Promoting Mathematical Thinking Among Secondary Learners

 

1. Introduction: The Nature and Importance of Mathematical Thinking

Mathematical thinking occupies a distinctive place among intellectual activities. As Zeeman (1972) argues, mathematics is unique among the sciences because its subject matter is governed not by empirical observation but by "elegance, intrinsic beauty, profundity, generality, simplicity, depth, subtlety and economy" — criteria that are subjective yet remarkably agreed upon among practitioners. This aesthetic dimension distinguishes mathematics from the natural sciences, where subject matter is determined by the question "What is there?" rather than by taste. The implication for education is significant: promoting mathematical thinking requires more than transmitting procedures; it demands cultivating a way of engaging with abstraction, pattern, and logical structure.

Cartwright (1970) traces the origins of mathematical thought from primitive peoples and young children through to advanced practitioners, arguing that "the power of complete abstraction comes very slowly" and drawing on Piaget's developmental research to show that the movement from concrete perceptions to abstract mathematical concepts is "a long and gradual process." This developmental insight underlines the challenge facing secondary educators: students at this level are often still transitioning from concrete to formal operational thinking, a fact that fundamentally shapes how mathematical reasoning can be promoted.

2. Defining Mathematical Thinking and Reasoning

Devlin (2012) draws a critical distinction between "doing math" — the application of procedures and symbolic manipulations — and "mathematical thinking," which he defines as "a specific way of thinking about things in the world" that includes "logical and analytic thinking as well as quantitative reasoning." He argues that secondary and high school education focuses almost exclusively on procedural mastery ("thinking inside the box"), while the real goal of mathematical education should be learning to "think outside the box" — tackling novel problems for which no standard procedure exists. Devlin traces this shift historically to a nineteenth-century revolution in which "the primary focus was no longer on performing calculations or computing answers, but formulating and understanding abstract concepts and relationships," a shift from doing to understanding.

The qualitative study by Sumiati and Agustini on junior high school students confirms that mathematical reasoning — defined as "a process for obtaining conclusions based on logical mathematical premises" — is inseparable from mathematical understanding: "mathematical material and mathematical reasoning are two things that are inseparable, mathematical material understood through reasoning and reasoning is understood and trained through learning mathematical material." Their case study of two eighth-grade students found stark differences between students with high and low mathematical abilities: high-ability students fulfilled all four indicators of mathematical reasoning (understanding the problem, planning, executing, and re-examining), while low-ability students "cannot fulfill the four indicators of mathematical reasoning" and were only able to partially understand the problem. The researchers attribute this in part to Piaget's developmental stages, noting that students aged 11–15 are in the "formal operational" stage and "can experience a transition step from the usage on a concrete operation into operation."

3. The Role of Heuristics and Problem Solving

The historical and epistemological literature emphasizes that problem solving lies at the heart of mathematical thinking. The entry on heuristics and the history of mathematics in mathematics education traces the foundational influence of George Polya, whose famous book How to Solve It (1945) established heuristics as "the study of means and methods of problem solving" — experience-based techniques such as "Think of a Similar Problem, Draw a Diagram, Working Backward, and Guess and Check" that help students understand, represent, and solve non-routine problems.

However, the same literature sounds a cautionary note: research from Begle (1979) to Schoenfeld (1992) consistently shows that "classroom teaching of problem-solving heuristics does little to improve students' problem-solving abilities." The constraints are threefold. First, heuristics are often taught as discrete strategies rather than as flexible tools, stripping them of their heuristic character. Second, as Schoenfeld (1992) noted, most heuristics "are really just names for large categories of processes rather than being well-defined processes in themselves," making them too general to guide novices effectively. Third, teaching heuristics demands considerable skill from teachers, who must "provide constructive and formative feedback to students' different approaches" and "assist students using the least possible help."

Schoenfeld's (1992) proposed solution — teaching specific rather than general heuristics, teaching metacognitive strategies alongside them, and improving students' beliefs about problem solving — remains a guiding framework for promoting mathematical thinking.

4. The History of Mathematics as a Pedagogical Resource

A rich body of scholarship argues that incorporating the history of mathematics into teaching can promote deeper mathematical thinking. The literature documents how, since the creation of the HPM (History and Pedagogy of Mathematics) Group in 1976, researchers have developed three interrelated contributions: epistemological, cultural, and didactical.

