ࡱ> `Sbjbjss AI  ///801 {|16D7D7D7L9Z9 f9֊؊؊؊؊؊؊$h_9 ?&9"H9?? D7D75FFF? D7 D7֊F?֊FFj^ ND71 C;/FAb2Z|K0{?2D?dNN? bn9&;:F`<\=Hn9n9n9@FXn9n9n9{???? J) J)   COUNTER-EXAMPLES IN LECTURING MATHEMATICS VALENTINA GOGOVSKA, RISTO MALCEVSKI Abstract: Adopting definitions and axioms, solving tasks and proofing theorems are considered as basic methods in Mathematics. When given the formulation: to prove that in most cases pupils understand following task: to find, that a given statement is correct. This is due to the fact that theorems, just like tasks aimed at proving a certain statement, are mainly connected with the demand to prove that a certain definition is correct, meaning that basically pupils are confronted with general confirmative statements, and very rarely with partially confirmative ones, while at the same time generally and partially denying statements are almost absent in the process of teaching mathematics. Their more and more frequent usage shall lead to improvements of the process of teaching mathematics, by providing access to more structural knowledge for the pupils, by promoting creativeness and motivating the pupils in that direction, as well by providing them with the possibility to self-assess their knowledge. Key words: Counter-examples, Creative Process, Creativity INTRODUCTION This document should be considered as a carefully designed attempt to introduce counter-examples in the process of teaching Mathematics. This should be seen, in first instance, as a contribution to the possibility to gain more sustainable structural knowledge by the pupils. At the same time, using counter-examples can promote processes of creative thinking by the pupils, as well to provide further motives in their process of ongoing education. However, such processes should be well planned, designed and put into practice. 1. MATHEMATICAL STATEMENT The statement is logical form of thinking by which some characteristics of given objects, appearances or relations are to be confirmed or denied. The statements in which we are meeting characteristics and relations of mathematical objects, we are naming as mathematical statements. 1) General confirmative statement that is expressed in the form: For any object x, if x has the characteristic S, than x has the characteristic P. Symbolic transcription of this statement will be  EMBED Equation.DSMT4 . Example 1. a) In every parallelogram the opposite angles are equal. b) Each natural number bigger than 1 is either common or complex. 2) Partially confirmative statement that is expressed in form There exist object x that has characteristic S, and also has the characteristic P. Symbolic transcription of this statement will be EMBED Equation.DSMT4 . Example 2. a) Some rectangle has alternate normal diagonals. b) In some triangles height and gravity line drown from the same stop of head are corresponding to each other. 3) Generally denying statement that is expressed in form: None object x that has characteristic S, has not the characteristic P. Symbolic transcription of this statement will be: EMBED Equation.DSMT4 . Example 3. a) None natural number in the same time is common and complex number. b) None triangle in the same time is rectangular and obtuse. 4) Partially denying statement that is expressed in form: There exists x that has characteristic S, and has not characteristic P. Symbolic transcription of this statement will be: EMBED Equation.DSMT4 . Example 4. a) Some trapeze has not equal diagonals. b) Some natural numbers has not more than two natural dividers. At this point we are going to consider three ways that are mostly used while assertion of some statement that is not true. First way for assertion that given statement is not true is the use of definitions, axioms and previously proven statements by which the conclusion is realized by use of the rule modus tolens. Example 1.1. If some number is divisible by 6, than it is divisible by 3. The number 736 is not divisible by 3 Conclusion: The number 736 is not divisible by 6. Second way that is used for assertion of the general denying statements is the use of total induction, as it is described in the following example. Example 1.2. Does exists three digit number A whose digits can be dislocated on such a way that the sum of given and new number to be 999? The total of three digit numbers is 900, so we can check all possibilities which will be an irrational procedure that will take a lot of time. The task can be solved if we are going to inscribe the three digit number in the form  EMBED Equation.DSMT4  and by relocation of the digits we can get following five numbers:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Then, for each of the numbers we have to check the required equation, for example  EMBED Equation.DSMT4 . By checking all five possibilities we can find out that none of the equations is true, so we have asserted that for any three digit number  EMBED Equation.DSMT4  and number  EMBED Equation.DSMT4  got by relocation of the digits, the equation  EMBED Equation.DSMT4 is not true. The third way is used while assertion of partially denying statements and imply to find element x that has characteristic S(x) and has not characteristic P(x). At the same time the element x is to be called counter sample. Such a way of assertion mathematical statement is known under name retraction of the statement by imposing counter sample. Example 1.3. Some natural numbers are not common. Because  EMBED Equation.3 , numbers 8 and 10 are complex and are not common. Hence, there are natural numbers (8 and 10) that are not common. 