The term “understanding” has been invoked several times in the previous sections. I now summarise and make explicit what understanding is and why it is an important concept from the perspective of mathematics. I do so because the term is at the heart of this investigation. Very few will contest the assertion that one of the most important goals of mathematics instruction is that learners should have an “understanding” of the concepts of mathematics. However, various meanings have been ascribed to “understanding” to the extent that defining it is not an easy task.
Some studies use “understanding” with the implicit assumption that there is universal agreement about its meaning. To complicate matters further, Machaba (2016) uses understanding and knowledge interchangeably. In spite of all these difficulties, an explicit attempt to define understanding is made by Sierpinska (1990) who proposes that understanding be regarded as ‘an act, but an act involved in a process of interpretation, this interpretation being a developing dialectic between more and more elaborate guesses and validations of these guesses’ (p. 26). This understanding is ‘acquired through years of watching, listening, and practicing’ (Lampert, 1990,
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p. 31). The sense I made out of these definitions was that what one thought is “understanding”
could in fact turn out to be a myth or a misconception.
To alleviate the multiple meanings ascribed to understanding, Holt (1966) developed a list of seven nonlinear but inexhaustive senses in which the term could be used in education. He suggests that understanding takes place when a learner can do some, at least, of the following about a concept or idea: state it in his or her own words; give examples thereof; recognise it in various guises and circumstances; make connections between it and other facts or ideas; make use of it in various ways; foresee some of its consequences; and state its opposite or converse. However, although the description of features of “understanding” is helpful, the concern with viewing understanding this way is that it does not distinguish between the different types of understanding.
I categorised “understanding” into “fundamental” to denote the type of mathematical understanding that is central to arguments permeating this study and “supplementary” to denote the type of mathematical understanding that enhances thinking about “understanding”.
2.2.1 Fundamental perspectives on types understanding
As already alluded to, mathematics education research has shown that most learners have serious difficulties with constructing proofs. Attempts to tackle this problem have focused on the widely known and useful distinction between the different types of understanding in mathematics; namely, instrumental, relational, logical, and functional (de Villiers, 1994). Skemp (1976) initially theorised the concept of understanding as either instrumental or relational.
Instrumental understanding refers to the learner’s ability to correctly and efficiently manipulate mathematical content by using rules without knowing why these rules work. This understanding is sometimes referred to as computational knowledge, computational skill, computational ability, procedural skill or procedural knowledge (Idris, 2006). In this type of understanding, a learner tends to memorise owing to the isolated nature of many rules. By way of example, instrumental understanding applies to the rule that “we flip and multiply when we divide fractions”. Skemp (1976) provides empirical evidence that material learnt relationally is, in a
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month, remembered seven times better than that which is learnt instrumentally. He adds that without understanding, a learner is mentally lost, anxious, and frustrated in mathematics.
Relational understanding refers to learner’s ability to deduce relationships between content and the underlying logic upon which these relationships are based. According to Idris (2006), relational understanding is used interchangeably with conceptual understanding or conceptual knowledge to denote not only knowing facts, rules and procedures, but also knowing why general principles and a network of ideas in mathematics work. According to Schäfer (2010), conceptual understanding relates to acquisition of knowledge that not only revolves around isolated facts but includes an understanding of the different contexts that frame and inform these facts and an understanding of why a particular mathematical idea is important. With this type of understanding, a learner would be able to adjust when a new and different task is introduced. For example, understanding that “the sum of interior angles of a triangle is 1800”, will be useful in proving deductively (informally) that “the angles of a quadrilateral sum up to 3600”.
Although relational understanding provides learners with a broader perspective of the mathematics discipline, the abstract nature of the subject requires further descriptions that go beyond making informal deductions (Idris, 2006). Skemp (1987) improved his theory by including
“logical understanding” to instrumental and relational understanding. In mathematics, logical understanding involves a learner’s ability to use an appropriate method to perform a task, knowing why the method works, and having mastery of the rhetorical demands of school mathematics in the appropriate context (Tirosh, 1999). By rhetorical demands is meant knowing how mathematical ideas are expressed or written and judged within the mathematics community. In other words, a learner has logical understanding if they are able not only to convince themselves, but being able to convince others when asked to reflect on the logic of the steps in working out a solution to a mathematical exercise or problem.
Resnick and Ford (1981) question the usefulness of Skemp’s (1976) dichotomy between instrumental and relational understanding. They point out that, for example, having instrumental understanding without attending to the relational, logical, or functional aspects is
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counterproductive. I concur with the sentiment in that strict adherence to one theoretical perspective at the expense of others is undesirable in mathematics education. However, as far as I could ascertain, there is no record of Skemp having advocated for this approach. I choose to read Piaget’s (1978) argument that instrumental understanding is not understanding at all charitably.
