Distributed cognition

Proceedings of the nd 2 National Conference on Graphing Calculators October 4−6, 2004 • 93−103

DISTRIBUTED COGNITION AND THE USE OF GRAPHING CALCULATORS IN THE LEARNING OF MATHEMATICS Pumadevi Sivasubramaniam Mathematics Department, Maktab Perguruan Raja Melewar, 70400 Seremban, Negeri Sembilan, Malaysia. e-mail: [email protected]

ABSTRACT Graphing calculators are powerful tools that eliminate the drudgery of tedious computations and algebraic manipulations. Despite the facilities provided by this tool, the acceptance of the use of graphing calculators in the mathematics classrooms are greatly impeded by false beliefs. This paper aims to dispel these beliefs and highlight the advantages of using graphing calculators in the learning of mathematics in the light of the theory of distributed cognition.

1 INTRODUCTION Electronic calculators were invented over 40 years ago. They were initially machines that performed four basic operations and today, they can execute highly-technical algebraic symbolic manipulations instantly and accurately, translate from one form of mathematical representation to another (eg. from equations to tables to graphs) and even produce mathematical representations concurrently with data collection from the real world. All these advances in a calculator have rendered the production of a breed of calculators which we know as graphing calculators or graphic calculators. The graphing calculator as its name illustrates is different from its ancestors, the calculator and later the scientific calculator because of its graphing facilities and all the other computation and manipulation facilities that it is capable of performing on command. With the advancement in the capabilities of the calculator, is the accompanying enhancement of the fear to use these machines in the mathematics classroom for the learning of mathematics. This unfounded fear stems from false beliefs about the learning of mathematics.

2 GENERAL BELIEF OF TEACHERS AND PARENTS ABOUT CALCULATORS Many teachers and parents believe that using technology deprives students from employing their brains to perform computations in mathematics. The problem being, in the old days when powerful tools such as calculators were unavailable to do the routine calculations, the mathematics education curriculum had to be structured to

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produce "human calculating machines", just to crunch numbers meaninglessly. In fact the four figure tables’ book employed at the time I was in school, was a tool used to do long multiplication and division. Many of us did not have a clue as to how the tables in the so called "log" book were created. But, no one complained because they felt that students were doing what their great grand fathers and their other ancestors before them did. Students were still adding, subtracting, multiplying and dividing – so that was fine. But with calculators – no one knows how these machines work except a specific few − like the "log" book, but this is different – students just pressed buttons. They are not doing the mathematics that was traditionally done. Everything is done by the calculator. Many believe that the traditional paper and pencil method should not be replaced. Parents and teachers who argue against the use of calculators, graphing calculators or computers in mathematics classes simply fear baselessly. They have recollection of their mathematics as consisting of drills and lots of algorithms which were meaningless to them [1]. The purpose to learn all these meaningless tedious computations and manipulations served no purpose except to perform the same feat in their examinations. Even till today children have problems tackling word problems related to the real world. The simple reason being mathematics is taught in isolation without the real world in context – just using numbers to do the four basic operations. We expect students to relate it to context. This results in poor performance and the claim that children lack the ability to tackle word problems. A child not knowing when and where to apply the mathematics learnt in school in the real world has not learnt any useful mathematics. Rote computations and tedious algebraic manipulations have always turned students away from mathematics. The traditionally taught mathematics, involves memorizing formulas and performing long tedious computations. Students who could perform these tasks quickly and accurately were said to be mathematically inclined, while those who did not were branded as poor in mathematics. A student who can do all the computations and manipulations in calculus is said to be excellent in calculus. Just ask them where and how they would use it to solve real world problems and their expertise in calculus would be revealed. In fact all context free mathematics are tools to solve real world problems. For example, if a child has learnt to solve question (a) below but is unable to solve question (b), then obviously the student has learnt a computation which is useless to him or her.

(a) 6 ÷ 2 = 3 (b) You need six nails. You have several bags of nails. Each bag has 2 nails. How many bags of nail do you need?

