3.3 MEANINGFUL LEARNING
3.3.3 Understanding
From a constructivist perspective on learning about Piaget (1970), learners construct knowledge by themselves not by swallowing ready-made knowledge from the environment. Furthermore, knowledge does not simply arise from experience; rather it arises from the interaction between a learner’s experience and his/her pre-existing set of knowledge. The learner is therefore not seen as passively receiving knowledge from the environment. The learner is an active
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participant in the construction of his knowledge (Hatano, 1996). However, the focus in this section is on the issue of learners’ understanding in Mathematics education.
Skemp (1976) distinguished two kinds of understanding and he called them instrumental understanding and relational understandings. By instrumental understanding, Skemp referred to rules/procedures without meaning/reasons.
This means that the learner would perform some computations/calculations without some justified explanation(s). In essence, this is some kind of understanding where rules, methods, or algorithms (mathematical procedures) are applied to mathematical problems which give some quicker results for the teacher in the short term (Skemp, 1976). During instrumental understanding, Skemp mentions that no attempt is made to link the new concepts with what has been learnt previously. By relational understanding Skemp referred to the understanding that is associated with many other existing ideas in a meaningful system of mathematical concepts and procedures. This means that the learner knows what to do and has reasons for doing that which is more beneficial in the long term and also aids motivation (Skemp, 1976). It can be argued that Skemp’s construct of relational understandings have some relevance on meaningful learning in learner-centred practices. The two constructs can be explained using the example below.
Suppose the learner is introduced to the concept of adding fractions with different names/denominators using the problem 2
3+1
4= _____ as an example.
The table below illustrates the differences in the different approaches.
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Table 3.2: Relationship between instrumental and relational understanding Instrumental understanding Relational understanding
2 3+1
4 =
To find the sum of the fractions, find the L.C.M. of the numerators (3 & 4).
Divide the L.C.M. by each of the
denominators and multiply each dividend by the corresponding numerator. Add the results and make sure they share the same denominator (the L.C.M.).
2 3+1
4
=8 + 3 12 =11
12
2 3+1
4 =
Learner converts the fractions into equivalent fractions with same denominators and adds them just like when adding fractions with same
denominators.
2 3+1
4= 8 12+ 3
12
=8 + 3 12 =11
12
From the above table, the learner who relies on instrumental understanding only would apply rules/procedures without meanings and explanation while the one with relational understanding would make links with other procedures or make conceptual connections (i.e. adding fractions with same names) to work out the problem. Skemp (1976) asserts that:
Learning relational mathematics consists of building up conceptual structure (schema) from which its possessor can (in principle) produce an unlimited number of plans for getting from any starting point within his schema to any finishing point (p. 20-26).
Skemp’s assertion implies that relational understanding involves connecting concepts. He also claims that instrumental understanding is useful when a learner knows how to do a specific task quickly, and is not too concerned about how this task fits into other concepts. This is what Olivier (1989) refers to as rote learning. Olivier argues that in rote learning the new knowledge is so different from any available schema such that it is impossible to link it to any existing schema i.e. neither assimilation nor accommodation is possible. Olivier elaborates that in rote learning; the learner creates what he terms a new ‘box’
and tries to memorize the new knowledge. Though the memorized knowledge may be used say to recall sequences of objects such as cell phone numbers, it can be argued that there is no understanding in rote learning because the current knowledge is not linked to any pre-existing knowledge. Also, this kind of learning (rote learning) does not serve learners well when they need to apply
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Mathematics to solve problems outside of school work or when they need to apply their mathematical knowledge to learn more advanced Mathematics (Feikes, Schwingendorf, & Gregg, 2009). On the same idea of current and pre- existing knowledge, Ausubel et al. (1978) contend that rote learning occurs if the learner lacks the relevant prior knowledge necessary for making the learning task potentially meaningful. However, they caution that rote learning can also incorporate new knowledge into the pre-existing structure but without interaction. It can thus be noted that there is no meaningful learning in either instrumental understanding or rote learning.
