What is Effective Teaching in Mathematics?

KEY FOCUS: Questioning to develop higher-order thinking and metacognition

Introduction (221/250 Words)

Effective teaching in mathematics is more than just helping students memorise content, it’s about developing the students’ ability to self-learn and adapt existing knowledge to unfamiliar situations. Husbands and Pearce’s report combines a breadth of educational research into nine evidence-based claims to describe great pedagogy [1]. Their seventh claim - that effective pedagogies “focus on developing higher order thinking and metacognition, and make good use of dialogue and questioning to do so” [1] - is especially relevant for mathematics education. Within Bloom’s revised taxonomy, another piece of educational research, higher-order thinking is comprised of not just remembering, but understanding, applying, analysing, evaluating, and creating [2]. Metacognition, as described by Flavell, a student analysing their own thinking and mental processes [3].

This essay argues that a combination of these ideas - higher-order thinking and metacognition - is the most important element of teaching, encouraged through dialogue and questioning, such that it then enables the student to better use other pedagogies. This claim is backed by constructivist learning theory, which states that students must actively try to find meaning, rather than just passively taking it in [10]: questioning creates the cognitive conflict which instigates longer-term learning. Pedagogical content knowledge (PCKs), mathematical misconceptions, and assessment for learning (AfL), all reinforce this claim as later discussed, albeit with some mentioned challenges implementing these techniques.

Discussion and Analysis (519/500 Words)

Among the nine claims posed by Husbands and Pearce, developing higher-order thinking through dialogue and questioning stands out by addressing the most fundamental aspect of teaching: improving a student’s cognitive abilities. Whilst the other claims focus on the conditions required to learn, this claim focuses on what learning should actually achieve. Schoenfeld’s research explains that metacognition is what separates expert mathematical problem-solvers from novices [4] - experts use a continuous strategy of self-questioning when solving problems, monitoring their own progress and changing techniques throughout depending on what they find. This aligns with the constructivist learning theory mentioned earlier: Kilpatrick et al. argue a similar need for “adaptive reasoning” and “productive disposition”, alongside the usual ability to methodically complete steps [10]. Teachers that develop their students’ abilities to analyse their own cognition not only equip them with content knowledge, but the methods to self-learn.

However, effective questioning relies heavily on pedagogical content knowledge (PCK) - the knowledge of a subject alongside the understanding of how students learn. Teachers must be able to anticipate how a student thinks in order to ask productive questions that would improve their understanding. Ball, Thames, and Phelps separate this “mathematical knowledge for teaching” from content knowledge alone [8]; including the ability to recognise student misconceptions and knowing how to represent concepts in a way that provides the most impact. Swan’s research demonstrates that purposefully testing misconceptions through questioning - for example, testing the students’ ability to find errors in solutions or compare conflicting methods - produces a deeper understanding than just going through the correct steps to a method [11]. PCK enables teachers to craft questions that bring up and address these misconceptions in a constructive way.

The seventh claim posed by Husbands and Pearce also naturally enables assessment for learning (AfL). Without metacognitive awareness, formative assessment just becomes another type of examination rather than a process of learning. The eighth claim, which discusses assessment for learning, emphasises the need to embed AfL within learning alongside good-quality feedback [1]; yet students must be able to reflect on this feedback for their understanding to grow. Mevarech and Kramarski’s IMPROVE method outlines that this metacognitive questioning - “What is the problem asking?” and “Does my solution make sense?” - improves both achievement and understanding [7]. Research also confirms that giving the student a longer time to answer particularly benefits lower-scoring students [5], suggesting that giving students time to process cognitive challenges - detached from their final answer - has a strong impact on their understanding. This is further confirmed by Hattie’s meta-analyses that show questioning yielding substantial effect sizes on achievement [6].

Critics might propose that subject knowledge or prior learning claims are more fundamental - without knowledge, how can thinking develop? This represents a false dichotomy: questioning simultaneously builds knowledge and develops thinking, rather than improving them discretely. However, effective questioning requires a substantial amount of teacher expertise, which often remains a challenge to build, yet does not diminish the claim’s importance at all. Instead, it emphasises that developing these questioning skills should be further prioritised in teacher education and professional development in the first place [8][9].

Conclusion (187/250 Words)

In summary, focusing on higher-order thinking and metacognition through questioning represents the most significant factor for effective teaching within mathematics. Constructivist learning theory describes the requirement for active engagement over passive reception for cognitive development [10], and questioning remains the single tool to instigate this. This claim is unique among the nine collated by Husbands and Pearce, enabling a variety of pedagogical principles. For example, scaffolding becomes meaningful when developing a students’ capability thinking, assessment for learning requires metacognitive awareness, and addressing misconceptions remains difficult without strategic questioning informed by PCK [1][8][11]. Teachers must also intentionally progress through Bloom’s cognitive levels [2], planning questions to step through the levels with enough wait time afterwards for genuine thinking [5], and design metacognitive questioning that students can internalise [7]. Assessment practices should place higher value on reasoning alongside correct answers [9]; effective mathematics teaching develops mathematical thinkers rather than mere mathematical doers. By making questioning for higher-order thinking a central practice, grounding in strong PCK and informed by learning theories, teachers can cultivate student ability beyond mathematics and into all areas of learning [1][10].