Learning approaches in mathematics

Critics argue that Piaget’s work is not a complete description of cognitive development (Eggen and Kauchak, 2003). Gelman, Meck and Merkin (1986) conjectured that Piagetian theory underestimates the abilities of young children; whereas Eggen and Kauchak (2003) criticise Piaget for overestimating the abilities of older learners. The notion of the possibility of explicitly being able to teach using a cognitive development approach was investigated by Adhami, Johnson and Shayer (1997). The research from the two projects Cognitive Acceleration in Mathematics Education (CAME) and Cognitive Acceleration in Science Education (CASE) demonstrated that the approach is able to accelerate pupil learning (Adhami, Johnson and Shayer, 1997; Shayer, 1999). Piaget

believed that not all pupils in a class are operating at the same cognitive development stage; therefore it might be a strong argument for self-

differentiating, open-ended tasks (appendix 50) as a way of enhancing a pupil’s learning of mathematics.

An alternative approach to the learning of mathematics might therefore be to have learners in mixed aged groups where pupils are at different cognitive developmental stages. There is a reasonable body of supporting research that support this view, albeit in small schools (Cornall, 1986; Galton and Patrick, 1990; Francis, 1992; Vulliamy and Webb, 1995). However, research by

Veenman, Lem and Roelofs (1989) concluded that there is no significant impact on pupil cognitive development when taught in multi-aged classes. Teachers

47 working in mixed-aged groups were often teaching as if they were two or more

different classes rather than considering all pupils to be at the same cognitive developmental stage.

Most abstract mathematics begins in secondary school and this is not solely confined to just algebra. Topics containing abstract ideas such as fractions, geometry and probability all have their beginnings in the early secondary school curriculum. New knowledge, according to Piaget, is built up through experiences which are then checked and validated against existing knowledge. He argues that new knowledge has to be assimilated and existing concept structures have to be reorganised or modified. Immature pre-operational thinkers can learn procedures or algorithms but do not develop conceptual understanding of abstract ideas. According to Burns and Silbey (2000, p. 55) “hands-on

experiences and multiple ways of representing a mathematical solution can be ways of fostering the development of this cognitive stage”.

The child at the early secondary school (ages 11 to 14) is operating in Piaget’s formal operations stage (mainly due to the ways in which they are taught) and should be capable of structuring concepts as a foundation for the development of more abstract thought patterns where reasoning is executed using symbols without the need to rely on manipulative materials. At this developmental stage the learner can solve 2x + 5x = 14 without having to refer to a concrete

representation. Contrastingly at the next cognitive developmental stage learners are capable of forming hypotheses and deducing possible consequences, to enable the child to construct their own mathematics.

This has implications in that the majority of children have developed a schema, or learning map, for the four mathematical operations on integers during the

concrete developmental stage (mainly during the ages 7 to 11 – Primary School Curriculum). Unfortunately pupils might then fail to apply these operation

schemas correctly to the rational numbers when trying to develop their own view of the next steps in learning mathematics (Tall, 2002). The extension of the four operations over the rational numbers is often begun with manipulative materials during the formal operations stage. The cognitive development of these four operations on fractions very quickly moves on to a much more abstract approach being taken by teachers. This is usually by considering the required, or formally

48 recognised, algorithms and procedures. The problem is then that pupils are often not able to link what has been previously learned with the abstractness of the procedures and algorithms. Research by Gabriel et al. (2013) who studied 21 mathematics textbooks, interviewed 24 teachers and analysed the results of 439 test scripts for 9 -11 year olds relating to fractions, found that the practice of focussing on procedures is not sufficient because “conceptual understanding is essential to ensure a deep understanding of fractions”(Gabriel et al., 2013, p. 9). The Vygotskian view of the social formation of the mind through scaffolded talk as a means of promoting understanding and reasoning has its basis in situated learning. Vygotsky's theories embody social interaction as a fundamental

element in the development of cognition where "all the higher functions originate as actual relationships between individuals" (Vygotsky, 1978, p. 57). With

situated learning embedded in the theories of the social learning and problem solving of Schoenfeld (1985) a general theory of knowledge acquisition can be seen to apply to the learning activities that focus on problem-solving skills. Eisenhart and Borko (1991) express meaningful learning in the form of active knowledge construction where learning occurs “as they [pupils] modify and elaborate their knowledge structures through a process of adaptation to the environment” (p. 142).

