Moreover, groups' written solutions, explanations and proofs were collected. During small group activities, the second author walked around the groups and tried to identify what they were doing. During whole class discussion, the camera focused on the whole classroom. The aim of this study is to examine the mathematical discussions of pre-service teachers about perimeter and area of a rectangle in terms of Lakatos' proofs and refutation method.
In this context, the data was examined using a framework included in Larsen and Zandieh's study. In the proofs and refutation process, we primarily focused on the pre-service teachers' responses and the outcome of their activities, like Larsen and Zandieh In particular, we analyzed whether the pre-service teachers' responses focused on the counter-examples, definitions, the conjecture or the proof, and whether the pre-service teachers' activity results in a modification to a definition or to the conjecture. Table 1 summarizes Larsen and Zandieh's framework for the method of proofs and refutations.
In this section, discussion of results of the pre-service teachers' work in groups was included. Thus, the situations of changing and improving the conjecture according to Lakatos' method are presented. In the classroom discussion, groups were named G1, G2, etc. A1: Our conjecture was "if the perimeter of a rectangle increases, the area always increases too.
After working separately, groups discussed their proofs, conjectures and examples in the classroom discussion. G1: If we increase two sides of a rectangle at the same time, area increases with perimeter. G6: If we keep fixed one of the lengths of an side and increase the other one, the area always increases. A1: Are there any cases in which while the perimeter increases, the area increases as well? G1: If we increase the length of sides at the same proportion, the area increases as well, yet our proof includes this case. G1 and G6 tried to confirm the conjecture and they proved that the conjecture is true for the following cases:.
First case: If we increase both sides of a rectangle at the same or different ratio , the area increases with perimeter.
Second case: If we keep side fixed width or length and increase the other one, the area increases with perimeter. The emergences of global counter-examples. G2: We found an example that if the perimeter increases, the area decreases. This example shows that the conjecture is not always true. For example, if the width of a rectangle is 7cm, the length is 8cm and its perimeter is 30cm, its area is 56cm 2.
Now, if we increase the length 3cm to 11cm and decrease the width 2cm, so we have width 5cm, the perimeter is 32cm and the area is 55cm 2. G7: We found another example when the perimeter increases but the area does not increase. That is, if the perimeter increases, the area can increase, reduce or as in our example it can remain the same.
G2 and G7 suggested two different counter-examples. Their counter-examples are global counter-examples, because they are directly intended for the conjecture and they do not examine the proof.
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It is observed that the pre-service teachers are surprised at this counter example and they believed that the conjecture is not true. A1: G2 and G7 put forward two crucial counter-examples that shake the accuracy of the conjecture. Now, is our conjecture collapsed completely?
For example, according to the G1and G6, our conjecture is true. I guess it seems there is no rejection Yet, whenever we decrease one of the sides, in this case despite the increased perimeter, the area reduces or does not change. So, when is our conjecture true? G8: Our conjecture is true when we increase the length of the sides or when we increase them while keeping one of them fixed.
We cannot guarantee it in other cases.
Lakatos' mitigated scepticism in the philosophy of mathematics
The conjecture of G8 as "if we increase all lengths of sides, its perimeter increases as does the area" limits the conjecture and at the same time it locks out all counter-examples. We categorized this response as monster-barring because the focus was on the counterexamples and students did not modify the conjecture.
After this process, it is observed that some pre-service teachers accepted the accuracy of the conjecture apart from these counter-examples; they did not try to improve the conjecture anymore. Yet, after the example given by A1 teacher, some pre-service teachers tried to change the conjecture by considering counter-examples. So, it becomes a transition to exception-barring. A1: We had a conclusion that satisfied us when we exclude counter-examples and rearrange it in accordance with G1 and G6 proofs.
But, I have an example in which, when the length of one side increases and the other one decreases, the area increases. Take a rectangle with width 4cm and length 8cm. If the length is reduced 1cm, and width increased 2cm, the perimeter changes from 24cm to 26cm, and the area increases from 32cm 2 to 42cm 2. G7: There are some cases in which it is not true. So, this is not an exact answer.
A1: Yes, but in some cases it is true. So, when is the conjecture true for these cases? After minutes, G2 explained their proof. G2: For this case we have such a proof. If we increase one of the sides as "n", and reduce the other one as "n-1", the area always decreases. A1: My example refutes the proof of G2. In the example I reduce the length 1 cm and increase the width 2cm. G8: We rearranged the proof of G2 as follows: "If the number added to the longer side is greater than the number removed from the shorter side, the area always increases when the perimeter increases.
G7: In your proof you do not use the difference is more than 1. Moreover, we discovered that your proof is also incorrect. For a rectangle measuring 3cm x 5cm, let's remove 2cm from the shorter side and add 4cm to the longer one.
So the rate of increase is 2cm. Yet, at the beginning, the area was 15cm 2 , but in the end it becomes 9cm 2. In this case G2 and G8 tried to improve the conjecture by considering the counter-examples; they did not exclude them. So, pre-service teachers focused only on the conjecture and counter-examples. For that reason, it is the result of exception-barring and not proof analysis, because the focus of their activities is only on conjecture and counter-examples and the proof is not perfect, yet.
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An improved conjecture is produced. G2: I guess we have solved the problem. Take a rectangle with side lengths "a" and "b". That is, the removed area "xb" must always be smaller than the added area "ay-xy". So, the area of final rectangle can always be greater than the first one.
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A1: Is there anyone who found a counter-example for the last proof of G2? So, what is the last form of our conjecture "if we increase the perimeter of a rectangle, the area increases, as well". G2: We have three cases for our conjecture:. First case: The increase of both sides of a rectangle. Second case: While the width or length of one side remains the same, the other one increases. Third case: On the condition that the amount of increase is greater than decrease.
Mathematics, Methodology, and the Man
Where the amount of increase is called as "x" and the amount of decrease is called as "y. The accuracy of the last proof is provided by G1 and G6's proof given at the beginning and improved proof towards G2 and G7's counter-examples. We can say that it is a result of proof-analysis which is different from exception-barring, because our primitive conjecture changes as in exception-barring. Yet, G2 changed the conjecture as the first steps of proof and the direction of their and G7's counter-examples.
And, it shows that the proof is working for the new conjecture. The pre-service teachers studied using a conjecture, proved it, tested their proofs, suggested definitions, created counterexamples, made mistakes and generalized conjectures; soon, they completed the steps of forming mathematical knowledge as suggested by Lakatos's assertions concerning the formalization of mathematics knowledge. Toumasis explained that NCTM standards suggest an environment in which students reason mathematically, communicate mathematically, and make mathematical connections.
Therefore, it is observed that there are many similarities between the environment proposed by NCTM standards and the environment prepared according to Lakatos's model. Moreover, Atkins explained that in Lakatos' environment, students can express their feelings, discuss their opinions and argue with the teacher and peers about their views.
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In this study, pre-service teachers discussed the relation between perimeter and area of a rectangle. They proved the conjecture, refuted it and discussed their proofs, opinions and examples with teachers and peers. Thus, Lakatos' method engages pre-service teachers in presenting mathematical discussions and testing the validity of those discussions in a social setting. Lakatos' method gives the students the opportunity to improve their knowledge by using an inductive method instead of the deductive method SRIRAMAN, While pre-service teachers were improving the primitive conjecture, they found proofs, counter-examples, lemmas and definitions and combined them during the process of proof-analysis.
That is, they tried to create a whole with the individual pieces. In this study, pre-service teachers examined area and perimeter of a rectangle. The perimeter and area of a figure are two different measures. Using counter-examples, they created new conjectures and definitions for two different concepts and identified the possible relationships between them.