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1 Solving problems in different ways: from mathematics to pedagogy and vice versa Roza Leikin Faculty of Education University of Haifa 19-10-2007, CET conference Mathematics in a different perspective

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 2 Solve the problem in as many ways as possible Multiple solution task -MST

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 3 Example 1: Presenting learners with different solutions A rectangle is inscribed in a circle with radius R. Find the sides and the area of a rectangle that has maximal area. 2R2R בני גורן (2001). אנליזה: חשבון דיפרנציאלי, טריגונומטריה, חשבון אינטגראלי/ עמוד 403, #41. 2R2R h h 2R2R x y 2R2R Solution 3: h altitude to the diagonal, Solution 4: Compare with a square 2R2R Solution 2: S( )=2R 2 sin Solution 1: x and y – the sides, Derivative of function S(x)… Solution 2: S( )=2R 2 sin

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 4 Pupils, what do they say? wise one The wise one, what does he say? Why don’t they teach us to do it this way? I can solve the problem that way [using calculus], but this way I can understand the solution. I can see it, I can feel it, and the result makes sense. wicked one The wicked one, what does he say? Of what service is this to you? (not for him) simple one The simple one, what does he say? What is this? one who does not know how to ask The one who does not know how to ask, what does he say? I do not understand anything

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 5 Teachers, what do they say? wise one The wise one, what does he say? Where can I implement this? Can I do this with any problem? How does this help students? Is this suitable for any student? Would they accept this in exams? Where can I find time for this? wicked one The wicked one, what does he say? Of what service is this to you? (not for him) simple one The simple one, what does he say? What is this for? one who does not know how to ask The one who does not know how to ask, what does he say? This will confuse pupils!

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 6 A researcher, what does he say? How to bridge between teachers and students? How to bridge between teachers and mathematics educators? to knowto understand What does it mean to know vs. to understand? Do MSTs develop knowledge (understanding) and how In pupils? In teachers?

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 7 MSTs that lead to equivalent results are essential for the developing of mathematical reasoning (NCTM, 2000; Polya, 1973, Schoenfeld, 1985; Charles & Lester, 1982). MSTs require a great deal of mathematical knowledge (Polya, 1973) MSTs require creativity of mathematical thought; some solutions may be more elegant/short/effective than others. (Polya, 1973; Krutetskii, 1976; and later Ervynck, 1991; and Silver, 1997) MSTs should be implemented in the area of curricular design MSTs can be used for the assessment and development of ones Knowledge Creativity Mathematics educators, what do they say?

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 8 Creativity : basic definitions Main components of creativity Main components of creativity according to Torrance (1974) are fluency, flexibility and novelty Krutetskii (1976), Ervynck (1991), Silver (1997), connected the concept of creativity in mathematics with MSTs. Examining creativity by use of MSTs my be performed as follows (Leikin & Lev 2007) flexibility flexibility refers to the number of solutions generated by a solver novelty novelty refers to the unconventionality of suggested solutions fluency fluency refers to the pace of solving procedure and switches between different solutions.

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 9 Tall, what does he say? From: Tall (2007). Teachers as Mentors to encourage both power and simplicity in active mathematical learning. Plenary at The Third Annual Conference for Middle East Teachers of Science, Mathematics and Computing, 17–19 March 2007, Abu Dhabi

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 10 Example 2: Examining students creativity half the timehalf the timehalf way half way Dor and Tom walk from the train station to the hotel. They start out at the same time. Dor walks half the time at speed v 1 and half the time at speed v 2. Tom walks half way at speed v 1 and half way at speed v 2. Who gets to the hotel first: Dor or Tom? Solution 2.1 – Table-based inequality Solution 2.2 – Illustration: Solution 2.3 – Graphing: Solution 2.4 - Logical considerations: If Dor walks half the time at speed v 1 and half the time at speed v 2 and v 1 >v 2 then during the first half of the time he walks a longer distance that during the second half of the time. Thus he walks at the faster speed v 1 a longer distance than Tom. Dor gets to the hotel first. Solution 2.5 – Experimental modelling (walking around the classroom) (From Leikin, 2006)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 11 Differences between gifted and experts The research demonstrated differences between the gifted and experts in the combination of novelty and flexibility. The differences between the gifted and experts are found to be task-dependent: processes vs. procedures (From Leikin, 2006; Leikin& Lev, 2007)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 12 In order to develop students’ mathematical flexibility teachers have to be flexible when managing a lesson Dinur (2003), Leikin & Dinur (2007)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 13 Example 3: 7 th grade mathematics lesson The problem: A slimming program plans to publish an ad in a journal for women. Which of the three graphs representing the measure of change would you recommend be chosen in order to increase the number of clients registering for the program? M-Ch 1 M-Ch 2 M-Ch 3 From “Visual Mathematics” program (CET, 1998).