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He or she may excel in the context free division section of an evaluation but has a useless tool. Likewise with solving a question on differentiation or constructing a graph or performing any other mathematical task in any other area of mathematics but being unable to solve a real world problem. To understand the role of graphing calculators in the learning process of mathematics, an understanding of distributed cognition will provide the reasons to allow routine mechanical tasks to be performed by machines. It will also illustrate to the reader why the use of technology such as graphing calculators will not impede the learning of mathematics.

3 DISTRIBUTED COGNITION The theory of distributed cognition was developed by Ed Hutchins and his colleagues at University California, San Diego in the mid to late 80s [2]. This theory claims that cognition is better understood as a distributed phenomenon, in contrast to the traditional view of cognition as a localised phenomenon that is best explained in terms of information processing at the level of the individual. Salomon, Perkins and Globerson [3] adopting this phenomenon summarize cognitive effects when using technology as "effects with technology obtained during intellectual partnership with it, and effect of it in terms of the transferable cognitive residue that this partnership leaves behind in the form of better mastery of skills and strategies". To explain distributed cognition as “effects with technology”, I turn to Döfler’s [4] view of cognitive processes (which also adopts Hutchins’ view of cognition). Döfler [4] suggests that cognitive processes be viewed as a system made up of the individual, the whole context and the multiple relationship between them. Thus, the cognitive system has the subject (the individual) and the available cognitive tools which would aid the thinking process. Cognitive tools can be paper and pencil, calculators, computers, graphing calculators, television, etc. Döfler [4] compared the thinking process to doing physical work stating that "There is no such thing as ‘pure’ work without using any tool." To attain the specific goals, one has to use tools in an appropriate organized manner. To illustrate the thinking process as a system, Döfler [4] used the artist as an analogy. Döfler stated: The skill and the intelligence of an artist like a painter are more appropriately viewed as being realized by the whole system consisting of the human individual and all his tools. These tools do not just express ideas and imaginations pre-existing in the mind of the artist and independently of the tools. Rather, the system of painter, brush, colours, canvas, etc. realizes the painting [4, p.173].

The thinking process can be explained in terms of "distributed cognition". Distributed cognition refers to the earlier described "system" – the individual and the available tools where cognition is viewed as distributed over them [4]. According to

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this view of thinking, to solve a given mathematical problem, the individual can employ the available tool and his or her own mind to solve it. To explain distributed cognition in mathematics, let us take for example the drawing of a straight line of a specific length, for example 5 cm in length. Using a ruler one can produce the 5 cm straight line. The individual need not have the skill of drawing a straight line unaided (that is free hand), nor does the individual need a mental representation of 5 cm. If the use of a ruler is not permitted, then fewer people will be able to draw a 5 cm straight line if they lack either the skill to draw a straight line or the appropriate mental representation of a length of 5 cm. With the use of a ruler, cognition is distributed in the process for producing a 5 cm straight line – the mental representation of 5 cm and the skill to produce a line which is straight are taken over by the ruler and the individual has to have knowledge about how to use a ruler to produce the 5 cm straight line. To explain the “effects of technology”, I turn to Pea’s [5] view that "intelligent" technology “offloads” part of the cognitive process as a result of distribution of cognition, allowing the user to focus cognitive resources elsewhere. Pea claims that over time the user will develop cognitive skills to accomplish many of the cognitive processes demonstrated when using technology and would be capable of demonstrating these skills without any longer requiring the aid of the technology. Let us refer again to drawing a straight line. Using a ruler enables an individual to draw straight sided shapes easily. Rather than expending cognitive processes on drawing straight sides of polygons, the individual can concentrate on studying the characteristics of the various polygons. Later the individual would be capable of recognizing the different polygons – a skill which obviously does not require the tool any more. Similarly using a graphing calculator to draw graphs based on given algebraic equations affords students more time to study the characteristics of graphs in relationship to their equations. This then would enable them to make a reasonable guess of the equation that a given graph represents without the aid of the graphing calculator. Now let us examine how the theory of distributed cognition can be used to explain the use of graphing calculators in the learning of mathematics.

