Connection between communication and reflection skills

The problem is how to learn the connection between communication and reflection skills to solve problems and how to understand mathematical concepts. Research findings about communication and reflection skills accentuate metacognitive instruction in solving mathematical authentic assignments. A characteristic feature when solving metacognitive assignments is that the approach is not immediately obvious (Darling-Hammond, 1992). In authentic assignments, students are confronted with excess information so that they have to make choices in order to get started (Cobb, 1994). The solution cannot be found through application of an algorithm or a standard solution (Prawat, 1998). It is difficult to establish links with problems already solved. Students find authentic assignments difficult (Kramarski, Mevarech & Libermann, 2001; Verschaffel, Greer & de Corte, 2000). According to Verschaffel et al., weaker students in particular have difficulty with abstraction into sub-problems, as they are unable to separate the information between relevant and irrelevant. They therefore soon give up, also due to algorithms which do not offer a solution (Anderson, 1990). These problems go beyond the skills of solving mathematical problems. They concern the skills of solving problems together with others, for example Cardelle-Elawar (1995).

Based on these findings, teaching methodology research aims at developing instruction methods to support teachers in training (TTs) in the activation of students’ metacognitive learning processes (Lester, Garofalo & Kroll, 1989; Mayer, 1987; Schoenfeld, 1987). Basic elements in the development of instruction methods for

students are metacognitive questions put to small groups of students, such as: (1) conceptual questions: What is it about? What is the question? What is the meaning of a mathematical concept? (2) relational questions: Does the question resemble…? Does the question differ from a problem already solved? Why? (3) strategic questions: What is the solution strategy? Why this strategy? How does this strategy work? and (4) reflective questions: What have I done? Was it purposeful? These metacognitive questions which students ask each other and answer jointly can be traced back to Polya's theories (1957). According to Polya, teachers gain insight into the way in which problems can be solved but also into how students can be supported, by analyzing various solution methods, communicating them to others, and reflecting on their effect. Metacognitive instruction is strongly related to cognitive units, connection and compression (Barnard & Tall, 1997). According to Pinto and Tall (2002), students’ cognitive constructions occur through reflective abstraction, in which a predicate with one or more variables is conceived as a mental process. Recent research findings indicate teachers’ role to stimulate reflective abstraction (Simon et al., 2004). The research study of Ainley and Lowe (1999) suggests different levels of understanding related to teacher interventions:

The ability to conceive and to manipulate cognitive units is a vital facility for mathematical thinking. Two complementary factors are important in building a powerful thinking structure: (i) the ability to make connections between cognitive units so that relevant information can be pulled in and out of the focus at will; and (ii) the ability to compress information to fit into cognitive units by communication. Compressibility of mathematical ideas relies on the nature of the connections from the focus of attention to other parts of the cognitive structure (Barnard & Tall, 2001).

Piaget (1972) emphasized the construction of meaning through different forms of abstraction. One of them, reflective abstraction, is a process focusing on mental actions and mental concepts in which the mental operations themselves become new objects of thought (Pinto & Tall, 2002). Later on Piaget (2001) described reflective abstraction as a process by which higher level mental structures could be developed from lower level structures, consisting of two phases: (1) a projection phase in which the actions at one level become the objects of reflection at the next level; and (2) a reflection phase in which a reorganization takes place. Simon et al. (2004) elaborate on Piaget’s reflective abstraction: (i) activity refers to a mental activity; (ii) activity sequence refers to a set of activism in an attempt to meet a goal; (iii) learners’ goals are not necessarily conscious; and (iv) effects are structured by assimilatory conceptions that learners bring to the situation.

Bereiter (1985) emphasizes that cognitive advance cannot be directly brought about; rather, teachers promote specific experiences for the development of the intended cognitive structure, a step-by-step outline of how to foster students’ reinvention of a particular process of the learner. Underlying is the idea that learnerAinley and Lowe (1999) defined four degrees of understanding: (1) no apparent understanding, students cannot make a start, no understanding could be identified; (2) procedural (instrumental) understanding, students know how to carry out a mathematical procedure but lack the deeper understanding to recognize when an algorithm had been misapplied or incorrectly remembered; (3) conceptual (relational) understanding, students recognize the constraints of answers, are able to comment constructively on their work; (4) proceptual understanding, students appreciate that the symbol ambiguously represents both the concept and the procedure. Teachers’ help interventions shall be categorized within these levels of understanding. No understanding (1) corresponds to teachers’ intervention: which concept to use (to find a unit)? The procedural (instrumental) understanding (2) corresponds to teachers’ intervention: how to use this concept (to find a connection, an algorithm)? The conceptual (relational) understanding (3) corresponds to teachers’ intervention: why to use this concept (to find related units)? The proceptual understanding (4) corresponds to teachers’ intervention: which choice of concepts is most effective to use (to find units and connections)?

The hypothesis is that (i) metacognitive instruction has positive effects on students’ learning results, rather than merely on the mathematical assignments; and (ii) these teachers’ interventions at the actual level of understanding aimed at the next level of understanding are effective (Mevarech & Kramarski, 1997).s impose mathematical relationships on the situation based on their available conceptions. Bereiter’s remarks indicate teachers’ role to stimulate reflective abstraction.

As indicated by critics, this type of research is methodically weak, crucial conclusions are insufficiently precise, it produces contrary results, is reported in incomprehensible jargon, does not lead to improved teaching results and must all take place much more thoroughly (Slavin, 2000). While good practices can certainly be identified and qualitatively described, the generalization questions (will it work for my subject / with these students / in our context / with a different teacher?), the reproducibility question (will it happen again tomorrow?) and the explanation question (what is the underlying causal relationship between “treatment” and results?) are seldom adequately answered (Van Keulen, 2006). Educational research in the form of action research is difficult but is certainly recommended for professionalization of teachers. The conclusions of the case study are challengeable because of the minimum of data.

Two arbitrarily cases have been analyzed. Therefore, the conclusions can not be generalized. Besides, a TT fulfilled the role of the teacher. She was strongly influenced by Tall’s theory of cognitive structures. She intended to attain mathematical concept development by questions at the highest level of abstraction. Her approach resulted in positive effects on high level students and negative results on low level students. She assumed the existence of some basic proceptual views. It is recommended that this approach to teachers’ intervention be repeated by questions in classroom practice at the actual individual level of understanding in collaboration with other students at lower or higher levels (Dekker & Elshout-Mohr, 2004). The results of this case study support the intention of such a type of research activities to design teachers’ intervention aimed at mathematical concept development. It also shows that despite the good results of students in the contests there is room for further development, by supporting teachers with focused classroom research – focused on essential elements of mathematics like communication and reflection.