Student’s intuition and reasoning in teaching and learning of mathematics through experiments with technology.
Abstract
In this post, some examples are given of the interrelation between intuitive and analytic thought in teaching and learning mathematics with technology. It is illustrated how the Dual Processes Theory in cognitive psychology (S1 and S2 processes) can be used as theoretical framework for the study and analysis of several didactic situations. The influence of experimentation, validation of conjectures and use of technology are studied in the context of intuitive versus analytic thought. We analyze the effects that technology produce and how it can be used to lead students arrive to correct conclusions by analyzing their answers from different points of view. Methods are suggested for generation of “good mental habits” in resolution of mathematics problems with technology. We illustrate how new technologies can contribute to the education of intuition, related to S1 processes, as well as to construction of “bridges” between both types of processes.
Key Words: Cognitive Theory; Dual Process Theory; Learning; Teaching Practice; Mathematics with Technology
INTRODUCTION
“Learning begins with actions and perception, proceeds hence to words and concepts and should end in good mental habits. This is the general aim of mathematics teaching, to develop in each student as much as possible the good mental habits of tackling any kind of problem”. (Polya, 1985)
Students often arrive at university without a background in abstract reasoning and with a limited experience in application of mathematical techniques. In order to acquire the necessary knowledge and the appropriate development of abstract reasoning; the students need first to gather evidence of mathematical phenomena.
Following experimental trend in teaching mathematics in our classes we encourage students to formulate and to verify conjectures, to discover patterns, to explore, to make a synthesis of computational results and to appreciate proofs and experiments as functional tools.
Implementation of experimental methods change the way we teach mathematics, imposing new demands to methodology, evaluation and curricula that should finally develop new abilities in the student, which in the past were not considered relevant such as – inductive thought, pattern’s discovery, mathematical intuition, critical analysis of experimental results, construction and validation of conjectures and experimental search for formal proof.
The transposition (Chevallard, 1985) into mathematical education of this new approach is not exempt of obstacles as Lagrange outlines in (Lagrange, 2004). In several activities of our didactic experiences in teaching mathematics we have built laboratories, exercises and assessments using technology. However, the answers we hoped to obtain from the students, frequently become in something totally unexpected. The obtained discrepancies often could be explained through important disarrangements between students’ intuition and the requirements of formal system of mathematics.
“Intuition comes to us much earlier and with much less outside influence than formal arguments which cannot really understand unless we have reached a relatively high level of logical experience and sophistication…”, (Polya, 1968).
The resolution of mathematical problems is a complex and demanding mental activity and often involves several associated tasks. People who are occupied in such activities are much more likely to respond to some of these tasks by blurting out whatever comes to mind. According to Kahneman and Tversky (2002) the rationality of thought is bounded by certain heuristic shortcuts which are applied in resolution of complex tasks and that in certain cases can lead to systematic errors.
We argue that cognitive psychology establishes a useful theoretical framework for the study and analysis of several didactic situations that appear during the implementation of some mathematical tasks. We use in this paper the so termed Dual Processes Theory (Kahneman, 2002; Stanovich, 1994) to analyze, on the base of examples, the interrelation between intuitive and analytic thought in mathematical problem – solving situations and how this often conflicting relationship can be affected by the introduction of new computational tools like CAS calculators.
According to Dual Process Theory, our cognition and behaviour operate in parallel in two quite different modes, called System 1 (S1) and System 2 (S2), roughly corresponding to our common sense notions of intuitive and analytical thinking.
Besides the typical errors by carelessness (mistakes in algebraic manipulations, sign’s change, etc.) or by simple ignorance of the subject matter, there is other kind of student’s errors, which could be related to the way in which the S1 and S2 systems work.
Due to the lack of experience and knowledge the students give solutions to a mathematical problem with fast and intuitive answers typical of the S1 system, without the controls and regulations that are characteristic of the S2 system.
Many authors are pointed out the persistence of student’s misconceptions with respect to specific topics and task (Confrey, 1990). Tirosh and Stavy (1999) state they have observed that students react in a similar way to a wide variety of conceptually non related problems which share some external common features. This fact allowed them to suggest that many responses that literatures describe as alternative conceptions (misconceptions) could be explained as evolving from common intuitive rules as “More of A – More of B”, “Same A – Same B”, “Everything can be divided”, “Overgeneralised linearity”, etc.
Often student’s intuition about certain concepts and ideas are not in line with accepted scientific frameworks in which are based the operation principles of computational tools used in mathematical experiments. In order to assure an intelligent dialogue with computational tools is necessary to impose a correct definition of objects and concepts used in mathematics. For example, student should clearly understand concepts such as list, vector, matrix, expression, function, equation, etc to appropriately work with sophisticate calculators, as Classpad300.
Example 1
The following example we took from Tirosh and Stavy (1999). They have associated the often erroneously answer given by students in this case with the intuitive rule “Same A – Same B”. The required activity was:
Activity 1: Consider a pentagon and a hexagon (see figures 1 and 2). All sides of the pentagon are equals. All sides of the hexagon are equal. The side of the pentagon is equal to the side of the hexagon.