4 DISTRIBUTED COGNITION AND THE GRAPHING CALCULATOR Distributed cognition is related to graphing calculators, computers and in fact any tool that is employed in the process of learning, including learning mathematics. To explain distributed cognition in relationship to graphing calculators let us focus mainly on one area of mathematics – graphing skills. A facility provided in all graphing calculators as its name suggests. Occasional reference to other areas may be made as deemed appropriate.

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When learning a skill in mathematics as in any other subject, it is important to know its purpose. Likewise when drawing graphs, whether employing the paper and pencil or the graphing calculator, the purpose of learning the skill is important. A graph is not drawn to produce a line (straight or curved or complex). A graph is drawn to represent the covariance of two variables. Now the next question arises: How do I produce the graph? To produce the graph in the paper and pencil medium, one has to have knowledge of scales, plotting points on a Cartesian coordinate system, joining the points, etc. to display the covariance of two variables. On the other hand, to produce a graph using a graphing calculator, one has to have a different kind of knowledge − which buttons to press, how to zoom in and zoom out, how to manipulate the scale of the axes, etc. In both cases the knowledge of how to organize the respective tools to produce the graph is essential but the process to producing the graph in the paper and pencil medium is more tedious while in the graphing calculator medium it is less tedious, less time consuming and more efficient and accurate. This is because when using the paper and pencil medium, only a small part of the cognitive process is taken over by the tools such as the square grid of the graph paper used provides for the equal spacing of points. Rulers and or curved rulers aid one to join the points to form the graph. The scaling, what goes on what axes, determining the range, plotting of points, etc. − a larger part of the cognitive process is carried out by the individual. When using the graphing calculator on the other hand, a larger part of the cognitive process is taken over by the tool such as scaling, plotting points, etc. – a lesser part of the cognitive process such as deciding what goes on what axes and determining the range, etc. are determined by the individual. Hence the graphing calculator is a more powerful tool compared to the traditional paper and pencil medium to construct graphs. What is employed to produce the graph is not of significance in the learning of graphing skills. What is important is that a graph is produced to display the covariance of two variables. The more powerful the tool, the greater part of the process leading to construction of the graph is taken over by the tool. Graphing calculators are such powerful tools. As Döfler [4] stated, to attain specific goals, one has to use tools in an appropriate organized manner. Thus in the construction of graphs, the goal is to display the covariance of the two variables. Scaling, plotting points, joining the points, etc. are all processes involved in attaining this goal. With the use of graphing calculators, students can concentrate mainly on the purpose of constructing the graph. When learning graphing skills traditionally, it is important to know how to produce the graphs using appropriate scales, plotting points, etc. These processes are taken over by the graphing calculator. Thus scales, plotting points, etc. are no longer skills required to be mastered to produce a graph, just as one need not have a mental representation of 5 cm to draw a 5 cm straight line.

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With the knowledge of what is meant by distributed cognition, let us examine the extent the taking over of a larger part of the cognitive processes in the learning of mathematical concepts and skills with the use of graphing calculators benefits the learning process in mathematics.