Whilst in the 1970’s Skemp had described knowledge outcomes in the teaching and learning of Mathematics in the education literature as instrumental and relational understanding, but since the mid 1980s the most predominant perspectives of knowledge outcomes have been conceptual knowledge and procedural knowledge (Star & Stylianides, 2013). The terms conceptual and procedural knowledge may be viewed as being extensions of Skemp’s original constructs of instrumental and relational understandings. According to Star &
Stylianides, conceptual knowledge denotes knowledge of concepts that involve a coherent of principles and definitions that learners can apply to different contexts. Furthermore, conceptual knowledge is related to meaning and making connections between different ideas. With regards to procedural knowledge, Star and Stylianides (2013) point out that it denotes knowledge of procedures that involve action sequences, rules and algorithms used to solve mathematical tasks or problems. Basically, procedural knowledge denotes the use of mathematical rules without necessarily knowing the reasons why or how the rules work.
Unlike in Skemp’s notion of instrumental and relational understanding, the constructs of conceptual and procedural are viewed as forming a knowledge web within each. This means that the knowledge within each one of them is interrelated either in principles (in the case of conceptual knowledge) or action sequences (in the case of procedural knowledge). However, the action sequences are such that doing one step triggers the next step in the sequence as described in Action, Process, Object and Schema (APOS) theory. Hence the knowledge web triggered by procedural knowledge is just a sequence of how different
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actions lead to the next. It can therefore be argued that the construct of conceptual knowledge has a strong connection with meaningful learning during learner-centred practices. It is conceptual knowledge that fosters meaningful learning. Conceptual knowledge would value the aspect of prior knowledge during the teaching and learning of Mathematics because of the inter-connection of knowledge within it.
On the same construct of understanding, Usiskin (2012) acknowledges Skemp’s arguments about instrumental and relational understanding but he differed slightly from the latter’s stand-point. He says that he agrees with Skemp that instrumental understanding and relational understanding are different but he disagrees that they are different objects. Usiskin argues that he views them as different aspects of understanding the same mathematical concept. He detailed five strands of understanding a mathematical concept from the learner’s perceptions and he called them dimensions of understanding a Mathematics concept.
We view there to be at least five aspects to this understanding.
In this view, a person has full understanding of a mathematical concept if he or she can deal effectively with the skills and algorithms associated with the concept, with properties and mathematical justifications (proofs) involving the concept, with uses and applications of the concept, with representations and metaphors for the concept, and with the history of the concept and its treatment in different cultures” (Usiskin, 2012, p. 19).
Now the question is: What does Usiskin mean by the dimensions of a mathematical concept? Below is a description of the five notions of Usiskin’s dimensions of understanding.
Dimension 1: Skills-algorithm understanding
According to Usiskin this understanding is where different learners may exhibit different ways of getting to a correct solution of a problem. The learners’
understanding may be influenced by prior knowledge that they possessed.
Dimension 2: Property-proof understanding
This is about identifying and using appropriate mathematical properties when working out a problem. For example, when learners are asked to work out the
50 problem 𝑦
2 4 ÷ 1
6𝑦3 , they need to use the concepts of reciprocal, the product rule of indices and simplifying common fractions. Working out the problem therefore is not arbitrary but requires the identification and use of different mathematical properties.
Dimension 3: Use-Application understanding
According to Usiskin, in Mathematics, a learner needs to deal effectively with understanding both the concept and its application. The application of the concept should not be viewed as higher order. Usiskin points out that application problems do not necessarily require higher order thinking and he strongly believes that such problems basically require a different kind of thinking.
Dimension 4: Representation-metaphor
Whilst Usiskin acknowledges the importance of the Dimensions of understanding 1, 2 and 3, however, he mentions that they do not carry the actual true understanding of Mathematics. His argument is that learners should represent mathematical concepts pictorially or display them concretely. For example when learners are given 7 + 5 to work out, they can make an illustration of the sum using diagrams or use counters to demonstrate their understanding.
Dimension 5: History of the concept
This is a dimension of understanding which according to Usiskin is about the history of the concept and its treatment in different cultures. Some mathematical concepts are understood as per their cultural origin, for example, the origin of Ethno Mathematics from different cultures. Also, different countries represent some mathematical symbols in a different way and one example is the way some countries represent coordinates. Some countries represent coordinates as (9, 8) whilst others as (9; 8). Both notations represent different dimensions of understanding.
Usiskin asserts that the five aspects of dimensions of understanding are connected when applied to a particular mathematical concept and they can be mastered independently of each other. According to him, learners come to understand a mathematical concept if they can deal effectively with all the five notions of dimensions of understanding. His constructs of dimensions of
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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.