Applying Learning Theories to Mathematics Education Glenn Mason-Riseborough (5/5/1997) Introduction Theories of learning are created by people in many different fields of research, professions, and cultures. While many of these theories are mutually exclusive or contradictory, it would be far too simplistic to say that there is only one absolute way of learning. Some theories emphasise neurophysiological or behavioural aspects while others come from a cognitive or social developmental perspective. One way to analyse these theories is to see how well they model real everyday situations. This essay will discuss an episode of learning mathematics, and then show how it is explained by one theory of learning. Since I have never taught mathematics in schools or otherwise, I will use a personal experience in which I learned an aspect of mathematics while I was at Primary School. I will then briefly describe aspects of Jean Piaget’s cognitive developmental theory, which proposes that cognitive conflict is resolved through the building and modifying of cognitive structures which Piaget called schemas. What will be discussed will by no means be a complete discussion of Piaget’s theory, or even of this particular aspect of his theory. This interpretation however, will be complete in such a way that it can explain my episode of learning. Thus the final part of this essay will describe what happened during my learning episode in terms of Piaget’s theory of cognitive structures as I have interpreted it. My episode of learning mathematics One of the few episodes in which I consciously remember not understanding an aspect of mathematics and then coming to an understanding, occurred when I was at Primary School. I cannot remember what year I was in, nor can I remember my teacher at the time since no-one else was directly involved in my learning experience. I estimate I was in about Standard 2 or 3 at the time so that would make me about 8 or 9 years old. I remember we used to frequently work out of textbooks, solving problems as set by the teacher. On one particular occasion we were using a new set of textbooks in which the problems were numbered differently from the previous textbooks we had used. These problems were numbered with the chapter first followed by a decimal point and the exercise number, for example chapter 3 problem 12 would be numbered 3.12. Thus the exercises were numbered (for chapter one) 1.1, 1.2, ..., 1.9, 1.10, 1.11, .... This numbering system confused me a great deal because I knew (or at least had been told) that 2.0 comes after 1.9 (one decimal place), and that 1.10 is equivalent to 1.1. I was also confused by the fact that 1.10 was called ‘one point ten’ as opposed to ‘one point one oh (or naught or zero),' as I thought it should be. Perhaps this is not strictly an episode of mathematics learning since my mathematics was correct, however it challenged my perceived usage of the decimal point. In response to the book numbering, I think I realised that there was some sort of distinction between it and regular decimals, and I was able to keep them separate in my mind. I did not understand why the same symbol should be used in a different context, but a textbook was powerful evidence that there was a distinction. As for the way in which the problem numbers were verbalised, I think I concluded that my classmates were wrong and I was right in how it was spoken. Over time I came to realise that the decimal point can and does get used in many different contexts including use as a separator, in decimal numbers, multiplication, and the logical AND. Each of these different contexts mean that the numbers to the left and right are treated and verbalised in a different way. Piaget’s theory of schemas and equilibration This aspect of Piaget’s theory states that we create mental frameworks called schemas or schemes that allow us to understand the world and perform tasks in a consistent manner. These schemas can be built upon and modified to adapt to new information. Learning and development occur in stages (1) in which an equilibrium is reached between environmental situations and the cognitive structures, processes and capabilities of the individual involved (Sternberg, 1994, pp 423 - 424). The process of returning to equilibrium is an active one which is called equilibration. When new information is received, it causes cognitive disequilibrium (or perturbations or cognitive conflicts) if it does not fit in with the existing schemas. Equilibrium is returned by either adding the new information to an existing schema (assimilation) or altering a schema to fit the fact (accommodation). Alternatively, both assimilation and accommodation can occur at the same time. This is called reciprocal assimilation and accommodation and it occurs when the new idea is seen differently, while at the same time the old schema is modified. There are three main reactions to the conflict; these are called alpha, beta and gamma (Irwin, 1997, pp 1 - 2). Alpha is the situation in which the schemas are not modified. This is either because (i) the fact was added, but it did not cause conflict because it was not fully understood, or (ii) the fact was ignored because there did not seem to be a way of incorporating it. Situation beta involves modifying the schema to fit the new information. Gamma occurs when the schemas are not modified because they are flexible enough to integrate the