Factors contributing to successful mathematical problem solving

I commence this section by referring to the social construct of ‘attitude’. McLeod (1992) identified attitude along with beliefs and emotions as one of three key affective paradigms in mathematics education. All practitioners can relate to classroom experiences where pupils display a range of different behaviourisms towards mathematical problem solving which are generally construed across a continuum of positive and negative dispositions. They can have an affective and emotional character, while on the other hand, are of cognitive origin. For many years, this phenomenon was surprisingly neglected by a lack of a theoretical framework and new methods of inquiry.

A seminal study in this area is the work of Di Martino & Zan (2010) who collected and analysed autobiographical narratives written by 1,662 Italian pupils whose school levels ranged from early primary to the end of secondary. The results of the study showed that almost all of the participants describe their relationship with mathematics along at least one of the following three trajectories:

 emotional disposition towards mathematics  vision of mathematics

 perceived competence in mathematics

Di Martino & Zan (2010) present a multidimensional model characterised by three strictly interconnected dimensions that pupils recognise as crucial in their development of their relationships with mathematics (Figure 2.11).

Figure 2.11 The three dimensional model for attitude (Adapted from Di Martino & Zan, 2010) EMOTIONAL DIMENSION VISION OF MATHEMATICS PERCEIVED COMPETENCE

Di Martino & Zan (2010) suggest the need for a new approach about the positive/negative portrayal of attitude and offer a definition of ‘negative attitude’ aimed at supporting teachers:

The multidimensionality of the model underlines the inadequacy of the positive/negative dichotomy for attitude referred to only to the emotional dimension (like/dislike), and rather suggests considering an attitude as negative, when at least one of the dimensions is negative. In this way, we can outline profiles of negative attitude, depending on the dimension that appears to be negative (p. 44).

In a study of 16 Belgium secondary children, Op’t Eynde, De Corte & Vershaffel (2006) examined the relationship between mathematical related beliefs, emotions and problem solving behaviour. They found that the nature and intensity of emotion experienced during problem solving fluctuated between participants. One significant aspect to emerge from the results was the level of confidence. Guven & Cabakor (2013) investigated factors influencing mathematical problem solving achievement of 115 Turkish secondary pupils. The researchers discovered that self-efficacy, beliefs and mathematical anxiety were noteworthy. However, the study suffers from poor external validity. In a study of 20 Israeli primary children, Prusak, Hershkowitz & Schwarz (2013) explored the culture of problem solving. They noted the success of their findings heavily relied on five principles such as encouragement to produce multiple solutions, creating collaborative situations; social-cognitive conflicts, providing tools for checking hypothesis and inviting students to reflect on solutions.

In a review of the locus of problem solving within mathematics curriculums of Australia, UK, USA and Singapore, Stacey (2005) asserts that successful mathematical problem solving depends upon many factors which have distinctly different characters, illustrated in Figure 2.12. A more comprehensive paper would include Scotland (since a UK curriculum does not exist) and non-English speaking countries. A number of scholars argue that pupils should

solve a wide range of types of problems and be regularly exposed systematically to planned problem solving instruction (e.g. Lester, Garafolo & Kroll, 1989; Lester, 1994, 2013; Boaler, 1998, Cai, 2003; Lesh & Zawojeswski, 2007). Schoenfeld (2011, 2013) maintains that learners require deep mathematical domain knowledge, heuristic strategies, metacognitive skills and relevant beliefs. Likewise, Goldin (1998) opines that beliefs systems are powerful facilitators of problem solving success, or otherwise, as obstacles to it. Finally, Lester (2013) points to the importance of intuition while Boaler (2016) advocates a growth mind set.