Personal experience suggests that instrumental understanding can be useful in building a foundation for relational understanding. If this is taken to be true, instrumental understanding does not seem inferior to relational and logical understanding. However, de Villiers (1994) points out that mathematical understanding cannot be described through these three perspectives of understanding only.
De Villiers (1990) identifies functional understanding to address the affect aspect which is embedded in doing mathematics. For this study it means understanding the role, function, purpose or value of proof in mathematics. He concludes that on the basis of extensive interviews with learners, most of their difficulty with proof seems not to lie so much with poor instrumental proficiency nor inadequate relational understanding as in poor functional understanding of proof (de Villiers, 1994).
2.2.2 Supplementary perspectives on types of understanding
Acknowledging the value of Skemp’s (1976) theory, Byers and Herscvics (1977) suggest an extension of understanding that includes “formal understanding” which relates to a learner’s ability to express mathematics in conventional forms of notation, and “intuitive understanding” which relates to a learner’s perception of a problem with little thought of the solution process. Bell, O'Brien, and Shiu (1980) provide examples of intuitive understanding. In this respect, rather than using a linear sequence of steps in solving 3𝑥 + 2 = 8, a learner spots that 8 is the same as 6 + 2 and hence 3𝑥 must equal 6 and so 𝑥 is in fact 2.
While agreeing with Skemp’s dichotomy on mathematical understanding of instrumental and relational understanding, Usiskin (2015) sees understanding of mathematical concepts in five independent and nonsequential (that is, can be learned in isolation from each other and in no
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particular order) types which he calls dimensions of understanding: skill-algorithm understanding, property-proof understanding, use-application understanding, representation-metaphor understanding, and history-culture understanding. A closer analysis of these forms of understanding revealed that they are an expansion of Skemp’s (1976) model that focused on understanding with respect to mathematical concepts only.
He describes skill-algorithm understanding as involving not only mastery of skill to obtain the right answer but also choosing a particular algorithm to obtain the right answer because it is more efficient than others. He suggests that being able to identify the mathematical properties that underlie why a certain method worked in obtaining the correct answer resembled property-proof understanding. As for the use-application dimension, he argues that this relates to individuals who know the uses of algorithms and the mathematical properties associated with a concept. He labels the ability to represent a concept in some way (for example, using manipulatives, pictorial representation or metaphor) representation-metaphor understanding.
In concluding his dimensions of understanding, he convincingly argues that understanding the cultural history of mathematical concepts is very important. For instance, he points out that some mathematical symbols are not the same everywhere; in some places, the fraction a/b is represented by a:b, while in other places the symbol a:b represents a ratio that is mathematically not identical to a fraction. However, Usiskin’s (2015) types of mathematical understanding are only limited to mathematical concepts; the problem is that learners may still see no value in learning these concepts.
A little further on Resnick and Ford (1981) emphasise that memorisation of certain facts and procedures is important not so much as an end in itself but as a way to extend the capacity of the working memory by developing automaticity of response and thus free up time to focus on understanding mathematical ideas. While I agree with their view, such a discussion is beyond the scope of this study. That notwithstanding, I am mindful of Tall’s (1978) suggestion that any useful classification of mathematical understanding must exhibits the reality that understanding is a dynamic process in the sense that understanding may take place for one week, forgetting and
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remembering the next. Before turning attention to research studies on the functions of proof, it is important to define what is meant by the term “reality”. Like Berger (1991), I view “'reality” to be a quality appertaining to phenomena that we recognise as being independent of our own volition;
things we cannot “wish away”.
2.2.3 Conflation of understanding, knowledge, and belief in mathematics education
Various definitions have been ascribed to “understanding”, “knowledge”, and “beliefs”. Pajares (1992) labels beliefs as a messy construct that travels under alias such as conceptions, perceptions, or understanding. Consistent with a dynamic view of mathematics, knowledge is viewed as contestable facts that are commonly shared among the mathematics community. Mathematical knowledge is indeed contestable given the discovery of non-Euclidean geometries which shattered the view that mathematics provides absolute certainty (Greiffenhagen & Sharrock, 2011). In contrast, beliefs are subsets of knowledge which are consciously held with varying degrees of importance and for which no social consensus regarding their validity is required (Philipp, 2007).
In other words, beliefs are ideas, views, assumptions, understanding, conceptions or perceptions, attached to mathematics, proof, and its functions, taken as true by the individual, and not readily amenable to, in Popper’s (1988) term, falsification12.