Figure 1
Figure 2
Circle your answer:
a. Angle 1 is greater than Angle 2.
b. Angle 2 is greater than Angle 1
c. Angle 1 is equal to angle 2.
d. It is impossible to determine.
Due to the equality of the side of the two polygons and to the overall similarity of the drawn figures many students claimed that the angles are equals “same sides/objects – same angles”.
Lacking of the highlighted attributes in both figures caused that students handled attributes such as: distance, size and similarity which automatically are registered by the S1 system (see next section). These attributes are denominated by Kahneman (2002) “natural assessments”. Again, in this case, the student didn’t adopt a critical position before giving their answer and threw the first thing that came to mind.
In the next section we will explain on the base of Dual Process Theory that frequently, student’s lack of experience and knowledge produces this kind of situations.
DUAL – PROCESS THEORY AND ITS APPLICATION TO TEACHING AND LEARNING MATHEMATICS
The distinction among intuitive and analytic way of thought is clearly established in cognitive psychology in the so called “Dual-Process Theory”. U. Leron and O. Hazzan (2006) write:
“According to this theory, our cognition and behaviour operate in parallel in two quite different modes, called System 1 S1 and System 2 S2, roughly corresponding to our common sense notions of intuitive and analytical thinking. These modes operate in different ways, are activated by different parts of the brain, and have different evolutionary origins (S2 being evolutionarily more recent and, in fact, largely reflecting cultural evolution) … Like perception, S1 processes are characterized as being fast, automatic, effortless, unconscious and inflexible (hard to change or overcome); unlike perception, S1 processes can be language-mediated and relate to events not in the here and-now (i.e., events in far-away locations and in the past or future). In contrast, S2 processes are slow, conscious, effortful and relatively flexible. The two systems differ mainly on the dimension of accessibility: how fast and how easily things come to mind.”
An important feature of this classification in cognitive psychology is that in the processes linked to S2 system are included monitoring functions. These functions belong to the effortful operations of S2 system and monitor the activities of S1 system. In the context of mathematical problem – solving situations we will associate these monitoring functions with check-up of the way in which problems are solved.
When a student erroneously answers to a question, or at least not in the way that it is expected, in general we blame the S2 processes of his failure. However, it is probable that the self – regulation mechanisms of S2 system has not had time of playing any role at all. The students often give erroneous answers in spite of having the enough knowledge and abilities (S2 related) to give a correct solution. The S1 processes are so fast that are immediately manifested. The problem is sometimes greater, when the student ignores important data from the exercise and assumes that his answer is correct without any intent to check-up not only the answer itself but the way in which the problem was solved.
“People are not accustomed to thinking hard, and are often content to trust a plausible judgment that quickly come to mind”, Kahneman (2002).
Strengthen the processes linked to S2 system is what the mathematical education and science in general have done, nevertheless seems to be required a better understanding of the interrelation between both systems in teaching and learning mathematics. A better understanding of the interrelation of the processes S1 and S2 will allow us, without doubts, to communicate to our students suitable heuristic strategies in the resolution of mathematical problems or what Polya called “good mental habits”. The mathematics is, first of all, to know how to do, is a science where the method usually is more important that the content.
A suggestion given by the authors Leron and Hazzan (2006) is the following:
“It is need to train people to be aware of the way S1 and S2 operate, and to include this awareness in their problem – solving toolbox”
The Polya’s proposal from the point of view of Dual – Process Theory.
The Polya’s work constituted an important motivation for one of current approaches in mathematical research, the so called experimental mathematics (Borwein; Bailey, 2004). Following the experimental line in mathematics education we would like to highlight in this subsection that Polya’s proposition for mathematical problem – solving strategies encourages an effective use of natural heuristics of thought.
Indeed, Polya’s model (Polya, 1957, 1968) proposes a set of four phases with some questions and suggestions that constitute a guide for the search and exploration of answer alternatives. In summary, the phases are the following ones: 1) Understand the problem; 2) Find the connections between the data and the unknown quantities; 3) Make and execute a plan and 4) Examine the obtained solution.
In the step 1 Polya stand out the important role of analytic thought related to S2 system to appropriately understand the problem before proceeding to solve it. In the step 2 he outlines the use of auxiliary, similar, or simpler problems related with the main problem, but more accessible. According to Kahneman (1983) often when we try to solve a complex problem happens what he denominates “attribute substitution”, which consists in that judgements are mediated by heuristic, when an individual values a specific real attribute of the object of judgement by means of another heuristic attribute which comes easier to mind. The real attribute is less accessible and another related attribute which is more available replaces the first one. This associative relation with real attribute is so close and fast that monitoring functions which are characteristic of the S2 system cannot be activated and who is solving the problem doesn’t notice that he is really responding to another question. Some of the former mentioned intuitive rules can be explained with this simple concept. In fact, to be aware of this fact and avoid possible mistakes on this stage, Polya includes in the step 1 the understanding of the problem, (using S2 related process). Only with this condition it is possible to use in an appropriate way the substitution of attributes which is characteristic of the S1 system so that start to work in searching for auxiliary problems that are more accessible and whose solutions in principle are known by the one who is trying to solve the problem. Also in the step 2, Polya proposes to enunciate the problem in another form, to create in this way the effect that Kahneman denominates “anchoring effect” in which judgement is influenced by a temporal increase of accessibility of some particular value of real attribute, i.e. Polya proposes to change formulation of the problem to highlight other aspects which had possessed a low accessibility in the previous formulation.
