Designing Mathematical Thinking Tasks

Abstract

Mathematical thinking is difficult to define precisely but most authors agree that the following are important aspects of it: conjecturing, reasoning and proving, abstraction, generalization and specialization. However, recent studies have shown that many sets of mathematical tasks produced emphasize lower level skills, such as memorization and the routine application of algorithms or procedures. In this paper we survey the literature on the design and use of tasks that aim to encourage higher level aspects of mathematical thinking in learners of mathematics at all levels. The frameworks presented here aim to guide task designers when writing a set of exercises.

Convergent and divergent thinking in task modification: a case of Korean prospective mathematics teachers’ exploration

“…In addition to the examination-centric education issue, mathematics teachers' reluctance or superficial implementation of creative teaching has been identified as being related to a lack of profound understanding of mathematics (Leikin et al 2013;Lev-Zamir and Leikin 2011), teachers' insufficient knowledge and experience in task design or task modification for teaching creativity (Breen and O'Shea 2010;Perkins 1985;Leung and Silver 1997;Haylock 1997;Hershkowitz et al 2017), and a lack of awareness and negative disposition towards creativity (Beghetto and Sriraman 2017;Craft 2005). In addition, avoidance of teaching creativity has been shown to be related to conflict between teaching creativity and teaching skills (Torrance 1987;Coil 2014) Abstract This study investigated how 38 secondary mathematics prospective teachers modified textbook tasks for convergent and divergent thinking while learning to teach mathematics during university coursework.…”

Preservice Mathematics Teachers’ Knowledge about the Potential of Tasks to Promote Students’ Mathematical Reasoning

“…Research has highlighted processes such as: explore and formulate conjectures and generalizations, requesting explanations and/or justifications of responses (Brodie, 2010;Thompson, 2012); identify counterexamples, and present and evaluate arguments, including mathematical argumentation of colleagues and teacher (Stein et al, 2008;Thompson, 2012); generate examples, evaluate mathematical statements, and use definitions to classify mathematical objects (Breen & O"Shea, 2019). It has also been suggested that tasks should include questions that encourage the formulation of conjectures and generalizations, involving students in purposeful or systematic observation of specific cases and in the search for similarities and differences among objects, as well as from their previous knowledge and other generalizations already known, but changing the conditions of the situation (Jeannotte & Kieran, 2017;Lin et al, 2011).…”

“…Mathematical reasoning (MR) has received increasing attention in curriculum documents as an essential ability to be developed by all students to acquire and communicate mathematical knowledge and to learn with understanding (Jeannotte & Kieran, 2017;NCTM, 2000;Stylianides & Stylianides, 2007). However, students" learning highly depends on the experience teachers provide them in mathematics lessons, particularly the nature of the proposed tasks and the way they are enacted in the classroom (Boston & Smith, 2011;Breen & O"Shea 2019;Stein, Grover & Henningsen, 1996). Considering that teaching for promoting students" mathematical reasoning is often a huge challenge for teachers, it is essential to highlight it as a focus of work in initial teacher education (Buchbinder & McCrone, 2020;Ponte & Chapman, 2015).…”

The Effect of Jumping Task and Collaborative Activity on Enhancement of Student Critical Thinking Ability

“…In the field of mathematics, Mason, and Johnston-Wilder argue that the purpose of the mathematical task is "to start mathematically fruitful activities" [6]. Furthermore, Breen and O'Shea, report that Mason and Johnston-Wilder also suggest, "The task must be involve a variety of possibilities and offer students the opportunity to discuss ideas" [7]. Some experts state that the task is what must be done by participants [8,9].…”