The notion of a learning schema affords theorists from differing epistemological standpoints a common language or model to describe cognitive constructions of learning. The term “schema” as applied to learning can be traced back to a study of memory by Bartlett (1932) which was then developed by Oldfield and Zangwill (1942a, 1942b, 1943) and Skemp (1962, 1971, 1979). Minsky (1975) introduced the notion of “frames” and this was developed by Tannen (1993) as a

methodology for the analysis of discourse. Schank (1975) had previously

developed the idea of “scripts” as a way of describing conceptual schemas. Both “frames” and “scripts” are closely linked and broadly similar to Bartlett’s schemas. However, whilst the term schema has been applied to mathematics education (Steffe, 1983, 1988; Davis, 1984; Dubinsky, 1992; Cottrill et al., 1996) there have been few attempts to define precisely what might constitute a schema in respect to all or even the majority of mathematical topics. Therefore, rather than viewing mathematics, and in particular operations on fractions, as a set of procedures or facts linked together by algorithms that need to be acquired, a constructivist’s

49 approach to the learning would be to create coordinated mental schemas

(Kieren, 1990; Kieren, 1994; Steffe, 1990). By considering the mathematical patterns and associated interlinked relationships learners might be able to self- construct these mental schemas.

Schematic learning when applied to mathematics education introduces the notion that new learning is assimilated, organised and interpreted with reference to past or prior learning (Skemp, 1979, 1986). It is therefore an important pedagogical task for the teacher to discover the schemas a child has internalised and that they are using. Being able to define new learning that builds naturally on the schemas that a pupil is already comfortable with and confidently using is at the heart of teaching. If a pupil has acquired or internalised a schema for division of integers then it would appear that Skemp is arguing that the learning of division of rational numbers should extend and build on the currently held schema, rather than deviating or introducing new schemas for specific cases. The introduction of a new schema for the division of fractions is often the case and this can result in confusion and poor recall. Building on prior schemas relies on the teacher having both solid subject knowledge and pedagogical insights into the pupils’ strengths, understanding and the ways they view and interact with mathematics.

If we view learning through the mediation and extension of schemas together with social interactions, then the learning of mathematics through collaboration has been shown to reduce peer competition, promote achievement and foster positive relationships. Research by Swan (2006) into the development of resources, using a model of collaborative discussion to reshape students’ existing knowledge when working towards public examinations, indicated that student-centred learning resulted in the greatest gains. According to the National Council of Teachers of Mathematics (NCTM, 1991) Teaching Standard 8,

learning should promote active learning and teaching; classroom discourse; and individual, small-group, and whole-group learning. Collaborative learning through classroom discussion can be the stimulus for active learning as noted by Swan (2006, p. 227) “discussion – based approach to learning is to encourage students to move from ‘passive’ learning strategies to more ‘active’ ones”. Ofsted (2009, 2012) noted that

pupils become confident learners as they develop skills in articulating their thinking about mathematics. They learn to make sense of ideas, and

50 reason and justify their methods and solutions because discussion is a

regular feature. Learning is therefore active and cumulative (Ofsted, 2009, p. 12).

They [the pupils] frequently told inspectors that in other subjects they enjoyed regular collaboration on tasks in pairs or groups and discussion of their ideas (Ofsted, 2012, p. 19).

Ingram, Sammons and Lindorff (2018) whilst reviewing the literatureon effective mathematics teaching found that learning mathematics is frequently described in terms of a collaborative activity. Mercer (2000) found that tasks which promote active involvement and encouraged critical, collaborative constructive discussion to be more effective than an uncritical acceptance of rules procedures, algorithms and methods. Swan (2006, p. 162) describes the distinction between a

transmission model and the active model of learning as “an individual activity based on watching, listening and imitating until fluency is attained”. The

transmission model of learning is in complete contrast to a collaborative learning approach where “learners are challenged and arrive at understanding through discussion”. Yet this transmission model is often seen in mathematics

classrooms and considered to be the definitive approach to teaching

mathematics. As Watson (2019, p. 2) observes “in-the-moment decisions can lead to a lesson becoming more traditional and teacher-centred even though the teacher may have the knowledge of and hold beliefs in reform-oriented student- centred approaches”.

The structure of a mathematics lesson is often “rigid, characterized by rote

learning and endless repetition of mechanical tasks” (Evans, 1994, p. 2). The use of practical elements, entailing “learning by doing” with “the additional feature of reflection upon both action and the result of action” is the key for experiential learning to take place (Capel, Leask and Turner, 2001, p. 252). Ellis (2007) suggests that practical and experiential learning are beneficial to all and are an integral part of the learning process, with Beard and Wilson (2006, p. 18) reminding us that “experiencing something is a linking process between action and thought”; by including experiential activities in the classroom, learners are encouraged to make logical links between theoretical models and real-life practice.

