Introducción Propósito

conceptual understanding; procedural fluency; strategic competence; adaptive reasoning and a productive disposition.

2.6.1 The strands of mathematical proficiency and teaching for mathematical proficiency

Kilpatrick, et al. (2001) identify mathematical proficiency as encompassing five interwoven and interdependent strands, namely:

 conceptual understanding – comprehension of mathematical concepts, operations, and relations

 procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

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 strategic competence – ability to formulate, represent, and solve mathematical problems

 adaptive reasoning – capacity for logical thought, reflection, explanation, and justification

 productive disposition – habitual inclination to see Mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (p. 5)

Thus, they are of the opinion that, in order for an individual to develop proficiency in teaching Mathematics one needs to develop:

 conceptual understanding of the core knowledge of Mathematics, students, and instructional practices needed for teaching;

 procedural fluency in carrying out basic instructional routines;

 strategic competence in planning effective instruction and solving problems that arise while teaching;

 adaptive reasoning in justifying and explaining one’s practices and in reflecting on those practices; and a productive disposition toward mathematics, teaching, learning, and the improvement of practice (p. 10).

In the subsequent paragraphs I discuss these strands in greater detail in addition to clarifying their link to the themes of enactivism employed in this research study.

Kilpatrick, et al. (2001, p .369) are of the opinion that teaching for mathematical proficiency entails a number of factors. Firstly, teachers need to have a fundamental understanding of what the goals of instruction are and what proficiency entails for the particular mathematical content they are teaching. In addition they need to have a core understanding of the mathematics that they are required to teach and how their students will build on it. Kilpatrick, et al. (2001, p. 369) maintain that “They need to be able to use their knowledge flexibly in practice to appraise and adapt instructional materials, to represent the content in honest and accessible ways, to plan and conduct instruction, and to assess what students are learning”. Of equal importance is the ability to analyse, hear and see their students’ mathematical ideas and respond in an appropriate manner to written work, reasoning and problem solving strategies. In addition, Kilpatrick, et al. (2001, p. 369) also state that teachers have

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the capacity to identify the possibilities of a task and adapt them in such a way as to meet the needs of all students. In order to support the acquisition of mathematical proficiency they will need to have a good understanding of the different trajectories along which mathematical ideas can develop.

To provide clarity regarding the link between enactivism and mathematical proficiency, it would be prudent to discuss Kilpatrick, et al.’s (2001, p.369) five strands in greater depth.

2.6.1.1 Conceptual understanding

According to Kilpatrick, et al. (2001, p. 118), to have conceptual understanding in Mathematics one must have an integrated and functional grasp of mathematical ideas. This will enable one to connect new ideas to existing ideas and to retain this new knowledge. In addition, conceptual understanding indicates that one recognises why an idea is important and in what contexts it would be useful. Characteristics of learners with good conceptual understanding are the ability to represent mathematical situations in a variety of ways and have the knowledge to identify which representations are best suited to a specific purpose. Furthermore, they are able to explain and justify the connections and consequences among concepts and procedures. This understanding is generally hierarchical in nature “with simple clusters of ideas packed into larger, more complex ones” (p. 120). Learning with understanding allows the learner to identify the relationships between their school and outside environment experiences and integrate, modify or adapt their skills to solve real problems efficiently. Therefore, in order for teachers to teach for proficiency within this strand they need to have an integrated and interconnected knowledge of Mathematics, the development of students’ mathematical understanding and a range of pedagogic practices that take into consideration the mathematical concepts being taught and how the students learn these particular concepts. Thus a teacher’s classroom practice must exhibit the effective use of both personal mathematical knowledge and knowledge of their students and how they learn (Kilpatrick, 2001, p. 381).

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2.6.1.2 Procedural fluency

Procedural fluency demands a sound knowledge of mathematical procedures and the ability to discern when and how to use them appropriately. A learner who demonstrates procedural fluency will be able to carry out procedures skilfully, with a high degree of flexibility, accuracy and efficiency. These learners are able to recognise the fact that well developed and structured procedures can be used for completing routine tasks (Kilpatrick, et al., 2001, p. 121). A further characteristic of this strand is well developed estimation skills. Sound procedural fluency is necessary for learners to deepen their understanding of mathematical concepts and identify relationships and patterns which in turn will enable them to develop the other strands of proficiency.

