematics-related investigations has dealt with teachers’ beliefs and conceptions of mathematics and mathematics teaching and learning (Hoyles 1992; Thompson 1992). In these studies the teachers’ mathematics-related beliefs and conceptions have not however been investigated specifically concerning some content domain in mathematics. Interest in teacher knowledge has also grown significantly in recent years (Connolly, Clandinin & He 1997). Much work on the development of a qualitative description of teacher knowledge and conceptions has been influenced by Shulman’s (Shulman 1986; see also Wilson, Shulman & Richert 1987; Grossman 1990) model for teacher knowledge. However, research on teaching and teacher education has, according to Ernest (1989a, 13) underemphasised this area, which by using Shulman’s (1986, 7) words has been called the ‘missing paradigm’ in research on cognitions. Research on teacher subject matter knowledge—which refers to ‘the amount and organisation of knowledge per se in the mind of the teacher’ (Shulman 1986, 9)—has been investigated in a large number of recent studies (e.g. Ball 1988; Graeber, Tirosh & Glover 1989; Ball 1990a, Ball 1990b, 1991; Borko & al. 1992; 

Even 1993; Simon 1993; Baturo & Nason 1996; Fuller 1997; Tirosh, Fischbein, Graeber & Wilson 1999; Ma 1999; Attorps 2002, 2003). The research results are essentially the same: teachers lack a conceptual knowledge of many topics in the mathematics curriculum. Current research on the relationship between teacher knowledge and teaching practice has also pointed out the need to carry out more studies involving specific mathematical topics. Furthermore, this research has shown that the way teachers instruct in a particular content is determined partly by their pedagogical content knowledge  which ‘goes beyond knowledge of the subject matter per se to the dimension of the subject matter knowledge for teaching’ (Shulman 1986, 9)—and partly by teachers’ mathematics-related beliefs (Brophy 1991a; Cooney & Wilson 1993).

Knowledge about students’ conceptions is one component of the teacher pedagogical content knowledge. Such knowledge has been gathered mainly in the last two decades of extensive cognitive research on student learning, which has yielded much useful data on students’ conceptions, preconceptions, and mistaken conceptions about specific topics in mathematics. Many studies have shown that students often make sense of the subject matter in their own way, which is not equal to the structure of the subject matter or the instruction. (Peterson 1988; Peterson & al. 1991; Kieran 1992; Even & Tirosh 1995). In numerous research reports (e.g. Stein, Baxter & Leinhardt 1990; Lloyd 1998; Bolte 1999; 357–363) a strong interdependence of conceptions about subject-matter knowledge and pedagogical content knowledge has been documented. Many teachers do not separate their conceptions about a subject specific topic from notions about teaching that topic. In many cases, teachers’ subject matter knowledge influences their pedagogical content-specific decisions (Even 1989; Even 1993; Even & Markovits 1993; Even & Tirosh  1995).

In the theoretical background of the research, different traditions of school mathematics learning and teaching are treated. By using theories of experiential learning, it has been possible to study such learning situations and experiences, which may lead to the development of subjective conceptions of mathematical concepts. In order to understand difficulties concerning the concept formation in mathematics the theory of the concept image and the concept definition as well as the theory based on the duality of mathematical concepts have been studied. The acquired experiences from school time seem to lay the basis of both the teachers’ subject matter and pedagogical content-specific conceptions and decisions. Different components in teacher knowledge base together with current research both in teachers’ subject matter and pedagogical content knowledge are therefore presented as the theoretical framework.