Engaging with the research presented here will deepen your understanding of principles and models for teaching mathematics, and effective mathematics teaching practices.


Teaching mathematics is more than teaching a set of procedures. This research explains that mathematics may be taught instrumentally or conceptually, and that conceptual teaching requires that teachers develop relational understanding themselves.

Richard Skemp’s research investigated the effects of teaching and learning Mathematics ‘Instrumentally’ and ‘relationally’, Instrumental understanding is defined as the possession of a rule, and ability to use it. Relational Understanding is defined as knowing both what to do and why. It could be argued that Instrumental Understanding is not understanding at all.

This report reveals that while a mathematical knowledge is essential to teaching mathematics, many teachers possess a limited knowledge of mathematics, because nowhere in their education have they had opportunities to develop relational understanding of mathematics. It further reveals that many teachers have a misconception that mathematics is a fixed body of facts and procedures, which influences the way they teach mathematics, concluding that the more a teacher understands, the better their teaching will be.

This article discusses the very real negative physical response that some adults and children experience to mathematics, emphasises going back to previous levels of conceptual understanding before moving students on. Its recommendations include teaching the unique language of mathematics, focusing formative assessment with on-the-spot monitoring and feedback, and giving students time to develop understanding.

This article discusses the key missing ingredient in traditional mathematics teaching – conceptual understanding, explaining that teaching for conceptual understanding has the benefits of less information having to be memorised, and increased student capacity to translate their knowledge to new situations. It explores the reasons why traditional teaching is still prevalent, including that teaching conceptually is harder and takes longer because we are teaching children to think mathematically which they are not used to doing, and because our current culture of testing means teachers and children are pressured to get the right answer ‘now’.

This review concludes that the pathway to improvement in the teaching of mathematics and student learning of mathematics is through treating the whole class as a community in which all students participate, with the teacher posing variations in task demand, arguing that the negative effects of achievement grouping can be avoided through the adoption of such approaches.

This review concludes the six principles for teaching mathematics are identifying key ideas, build on what students know, engage students in learning with appropriate challenge, differentiate learning, structure lessons and build fluency on understanding. (Section 5)

Peter Sullivan, and Judith Mousley’s research proposes a model which links the components of quality mathematics lessons including a teacher who is in control of the teaching, who has a clear purpose for lessons, who facilitates learning, who encourages independence, who is interesting, and who provides challenging and mathematically important material, allowing students to think for themselves.

This publication identifies effective mathematics teaching practices including establish mathematics goals to focus learning, implementing tasks that promote reasoning and problem solving, using and connecting mathematical representations, facilitating meaningful mathematical discourse, posing purposeful questions, build procedural fluency from conceptual understanding, supporting productive struggle in learning mathematics, and eliciting and using evidence of student thinking.

This project finds that instead of cursory and repeated treatments of a topic, the curriculum should be focused on important ideas, allowing them to be developed thoroughly and treated in depth. Rote learning of arithmetic procedures no longer has the clear value it once had because of the widespread availability of technological tools for computation, but people are much more exposed to numbers and quantitative ideas and so need to deal with mathematics on a higher level than they did just 20 years ago.

Anthony and Walshaw’s booklet focuses on effective mathematics teaching. Drawing on a wide range of research, it describes the kinds of pedagogical approaches that engage learners and lead to desirable outcomes, including classroom environment, building on student’s thinking, making connections, assessment for learning, mathematical communication, mathematical metalanguage, and teacher knowledge.

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