Finally in the steps 3 and 4 Polya encourage critical reasoning regarding the resolution of the problem, highlighting this way the role of monitoring mechanisms of the system S2.
Example 2
The following activities (Preiss, 2000) were applied to students of Calculus with knowledge on analytic geometry, algebra and trigonometry; also to a group of university mathematics teachers with no prior experience using technology for educational purposes. The requested activities were:
Activity 2: Obtain the graphic of the function, given in polar coordinates r(theta) = 3cos(theta)+2sin(theta), using classpad330 with a standard visualization window and determine to what known curve corresponds.
Solution The obtained graph is given in figure 3
Figure 3
Figure 4
Figure 5
The almost unanimous opinion of all, teachers and students, was that the given polar function represents an ellipse. As a next step was suggested to revise the visualization window used to obtain this graphic (see figure 3).
It was communicated by the expert that the scale of visualization window not corresponds to a 1:1 scale in relation to the calculator screen. It was informed that relationship between screens width and high of graphic calculators is usually 2:1, that is, screen width double its high. Therefore, if we want to see a 1:1 scale graphic in calculator screen the units assigned to – axis should be twice of the units assigned to – axis. A new graph is requested, but now with a visualization window given in figure 5.
The participants, students and teachers, doubt this time of their previous conclusion and conjecture that the drawn graphic is really a circumference.
Also, the expert made notice a strange fact that although the graphic of the curve is complete, the calculator continued calculating and it delayed a little more time in finishing. This fact worried them because they didn’t know how to determine the cause of this additional time seemingly unnecessary used in the construction of the graphic.
It is requested to participants to mathematically prove this new conjecture.
Example 3
Let us consider again the example given by Tirosh and Stavy (see example 1) and use technology to highlight other attributes in the figures different from the natural ones. That is, following Polya, we will change formulation of the problem to highlight other aspects which had possessed a low accessibility in the previous formulation. This way we will create what Kahneman denominates “anchoring effect” in which judgement is influenced by a temporal increase of accessibility of some particular value of real attribute. One way to do this is draw together three of each type of polygons with a common vertex and without overlapping (see figures 6 and 7). In this case, we can give to students the following activity:
Activity 3 Consider a pentagon and a hexagon. All sides of the pentagon are equals. All sides of the hexagon are equal. The side of the pentagon is equal to the side of the hexagon. Keeping in mind the property of covering the whole plane with polygons used as tiles, what are you able to say regarding the angles of these polygons?
Circle your answer:
e. Angle 1 is greater than Angle 2.
f. Angle 2 is greater than Angle 1
g. Angle 1 is equal to angle 2.
h. It is impossible to determine.
Figure 6
Figure 7
Another way is to consider polygons inscribed in the circle (see figures 8 and 9) and the property of these polygons to tend to a circle when the number of sides becomes very large.
Figure 8
Figure 9
As a preliminary result, we received in these cases more correct answers than in the previous formulation of the question (see example 1).
CONCLUSIONS
To consider a successful transposition of experimental methods into teaching and learning mathematics we need on one hand a better understanding of the effects that new technologies produce and on the other hand we need a careful study of mental process involved during the resolution of mathematical problems.
The implementation of computational tools in mathematics education requires that cognitive functions should be distributed in an optimal way between student and tool. In the old paradigm, teaching mathematics was based on repetitive practices until perfecting certain knowledge, with the use of technology these practices should be change in an intelligent way. Regarding this, we propose to consider the different effects that technology produce when it is used for intellectual amplification during teaching – learning process of mathematics.
We should worry not only about mathematical knowledge that student should possess, but also on meta-cognition tools he needs to manage. It is fundamental to know and to understand in a better way the natural heuristic strategies of thought when it faces resolution of complex tasks, as it is a mathematical problem.
The basic objective in our work when implementing technology in mathematics education is to generate “good mental habits of tackling any kind of problem”, based on effective processes of thought (natural heuristics of thought) that don’t become obsolete quickly and that don’t depend on vertiginous change related whit technology.
In this paper we use the Dual Processes Theory as useful theoretical framework to analyze, on the base of examples, the interrelation between intuitive and analytic thought in mathematical problem – solving situations and how this often conflicting relationship can be affected by the introduction of new computational tools like CAS calculators.
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