In this study, I make explicit the meaning of these terms consistent with Lloyd’s (2002) line of argument to ease communication. Similar to Knuth (2002), she defines understanding as one’s general mental structure encompassing beliefs and knowledge. Beliefs are defined as understanding that are experiential or fantasy in origin and thus disputable while knowledge is defined as understanding which are compatible with consensually held information within the mathematical community. However, beliefs are crucial in that they are thought to influence the application of knowledge in the classroom (Leder, Pehkonen, & Törner, 2002). Therefore, learners’ understanding of proof is a manifestation of their experiences with proof instruction,
12 The act of deliberately seeking counter-examples to disconfirm a theorem thereby strengthening its truth if it survives such act (Cohen, Manion, & Morrison, 2011).
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other social interactions in their environment including simply their own fantasy about proof. In this study, I also used the terms beliefs, understanding, and views interchangeably.
The investigation in this study was meant to determine learners’ understanding of the functions of proof which, unlike knowledge, is subject to corrections if inconsistent with accepted mathematical interpretation. Thus, viewing understanding as an ideal to be attained by learners, a model that precisely defines understanding is necessary. De Villiers’ (1994) model provides an ideal definition of understanding by taking into account the depth of understanding a learner has experienced. For this reason, I focused only on learners’ attainment of functional understanding of proof.
2.2.4 Learners’ understanding of the verification function of proof
[h]aving verified the theorem in several particular cases, we gathered strong inductive evidence for it the inductive phase overcame our initial suspicion and gave us a strong confidence in the theorem.
Without such confidence we would have scarcely found the courage to undertake the proof which did not look at all a routine job. When you have satisfied yourself that the theorem is true, you start proving it. (Polya, 1954, pp. 83-84)
Euclidean geometry is the place for learners to “see” the functions of proof in mathematics.
However, of all the five functions of proof invoked in this study, studies have shown that the verification function is persistently pervasive. Learners are under the misapprehension that making empirical arguments is justification (proof) for the truth of a proposition; hence, as already pointed out, this function occupies a low status and therefore regarded as being naïve among the functions of proof. But, why is this belief resistant? This is the question that will be answered shortly.
Kunimune, Fujita, and Jones (2010) suggest instructional practices to make learners understand:
the generality and universality of proof, the roles of figures, and the difference between formal proof and experimental verification. By constructing formal proofs, learners come to understand that the conjectures that they have found to be true in one context are always true. Thus, they will need to understand that proof is required to achieve generality of mathematics propositions.
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Indeed, very few will contest that Michael de Villiers has made an outstanding contribution in the field of Euclidean proof. In his book in which he introduces a DGS, Rethinking Proof with the Geometer's Sketchpad, he briefly describes activities wherein learners make and verify conjectures using sketches and engage in activities that reflect the various functions of proof at the van Hiele levels lower than 3 (known as informal deduction). However, a serious shortcoming of the van Hiele theory is that it introduces only one function of proof, systematisation, at Level 3.
Mudaly’s (1999) finding that functions such as verification, explanation, and discovery can be meaningful give support to this criticism. This is why de Villiers (2012) argues that it is far more meaningful to introduce proof within a dynamic geometry context, not as a means to verify, but rather as a means to explain, systematise, and discover prior to engaging in formal proof.
However, the potential risk associated with dynamic geometry is that both learners continue not see deductive proof as the ultimate means of verification (de Villiers, 2006) that provides assurance that there cannot be counter examples to refute a conjecture. Further, Laborde (2000) argues that the opportunity offered by DGS to “see” properties of geometric figures ‘so easily might reduce or even kill any need for proof and thus any learning of how to develop a proof’ (p. 151). This is in contrast to Chazan’s (1993) finding that even extensive use of DGS or measuremnt of examples in geometry classes would not hinder learners' appreciation of mathematical proof. My view is that empiricist (proving propositions by providing specific examples) behaviour persists because of learners’ inability to distinguish between inductive and deductive arguments. More broadly, I argue that this behaviour is symptomatic of a lack of functional understanding of proof in mathematics.
Learners are definitely not alone in relying on verificstion. Weber and Mejia-Ramos (2011) point out that the learners’ tendency of verifying theorems with examples is akin to how mathematicians gain full confidence that a proof is completely correct. That is, mathematicians do not solely gain confidence by inspecting the logic of the proof line-by-line; they use examples to increase their conviction in, or understanding of, a proof. Weber and Mejia-Ramos (2011) further
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caution that learners should be aware of the limitations of empirical reasoning and the generality of a deductive proof. It is my view that the behaviour of both mathematicians and learners towards the use of proof as a means to verify the truth of a conjecture is a natural everyday way of gaining evidence by observation.