60

61

PROBLEM POSING IN A DYNAMIC GEOMETRY ENVIRONMENT

AND THE DEVELOPMENT OF MATHEMATICAL INSIGHTS

Atara Shriki Ilana Lavy

Oranim Academic College of Education Emek Yezreel Academic College

ABSTRACT

Engagement in problem posing activities is among the significant activities of teaching and learning

mathematics. This engagement might become richer when technology is involved. In this paper, we

describe the experience of teachers who were engaged in problem posing within a dynamic geometry

environment, present some of their works, and discuss the prominent issues that emerged as a result of

their experience.

Keywords: Mathematics education; Problem posing; “What If Not?” strategy;

Dynamic geometry software.

Introduction

Teachers often engage their students in problem solving activities. However, as Einstien and Infeld (1938)

(in Ellerton & Clarkson, 1996) believe: "The formulation of a problem is often more essential than its

solution, which may be merely a matter of mathematical or experimental skills. To raise new questions,

new possibilities, to regard old questions from a new angle, require creative imagination and marks real

advance in science" (p. 92).

In order that teachers will be able to integrate problem posing activities in their mathematics classes, they

have to gain the required pedagogical knowledge and confidence, and acknowledge the benefits and

shortcomings of this approach through experiencing it first by themselves (Cunningham, 2004). In this

paper we present the approach we apply in our work with mathematics teachers who were M.Ed. students,

while engaging them in problem posing activities. This approach is based on Brown & Walter's (1969;

Issue 5, pp.61-70

62

1990; 1993) “What If Not?” strategy, and is implemented within a dynamic geometry (DG) environment.

We then present some examples taken from middle and high school mathematics teachers' works, and

discuss the main insights they gained as a result of their experience.

Literature Background

In this section, we present a concise literature background concerning problem posing and its connection

to the development of mathematical knowledge. In particular, we focus on the “What If Not?” (WIN)

strategy, and how it can be employed for posing new problems based on a given problem. In addition, we

refer to the advantages of using dynamic geometry software (DGS) while engaging in posing geometrical

problems.

Problem Posing

Teachers frequently tend to focus on verification and undervalue exploration (De Villiers, 1998).

Explorations can be implemented through engaging learners in problem posing activities, which are

among the significant activities of teaching and learning mathematics (NCTM, 2000). When students are

engaged in problem posing activities they generate new problems and questions, reformulate given

problems, and explore the new generated problems (Silver, 1994), and as a result they enrich and

strengthen their knowledge, promote diverse and flexible thinking, improve their problem solving skills,

change their view of mathematics, and develop new insights regarding the nature of mathematics (Brown

& Walter, 1993; English, 1996; English, 2003). Moreover, this engagement reduces students' reliance on

their teachers and textbooks, thus providing them with the opportunity to be responsible for their own

learning (Cunningham, 2004), and become active learners through adjusting the problems they pose to

their own domain of interest and cognitive abilities (Mason, 2000). Considering the above, problem

posing should be an integral part of the mathematics curriculum (Brown & Walter, 1993; NCTM, 2000).

The "What If Not?" Strategy

The “What If Not?” strategy is a helpful approach for applying problem posing activities, as

suggested by Brown & Walter (1969; 1990; 1993). This strategy is based on the modification of

at least one of the components of a given problem to yield new problems that might lead to the

discovery of an interesting and unexpected regularity. The WIN strategy is a three-stage process:

At the first stage, the learner has to produce a list of the problem’s attributes. At the second

stage, the learner has to relate to each attribute, address the WIN question, and then suggest

alternatives to it. At the third stage, the learner poses new problems and questions, stimulated by

63

the alternatives. According to Brown & Walter (1993), the implementation of the WIN strategy

enables teachers to move away from the rigid teaching format that convey the impression that

there is only one ‘right way’ to refer to a given problem. As a result, instead of focusing on

finding a solution, the students develop insights regarding the meaning of the problem.

Our experience with prospective teachers who were engaged in mathematical exploration (Lavy & Shriki,

2010) indicates that employing the WIN strategy provided them with a 'safe environment' and concrete

guidelines to follow, and consequently they were able to pose meaningful and rich problems. Dismantling

a problem into its components enabled them to recognize the fact that there are logical connections

between the givens, and while being grouped together they create a coherent mathematical situation. In

other words, the components of mathematical problem are not arbitrarily 'placed together'. In addition,

implementing the WIN strategy motivated them to rethink mathematical objects and concepts, and to

develop an insight regarding the meaning of an 'interesting mathematical regularity'.

