3.3 MEANINGFUL LEARNING
3.3.4 Intellectual need
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understanding are more of an extension of Skemp’s notions of instrumental and relational understanding in the sense that they can be applied to a mathematical concept coherently. It can be argued that Usiskin’s dimensions of understanding have a strong link to the construct of meaningful learning because of their inter- connectedness when applied to a mathematical concept. This is because core to meaningful learning is pre-requisite knowledge and new knowledge which should be integrated for effective learning to occur.
It can be pointed out that there is no clear cut meaning of the construct of understanding in Mathematics education hence the construct of understanding is an ongoing debate. Nonetheless, in this study, understanding is considered as a process where learners make appropriate connections between their experiences and new knowledge.
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43 ÷ 43 = 4(3 – 3) (rule of division of powers of the same base) = 40 =?
But 43 ÷ 43 = 4 × 4 ×4
4 × 4 ×4 (using ordinary expansion) = 1 (cancelling)
∴ 40 =1, hence a 0 = 1 Category 2: Need for causality
This need, as Harel points out, goes beyond achieving certainty. It seeks to explain or justify mathematical ideas rather than removing doubts as in the need for certainty. In brief, the need for causality provides a reason or causes for truth – the cause that makes the idea true. If we say some idea is true, then the question is: Why? Learners must therefore seek to understand the explanation within the mathematical discipline. As an example, let us consider the following different workings (A, B and C) to finding a solution to the problem 2
3 + 1
5= Working A: 2
3 + 1
5
= 2×5+1×3
3×5 (cross multiply numerators & denominators;
multiply denominators together)
= 10+3
15
= 13
15
Working B: 2
3 + 1
5
= (15÷3)×2+(15÷5)×1
15 (find LCM of 3 & 5; divide LCM by each of the denominators and multiply the quotient by the respective numerator)
= 10+3
15
= 13
15
Working C: 2
3 + 1
5
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= 2
3×5
5+ 1
5×3
3 (converting fractions to equivalent fractions with the same denominators)
= 10
15+ 3
15
= 10+3
15 (add the numerators & keep the denominator as in addition of fractions with same denominators)
= 13
15
All the workings above are mathematically correct, but Working C reveals the reason or rather cause for why convert the fractions to equivalent fractions with same denominators. This reason has more intellectual value than the algorithms in Workings A and B hence provides both certainty and understanding of cause.
Category 3: Need for computation
This is where symbolic algebra is used to quantify and calculate values of quantities and relations (Harel, 2013). According to Harel, in the need for computation, learners represent everyday situations or experiences into symbols. They then manipulate the symbols as if they have a life of their own and use them to perform some computations. Basically, the need to compute refers to the learners’ desire to quantify, manipulate, and compute by means of symbolic algebra. Harel’s concept of the need for computation is strongly linked to the Realistic Mathematics Education (RME) notion of horizontal and vertical mathematization where learners solve mathematical problems situated within everyday situations or problems within Mathematics as a subject discipline.
Category 4: Need for communication
According to Harel, this need is divided into two reflexive needs and calls them the need for formulation and the need for formalization. In essence, the need for communication occurs in a Mathematics discourse. A mathematics discourse includes how we use language to listen to Mathematics, act in a Mathematics class and use the Mathematics register (Gee, 1996). The Mathematics discourse develops out of formal and informal communication of mathematical ideas. By the need for formulation, therefore, Harel refers to transforming spoken language into algebraic expressions. One example of the
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need for formulation is the use of “the difference between eight and three” in spoken English which can be translated into the mathematical expression 8 – 3 and can be easily computed. Whilst by formalization Harel refers to the act of externalizing the exact intended meaning of an idea or concept. In this way the learner gains control over the idea so as to be able to talk like a mathematician (Pimm, 1987). Pimm calls this a Mathematics register. According to him a Mathematics register is a set of meaning that belongs to language of Mathematics. This set of meanings that constitutes a register does not refer only to words and structures but also to the styles of meaning and modes of argument.
For example the phrases/words ‘take away’, ‘top heavy’, ‘combine’, ‘divide into’, etc. have an everyday usage or meaning and an altered meaning or grammatical meaning in a mathematical discourse. When learners are learning Mathematics in school they are therefore attempting to acquire communicative competence in the Mathematics register.
Category 5: Need for structure
Harel mentions that the need for structure includes the need to reorganize learnt ideas or concepts into a logical structure. This need resonates with Piaget’s interrelated concepts of assimilation and accommodation as discussed earlier in the chapter. Harel’s notion of ‘reorganize’ is synonymous to accommodation which is the process by which a learner’s existing schema is modified to fit incoming ideas or concepts. The verb ‘organize’ in ‘reorganize’ means there are some ideas or concepts already existing as in Piaget’s pre-existing schema.
From the discussion thus far, what stimulates intellectual need depends on learners’ reinvention of mathematical ideas with the guidance of the teacher which is analogous to the notion of RME. Here, the teacher needs to facilitate and guide the learners as they attempt to reach conceptual understanding. And all Harel’s constructs of intellectual need would lead to meaningful learning.