Epistemologically, history demonstrates that mathematical concepts were invented through problem solving and that "the development of proofs is concomitant with the construction of mathematical objects and the construction of mathematical rationality." This counters the common classroom assumption that logical rules are prerequisites for proving, and instead presents mathematics as a living process of questioning and discovery. Culturally, history "provides a different image of mathematics both to teachers and — more importantly — to students, on which their more positive relation with mathematical knowledge can emerge." Reading original historical texts produces what has been called "epistemological astonishment" — a "cultural shock" that questions knowledge and procedures "taken for granted" and invites students and teachers alike to ask why certain ideas were difficult for contemporaries and remain difficult for learners today.

5. Cooperative Learning as a Vehicle for Mathematical Thinking

Johnson and Johnson's comprehensive work on cooperative learning provides a strong theoretical and empirical foundation for structuring classroom environments that promote mathematical thinking. They define cooperative learning as "the instructional use of small groups so that students work together to maximize their own and each other's learning," grounded in Social Interdependence Theory and Structure-Process-Outcome Theory.

The research base is extensive: over 1,200 studies confirm that cooperative learning produces higher achievement and greater productivity than competitive or individualistic approaches, with the superiority increasing "as the task became more conceptual, the more higher-level reasoning and critical thinking was required, the more desired was problem solving, the more creativity was desired." This finding is directly relevant to mathematical thinking, which demands precisely the kinds of higher-order cognitive engagement that cooperative structures promote. For cooperation to be effective, five elements must be present: positive interdependence, individual accountability, promotive interaction, social skills, and group processing.

The link between cooperative learning and mathematical thinking is further supported by the constructive controversy mechanism, in which "one person's ideas, information, conclusions, theories, and opinions being incompatible with those of another" leads to "higher quality decision making and achievement, greater creativity, higher cognitive and moral reasoning." This mirrors the mathematical process of conjecture, critique, and proof that lies at the heart of mathematical reasoning.

6. Instructional Models: The SSCS Framework

Empirical evidence from the study on the Search, Solve, Create, and Share (SSCS) learning model offers a concrete example of how structured instructional frameworks can promote mathematical reasoning among secondary students. The quasi-experimental study of 102 eighth-grade students found that the SSCS model produced an effect size of 0.97 on mathematical reasoning abilities — a substantial impact — with the experimental group achieving a post-test mean of 81 compared to 60 for the control group using direct instruction. The SSCS model proceeds through four phases: searching (identifying problems), solving (planning and gathering information), creating (executing solutions and formulating outcomes), and sharing (presenting findings to the class).

The model's effectiveness is attributed to its emphasis on active student involvement: "students are given the opportunity to explore concepts and problems independently or in groups, allowing them to build their own understanding, beyond just memorizing." This aligns with the broader literature on cooperative learning and active engagement, and with Devlin's argument that mathematical thinking develops through grappling with problems rather than following prescribed procedures.

7. Synthesis and Implications

Taken together, these sources converge on several key themes for promoting mathematical thinking among secondary learners. First, mathematical thinking is fundamentally distinct from procedural fluency — it involves reasoning, abstraction, and the capacity to approach novel problems, as argued by Zeeman (1972), Cartwright (1970), and Devlin (2012). Second, the development of mathematical reasoning is mediated by cognitive development (Piaget's formal operational stage) and is strikingly uneven across students, as the qualitative case study demonstrates. Third, while heuristics remain valuable descriptive tools, their effective teaching requires specificity, metacognitive support, and skilled pedagogy — merely naming strategies is insufficient. Fourth, the history and epistemology of mathematics can serve as a powerful pedagogical resource, providing both motivation and a more authentic image of mathematical activity. Fifth, cooperative learning structures and active instructional frameworks like SSCS create the conditions — dialogue, critique, shared problem solving — under which mathematical thinking can flourish.

The challenge for secondary educators remains translating these insights into sustained classroom practice, particularly given the developmental variation among learners and the pedagogical demands that teaching for mathematical thinking places on teachers themselves.