Problem Posing within a Dynamic Geometry Environment

"There is no question that computer technology has fundamentally changed the face of mathematics over

the past 50 years so" (De Villiers, 2006, p. 46), thus, involving students in problem posing activities may

become richer and thoughtful when technology is involved (Aviram, 2001). In our case, the teachers were

engaged in problem posing within a DG environment. Posing problems within a DG environment

supports the development of visual reasoning, helps learners to raise assumptions and validate/refute

them, and thus increases the likelihood of discovering regularities and generalizations (Laborde, 1998). In

fact, DG environments can change the teaching and learning of mathematics by turning mathematics into

an experimental science rather than focus on computation and symbolic manipulation (Olive, 2002). The

crucial role of dynamizing is not only to enable one to discover a theorem, but also to verify conjectures

(Landau, 2005), develop students' ability to bridge between geometrical construction and deduction, help

them to formulate deductive explanations, and develop ideas of proof and proving (Jones, 2000).

There are also technical advantages to the use of DGS, as the technical part of the work (drawing,

graphing, and calculating) is completed rapidly and efficiently (Aviram, 2001). The interaction with

dynamic software, "its accuracy, immediate visual feedback and the ability to check various cases in a

short space of time" (De Villiers, 2006, p. 48) enable to focus on various routes of investigations and

explorations, and discover interesting problems and relationships, without wasting time and efforts on

technical aspects (Sinclair, 2004).

64

Examples of Teachers' Works

Since the teachers who participated in our course had no previous experience with problem posing or the

use of DGS, they started, as recommended by De Villiers (2006), with exploring simple and basic

concepts, such as angle bisector, median and midsegment of a triangle. At the first session we exposed the

teachers only "to the specific software skills necessary for a particular learning context" (p. 47). Namely,

we demonstrated how to construct objects, drag unfixed objects, measure distances and areas, and create a

graphical representation. During their experience, for a period of two months, the teachers documented

their entire course of work, including their false trials, indecisions, thoughts and insights within a

portfolio. The following examples and excerpts are taken from these portfolios1.

Exploring Basic Concepts

The teachers' first assignment was to select one of the special lines of triangle, state its definition, and

some of its associated theorems. Then they had to focus on one of the theorems, and explore it through

the implementation of the WIN strategy and the use of the DGS.

The first example and Figure 1 are taken from Allan's portfolio:

"In any triangle, each median divides the triangle into two triangles having the same area. I always

present it to my students as an interesting theorem, but employing the WIN it was only natural for me to

ask what would happen if instead of connecting the mid-point of a side to its opposite vertex, I'll choose a

point that divides the side in a 1:2 ratio, 1:3, and so on. Actually, I imagined a point moving along a side

instead of a fix point as implied by the theorem…It was so amazing to think of mathematics in terms of

dynamism!…I marked a point D on AC and dragged it. The measures of DC

AD and

CBD

ABD

S

S indicated that

they were the same! Of course, it is so obvious! …Actually, there is a linear relation between the two. If

I'll takeDC

AD to be the independent variable and

ACD

ABD

S

S

as the dependent, I'll obtain the function

xxf )( . I graphed it, and while dragging D in order to create the graph, it suddenly occurred to me

that the original theorem is actually the point (1,1). This was really an "Aha!" moment for me…I realized

that I always present my students just a point on a graph".

1 The utterances were originally stated in Hebrew. In translating them to English we made efforts to preserve their essence.

65

The second example and Figure 2 are taken from Gail's portfolio:

"I decided to explore the midsegment. So instead of dividing two sides into two equal segments, I divided

them into thee such segments, and measured the ratio between the area of the original triangle and the

little triangle. It was 9…It took me a moment, and then it was so clear-the triangles are similar, so easy to

prove it…So I constructed a line parallel to one of the sides, and just dragged it. The measures indicated

this similarity ratio…When I graphed the results, I could clearly observe the parabola…it was so nice…I

could clearly see on the graph the point from which I have started".

The third example and Figure 3 are taken from Irina's portfolio:

"The angle bisector generates some 'mysterious' relation between sides, not something simple as the other

special lines, so I decided to examine it closer. I marked a point D on AC, and connected it with vertex B.

I took two measurements: DBC

ABD

and

CD

BCADBA

. The numbers didn't make any sense to me. I made a graph,

and I thought that perhaps this would give me some hint… the graph looked 'smooth', not some random

points, so there must be some pattern behind it…Then I thought that if I'll drag the vertices the graph

would change. But no! And this was the amazing part, because I was certain that the pattern is related to

Figure 1

Figure 2

66

the measures of the angles of the original triangle. But the graph showed that only the relations between

sides were important…For the moment I am still working on finding how to generalize it, because the

graph indicates that there is some generaliztion".