2. CREATING SKILLS OF THE STUDENTS FOR USE OF COUNTER EXAMPLES Creating skills of the students for use of counter examples as a mean of judgment can be reached on two ways: By participation of a student into collective thinking, governed by the teacher in the cases where the teacher is using counter examples and By single minded training of the students while solving tasks that are connected with the certain teaching content during the entire academic year. Considering first way (a), we are going to indicate few possibilities for successful use of counter examples. a1) While discovering imperfection in formulation of the statements. Some times statements might be incomplete in formulation, to contain unnecessary conditions, contrary conditions or to have irregular logical structure. Example 5. Lets considerate the statement: If the legs of an angle are parallel to the legs of other angle, then either both angles are equal, or their sum is equal to 180 . We can notice that while formulation of above statement it was used exclusive disjunction i.e. statement has irregular logical structure. Hence, it is better to draw two right angles with parallel legs that will indicate existing of a sample that is fulfilling the condition from the statement, but not the conclusion. In this case angles are equal and their sum is 180 . a2) While establishing that opposite statement is not a theorem. Example 6. For the theorem: If two angles are neighboring, then their sum is equal to 180 . The opposite statement will be: If the sum of two angles is equal to 180 , then those angles are neighboring. It is clear that opposite statement is not theorem and the students can make that conclusion by their selves if you are going to indicate them two opposite angles in rectangle. a3) While analyzing of the consistency of the conditions in some theorem i.e. to establish that conclusion in some theorem is not valid if some of the conditions is omitted. Example 7. Lets consider the Roll Theorem: If the function f is fulfilling following conditions: a)  EMBED Equation.DSMT4  b) for every EMBED Equation.DSMT4 , there is EMBED Equation.DSMT4  and c)  EMBED Equation.DSMT4  then exist at least one point  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 . To show that conditions in above theorem are concise, it is enough to consider the functions shown below:  Namely, above drawings are diagrams of three functions determined into interval [-1, 1]. For each of the functions only one of the three conditions is not fulfilled according the Roll Theorem, so there is no point  EMBED Equation.DSMT4  such that EMBED Equation.DSMT4 . So: For the function from diagram a) point x0 = 1 is point of interruption, For the function  EMBED Equation.DSMT4  (drawing b)) we have: f is uninterrupted on interval [-1,1],  EMBED Equation.DSMT4 , but there is no point  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4 , because  EMBED Equation.DSMT4 . We have to mention that besides this function is reaching its minimum at the point  EMBED Equation.DSMT4 , the Roll Theorem is not valid and the reason for that is absence of  EMBED Equation.DSMT4 . For the function (drawing c)) we have: f is uninterrupted on interval [-1,1], for each  EMBED Equation.DSMT4  exists  EMBED Equation.DSMT4  , but there is no point  EMBED Equation.DSMT4  such that  EMBED Equation.DSMT4  because  EMBED Equation.DSMT4 . a4) While detecting errors in defining the concept. In the above case most common errors are unnecessary conditions, incomplete conditions, wrong choice of concept etc. Example 8. In the definition: The medial line of a triangle is the line that is connecting stop of head of the triangle with the middle of the side of the triangle. This is a case of incomplete condition and wrong choice of concept. Namely, instead of segment 1, it is said line, and instead the middle of the opposite side it is said the middle of the said. The students shall easily realize that it is mistake in definition of the concept. Example 9. In the definition: Parallelogram is a polygon that has two by two sides being parallel. In this case there is wrong choice of concept that should be quadrangle instead polygon. The students can easily recognize the mistake if they will be given the counter sample hexagonal. Prior getting over to further considerations, we are going to mention that detecting of mistakes in formulating statements or definitions is connected to describing of their structure. a5) While detecting mistakes in formulation of the tasks. Example 10. In the task : Prove that  EMBED Equation.DSMT4  is divisible to 100 where a is natural number which is not divisible to 5 the mistake has been made in formulation. Namely, instead the expression  EMBED Equation.DSMT4  it should be  EMBED Equation.DSMT4 . The mistake can be noted by using successive  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  and the expression  EMBED Equation.DSMT4  is reaching the values 8, 308, 6808 and 66308 all numbers that divided to 100 are giving rest 8. In case such samples are not to be considered with the students, it is naturally to expect that students may suspect in their knowledge i.e. to start thinking that they had made mistake while solving the task. a6) To come to conclusion that some classes of mathematical objects are inco Example 11. The class of uninterrupted functions and the class of functions with limited variation are mutually incomparable. From one side monotonous functions are functions with limited variation, but we also know that monotonous function is not necessary to be uninterrupted. On the other side the function  EMBED Equation.DSMT4  for  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  is uninterrupted in the interval [0,1], but it is not function with limited variation. Really, for the division we are getting  EMBED Equation.DSMT4  when  EMBED Equation.DSMT4 . The above examples are indicating on the necessity of using counter samples in case b) i.e. for training of the students to use counter samples while solving the tasks that are connected to the teaching content during the entire academic year. The task should be chosen on such a way providing that following additional goals should be reached. Example 12. For the quantity  EMBED Equation.DSMT4  which of the following statements are true: Each number thats belongs to A is detachable to 7. Quantity A contains at least one even number. Quantity A contains only one number that is detachable to 3. True statements are statements 1 and 2. b2) Negation of the statement expressed