Problem solving in mathematics has had a great deal of scholarly attention over the last few decades. Schoenfeld (1992) stated that problem-solving is a process in which students are asked to find a solution to a mathematical problem for which they know no immediate solution, and no algorithm that they can directly use to find one. Students are at the centre of the process and are responsible for reading the mathematical problem carefully, analysing the information in the question, and examining their own mathematical knowledge to discover a strategy to find a solution.
Many scholars and educators have argued that teaching through problem-solving approach benefits students more than teaching through other approaches. Empirical research has confirmed this claim (see, for example, Ali, Akhter & Khan 2010; Gallagher, Stepien, Sher & Workman, 1995; Major, Baden & Mackinnon; 2000; Okereke, 2006).
Okereke (2006) for example, argued that mathematics should be taught using problem solving because the approach is centred on the students, and is capable of promoting active and motivated learning as well as the acquisition of problem-solving skills. This argument resonates with the statement by Major et al. (2000) that students take much more responsibility for their learning in classrooms that use problem-solving instructional processes. In addition, Gallagher
et al. (1995) stated that a problem-solving approach made students act as professional
mathematicians, and enabled them to tackle challenging problems.
Although the literature proposes that problem solving should be a fundamental goal of teaching mathematics, Stacey (2005) stated that, although it is “one of the most fundamental goals of teaching mathematics, [it is] … also one of the most elusive” (p. 341). However, many researchers (for example, Isoda, 2010; Shimizu, 1999; Stigler & Hiebert, 1999; Takahashi, 2008; Takahashi, Lewis & Perry, 2013) have noted that the typical mathematics lesson in Japan is taught using this approach, with Stigler and Hiebert (1999) summarising several features of Japanese mathematics lessons and labelling these lessons as “structured problem-solving”:
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In Japan, teachers appear to take a less active role, allowing their students to invent their own procedures for solving problems. And those problems are quite demanding, both procedurally and conceptually. Teacher, however, carefully design and orchestrate lessons so that students are likely to use procedures that have been developed recently in class. An appropriate motto for Japanese teaching would be “structured problem-solving”. (Stigler & Hiebert, 1999, p. 27)
Takahashi (2000) stated that Japanese structured problem-solving lessons were designed to create interest in mathematics and stimulate creative mathematical activity in the classroom through students’ collaborative work. The focus was to enable students to develop mathematical concepts, skills, and procedures (Takahashi, 2006). Similarly, Lewis (2011) stated that the goal of Japanese structured problem-solving lessons was to build students’ mathematical knowledge, their mathematical practices, and habits of mind – such as sense- making, perseverance, constructing and critiquing arguments, modelling with mathematics, keeping track of data.
In contrast, some scholars (for example, Lesh & Zawojewski, 2007; Takahashi, 2006) have argued that problem-solving lessons outside Japan, such as those taught in the USA, focus on developing problem-solving skills and strategies by showing students how to solve a problem and asking students to practise the solution method. Lesh and Zawojewski (2007) stated that problem-solving lessons outside Japan could take various forms, such as practising of previously taught content using word problems, a universal set of steps, a collection of heuristics, or the mobilisation of mathematical knowledge, strategies, dispositions, and beliefs, to investigate unique problems.
According to Shimizu (1999), a typical mathematics lesson in Japan, comprises four steps: (1) Presentation of a problem, (2) Individual problem-solving by students, (3) Whole- class discussion about the methods for solving the problem, and (4) Summing up by the teacher. Shimizu (1999) defines and names the four pedagogical terms commonly used in relation to these four steps of the lesson: Hatsumon, Kikan-shido, Neriage, and Matome.
Step 1: Hatsumon refers to “the [single] key question that provokes students’ thinking at a particular point in the lesson” (Shimizu, 1999, p. 110). At the beginning of the lesson, the teacher might pose a question to investigate or encourage students’ understanding of the problem. Doig, Groves and Fujii (2011) emphasise the care with which this single problem or
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task is selected for the problem-solving activity. They state that a task is chosen through
kyozaikenkyu – an intensive and complex investigation of a range of instructional materials.
Fujii (2014) states that Japanese educators teach mathematics through solving the task, and if chosen well, “a single task allows for the important new mathematical ideas to emerge in the discussion, and additional tasks are unnecessary” (p. 5).
Step 2: Kikan-shido refers to “instruction at students’ desk and includes a purposeful scanning by the teacher of the students’ individual problem-solving processes” (Shimizu, 1999, p. 110) or as Becker, Silver, Kantowski, Travers and Wilson (1990) put it, “purposeful scanning” (p. 15). Kikan-shido includes the teacher performing two critical activities that are firmly attached to the entire class discourse that will take place after the individual work. To start with, the teacher evaluates students’ problem-solving progress. Second, the teacher carefully considers which students utilized the expected approaches (outlined in the lesson plan) and which students used different approaches to the problem (Shimizu, 1999).
Step 3: Neriage in Japanese means kneading or polishing up. According to Shimizu (1999), neriage is “a metaphor for the process of polishing students’ ideas and of developing an integrated mathematical idea through the whole-class discussion. Japanese teachers regard
neriage as critical for the success or failure of the lesson” (p. 110). According to Takahashi
(2006), neriage denotes the teacher “facilitate[ing] mathematical discussion after each student comes up with a solution” (p. 42).
Step 4: Matome in Japanese means summing up. Japanese teachers think that this stage is indispensable for a successful lesson. In this stage, the teacher reviews what students have discussed in the whole-class discussion and summarizes what they have learned during the lesson (Shimizu, 1999). According to Fujii, Kumagai, Shimizu, and Sugiyama (as cited in Fujii, 2016), the teacher could just say which strategy is the most fruitful or correct, and why; however, matome ought to transcend that to incorporate comments by the teacher regarding the mathematical and academic value of the task and lesson. Fujii (2016) stated that while the
matome should be indicated in the lesson plan, for a lesson to incorporate successful matome
the “task should be understandable by the students with minimal teacher intervention; it should be solvable by at least some students (but not too quickly), and it should lend itself to multiple strategies” (Fujii, 2016, p. 414).
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most teachers in the USA did not include matome at the end of a lesson, a situation that, according to Fujii (2015), leaves the students feeling unsatisfied with what was presented to them.
2.4.1 Implications for this research
This section has described Japanese structured problem-solving lessons that are typically used in mathematics research lesson in Japanese Lesson Study. This study will investigate the extent to which research lessons in mathematics in Zambia are both expected to follow the structured problem-solving approach and do so in practice in order to gauge the adaptations of Japanese Lesson Study in Zambia.