Moon (2009), a researcher in experiential learning, suggests that learning from experience and situations often occurs independently of a teaching process. She

51 also reminds us that the literature on experiential learning (learning from

experience) is diverse in nature with no common consensus as to meaning, and each author developing a variant to suit the context in which they are

researching. Usher and Edwards (1994, p. 201) argue that when quoting experiential learning as a learning theory “different groups give it their own meanings and construct it in their own ways”. For example, Usher and Soloman (1999) see learning from experiences as being placed in everyday contexts of the real world and is therefore always situated.

If experiential learning implies learning by doing, and this approach requires pupils to actively participate in the learning, then just being a passive learner might not suffice. Boydell (1976) takes the view that “learning is a dynamic, active process, so the trainee learns best by participation. If only a man’s ears are involved (e.g. lecture), much less is learned than if his eyes, muscles, thinking processes and feelings are involved”. Kolb (2015) argues that experiential learning theory describes how real life experiences play an important role in the acquisition of new knowledge. Contrastingly Boaler (2009) argues that “students need to be actively involved in their learning as well as needing to be engaged in a broad form of mathematics, using and applying methods, representing and communicating ideas” (p. 76) implying mathematics is much more than real-life problems.

Brumbaugh and Rock (2013), mathematics educationalists and authors of Teaching Secondary Mathematics, explain that learning by doing or discovery is “a method of indirect instruction where the teacher organises the learning

environment, enabling the learner to develop conclusions” (p. 202). Weegar and Pacis (2012, p. 7) support Brumbaugh and Rock suggesting that where learning employs constructivist strategies “students learn by discovering on their own, to students collaborating with others”. Henson (2013) provides us with a working definition for learning by discovery as “intentional learning through problem- solving and under the supervision of the teacher” (p. 101). The use of practical equipment such as manipulatives gives pupils a sense of semi-autonomy or what Brumbaugh and Rock describe as guided discovery leading “the learner in a particular direction toward a desired conclusion” (Brumbaugh and Rock, 2013, p. 144). Guided discovery as a vehicle for pupil learning allows the freedoms and

52 legitimises discussion as well as the construction of a shared understanding of

the mathematics.

Mathematics teaching is constantly changing, albeit slowly, in light of developing learning theories and technological advances. Consequently there is no longer a fundamental need to equip learners with prescriptive methods, procedures and algorithms as we are less likely to think of learning defined in terms of a

behavioural change or the acquisition of knowledge but more in terms of a social activity (Jarvis, Holford and Griffin, 2003). Hiebert and Grouws (2007, p. 373) argue that “within mathematics, theories of learning have been more clearly articulated than theories of teaching. Although theories of learning provide some guidance for research on teaching, they do not translate directly into theories of teaching”. They go on to put the viewpoint that theories which “specify the ways in which the key components of teaching fit together to form an interactive, dynamic system for achieving particular learning goals have not been sufficiently developed (ibid, p. 373). The argument here is that there is interdependence between learning theory and the content specification of teaching sequences which are used to support the learning theories. The implication being as learning theories are constantly developing then the associated theories of teaching need to adapt and the approaches to teaching mathematics need to change.

Having examined the relevant learning theories the core constructivist principles that are effective for mathematics learning can be summarised as being

Learning mathematics is active and often situated in a context Learning involves prior knowledge, experience and discovery Learning requires social interactions mediated through language

However, whether pupils are taught using a cognitive or constructivist approach utilising real life problems through guided discovery or by the more traditional didactic approach the resulting impact on the learner would have profound effects for their view of mathematics. As Hoyles (1982) found, secondary school pupils tend to associate their mathematical experiences with feelings of anxiety, shame, and a sense of failure and this is often linked to the teaching approaches taken to engage learners. This theme is explored in the next section.

The literature has led me to the belief that the learning of mathematics is a complex process and more than an individual cognitive process (Piaget, 1952).

53 Learning is more likely to occur if lessons are designed and based around real-

life situations with pupils and teachers working in what Lave and Wenger (1998) call a community of practice. Furthermore knowledge is constructed through social interactions and is mediated through language and discussion and that these should be features of a lesson design. I therefore decided to test these theories by designing, for this research study, a lesson using these concepts where pupils were learning the division of fractions.