2 October 2024: Yesterday, AERO released their ‘explainer’ on Developing Maths Proficiency, prompting me to reflect on how a balanced approach to the strands of mathematical proficiency is essential for helping students succeed. I’ll come back to AERO at the end.
A few days ago, I asked a group of mathematics educators to define ‘mathematical problem solving’. I’ve run this exercise with hundreds of participants, and the responses are consistently diverse. Here are a few examples:
- Applying diverse ways of thinking.
- Using maths to address real-world issues.
- Being creative in finding the answer to a question or a problem.
- Applying a variety of effective and efficient strategies to solve a problem in multiple ways.
- Relying on knowledge learnt in the past to solve problems.
- Using prior or newly learnt skill set to break down a problem.
- Solving problems by using a range of mathematical skills, methods, and concepts.
- Transferring mathematical knowledge to find a solution.
- Tackling new or unknown problems that students can’t solve instantly.
- Using strategies like trial-and-error or working backwards to solve complex problems.
Together, these responses offer a well-rounded perspective on problem solving, aligned with the AC:M definition. According to this (lightly edited) definition, problem solving is a process for tackling both mathematical and real-world problems. These problems may be either routine (familiar) or non-routine (unfamiliar)1. Importantly, students are presented with problems for which they do not immediately know the answer. They make mathematical decisions, work through a process of planning, applying strategies and heuristics, and draw on previously learned concepts, skills, and procedures, to solve, review, and analyse problems, and evaluate, interpret, and communicate their solutions. In short, the AC:M emphasises teaching for, about, and through problem solving.
However, the diversity of the individual responses reveals a potential issue: as educators, we may unconsciously prioritise certain aspects of the problem-solving proficiency. This can result in a focus on specific elements while inadvertently overlooking others. For instance, some responses, like 5, 7, and 8, highlight the reliance on prior knowledge. Responses 4 and 6 emphasise efficiency and strategy, while others, like 1 and 3, focus on creativity and exploration. Responses 9 and 10 underscore the need to tackle unfamiliar and complex problems, while 2 frames problem solving specifically as a tool for addressing real-world issues.
Each of us views mathematics education through our own lens, shaped by beliefs, values, and experiences. This personal perspective may lead to emphasising aspects of problem solving we value most—whether that’s creativity, strategy, efficiency, or knowledge application. For example, I believe problem solving equips students with a set of mathematical skills, strategies, and attitudes needed to face any type of problem, especially unfamiliar and challenging ones. I want students to experience the intellectual satisfaction of figuring things out for themselves. This aligns with my learner-centred orientation2 towards teaching and learning so, if I need to choose, I’ll prioritise this. However, I also recognise that developing fluency through routine problem solving, grounded in a strong knowledge base, is essential preparation for students to confidently tackle more complex, non-routine problems.
As educators, we have a responsibility to provide students with well-rounded problem-solving experiences. Choosing only the elements of problem solving that align with our beliefs limits students’ learning. Over-emphasising creativity, for example, may lead students to struggle with fluency in solving routine problems. Conversely, focusing solely on applying prior knowledge without encouraging exploration may leave students ill-prepared to tackle unfamiliar, complex challenges. Our role is to ensure students build a comprehensive set of problem-solving skills that enable them to approach both familiar and novel problems with confidence and flexibility.
The need for balance applies within all the proficiency strands — understanding, fluency, problem solving, and reasoning — which were carefully articulated by Kilpatrick et al. (2001) and later adapted by ACARA to describe the many facets of what it means to be mathematically proficient. This might be the perfect time to revisit those strands with fresh eyes. Are there aspects you’re unintentionally prioritising while allowing others to fall into the background?
The interwoven nature of proficiency strands
In presenting the proficiency strands, Kilpatrick et al. (2001, p. 116) emphasised that “the most important observation we make … is that the five strands are interwoven and interdependent in the development of proficiency in mathematics”, meaning that they develop simultaneously and reinforce each other. These strands work together to build mathematical proficiency. As noted in the NSW Mathematics K-10 Syllabus Overview, “Students learn to work mathematically by using these processes in an interconnected way. The coordinated development of these processes results in students becoming mathematically proficient.” Effective teaching should incorporate all strands in flexible combinations, depending on the goals of the lesson and the needs of the students.
Research supports the idea that students benefit from varied pathways to proficiency. For instance, Rittle-Johnson et al. (2015), in studying the bidirectional relationship between conceptual and procedural knowledge, notes that “there is not an optimal ordering of instruction, but rather multiple routes to mathematical competence,” a concept that extends to the other proficiencies as well. The understanding → fluency → problem solving sequence is fairly common, so let’s explore a few different examples of how the proficiency strands can work together. (The omission of the fourth proficiency in each example is simply for brevity.)
Fluency → Reasoning → Understanding
A well-designed sequence of problems allows students to practice procedures while simultaneously reasoning about patterns and relationships. Consider a problem string3 like 26 – 10, 26 – 9, 55 – 20, 55 – 19, and 42 – 17. Students could begin by practicing the subtraction algorithm, reinforcing their procedural fluency. Through strategic questioning, the teacher can guide students to notice the small variations between problems, prompting them to reason that 26 – 9 can be approached as 26 – 10 + 1, leading to an understanding of the compensation strategy. This progression moves students from practicing a procedure to understanding the underlying concept through reasoning about patterns they observe.