new fact. Analysis of my learning episode My episode of learning as described in this essay can be described in terms of the above interpretation of Piaget’s cognitive development theory of schemas. It can be seen that cognitive disequilibrium occurred when I saw the decimal point used in a context with which I was not familiar. I was expecting the decimal point to be used as an indicator of decimal numbers. This usage would have conformed with the schema containing rules of decimal numbers that I had been taught in the past. Instead, these numbers seemed to follow a different set of rules, and I was unable to fit these rules to the current schema. There was actually two separate pieces of information to analyse before equilibrium could be returned. The first new fact, was the observation of the textbook. For this fact I reacted in a beta manner -- I compartmentalised the fact by realising it applied to a particular context. Accommodation occurred because I was able to change my schema of decimal points to include textbook problems. Instead of saying decimal points are used in decimal numbers, I now understood decimal points were used in decimal numbers and also textbook problems. I also understood that textbook problems did not have to comply with normal decimal ordering. The second new fact was the verbal one, regarding how the number should be read. For this I reacted in an alpha (ii) manner. I neither used assimilation nor accommodation to return to a state of equilibrium. Instead I ignored the problem by thinking that my way was right -- I did not have a problem. I thought everyone else was wrong, that it was them with the problem. At the time my schemas were not flexible enough to integrate the information fully. As my mathematical knowledge increased, my schemas became more flexible and I was able to look back at this memory and integrate the knowledge without changing the schemas (gamma reaction). In her paper Irwin (1997) discusses Piaget’s theory from an educational point of view. She states that the gamma reaction could be seen as the goal of learning. This is because the schemas have become flexible enough to integrate the new information in a particular domain. It is the beta reaction however, that is the most useful in a classroom. This is the reaction that through conflict, leads to a better understanding of the concepts. Alpha reactions should be avoided if possible. They may provide the learner with self-confidence but they do not extend the learner’s knowledge. In this sense the theory is an indicator of the usefulness of activities in the classroom. If we can pinpoint the type of reaction that occurred from a certain activity then we can analyse the usefulness of the activity. Looking at my learning episode, I had all three reactions to the information (although gamma occurred much later). This does not necessarily occur, but it shows that even in a so called ‘learning episode,’ there may be ‘sub-episodes’ in which some concepts are understood and others are not. Had my teacher been aware of my confusion (through either observation or me asking for help), he (2) may have been able to explain why the numbers are verbalised as they are. He may have been able to create a beta reaction to the conflict instead of alpha (ii), and I would have thus come to an understanding quicker. Conclusions Piaget’s cognitive developmental theory is able to explain my episode of learning in terms of schemas. It does so in such a way that we can see how my understanding improved as I was introduced to new and conflicting information. It showed that I was able to absorb some information, but other information was ignored until my knowledge base was wide enough to accept it. Piaget’s theory also describes the goals of learning from an educational perspective. Long term goals can be seen as creating flexible schemas that can absorb a wide variety of new information. A short term goal is to create cognitive conflict within individuals to enable their schemas to be modified into more flexible frameworks. Denial of a fact is undesirable except to increase the learner’s self-confidence. Having said that schemas have strong explanatory power in terms of long term memory acquisition (3), there are a number of problems associated with them (Eysenck & Keane, 1995, pp 261 - 270). One such problem is that we cannnot know the contents of these schemas or even how the schemas are formed. This gives us a theory that can explain results in an ad hoc fashion, but is not very predictive. References: Eysenck M. W. & M. T. Keane (1995). Cognitive Psychology: A Student’s Handbook (3rd edition). Psychology Press. Irwin, K. (1997). Paper prepared for the Twentieth Conference of the Mathematical Education Group of Australasia, Rotorua, 7 - 11 July 1997. Sternberg, R. J. (1994). In Search of the Human Mind. Harcourt Brace. Footnotes: 1 These stages are distinct from Piaget’s four discrete cognitive developmental stages; namely the sensorimotor, preoperational, concrete- operational and formal-operational stages. 2 While it is normal to speak of an unknown person as he/she or she/he, in this situation I am sure my teacher was male since I had male teachers in both Standard 2 and Standard 3. 3 There is strong evidence to suggest that schema-like knowledge structures do exist. This evidence comes from a wide range of research areas (see for example Eysenck & Keane, 1995, p 265).