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

1

Multiple solution task -MST

Solve the problem in as many ways as possible

19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 2

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.

.

41 # , 403 דומע / ילארגטניא ןובשח , הירטמונוגירט , ילאיצנרפיד ןובשח : הזילנא .) 2001 ( ןרוג ינב

Solution 1: x

and

y

– the sides,

x

2 

y

2  4

R

2 

y

 Derivative of function

S(x)… x



y

 4

R

2 

x

2 

S

(

x

) 

x

 2

R

,

S

 2

R

4

R

2 

x

2 a

)=2R 2 sin

a , , 1 , ,

x y

2R

Solution 4:

2R

Compare with a square

Solution 3: h h

altitude to the diagonal, max 

R



S

max  2

R

2

S

(

h

)  2

R



h

19-10-2007 Roza Leikin, University of Haifa

h

2R

h

Multiple Solution Tasks: From mathematics to Pedagogy and vice versa

2R

3

Pupils,

what do they say?

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.

The wicked one, what does he say?

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

The simple one, what does he say?

What is this?

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

I do not understand anything

19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 4

Teachers,

what do they say?

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?

The wicked one, what does he say?

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

The simple one, what does he say?

What is this for?

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

This will confuse pupils!

19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 5

A researcher,

what does he say?



How to bridge between teachers and students?



How to bridge between teachers and mathematics educators?



What does it mean to know vs. to understand?



Do MSTs develop knowledge (understanding) and how



In pupils?



In teachers?

19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 6

Mathematics educators, what do they say?

 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 should be implemented in the area of curricular design MSTs can be used for the assessment and development of ones  Knowledge  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) 19-10-2007 Roza Leikin, University of Haifa  Multiple Solution Tasks: Creativity From mathematics to Pedagogy and vice versa 7

Creativity : basic definitions

 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 refers to the number of solutions generated by a solver  novelty refers to the unconventionality of suggested solutions  fluency refers to the pace of solving procedure and switches between different solutions. 19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 8

Tall, what does he say?

19-10-2007 Roza Leikin, University of Haifa 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

Multiple Solution Tasks: , 17 –19 March 2007, Abu Dhabi From mathematics to Pedagogy and vice versa 9

Example 2: Examining students creativity

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

:

Tom Dor

v

1

1 / 2 S v

1

Solution 2.4

-

Logical considerations

:

v

2

1 / 2 S v

2

Solution 2.3

–

Graphing:

s

S 1 / 2

s

S

2 /

v

1

1 / 2 T

/ t 2

v

2

Dor

v

2

1 / 2 T

t T

Tom

x

t

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) 19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa (From Leikin, 2006) 10

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

19-10-2007 Roza Leikin, University of Haifa (From Leikin, 2006; Leikin& Lev, 2007) Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 11

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

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?

19-10-2007 Roza Leikin, University of Haifa

M-Ch 1 M-Ch 3 M-Ch 2

From “Visual Mathematics” program (CET, 1998). Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 13

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 19-10-2007 Roza Leikin, University of Haifa Dinur (2003), Leikin & Dinur (in press) Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 14

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.

an expected mistake appeared to be a correct answer!

Dinur (2003), Leikin & Dinur (2007) 19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 15

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) 19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 16

Why being flexible?

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

Teachers learn themselves when teaching multiple solutions

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

Problem in the test

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

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?

E F

19-10-2007 Roza Leikin, University of Haifa

C B D A

From: Applebaum & Leikin (submitted) Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 20

Formal solution

D E C K A B F

19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 21

Process

vs. procedure

E F C B D E F A D

19-10-2007 Roza Leikin, University of Haifa

C B A A 1

Multiple Solution Tasks: From mathematics to Pedagogy and vice versa

B 1

22

In MSTs For the development and identification of

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

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..

19-10-2007 Roza Leikin, University of Haifa Multiple Solution Tasks: From mathematics to Pedagogy and vice versa 24