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When a larger part of the cognitive process is taken over by the graphing calculator, the following effects may be realised: 5.1 More Individual Will Be Able to Perform Mathematical Tasks A research carried out in three secondary schools in Negeri Sembilan involved Form 1 students who had difficulty constructing graphs in the paper and pencil medium. The difficulty was found to stem from their lack of understanding scales involving decimals and fractions. The same group of students had no problem constructing graphs using the graphing calculator. They also could read points on the graph using the trace facility which again did not demand the understanding of scales [6]. This study supports the claim that more students would be able to perform mathematical tasks with the use of graphing calculators. As was stated in the previous section, cognition is distributed in that the cognitive processes required in using the appropriate scales, etc. is carried out by the graphing calculator, the purposes to display the covariance of the two variables and which variable is to be represented by which axis, etc. is decided by the individual and the product – the graph is displayed on the graphing calculator screen. Hence difficulties in understanding scales, plotting points and difficulties assimilated from other related concepts of mathematics for an individual will not deter him or her from constructing graphs using the graphing calculator nor reading points on the graph. The distribution of cognition when using graphing calculators, enable more pupils to construct graphs, just like the distribution of cognition with the ruler enables more people to construct a 5 cm straight line. Hence distribution of cognition illustrates that using a more powerful tool like the graphing calculator in the learning of mathematics enables more individuals to learn new mathematical skills irrespective of their incompetency in other related areas of mathematics. 5.2 Enables More Individuals to Understand the Purpose of Performing a Mathematical Task and to Apply Mathematics As stated earlier, that the major advantage with a larger part of the cognitive process for construction of graphs being taken over by the graphing calculator is that it affords students more time to focus attention on the purpose of drawing graphs. In contrast, in the traditional paper and pencil medium attention is first focused on construction of the graph which itself makes great cognitive demands and is time consuming, leading one to belief that it is an end in itself. To go on to focus attention

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on understanding the purpose of drawing the graph requires students to switch attention from construction to interpretation. The act of switching attention to an event according to Norman [7] may both "blur perception and cause confusion in one’s judgement of its temporal properties". Hence, the distribution of cognition in the graphing calculator medium requires pupils to focus attention only on one aspect and hence enhance the understanding of the purpose of performing the mathematical task. Several small scale studies by Monaghan and Etchells [8] reports that students using technology with CAS (Computer Algebra System) have a better understanding of integration than students following a traditional course and that they are able to use this understanding to reason correctly when integration is applied to new areas in mathematics. Similar claims have been stated by Waits and Demana [9] about the use of graphing calculators in understanding algebra. This is also true of other computations which demand a long pencil and paper process to perform compared to using a graphing calculator which speeds up the computation process. Using a graphing calculator makes less cognitive demands on individuals hence enabling them to focus attention on the problem to be solved rather than the routine mechanical manipulations that require the switching of focus of attention from the problem to the computation and then back to the problem. An example of a numerical calculation which is tedious but is crunched instantly with the use of graphing calculators or computer software is the calculation of the t-value. This very lengthy and time consuming calculation which is used frequently today to analyse research data is an example of a numerical value whose interpretation plays an important role when inferring about relationships between sets of data, which in the past would have been dared only by a minority of individuals claimed to be mathematically inclined. Today anyone who decides to use it may employ it without having a clue of the immense computation behind it by employing a statistical software package. Hence with calculators taking over tedious computations and other mathematical tasks that discourage many students from learning mathematics, calculators will make it possible to allow for more students to work application problems with numerical values that would otherwise have been virtually impossible and appreciate the application of mathematics. 5.3 Provides Concrete Models of Abstract Mathematical Concepts Another advantage of calculators being able to take over a large part of the cognitive process is related to the view that mental processes have concrete and imaginistic basis. Dofler [4] claims that there is empirical evidence that construction of mental representations in a concrete form enables an individual to solve mathematical problems. Cognition was and is till today supported with the use of diagrams, drawings, physical models, etc. Dofler [4] states that experts have more effective and adequate mental representations which the expert can apply to a given problem situation. The graphing calculator can be used to help the individual learner construct these mental models. Graphic systems, simulations and representations can act as 99