According to Kilpatrick, et al. (2001, p. 382), a teacher who demonstrates proficiency in teaching Mathematics has a repertoire of instructional routines from which to draw on. These routines encompass both mathematical pedagogy and classroom management. To implement these routines and strategies effectively the teacher will need to have the confidence to select the appropriate strategies, apply them flexibly and adapt them to suit a particular context and/or situation as it arises. Thus the proficient teacher would know how to respond appropriately to any given situation that may arise and have a wide variety of strategies to draw from, should a particular approach not address a given situation.

2.6.1.3 Strategic competence

Kilpatrick, et al. (2001, p. 124) define strategic competence as “the ability to formulate mathematical problems, represent them, and solve them”. They emphasise that this strand entails a learner having the skill to understand a problem sufficiently well to be able to generate a suitable mathematical problem model encompassing core mathematical elements in order to arrive at a workable solution. Furthermore, competency in this strand implies the capacity to recognise shared common mathematical structural relationships that may indicate possible solutions. Characteristics of learners exhibiting strategic competence are the ability to detect mathematical relationships and develop flexibility in solving non-routine problems that require productive rather than reproductive thinking, for example, being able to

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identify several approaches to solve non-routine problems. An indication of growing strategic competence is the ability to replace cumbersome procedures with more efficient concise procedures and to differentiate and select the most appropriate procedure for any given problem.

A teacher who teaches proficiently will demonstrate strategic competence through his/her ability to analyse instructional problems and to address them in a reasonable and intelligent manner (Kilpatrick, et al., 2001, p. 383).

2.6.1.4 Adaptive reasoning

According to Kilpatrick, et al. (2001, p. 129), adaptive reasoning is “the capacity to think logically about relationships among concepts and situations”. It entails the use of deductive reasoning to consider the many facets of a situation, consider all the alternatives and arrive at a justifiable solution or conclusion. Thus, the essential element of adaptive reasoning is ensuring that one’s reasoning is indeed valid. Adaptive reasoning should also encompass “intuitive and inductive reasoning based on pattern, analogy, and metaphor” (Kilpatrick, et al., 2001, p. 129). According to Kilpatrick, et al. (2001), when determining the authenticity and appropriateness of their proposed strategies, students will need to make use of adaptive reasoning. Developing adaptive reasoning proficiency requires that a teacher engage in reflective practices. This reflexivity would need to focus on teaching strategies and how they relate to the mathematical concepts being taught and the difficulties that learners may have in the learning process. In addition, it would need to consider learners misconceptions and conceptions regarding Mathematics and what “representations are the most effective in communicating essential ideas” (Kilpatrick, et al., 2001, p. 384).

2.6.1.5 Productive disposition

A characteristic of an effective learner of Mathematics is a productive disposition which is the capacity to appreciate the sense and meaningfulness in Mathematics (Kilpatrick, et al., 2001). These learners have a belief in their own efficacy and are generally confident about their ability to do Mathematics, recognising that with effort

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and experience they are able to learn. In order to develop productive disposition learners need to be exposed to opportunities that will enable them to make sense of Mathematics and appreciate the benefits of persevering in order to experience the resulting rewards. A contributing factor in experiencing mathematical success relates to learners’ disposition and their capacity to view their ability as expandable and themselves as capable of understanding Mathematics (Kilpatrick, et al., 2001).

The role of the teacher is critical in providing a mathematics environment that develops supports and maintains in the students a positive disposition towards Mathematics. “Effectiveness depends on enactment, on the mutual and interdependent interaction of the three elements – mathematical content, teacher, students – as instruction unfolds” (Kilpatrick, et al., 2001, p. 9).

Teachers who exhibit productive disposition proficiency know that “Mathematics, their understanding of children’s thinking, and their teaching practices” are interconnected in such a manner as to be meaningful and make sense (Kilpatrick, et al., 2001, p. 384). Furthermore, they realise that by analysing their own teaching and classroom practice they will learn more about “Mathematics, student mathematical thinking, and their own practice” (Kilpatrick, et al., 2001, p. 384) and in this way have control of not only their continued development but their teaching practice.

In this study the role of the lecturer is thus significant with regard, firstly, to facilitating a mathematics environment that creates an awareness of identity and develops the student teachers’ positive disposition towards Mathematics and the teaching thereof. Secondly the lecturer will take on the role of the observer, as described by Maturana (1987, p. 65) in order to notice demonstrations of “adequate conduct”.

Using these five strands as a lens I analysed the outcome of underpinning the mathematics module with the themes of enactivism and the perceived increase and enrichment in teaching for mathematical proficiency of pre-service teachers.

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