Exploring Problems from Textbooks

After the teachers completed their exploration of the special lines, they were asked to select a problem

from their textbook, and repeat the process.

The fourth example and Figure 4 are taken from John's portfolio:

"Several weeks ago I assigned my students the following problem: ABCD is a square. Point E lies on side

BC, such that 2EC

BE . Find the ratio of areas: CEDS

BEDAS. I remembered that the answer was 5… I was

curious to see what will happen if I drag point E along BC…I measured CE

BE and CEDS

BEDAS, and except for

the case of the original problem, these measurements provided me merely with a vague idea of what was

going on there…Since in the case of the medians I used the option of graph, I thought it might be helpful

to do it again…Wow! I was surprised when I found again a linear relation…The point (2,5) suddenly

appeared as meaningless, just a point…Realizing that there is some linear connection, I was curious to

discover the algebraic pattern (which I assumed was going to be easy, as all I had to find was a linear

equation)…Indeed, as you can see, if |EC|=a, |EB|=ka, than 12)( kkf ".

Figure 4

Figure 3

67

Figure 5

The fifth example and Figure 5 are taken from Sara's portfolio:

"Since in the previous phase I examined the midsegment, I thought it might be interesting to examine the

following problem: ABCD is a parallelogram. E, G, I, K are the midpoints of AB, BC, CD, DA,

respectively. Find the ratio: EGIK

ABCD

S

S. If you think about it, this problem is actually a combination of two

negations of the triangle's midsegment- What-if-not one midsegment? What-if-not a triangle?...I had to

construct 4 movable points, but I wasn't sure how to do it in a way that would enable me to drag them so

that the ratio of the segments on each side will change simultaneously…After some false trials I realized

that it was exactly like in the case of the triangle, as there are 4 triangles created by the 2 diagonals! [AC

and BD]...The graph looked strange at first glance, not a parabola like before, but it certainly indicated

that there was some rule…So with some trigonometric calculations it turned out that the lengths of the

sides are omitted, as well as the angles of the parallelogram, and only the ratio between the segments

counts…I believe that applying the WIN again (for example- a general quadrangle, pentagon, hexagon,

and so on) would also yield some regularity, and perhaps there is a 'meta rule' that governs everything".

Conclusions and Discussion

The problem posing approach embedded in the WIN strategy enabled each teacher to engage in problems

that relate to his/her own domain of interest (Mason, 2000). As can be seen from the teachers' works, the

dynamism feature of the DGS and the WIN strategy are interwoven in a natural way, and in fact stimulate

and nurture each other. Specifically, the WIN question and the need to determine which components of a

figure are fixed and which are unfixed appear to go hand in hand, since in order to progress in their work,

the teachers had to acknowledge that a figure consists of relationships among its components, and that a

functional dependency exists among them (Hölzl, Healy, Hoyles & Noss, 1994).

68

Employing the WIN strategy within a DG environment supported the development of the teachers' new

understandings and insights. The main insight they gained was realizing that 'mathematical situations' do

not necessarily describe constant states, but rather dynamic ones. This recognition, in fact, changed their

perception of the nature of mathematics, and led them to perceive mathematical objects as subjected to

some 'meta-rules' or surprising order that can be generalized and described by algebraic patterns. The new

insight was gained as a result of a self discovery, with no external instruction, and as a result the teachers

were also motivated to prove their obtained results. It should be noted that commonly, students are asked

to prove theorems whose validity is not questionable. Theorems are often stated in textbooks as complete

facts, and students only have to justify them. In such cases, there is no genuine inner motivation to prove

a given statement. However, when one discovers a phenomenon or a certain regularity, he/she is

motivated to prove the conjectures (Lavy & Shriki, 2010).

As evident from the teachers' works, they all employed three of the representations available in the DGS,

namely- figural, numerical, and graphical. Obviously, the drawings were essential for starting the

investigation. In order to carry out the investigation, through implementing the WIN strategy, the teachers

had to dismantle the geometrical objects, and make some inferences concerning their fixed and unfixed

components (De Villiers, 2003). The diverse representations of the DGS enabled the teachers to identify

connections among various areas of mathematics (for example- geometry, algebra, and trigonometry),

where one area reinforces the understanding of the other. Moreover, the multiple representations

supported the development of the ability to move between representations, choosing the most appropriate

one. In addition, and while observing patterns and arriving at generalizations they were not previously

aware of, they enriched their mathematical knowledge and insights (Brown and Walter, 1993; English,

1996; English, 2003).

