In this study, mathematical inquiry is defined as an instructional strategy of teaching and learning of mathematical objects through “problems”. One important aspect in which mathematical inquiry classrooms differ from conventional classrooms is in the treatment of “problems”. The nonhomogeneous nature of schools notwithstanding, instructional practices in most mathematics classrooms are plagued with structured problems, questions and activities that take only a few minutes to respectively answer. Makar (2014) points out that such instructional practices contrast
The review of literature Mathematical inquiry: Experimentation and conjecturing
sharply with mathematical inquiries which often involve problems that could take days or even weeks to solve and whose solutions often contain a number of ambiguities.
Various meanings are ascribed to the term “problem”. Some people use it to define mathematics while others use it to refer to routine exercises designed to yield mastery of procedural skills. According to the American policy document Principles and Standards for School Mathematics, a problem requires engagement in an activity ‘for which the solution method is not known in advance’ (National Council of Teachers of Mathematics [NCTM], 2000, p. 52). Here, I adopted Makar’s (2014) definition of a problem in the context of mathematics. She claims that in the context of mathematical inquiry a “problem” does not refer to the task as in a textbook, but rather to the larger contextual issue to which there is no readily available procedure for finding the solution. But, what is important in inquiry is that the solution to the problem needs to be strongly underpinned by mathematics. In other words, problems in mathematical inquiry require mathematisation, that is, application of mathematics to an authentic and illstructured contextual problem.
According to Makar (2014), mathematical inquiry is a process of solving illstructured problems – that is, problems whose solutions were typically not “right” or “wrong” but require the learner to justify their conclusion, including the process used to reach it – that significantly relies on mathematics in the solution process. However, most problems in school mathematics are well structured in that they are clearly defined and learners enter the solution process with a limited number of pathways to reach a successful solution. This process involves connecting all four of these elements, purpose-question-evidence-conclusion (Makar, 2014).
Mathematical inquiry, which is very different from discovery learning where learners are expected to “discover” the mathematics they need and the teacher provides little input during investigation, requires high quality scaffolding and expertise from the teacher in knowing how to balance when to step in and when to allow learners to wrestle with challenging ideas. Judging by the features of mathematical inquiry, argumentation aligns closely with mathematical inquiry (Hunter, 2006; Makar, 2014). In this study, the learning activities in which learners conduct
The review of literature Mathematical inquiry: Experimentation and conjecturing
investigations, make conjectures, perform measurements and constructions in authentic everyday problems that can be mathematised constitute mathematical inquiry. Makar (2014) points out that in mathematical inquiry, learners are provided with multiple opportunities to use their contextual understanding in building mathematical concepts and structures that underpin the problem and create a need for learning mathematics. Seen in this light, mathematical inquiry need not to be seen as a learning and teaching approach geared towards fulfilling a utilitarian perspective of mathematics but as using real-world problems as a context to the application of mathematical concepts.
In respect of the distinction between proof and proving, the former is an object, a product, and the latter the activity associated the search for a proof (A.J. Stylianides, 2007). However, it is important to mention that proving may involve arguments which ultimately do not lead to proof as defined in the foregone subsection. Also, when referring to nonproof or the colloquial sense of proof, I used the term “empirical argument” to refer to inductive proof or proof by examples or rather put the word proof in inverted commas to refer to its nontechnical meaning. Ideally, proof as a product begins with experimentation involving construction, measurement and observation.
This experimentation can either be done by hand or with the aid of DGS. Experimentation is closely associated with, in Felbrich, Kaiser, and Schmotz’s (2012) terms, an individualistic culture where the individual participates in the generation of mathematical ideas rather than merely fitting in what authority transmitted. For an elaborated description of the different cultural notions, the reader is referred to Hofstede (1986).
However, experimentation is a result of investigations triggered by the need to prove the truth of a conjecture. Flowing from experimentation is inductive reasoning, usually resulting into some unproved generalisation, a conjecture. On the one hand, a conjecture is a mathematical proposition whose veracity has not been established yet. On the other hand, a generalisation is a proposition that has been accepted as true by a social group (Reid, 2002). This distinction notwithstanding, I used “generalisation” to refer to a conjecture whose truth arose from observation or experimentation with a few or selected cases forming a pattern but lacked a
The review of literature Mathematical inquiry: Experimentation and conjecturing
deductive proof to work for all cases exhaustively. The impression I gathered from making a generalisation, as a product that reveals a pattern of mathematical objects was that it is crucial for proof construction. For instance, in an examination of the mathematical practice, Lakatos (1976) points out that conjecturing precedes a proof. Personal experience suggests that such a practice is foreign to school mathematics classrooms where the proving environment begins with the use of axioms and definitions.
According to the CAPS document, it is statutory that investigations be an integral part of instructional practices in mathematics classrooms. The emphasis on investigations reflects two important notions; conjecturing and proving:
Investigations are set to develop the skills of systematic investigation into special cases with a view to observing general trends, making conjectures and proving them. (Department of Basic Education [DBE], 2011, p. 51)
The emphasis on these activities is a direct reflection of clear evidence that the South African school mathematics curriculum is preparing learners for future success. The emphasis on these activities is a direct reflection of clear evidence that the South African school mathematics curriculum is preparing learners for future success. The focus on investigations further substantiated my claim that the South African society and education system embraced a collectivist culture where lack of success is attributed to a lack of effort on the part of the individual learner.
Also, that learners were encouraged to engage in investigations of mathematical objects on their own reflected a mathematics curriculum that promoted a dynamic view of mathematics.
On the basis of the arguments invoked so far, I am inclined to assume that conjecturing from the given diagram implied ability to engage in proof activity. Further, the view that for learners to do proof with understanding, effort must be devoted to ensuring that they develop appropriate understanding of the significance of proof was justified. I further argued that an understanding of the functions of proof facilitated the understanding of the coherent nature of mathematical knowledge. Hence, an informed understanding of the significance of proof must
The review of literature The notion of argumentation
include a consideration of proof in each of these five functions it performs in mathematics (Knuth, 2002).
This probably prompted Balacheff (1991) to suggest that there is a long distance between these functions of proof and their manifestation in school mathematical practices. In particular, most challenging is finding more effective ways of using proof for explanatory purposes (Hanna, 2000). However, this need not to be construed as seeing the explanatory function of proof as more important than the others. Hence, I used de Villiers’ (1990) multidimensional framework to organise the discussion of the literature on the functions of proof in mathematics. It is in this framework that investigations and conjecturing, enshrined in the CAPS, aligned well with mathematical inquiry which in turn aligned with argumentation. I now turn to defining what I meant by the term “argumentation” in this study.