Figure 2.12 Factors contributing to successful problem solving (Adapted from Stacey, 2005)

Solving problems successfully requires a range of skills Deep mathematical knowledge Personal attributes e.g. confidence, persistence, organisation

Communication skills _{e.g. orientation to}Helpful beliefs ask questions

General reasoning abilities

Heuristic strategies

Abilities to work with others effectively

I now turn my attention to the multifaceted role of the teacher (Lortie, 1975). From my professional experience, the selection of a mathematical problem is critical to the successful outcome of any lesson. To ensure equitable engagement of all levels of ability, suitable problems must present opportunities to be solved or at least partly attempted by low confident learners. Accessible problems should integrate enabling prompts for pupils experiencing difficulty and extending prompts for pupils who have completed the tasks (Hiebert et al., 1997; Sullivan, 2011; Van de Walle, Karp & Bay-Williams, 2014). An overarching ability to choose appropriate problems is interrelated to content knowledge and proficiency of solving

mathematical problems including an understanding of how young people think about problem solving (Chapman, 2015). If teachers are unable to relate first hand to the tension and triumph of discovery engendered by solving problems, they are unlikely to be adept at fostering mathematical problem solving. During classroom discourse, practitioners should solicit questions that unpack pupils’ thinking and press for them to explain their reasoning behind the process (Rigelman, 2007). Similarly, a resilient dexterity to identify common misconceptions is essential. Schoenfeld (1992) encapsulates widely recognised pupil beliefs about mathematics which without approach, form a barrier to the effective learning of mathematical problem solving:

 Mathematics problems have only one and only one right answer.

 There is only one correct way to solve any mathematics problems – usually the rule the teacher has most recently demonstrated to the class.

 Ordinary students cannot expect to understand mathematics; they expect simply to memorize it, and apply what they have learned mechanically and without understanding.

 Students who have understood the mathematics they have studied will be able to solve any assigned problem in five minutes or less.

 The mathematics learned in school has little or nothing to do with the real world.  Formal proof is irrelevant to processes of discovery or invention (p. 359).

2.10 Summary

The centrality of problem solving in mathematics is incontrovertible. It can promote deep conceptual understanding, critical and independent thinking, habits of persistence and curiosity, confidence in unfamiliar situations that will serve pupils greatly in everyday life and in the future workplace (Lester, 1985; NCTM, 2000, Cai, 2010). No universally accepted definition of mathematical problem solving exists or the imminent prospect of a construct being agreed (English & Gainsburg, 2016). Mathematical problems encompass many characteristics and are classified in different ways. The learning of problem solving is extremely complex and multidimensional with much interplay rooted in the field of cognitive science. It can nurture creativity, flexibility and mental fluency (Silver, 1997; Guberman & Leikin, 2013). Considerable research has focussed around the theoretical framework introduced by Polya (1957). Schoenfeld (1985) established that resources, heuristics, metacognitive control and beliefs systems are fundamental mechanisms of successful mathematical problem solving.

The role of the teacher is instrumental in supporting learners to develop higher order thinking skills through generating multiple solutions and providing rich opportunities for comparing

and reflection. Continued support exists for teaching mathematics using problem solving as a vehicle (Lester & Cai, 2016). There is little evidence to suggest that demonstrating heuristics to pupils leads to greater success in solving problems (Lester, 2013) although some strategies have merit. Although there is no obligation for practitioners to be expert problem solvers, a degree of mathematical proficiency blended with skilful pedagogical knowledge is required (Lester, 2013; Chapman, 2015). Concomitantly, tension of high-stakes national mathematics examinations which exclusively concentrate on assessing basic skills place educators in an undesirable position inconsistent with curriculum objectives (English & Sriraman, 2010). Such a misalignment with classroom practice suggests that a review of the philosophy of external assessment within CfE may be desirable.

New directions and perspectives emerging from the literature (e.g. English & Gainsburg, 2016) has proposed that future mathematical problem solving research be converged on modelling. Whilst I welcome such a move, it is debateable if modelling is a division of problem solving or a separate entity that requires a diverse set of skills. Likewise, there is a request for the recontextualisation of school mathematical problems so as to offer more cognitively challenging dynamic tasks that authentically simulate demands of 21st century work and life.

However, I believe that in order to advance the mathematical problem solving skills of all of our young people, research has to coalesce within two interrelated domains. Firstly, that of mathematical problem posing due to the valuable learning benefits that subsist. Secondly, teachers’ beliefs since they appear to significantly impact on what takes place in classrooms.