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 14 When planning the lesson Anat considered the graphs corresponding to each of the measures of change given in the problem, and constructed a graph for each function Dinur (2003), Leikin & Dinur (in press)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 15 During the lesson Maya -- as expected: If the horizontal axis represents time and the vertical axis represents weight, then somebody loses weight if the measure of change is negative. First they lose weight quickly and then slower. [The corresponding graph of weight was drawn on the blackboard] Aviv – unexpected solution I chose the second [measure of change] because you may take the rate of losing weight, of slimming, instead of the rate of changes in weight. They [the task] did not say it should be weight. Other students (together): It is the number of kilos lost. Dinur (2003), Leikin & Dinur (2007) an expected mistake appeared to be a correct answer!

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 16 On the complexity of teaching On the one hand, the teacher follows students' ideas and questions, departing from his or her own notions of where the classroom activity should go. On the other hand, the teacher poses tasks and manages discourse to focus on particular mathematical issues. Teaching is inherently a challenge to find appropriate balance between these two poles. (Simon, 1997, p. 76)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 17 Why being flexible? From Leikin & Dinur (in press)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 18 Teachers learn themselves when teaching multiple solutions Teachers’ Mathematical Knowledge & Pedagogical Beliefs Teachers’ mathematics noticing & curiosity Understanding of students’ language encouraging multiple solutions by students Students knowledge & classroom norms Students production of multiple solutions Development of Teachers’ Mathematical Knowledge & Pedagogical Beliefs Conviction

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 19 Problem in the test Lev (2003); Leikin & Levav-Waynberg (submitted)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 20 Example 6: Communication among collegues On the face ABE of the quadrangular right pyramid ABCDE tetrahedron ABEF is built. All the edges of the tetrahedron and the pyramid are equal. This construction produces a new polyhedron. How many faces does the new polyhedron have? A B FE D C From: Applebaum & Leikin (submitted)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 21 Formal solution A B FE D C K

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 22 Process vs. procedure A B F E D C F A D B E C B1B1 A1A1 From: Applebaum & Leikin (submitted)

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 23MSTs In Instructional design Teacher education Teachers’ practice Pupils’ learning Research For the development and identification of Knowledge / beliefs/ Skills Creativity Critical thinking ……

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19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 24 Publications related to MSTs -- Leikin R. Leikin, R., Berman, A. & Zaslavsky, O. (2000). Applications of symmetry to problem solving. International Journal of Mathematical Education in Science and Technology. 31, 799-809. Leikin, R. (2000). A very isosceles triangle. Empire of Mathematics. 2, 18-22, (In Russian). Leikin, R. (2003). Problem-solving preferences of mathematics teachers. Journal of Mathematics Teacher Education, 6, 297-329. Leikin R. (2004). The wholes that are greater than the sum of their parts: Employing cooperative learning in mathematics teachers’ education. Journal of Mathematical Behavior, 23, 223-256. Leikin, R. (2005). Qualities of professional dialog: Connecting graduate research on teaching and the undergraduate teachers' program. International Leikin, R., Stylianou, D. A. & Silver E. A. (2005). Visualization and mathematical knowledge: Drawing the net of a truncated cylinder. Mediterranean Journal for Research in Mathematics Education, 4, 1-39. Leikin, R., Levav-Waynberg, A., Gurevich, I. & Mednikov, L. (2006). Implementation of multiple solution connecting tasks: Do students’ attitudes support teachers’ reluctance? FOCUS on Learning Problems in Mathematics, 28, 1-22. Levav-Waynberg, A. & Leikin R. (2006). Solving problems in different ways: Teachers' knowledge situated in practice. In the Proceedings of the 30th International Conference for the Psychology of Mathematics Education, v. 4, (pp 57-64). Charles University, Prague, Czech Republic. Leikin, R. (2006). About four types of mathematical connections and solving problems in different ways. Aleh - The (Israeli) Senior School Mathematics Journal, 36, 8-14. (In Hebrew). Levav-Waynberg, A. & Leikin, R. (2006). The right for shortfall: A teacher learns in her classroom. Aleh - The (Israeli) Senior School Mathematics Journal, 36 (In Hebrew). Leikin, R. & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks. Educational Studies in Mathematics, 66, 349-371. Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. The Fifth Conference of the European Society for Research in Mathematics Education - CERME-5.Leikin, R. & Dinur, S. (in press). Teacher flexibility in mathematical discussion. Journal of Mathematical Behavior Leikin R. & Levav-Waynberg, A. (accepted). Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers' knowledge. Canadian Journal of Science, Mathematics and Technology Education. Applebaum. M. & Leikin, R. (submitted). Translations towards connected mathematics..

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