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substitutes for human imagination and help to enlarge and reorganize the cognitive system. For example, a research done by Thornton and Sokoloff [10,11] claims that real time graphing which translates actions in the physical world directly to graphical representations, lead not only to understanding of graphs but also aids understanding of physical concepts. Similar claims are made by other studies [12−16]. Pupils who have problems grasping graphing skills may not have adequate mental representation of how the co-variance of the variables in the physical system relates to graph structure. The use of graphing calculators and real-time graphing can compensate for the human imagination or inadequacy of the mental representation to link the concrete co-variance of the variables in the physical context to the abstract graphical representation. By compensating for human mental inadequacy the graphing calculator through distribution of cognition is able to evoke knowledge construction, which is more meaningful – a distinct feature of constructivism. Hence the graphing calculator taking over a large part of the cognitive process is able to provide more students with the appropriate concrete model of abstract mathematical concepts. 5.4 Supports Discrimination Learning The graphing calculator as discussed in the previous section is able to provide experts representation resulting in meaningful learning. The next stage to consider in the learning process is retention of what has been learnt for a longer period of time. The distribution of cognition with the larger part of the cognitive process being taken over by the graphing calculator results in more of a topic to be covered in a shorter span of time. This is contrary to the traditional learning process which requires mastery of a small part of a topic before learning the next part. For example, when students learn the topic graphs of functions, they learn linear graphs first, then quadratic and finally cubic graphs in our curriculum up to the upper secondary level. This would take over a week to complete. Often after all the lessons have been completed, to the disappointment of teachers, most of their students are unable to predict the equation of a given graph or vice versa. A study [16] in Negeri Sembilan involving three secondary schools revealed a statistically significant difference between the group using graphing calculators and the group using scientific calculators in determining functions of given graphs and vice versa in the paper and pencil medium without the aid of the calculators. The scientific calculator group took three days of two hour sessions per day to perform the same set of tasks of constructing linear and quadratic graphs as the graphing calculator group which only took two one and a half hour sessions for the same set of tasks. This study reveals that speed, accuracy and the systematic dynamic manner in which the graphs are produced by the graphing calculator enabled students using it to be more observant and in turn retain longer the characteristics of the respective graphs. It also had developed a skill in these students to observe graph structures in relation to the equations of the functions and vice versa – the effects of working with graphing calculators.

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This is in agreement with Klix’s [17] claim that knowledge may be perceived through observation of an event and this knowledge can be stored in the human memory. The knowledge stored can only be reproduced to the degree to which the observation was made. In the early stages of a child learning, singular objects or pictures are displayed to the child and their phonological properties, that is words or phrases are used to represent it. This form of learning is also used in learning among older children and even adults. For example, the following graph structure may be described by the phrase "y increases by greater and greater amounts with constant increases in x". If the pupil perceives the graph structure as a sign to replace its phonological properties – the phrase, then the next graph structure encountered will be perceived as a separate entity from the first graph structure.

y

x

Figure 1. Graph of phrase "y increases by greater and greater amounts with constant increases in x"

On the other hand, if the graph structure is comprehended as structures related to the changes in gradient then it will enable the student to retrieve the structure because the phrase is related to the structure meaningfully. For this to occur, Estes [18] states that attention must be shifted from irrelevant components to relevant components of the event. This is a form of learning called discrimination learning. Estes [18] claims that short intervals between experiences enhances this shift. The individual is more likely to note the relationship between the two experiences if the interval between the experiences are short. For example, introducing one graph structure followed by the next provides opportunity for the individual to note the relationship between the structures and phrases used to describe them. Hence, once again the distribution of cognition with the larger part being taken over by the graphing calculator enhancing efficiency and hence producing experiences in a shorter interval of time is more likely to enhance retention through discrimination learning compared to the paper and pencil medium.

6 CONCLUSION The theory of distribution of cognition has highlighted the role of the graphing calculator in the learning of mathematics. However, it is important to mention that the facilities and capabilities of the graphing calculator or technology as a whole 101

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cannot replace the human mind. In the learning of mathematics, the graphing calculator should be viewed as a powerful tool that helps to solve mathematical problems not as a tool that solves mathematical problems. It must also be mentioned that the skills attained by individuals may also be attained without the use of technology, but technology proves to accelerate the process of attaining these skills for many individuals.

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