Blog post

# Making connections when teaching mathematics: The role of teacher knowledge and beliefs

Mathematical ideas, concepts or procedures are interconnected; they therefore cannot be understood well in isolation. Mathematical connections made in the classroom give students opportunities to relate mathematical ideas, concepts, meanings and procedures to each other, and to see mathematics as an integrated whole (see Coxford, 1995). Not all mathematics classroom practices, however, satisfy this desire.

My recent research reveals that teachers who have a deeper understanding of mathematics make more connections in their classroom instruction (Hatisaru, 2022). A mathematical connection can be considered as a relationship between two or more mathematical ideas (Businskas, 2008). For example, a function is a specific type of relation, and all functions are relations. Mathematics teachers make these connections by relating mathematical and non-mathematical contexts (such as connecting musical rhythms with mathematical patterns) and/or relating ideas, concepts, procedures or representations within mathematics (for instance connecting fractions with decimals).

**Teacher knowledge and making connections**

It is widely acknowledged that a teacher’s ability to recognise and make connections, and/or their beliefs about how mathematics is taught and learned, are linked to their knowledge of mathematics (see for example De Gamboa et al., 2020). There is, however, limited empirical evidence showing these associations.

In a larger qualitative exploratory study, I investigated how teacher mathematical knowledge impacts student learning by observing the instruction of two secondary mathematics teachers teaching the concept of function – a sub-domain of algebra – to year nine students (aged 15–16) (Hatisaru & Erbas, 2017). I selected two teachers (Fatma and Ali, both names are altered) – the former with stronger and the latter with weaker mathematical knowledge – to observe the differences between student learning in each teacher’s class. In the current research, I aimed to understand the associations between teacher mathematical knowledge and ability to make connections. Types of mathematical connections include part-whole connections (PWC); feature/property (F/P); analogical connections (AC), different representations (DR); if-then connections (I-T); instruction-oriented connections (IOC); and procedural connections (PC) (see Hatisaru, 2022).

The graph in figure 1 captures the types and frequency of mathematical connections made in each of the two teachers’ instruction, while the graph in figure 2 makes a comparison between the two teachers.

*Figure 1:** Frequency (f) of mathematical connections identified in the 24 audiotaped* *lessons*

A total of 485 connections were identified in a sample of 24 lessons observed (12 lessons per teacher). Some of the connections made are very interesting as they connect an abstract mathematical concept (function) with a concrete, real-life situation. That is, a function is a relationship between two variables, and often is described as an input–output relationship. For every input there is *only* one output, and that means for an input there cannot be two different outputs. Fatma established a connection using an analogy of cinema tickets, and this characteristic of functions, and made the following connection:

*‘Normally, a cinema ticket is for one seat; for instance, Person A has Row K-3; Person B has Row L-11. One ticket is not sold to more than one person. That is, each element in *

The teacher with stronger knowledge (Fatma) produced far more connections (*f *= 317) than the teacher with weaker knowledge (Ali) (*f *= 168) (see figure 2). Moreover, the teacher with stronger mathematical knowledge made relatively more sophisticated types of connections (for example PWC, F/P, AC, DR and I-T) more often than the teacher with weaker knowledge.

*Figure 2:** Frequency (f) of mathematical connections identified in the audiotaped lessons of two teachers*

‘The teacher with stronger mathematical knowledge made relatively more sophisticated types of connections more often than the teacher with weaker knowledge.’

**What do findings tell us?**

I acknowledge that the study findings are limited to classroom observations of two teachers teaching the concept of function. The findings thus may not apply to mathematics teachers in different contexts and/or teaching other mathematical concepts. The deeper insights gained into the mathematics teachers’ capability to make connections, nevertheless, may inform teacher education programmes and teacher development initiatives to better support teacher learning and encourage reflection among teachers of mathematics on their own practices. As this capability appears to be mediated by teacher mathematical knowledge, future programmes could focus more on the enhancement of mathematical knowledge needed for teaching in teachers.

The observed ‘connections gap’ might reflect differences in two teachers’ knowledge and/or in their beliefs about how mathematics is taught and learned. In turn, it implies that the students in each of their classes had different opportunities to learn mathematics. We need to investigate how student learning is influenced by the connections experienced in the classroom.

**References**

Businskas, A. M. (2008). *Conversations about connections: How secondary mathematics teachers conceptualize and contend with mathematical connections* [Unpublished doctoral dissertation, Simon Fraser University, Canada].

Coxford A. F. (1995). The case for connections. In P. House & A. F. Coxford (Eds.), *Connecting mathematics across the curriculum* (pp. 3–12). National Council of Teachers of Mathematics [NCTM].

De Gamboa, G., Badillo, E., Ribeiro, M., Montes, M., & Sánchez-Matamoros, G. (2020). The role of teachers’ knowledge in the use of learning opportunities triggered by mathematical connections. In S. Zehetmaier, D. Potari, & M. Ribeiro (Eds.), *Professional development and knowledge of mathematics teachers* (pp. 24–43). Routledge.

Hatisaru, V. (2022). Mathematical connections established in the teaching of functions. *Teaching Mathematics and its Applications: An International Journal of the IMA*. Advance online publication. https://doi.org/10.1093/teamat/hrac013

Hatisaru, V., & Erbaş, A. K. (2017). Mathematical knowledge for teaching and student learning outcomes. *International Journal of Science and Mathematics Education*, *15*(4), 703–722. https://doi.org/10.1007/s10763-015-9707-5