by means of quantifications. Example 13. For the quantity  EMBED Equation.DSMT4  make negation on the following statements and than determine which one of the three statements is true and which one is false. Quantity E contains at least one number that ends on digit 7. Quantity E contains at least one number that is divisible to 5. Quantity E contains at least one number in which digit of tens is 5. Answer: In the quantity E there is no number ending on digit 7. In the quantity E there is no number divisible to 5. In the quantity E there is no number in which digit of tens is 5. Only the statement 2.1 is true one. b3) Analyzing the text of the task, information in it and formation of the answer. Namely, in order to enable students to acquire skills for use of counter samples as a mean for solving tasks, it is recommended on classes to solve tasks that will guide the students towards different alternatives, assertion of the accuracy of the statement or its retraction. We shell consider few samples of this kind. Example 14. Let  EMBED Equation.DSMT4  and P is the quantity of common numbers. Whether is true the following statement: Quantity M is sub-quantity to the quantity P. While solving this task it is recommended to use the following table:  EMBED Equation.DSMT4 123456... EMBED Equation.DSMT4 51117232935... From the table we can note that for  EMBED Equation.DSMT4 , result is  EMBED Equation.3  that is not common number, and according to that, the above statement isnt true. Example 15. Let  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Retract the statement: For each  EMBED Equation.DSMT4  the number of type  EMBED Equation.DSMT4  is common number While solving this task the following table can be used:  EMBED Equation.DSMT4 123456... EMBED Equation.DSMT4 51119294155... From the table above it is obvious that for  EMBED Equation.DSMT4  the number  EMBED Equation.3  is complex number i.e. exists  EMBED Equation.DSMT4  such that makes number  EMBED Equation.DSMT4 to be complex one by which the statement has been retracted. Example 16. Is it true the statement: For each  EMBED Equation.DSMT4  it is valid  EMBED Equation.DSMT4  It is obvious that for each positive integer n, the above inequity has been fulfilled. But if  EMBED Equation.DSMT4  then the table bellow shows  EMBED Equation.DSMT4 -1-2-3-4-5... EMBED Equation.DSMT4 6335153-1... That for  EMBED Equation.DSMT4  inequity has not been fulfilled. Hence, the answer to the question is: The statement is not true. Example 17. To prove or to retract the statement: Each natural number for which the sum of the divisors is odd number, is accurate square. The natural number 2 has two divisors: 1 and 2. Their sum is 1+2 = 3, and the same number 2 is not the accurate square. Hence, the above statement is not true. On the end of this part we have to mention that skilful use of counter examples while teaching mathematic to a grate extend contributes to treat the quality of thinking. It is also necessary to note that searching for the counter examples, very often is a serious task that prefers special development of thinking and demands adopted deepened mathematical knowledge. Besides that, in lot of cases, the finding of counter examples is connected to informal approach in judgment for given problem and is guiding the students toward elementary heuristic activity. 3. CONCLUSION In foregoing considerations we put the stress on the situations where counter examples were used, and also on creation of skills to the students for use of the counter examples. Counter examples in teaching mathematic has a particular didactic place and they are of special importance for overall development of each student, especially those that are gifted for mathematic. In addition to the above establishment, there are following undeniable facts: Skilful use of counter examples while teaching mathematic is of a special importance for tending the capacity of thinking, especially deepness, creativity and crucial of the thinking process, Searching for counter examples, very often is a serious task that prefers special development of thinking, and also can be an effective method for development of the same, Finding , the finding of counter examples is connected to informal approach in judgment for given problem and is guiding the students toward elementary heuristic activity which is of grate importance to the development of the students gifted for mathematic, The use of counter examples and training the students for independent finding of same, can be effective means for realization and differentiation of the students and that is effective method while identifying the level of gift of each student and The use of counter examples unequivocally leads to a depended knowledge of mathematic, skills and abilities that are of crucial meaning for the students gifted for mathematic. REFERENCES Klarin, M. Pedagogic Technology in Curriculum Process Skopje, 1995 Poljak, V. Didactic Zagreb, 1970 Poposki, K. MA: Contemporary Thoughts for Examining and Assessing Students Achievements Skopje, 1996 Popovski, K. Educational Process Programming Skopje, 2000 Vilotijevic, M. Evaluation of the Pedagogic Work in Schools Belgrade, 1992 Whitcombe, A. (1988). Mathematics: Creativity, imagination, beauty. Mathematics in School, 17, 13-15 Heibert, J., Carpenter, T. P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., et al. Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann (1997).. Handbook of Gifted Education (2nd Edition) by Nicholas Colangelo, Gary A. Davis, Allyn & Bacon (2003) ABOUT THE AUTHORS Valentina Gogovska, Master of mathematics Teaching and Researching Assistant, Institute of Mathematics-Skopje, Faculty of Natural Science Gazi Baba bb 1200 Skopje F.Y.R.O.M. Cell phone: +389 70 240 999 -mails: valet@iunona.pmf.ukim.edu.mk ;  HYPERLINK "mailto:valentinagogovska@gmail.com" valentinagogovska@gmail.com Risto Malcevski, Professor, Ph.D. Faculty of Informatics Europian University- Republic of Macedonia Skopje F.Y.R.O.M. 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