In my position as a curriculum leader and a researcher, I have heard some teachers say, ‘I have tried everything, but my students do not have the mathematics brain’. Contrastingly, cognitive scientists emphasise that, just like in any other subject, some students may be better than others, but every student is capable of learning K-12 mathematics (Willingham, 2010).

So, how can we support the teaching and learning of mathematics in ways that can enhance student engagement, participation and thus learning? Over the past 3 years I have led a study through James Cook University that focused on supporting the teaching and learning of maths in Queensland secondary schools. The research involved 16 senior secondary teachers across 12 schools; student artefacts (including concept maps and procedural flowcharts) were also collected from the participating schools.

The Queensland Curriculum and Assessment Authority has identified building new knowledge from prior knowledge, and the ability to represent mathematical knowledge from one form to another, as vital for effective mathematical teaching and learning. Importantly building from what students already know is key in enhancing participation and engagement. Furthermore, research has identified that teaching and learning that involves the use of visual representation can enhance student engagement, knowledge retention, and development of deeper understanding.

### Research findings

The study found mathematics teachers need to engage the following to support students’ participation, engagement and learning.

**Linking junior level concepts to new concepts in senior secondary**

Students do not come into the learning space blank; they bring valued experience and prior knowledge from previous levels. The study found the key, and mostly undervalued, stage of mathematics teachers’ planning is content sequencing (Chinofunga et al., 2022a; 2023a). Content sequencing provides the opportunity for teachers to hypothesise how students will develop their mathematical understanding, building from what they already know. Moreover, it provides a pathway that can be activated once the teacher has identified students with limited prior knowledge, as it supports backward mapping to foundational concepts that are key to understanding new concepts. When teaching and learning starts from what students regard as familiar it increases the chances of engagement and conceptual development of mathematical knowledge.

**Include visual representation in developing mathematical knowledge**

Mathematical knowledge is generally classified as conceptual and procedural. The 2 are complementary and neither should be developed at the expense of the other for effective learning. This study found that mathematics teachers should include concept maps and procedural flowcharts to support the development of conceptual and procedural knowledge. Representing mathematical knowledge in non-linguistic visual representation can help some learners in deepening their mathematical understanding (Chinofunga, 2022).

Concept maps are key in linking concepts and identifying essential concepts that are foundational to the development of new knowledge (Chinofunga, 2022; Chinofunga et al, 2023b; 2022c). The importance of promoting and demonstrating the interconnection of mathematics concepts to students is that it helps them to understand the significance of retention as mathematics cannot be effectively understood in chunks but as a network of concepts. Likewise, procedural flowcharts as a visual representation of steps that need to be followed to achieve a specific outcome develop students’ procedural fluency.

Instead of just routine practice through solving questions, which is common in mathematics classes across Australia, teachers can ask students to develop procedural flowcharts. This can help them to summarise and generalise how they have been solving the questions. Developing procedural flowcharts will deepen students’ understanding as the process promotes exploration, synthesis and evaluation of their understanding (Chinofunga, 2022; Chinofunga et al., 2022b).

**Include visual representation during problem-solving**

The new Australian Curriculum Version 9.0: Mathematics places problem-solving at the centre of mathematics knowledge development. The study noted that when students are faced with an open-ended problem-solving task which requires them to use multiple procedures, use of procedural flowcharts can support their problem-solving skills (Chinofunga, 2022). In this case, the procedural flowchart can be a visual representation of how the student plans to solve the problem. It can also be a visual presentation of the extent of how different procedures can address the problem. Both teachers and students may benefit from the use of procedural flowcharts because feedback can be provided quickly as visuals are easy to process.

### A way forward

Firstly, this study found that content sequencing from junior- to senior-level concepts is key to mathematics planning that supports student engagement. The use of a research-informed framework that backward maps concepts will help teachers and students to gradually develop concepts from familiar to unfamiliar in a way that promotes greater chances of transfer.

Secondly, the use of concept maps and procedural flowcharts should be encouraged in mathematics classes as they provide an alternative way of representing students’ mathematics knowledge. Developing such resources during learning will deepen students’ understanding, promoting student engagement and participation during teaching and learning of mathematics.

**References and related reading**

Chinofunga, M. D. (2022). An investigation into supporting the teaching of calculus-based Mathematics in Queensland. PhD thesis. https://researchonline.jcu.edu.au/80691/

Chinofunga, Musarurwa; Chigeza, Philemon; Taylor, Subhashni. (2022a). A framework for content sequencing from junior to senior mathematics curriculum. *Eurasia Journal of Mathematics, Science and Technology Education, 18* (4). em2100. https://doi.org/10.29333/ejmste/11930

Chinofunga, M. David., Chigeza, P., Taylor, S. (2022b). Procedural Flowcharts can Enhance Senior Secondary Mathematics. In: *Mathematical Confluences and Journeys: proceedings of the 44th Annual Conference of the Mathematics Education Research Group of Australasia, July 3-7).* pp. 130-137. Launceston, TAS, Australia. https://merga.net.au/common/Uploaded%20files/Annual%20Conference%20Proceedings/2022%20Annual%20Conference%20Proceedings/Research%20Papers/Chinofunga%20RP%20MERGA44%202022.pdf

Chinofunga, M.D., Chigeza, P., Taylor, S. (2022c). Concept Maps as a Resource for Teaching and Learning of Mathematics. *In Mathematical Confluences and Journeys: Proceedings of the 44th Annual Conference of the Mathematics Education Research Group of Australasia, July 3-7)*, p 584. Launceston, TAS, Australia. https://merga.net.au/common/Uploaded%20files/Annual%20Conference%20Proceedings/2022%20Annual%20Conference%20Proceedings/Short%20Comms/Chinofunga%20SC%20MERGA44%202022.pdf

Chinofunga, M. D., Chigeza, P., & Taylor, S. (2023a). Teachers’ perceptions on how content sequencing enhance planning, teaching and learning of senior secondary mathematics. *Eurasia Journal of Mathematics, Science and Technology Education, 19(4).* https://doi.org/10.29333/ejmste/13108

Chinofunga, M. D., Chigeza, P., & Taylor, S. (2023b). Teachers’ perceptions on how concept maps can enhance teaching and learning of mathematics at senior secondary level. *International journal of innovation in science and mathematics education (IJISME).* https://openjournals.library.sydney.edu.au/CAL/article/view/16431

Willingham, D. T. (2010). Ask the cognitive scientist: Is it true that some people just can’t do math? *American Educator*, *33*(4), 14-19. https://www.aft.org/ae/winter2009-2010/willingham

On your own, or with a group of colleagues, look at the 3 areas David Chinofunga shares in this article as being important for mathematics teachers to engage to support students. Which are areas of strength in your own practice, and which need more development? Which one are you currently not doing that you could start to include in your own classroom practice?