Academic journal article Perspectives in Education

Opportunities to Develop Mathematical Proficiency in Grade 6 Mathematics Classrooms in KwaZulu-Natal

Academic journal article Perspectives in Education

Opportunities to Develop Mathematical Proficiency in Grade 6 Mathematics Classrooms in KwaZulu-Natal

Article excerpt

Introduction: Mathematical proficiency in learning and teaching

Kilpatrick, Swafford & Findell (2001: 116) contend that the term mathematical proficiency (MP) was chosen to "capture what we believe is necessary for anyone to learn mathematics successfully". Proficiency in school mathematics was characterised in terms of five strands:

* Conceptual understanding

* Procedural fluency

* Adaptive reasoning

* Productive disposition

* Strategic competency

The expectation is that a successful mathematics learner is proficient in mathematics if s/he 'possesses' the five component strands in such a way that they can be brought to bear on different situations. These strands are interwoven and interdependent. While many teacher educators may regard conceptual understanding as superior and in contrast to procedural fluency (Bossé & Bahr, 2008), we stress Kilpatrick et al.'s point that the strands support rather than limit each other (2001).

This particular explication of MP was a result of the National Research Council of the USA convening a group of experts to review research on effective mathematics learning, in 1999. Since then, questions have been asked about how to assess for MP, and the strands of MP have been used as a framework for assessing the impact of teaching approaches (Langa & Setati, 2007; Moodley, 2008; Pillay, 2006; Samuelsson, 2010; Suh, 2007; Takahashi, Watanabe & Yoshida, 2006), or have informed classroom practice (Suh, 2007). In South Africa, the MP notion has been used to analyse the curriculum (Sanni, 2009) and inform initiatives for teachers (government gazette of 14th March 2008).

While "designing classroom environments and teaching pedagogies that effectively promote this vision, has proven more elusive" (Pape, Bell & Yetkin, 2003: 180), the five strands of proficiency are not useful descriptors of teaching or instructional situations. If the aforementioned attributes of proficient learners are to be developed over time, classroom teaching needs to be constructed so that each strand is promoted. This is in line with the notion of mathematical knowledge for teaching (MKT) as defined by Ball, Hill and Bass (2005), who regard MKT to be about disciplinary knowledge with a view to assisting learners and students in their development of MP.

When Kilpatrick and his team wrote about developing proficiency in teaching mathematics (Kilpatrick et al., 2001: Chapter 10), they did not engage indicators of teaching for each strand of MP. Instead, they discussed the knowledge base for teaching mathematics including three types, namely mathematical knowledge, knowledge of students, and knowledge of instructional practice. (This is in line with the typologies for mathematics teacher knowledge proposed by others. See, for instance, Adler & Patahuddin, 2012; Ball, Thames & Phelps, 2008; Christiansen & Bertram, 2012; Krauss & Blum, 2012; Shulman, 1986). Kilpatrick et al. (2001: 380) then went on to list five components of proficient teaching of mathematics, which are, however, not linked directly to the strands of MP. Thus, they do not serve as indicators of the extent to which teaching promotes MP.

Some years later, Schoenfeld and Kilpatrick (2008: 322) produced a different list of requirements for proficiency in teaching mathematics, but also not directly linked to the strands of MP. It included general competencies such as crafting and managing learning environments, developing classroom norms, and supporting classroom discourse as part of "teaching for understanding", and so on.

They also mentioned that proficient mathematics teachers have both deep and broad knowledge of school mathematics, including representations (cf. Suh, 2007) and conceptual connections (cf. Hattie, 2003). A recent case study in South Africa took up the latter, finding that high school teachers displayed faulty or superficial mathematical connections (Mhlolo, Venkat & Schäfer, 2012). …

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