1 Solving problems in different ways: from mathematics to pedagogy and vice versa Roza Leikin...

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Solving problems in different Solving problems in different ways: ways:

from mathematics to from mathematics to pedagogy and vice versa pedagogy and vice versa

Roza LeikinFaculty of EducationUniversity of Haifa

19-10-2007, CET conference

Mathematics in a different perspective

19-10-2007Roza Leikin, University of Haifa

Multiple Solution Tasks: From mathematics to Pedagogy and vice versa

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Solve the problem Solve the problem in as many ways as in as many ways as

possiblepossible

Multiple solution task -MSTMultiple solution task -MST



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Example 1:Example 1: Presenting learners with different Presenting learners with different solutionssolutions

A rectangle is inscribed in a circle with radius R.Find the sides and the area of a rectangle that has

maximal area. 2Rגורן ) / 2001בני , , : עמוד(. אינטגראלי חשבון טריגונומטריה דיפרנציאלי חשבון .#41, 403אנליזה

2Rhh

2Rx

y

2R

hRhS 2)(Solution 3:h altitude to the diagonal,

2maxmax 2RSRh

Solution 4:Compare with a square

2R

Solution 2:S()=2R2sin 2

0

22,1sin2

RS

2222222 4)(44 xRxxSxRyRyx

RSRyx 2,2

Solution 1:x and y – the sides,

Derivative of function S(x)…

Solution 2:S()=2R2sin 2

0

22,1sin2

RS



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Pupils, Pupils, what do they say?what do they say?

The wise onewise 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.

The wicked onewicked one, what does he say?

Of what service is this to you? (not for him)

The simple onesimple one, what does he say?

What is this?

The one who does not know how to askone who does not know how to ask, what does he say?

I do not understand anything



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Teachers, Teachers, what do they say?what do they say? The wise onewise 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?

The wicked onewicked one, what does he say?

Of what service is this to you? (not for him)

The simple onesimple one, what does he say?

What is this for?

The one who does not know how to askone who does not know how to ask, what does he say?

This will confuse pupils!



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A researcher, A researcher, what does he what does he

say?say? How to bridge between teachers and students?

How to bridge between teachers and mathematics educators?

What does it mean to knowto know vs. to understandto understand?

Do MSTs develop knowledge (understanding) and how In pupils? In teachers?



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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? Mathematics educators, what do they say?



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Creativity : basic definitionsCreativity : basic definitions Main components of creativityMain components of creativity according to Torrance (1974)

arefluency, 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)

flexibilityflexibility refers to the number of solutions generated by a solver

noveltynovelty refers to the unconventionality of suggested solutions

fluencyfluency refers to the pace of solving procedure and switches between different solutions.



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Tall, what does he say? 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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Example 2: Examining students Example 2: Examining students creativitycreativity

Dor and Tom walk from the train station to the hotel. They start out at the same time. Dor walks half the timehalf the time at speed v1 and half the timehalf the time at speed v2. Tom walks half half wayway at speed v1 and half wayhalf way at speed v2. 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 v1 and half the time at speed v2 and v1>v2 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 v1 a longer distance than Tom. Dor gets to the hotel first.

Solution 2.5 – Experimental modelling (walking around the classroom)

Tom Dor

v1 v2

1/2S v2

1/2S

v1

s

t

s/2 t

/2

T x

Dor Tom

1/2S

S

v1 v2

v2

t 1/2T

1/2T

)From Leikin, 2006(



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Differences between gifted and Differences between gifted and expertsexperts

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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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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Example 3: 7Example 3: 7thth grade mathematics grade mathematics lessonlesson

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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When planning the lessonWhen 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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During the lessonDuring the lessonMaya -- 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 solutionI 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!an expected mistake appeared to be a correct answer!



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On the complexity of teachingOn 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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Why being flexible?Why being flexible?

From Leikin & Dinur (in press)



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Teachers learn themselves when Teachers learn themselves when teaching multiple solutionsteaching multiple solutions

Teachers’ Mathematical Knowledge & Pedagogical Beliefs

Teachers’ mathematicsnoticing & 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

Convic

tion



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Problem in the testProblem in the test

Lev (2003); Leikin & Levav-Waynberg (submitted)



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Example 6: Communication among Example 6: Communication among colleguescollegues

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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Formal solutionFormal solution

A

B

FE

D

CK



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ProcessProcess vs. vs. procedureprocedure

A

B

FE

D

C

F

AD

B

E

C B1

A1From: Applebaum & Leikin (submitted)



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MSTsMSTs

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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Publications related to MSTsPublications related to MSTs -- -- Leikin R.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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