The role of teachers’ knowledge in the use of learning opportunities triggered by mathematical connections
Cenaro de Gamboa, Edelmira Badillo, Miguel Ribeiro, Miguel Montes
The making of connections is a linchpin of mathematics education, as it is related to the development of a broader and deeper knowledge of mathematics (Skemp, 1971; Triantafillou & Potari, 2010). Mathematics, and in particular mathematics problem-solving, is characterized by the interconnectivity between different content areas (e.g. algebra and geometry), between different representations or different procedures and between mathematics and outside-mathematics situations. Therefore, connections play a major role at all educational levels, especially in primary school, where traditionally formal mathematical education begins.
Following a research trend in recent years, several perspectives on connections have been developed, both regarding the ways connections occur in mathematics (e.g. Zazkis & Mamolo, 2011) and the way connections are established in the classroom (e.g. De Gamboa & Figueiras, 2014; Montes, Ribeiro, Carrillo & Kilpatrick, 2016). We focus on the latter perspective and on the categorizations that emerge which describe connections as a complex system of relationships in which outside-mathematics situations, systems of representation and/or heuristics are linked.
Making connections in the classroom can help students to identify new applications of mathematics to real-world problems and to give meaning to such problems in school contexts. It may also make it easier to use different representations when solving problems, as well as to reinterpret and rebuild connections between concepts, properties and/or procedures. Consequently, promoting the emergence of connections in the classroom has the potential to trigger a wide array of learning opportunities. The usefulness of these opportunities depends on teachers’ knowledge and on their ability to identify, interpret and promote learning opportunities stemming from connections, and to make decisions during the classroom activity that help students to build up a broader and deeper mathematical knowledge.
The presence of connections when conceptualizing teachers’ knowledge - assuming a practice-based perspective - reveals a relationship between teachers’ knowledge and the way connections are established and used in the classroom. For instance, Rowland, Turner, Thwaites and Huckstep (2009) consider connections as a domain of teachers’ knowledge that refers to teachers’ ability to anticipate complexity, make decisions about sequencing, make connections between procedures and make connections between concepts. As regards Ball, Thames and Phelps (2008), connections are related to teachers’ awareness of how mathematical concepts are connected throughout school years.
In managing the development of students’ understanding, teachers need to mobilize their knowledge (both mathematical knowledge and pedagogical content knowledge) in a very specialized way. In order to capture the nuances of that knowledge and to characterize the specialized features of mathematics teachers’ knowledge in terms of what is mobilized when promoting and exploring mathematics learning opportunities arising from mathematical connections, we consider such specialization in the sense of the framework of the Mathematics Teachers’ Specialized Knowledge (MTSK; Carrillo et al., 2018).
With a view to deepening our understanding of connections and their role in practice (potentialities and constraints), and to conceptualize ways of improving the effectiveness and utility of such connections in terms of teacher knowledge, the concretization of the aforementioned categorizations of connections need to be studied and expanded in relation to several mathematical topics. Amongst the diversity of topics in school mathematics, measurement is perceived as a crucial element in pupils’ development of mathematical understanding and knowledge (e.g. Sarama, Clements, Barret, Van Dine & McDonel, 2011). It is also a rich environment in relation to the emergence of mathematical connections, as it represents a natural linkage between numbers and operations, geometry and real-world problems. In particular, the introduction of the measurement of length - its different dimensions (Clements & Sarama, 2007) and properties - can enact several aspects of mathematical connections related to natural and rational numbers, different representations of numbers, the procedures related to the measurement of length and different units of measurement (e.g. Szilagyi, Clements & Sarama, 2013).
With the aim of gaining a better understanding of what features of teachers’ knowledge can help teachers to foster the making of connections in the classroom and to make the most of these connections in terms of exploiting the learning opportunities stemming from them, in this chapter we present the case of Carla, a prospective teacher developing an introduction to standard length units in the second grade of primary school. We start by characterizing mathematical connections that emerge during Carla’s lesson, and we identify and discuss the learning opportunities stemming from those connections. Then, we analyse what features of her knowledge are related to the use of connections, and what other features of teacher knowledge would have helped her to effectively use connections to build a deeper and broader knowledge of length measurement.