A common understanding of the term exercise, when related to mathematics teaching, might be that of a number of textbook problems to develop a particular mathematical skill or set of skills. However the word exercise does have a
multiplicity of meanings such as a physical exercise, the exercise to practice a skill or an exercise as a piece of work intended to test knowledge. Even when we consult a single source such as the 2002 Proceedings of the Annual Meeting of
Purpose / Definition Researcher(s) Characteristics / Features
Generic definition in five stages
Dreyfus and Dreyfus (2004)
Hierarchical list of
development features Novice, Advanced Beginner,
Competent, Proficient and Expert
Professional skills Knowing “that” and “how”
Lowe and Muller (2010)
Tested in examinations
Soft Skills DfBIS (2016)
Skillsforyou (2017)
Eg. Communication, Decision making, Self-motivation, Leadership Team-working, Thinking Time management. Defined in terms of mathematical content NCSM (1977) Cockcroft (1982) Smith (2004) Vorderman et al. (2011) Mathematics National Curriculum (2013) Eg Computation, geometric interpreting graphs and tables.
Learning intentions AfL (2017) Mathematics they know and
understand Conceptual and
procedural knowledge
96 the Canadian Mathematics Education Study Group four contributors use the word in different ways for example:
Once found, it is easy to prove in Mathematica, in Maple or by hand— and provides a very nice calculus exercise (Borwein, 2002, p. 24). Participants are first asked to make note of geometric shapes that they notice around them—a task that has consistently and reliably given rise to lengthy lists of Euclidean forms. That list is pushed aside during the fractal cards activity, after which participants are invited to repeat the
exercise of looking for geometric shapes (Gerofsky, Sinclair, and Davis,
2002, p. 37).
This led the group to consider if there is an empirical way to study this question of emotions in mathematics? For example, divided-page
exercises. Give them a problem and on one side they do the problem
and the other side they write thoughts and ideas. Those teachers were using the exercises to help them work on their emotions (Pallascio and Simmt, 2002, p. 49).
My goal is to pursue this and investigate conceptualizations across contexts, such as geometric, graphical, analytical, and others, and also across approaches, such as responding to situations, discovery
exercises, constructions and counterexamples to force focusing on
properties and definitions, and to look for themes and patterns (Brown, 2002, p. 73).
Borwein (2002) appears to be using the term exercise to describe a learning episode related to a single problem that investigates a mathematical concept. This is in contrast to Gerofsky, Sinclair, and Davis (2002) who are using the term to describe a repetition of a piece of learning once additional mathematical
information has been added. For Pallascio and Simmt (2002) the term exercise is used to describe a method of presenting a mathematical solution and the effect that the resulting presentation has on the emotions of the learner. Brown (2002) is focusing on the use of the term exercise as a means of a learning episode described through mathematical discovery.
All four researchers would therefore appear to be using the word exercise in different ways. I decided to consult more sources and found that teachers and textbooks tend to use the term exercise as a means of demonstrating
mathematical procedures, algorithms or worked examples as a precursor to the setting of a repetitive sequence of questions for pupils to undertake as reported by Post et al. (1993, p. 1) view of textbooks as
Page after page of drill and practice exercises are still the norm rather than the exception; problem solving seemingly has more to do with the
97 existence of words than it has to do with the presence of a problematic
situation for which the person involved has no readymade response patterns - the more or less standard definition of problem solving. The presence of real-world problem situations that will require extended and repeated periods of contemplation are virtually non-existent.
Hattie (2012) argues that one of the roles of a school and teachers is to teach pupils the value of deliberate practice (exercises that involve challenge, concentration, monitoring and instant feedback). This is in comparison to the more normal view of an exercise as the practising of a set of repetitive similar questions (Rasmussen et al., 2005). Here Hattie is not trying to define the term excises but simply to describe an exercise in terms of its features and qualities. Similarly the features of a mathematical exercise were investigated by Lithner (2003) who found that of 600 exercises the majority were possible to solve using method such as identifying similarities in between questions which he calls “Identification of Similarities” (ibid, p. 35). However, on closer inspection the research uses the word “exercises” to mean mathematics questions that are either designed by teachers or taken from textbooks. Clements (2000) presents the more traditional view of an exercise as “drill-and-practice” (p. 10) but it is duly noted that “students need more than drill and practice; they need to understand the mathematical concepts beyond the practice exercises (Davis et al., 1990). So to summarise the various uses of the term exercise:-
Purpose / Definition Researcher(s) Characteristics / Features
Investigating mathematics
Borwein (2002) Engagement
Repetition of learning Gerofsky, Sinclair, and Davis (2002)
Additional information being added to create deeper levels of understanding Presenting a
mathematical solution
Pallascio and Simmt (2002) Emotional Discovering mathematics Brown (2002) Discovery Demonstrating mathematics
Post et al. (1993) Procedures, algorithms or worked examples
Deliberate practice Hattie (2012) Challenge, concentration,
monitoring and instant feedback
Drill-and-practice Davis et al. (1990); Clements (2000)
Practising and creating a deeper mathematical understanding of concepts Identifying similarities Lithner (2003) Designed by teachers
98 With a number of differing views in the literature as to what constitutes or defines a mathematical exercise I decided that because the study would involve
practising teachers I would need to use a definition that would be both
recognisable to them and be true to the aim of the study. I therefore decided that Clements’ (2000) view of an exercise as drill and practice would be recognisable to teachers and that Brown’s (2002) view of learners constructing learning by discovery would be the feature that this study would investigate. The idea of pupils discovering mathematics by constructing the learning through posing their own questions is seen by a number of mathematics education researchers as an important aspect of learning (Kilpatrick, 1987; Silver, 1994; Chin and Kayalvizhi , 2010; Abramovich, 2015; Wong, 2015 ). At the point where the learner takes control of the learning the teacher’s role is to loosely guide rather than direct the learner. In this respect it is an open piece of learning which requires little
exposition and explanation and usually follows pathways which the learner is interested in pursuing. An exercise therefore has its basis in self-directed learning and experiential learning theory (see section 2.2) where the exercise allows pupils to socially construct learning (see section 2.2) based on their experiences. The literature therefore leads me to my working definition of an exercise using as the basis Clements (2000) and Silver (1994) and is defined as follows
An Exercise: is an aspect of pedagogy designed by a teacher to encourage
learning and is seen as an extension to drill and practice type questions to gain an understanding of a mathematical concept. Additionally the newly acquired piece of learning is explored through a limited number of teacher prescribed questions and importantly extended by an additional set of learner generated questions.