Problem solving → Reasoning → Understanding
Another approach starts with a problem-solving challenge, which can naturally lead to reasoning and deeper understanding. Consider this scenario: “A farmer wants to build a rectangular garden with 24 metres of fencing. What dimensions would give the largest possible area?” Students experiment with different combinations of length and width, calculating areas for shapes like 6m by 6m, 8m by 4m, and 10m by 2m. Through trial and error, they explore multiple possibilities, though they may not yet understand the optimal relationship between perimeter and area.
Guided by the teacher, students begin to reason about the results. They notice that the square (6m by 6m) provides the largest area for a given perimeter. Finally, they reach an understanding when they learn that a square maximises area for a fixed perimeter because it distributes the dimensions evenly. This progression from problem solving to reasoning, and ultimately to understanding, helps students grasp how perimeter and area are connected.
Reasoning → Understanding → Fluency
Instead of directly telling students a fact, we can structure learning so they investigate for themselves. For example, rather than telling them that the angles in a triangle add up to 180 degrees, ask students: “If you measure the three angles in various triangles, what patterns do you notice?”
Students might use tools like protractors or interactive geometry software to measure angles in various triangles. Through effective teacher questioning that promotes reasoning, students will notice that the sum is consistently 180 degrees. This reasoning fosters a deeper conceptual understanding of the general rule. Once students develop this understanding, they can practice with problems involving angle calculations, reinforcing fluency and application.
Of course, other sequences are possible as well. No single sequence works best in all situations, and an overemphasis on one at the expense of others limits the breadth and depth of students’ learning. The key is balance and flexibility, all carefully structured by the teacher.
So what?
As educators, we know that no two students learn in exactly the same way, and that different concepts often require different teaching approaches. In short, we need to be pedagogical pragmatists—drawing from a diverse body of educational research and strategies to tailor our teaching practices to meet the unique learning needs of our students.
This is why AERO, as an influential voice in education, must be held to a higher standard. Just as teachers are expected to adapt their methods to diverse learners, AERO, by positioning itself as an ‘Australian Educational Research Organisation’, is obligated to acknowledge a broader spectrum of educational philosophies and bodies of research. While their approach aligns with a social efficiency perspective—prioritising measurable outcomes and workforce readiness—the complexity of the educational system requires recognising the diverse perspectives and needs of all students. Though I have several criticisms of AERO’s document, I will focus on two key issues.
First, although AERO acknowledges that mathematical proficiency comprises “multiple strands, which are interrelated, develop simultaneously, and are mutually reinforcing,” their document contradicts this vision by presenting a single, hierarchical progression for the development of proficiency that aligns with the model of teaching and learning they advocate, that is, understanding → fluency → problem solving → reasoning.
However, this creates a rigid framework that neglects the flexibility needed to foster genuine mathematical proficiency. For instance, some students may benefit from engaging with reasoning before fluency or from tackling problem solving earlier in their learning process. Different concepts also naturally lend themselves to varied instructional sequences.
Second, AERO appears to cherry-pick from the proficiency strands to support its model of teaching and learning. Instead of embracing the full definitions and rich interconnections among the strands, AERO’s selective focus leads to a narrow interpretation of what it means to be mathematically proficient. For example, problem solving is far more than applying known strategies to solve a problem; as noted earlier, it can initiate the entire learning sequence.
Starting with a fixed view of teaching and learning and then trying to fit the mathematical proficiencies into it is backwards. Instead, the proficiencies should guide the teaching approach. If a model can’t accommodate the integrated nature of the proficiencies, it’s a sign that the model needs to evolve or that multiple models may be needed.
As educators, we must ensure that students develop mathematics proficiency by engaging with all the strands—understanding, fluency, problem solving, and reasoning—without confining ourselves to a single, rigid sequence. True mathematical proficiency emerges from the coordinated development of these strands, working together. Cherry-picking elements of proficiency, whether as educators or institutions, undermines this goal and does a disservice to our students. Our challenge is to embrace the full complexity of mathematical learning and respond accordingly. Our students deserve nothing less.
Selected further reading
Brodie, K. (Ed.). (2010). Teaching Mathematical Reasoning in Secondary School Classrooms. Springer US.
Kilpatrick, J., Swafford, J., Findell, B., & National Research Council (U.S.) (Eds.). (2001). Adding it up: Helping children learn mathematics. National Academy Press.
Lester, F. K., & Cai, J. (2016). Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and Solving Mathematical Problems: Advances and New Perspectives (pp. 117–135). Springer International Publishing.
Rittle-Johnson, B., Schneider, M., & Star, J. R. (2015). Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics. Educational Psychology Review, 27(4), 587–597.
Photo by Kevin Grieve on Unsplash
- I have clarified that routine and non-routine problems differ in familiarity, in line with Kilpatrick et al. (2001). The AC:M inexplicably distinguishes between them based on the number of valid solutions. ↩︎
- Learner Centred is one of four curriculum ideologies, identified by James Schiro (2013). Each ideology serves as an archetype, representing a distinct vision of education. The Scholar Academic ideology focuses on the transmission of established knowledge from academic disciplines, emphasising intellectual development. The Social Efficiency ideology aims to prepare students for specific societal roles, valuing efficiency and productivity in education. The Learner Centred ideology prioritises individual student growth, tailoring the curriculum to meet learners’ needs and interests. Finally, the Social Reconstruction Ideology views education as a tool for social change, encouraging students to address societal injustices and work toward reform. Educators, and those with a stake in education more broadly, tend to cluster around the ideologies that resonate with their own values and beliefs. ↩︎
- https://www.mathisfigureoutable.com/blog/aproblemstringisnot ↩︎
Wonder in Mathematics 