As can be seen, the numerical representation (measurements of lengths, areas, and angles) were not

helpful, as they did not provide any 'sense' as to the general behavior or pattern emerged. The graphical

representation was the most meaningful in terms of implying on a possible generalized pattern, and

enabled the teachers to realize that the conditions of the original problem are represented merely as a

point on the resulted graph, thus concluded the existence of a generalized rule or pattern. Namely, the

given situation was no longer the main issue; it was not as interesting as the generalized phenomenon- the

entire graph. In fact, viewing the entire they had no need to prove the existence of a rule, but rather to

identify its pattern, and in order to find the pattern they were motivated to engage in problem solving,

without being forced to.

69

To conclude, computer software provides a way to visually represent situations, and therefore enable to

identify patterns (Mckenzie, 2009). In addition, they allow the learners to conjecture and explore

interrelations among concepts rather than spending time on calculation (NCTM, 2000). Therefore, we

recommend the implementation of the WIN strategy in computerized environment.

Obviously, this combined approach is not limited to mathematics. Teachers from all disciplines can

benefit from implementing this approach in their classes, developing their own disciplinary knowledge, as

well as their students' knowledge and insights, and their ability to pose and solve problems.

References

Aviram, A. (2001). From “computers in the classroom” to mindful radical adaptation by education

systems to the emerging cyber culture. Journal of Educational Change, 1, 331-352.

Brown, S. I. & Walter, M. I. (1969). What if not? Mathematics Teaching, 46, 38–35.

Brown , S. I. & Walter, M. I. (1990). The art of problem posing (2nd ed.). Hillsdale, NJ: Lawrence

Erlbaum Associates.

Brown , S. I. & Walter, M. I. (1993). Problem Posing in Mathematics Education. In Stephen I. Brown

& Marion I. Walter (Eds.) Problem posing: Reflection and applications, Hillsdale, New Jersey:

Lawrence Erlbaum Associates, 16-27.

Cunningham, R. F. (2004). Problem posing: an opportunity for increasing student responsibility.

Mathematics and Computer Education, 38(1), 83-89.

De Villiers, M. (1998). An alternative approach to proof in dynamic geometry, in R. Lehrer

& D. Chazan (Eds.), Designing learning environments for developing understanding of

geometry and space, Lawrence Erlbaum Associates, Hillsdale, N.J., 369–393.

De Villiers, M. (2003). Rethinking proof with Sketchpad 4. Emeryville, CA: Key Curriculum Press.

De Villiers, M. (2006). Some pitfalls of dynamic geometry software. Learning and Teaching

Mathematics, 4, 46-52.

Ellerton, N. F. & Clarkson, P. C. (1996). Language factors in mathematics teaching and learning. In J.

Bishop, K. Clements, C. Keitel, J, Kilpatrick and C. Laborde(Eds.), International handbook of

mathematics education. Kluwer Academic Publishers. 987-1033

English. L. D. (1996). Children’s problem posing and problem solving preferences, in J.

Mulligan & M. Mitchelmore (Eds.), Research in early number learning. Australian

Association of Mathematics Teachers.

70

English. L. D. (2003). Problem posing in elementary curriculum. In F. Lester & R. Charles

(Eds.), Teaching mathematics through problem solving. Reston, Virginia; National Council of

Teachers of Mathematics.

Hölzl, R., Healy, L., Hoyles, C. & Noss, R. (1994). Geometrical relationships and dependencies in Cabri.

Micromath, 10(3), 8–11.

Jones, K. (2000). Providing a foundation for deductive reasoning: Students’ interpretations when using

dynamic geometry software and their evolving mathematical explanations. Educational Studies in

Mathematics, 44, 55–85.

Laborde, C. (1998). Visual phenomena in the teaching/learning of geometry in a computer-based

environment. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the

21st century. The Netherlands: Kluwer Academic Publishers, 113-122.

Landau, G. T. (2005). Connecting arguments to actions – Dynamic Geometry as means for the attainment

of higher Van Hiele levels, ZDM, 37 (5), 361-370.

Lavy, I. & Shriki, A. (2010). Engaging in problem-posing activities in a dynamic geometry

setting and the development of prospective teachers' mathematical knowledge. Journal of

Mathematical Behavior, 29, 11-24.

Mason, J. (2000). Asking mathematical questions mathematically. International Journal of Mathematical

Education in Science and Technology, 31(1), 97-111.

Mckenzie, J. (2009). Beyond cut-and-paste- engaging students in making good new ideas. FNO Press.

NCTM - National Council of Teachers of Mathematics (2000). Principles and standards for

school mathematics. Reston, VA: NCTM.

Olive, J. (2002). Implications of using dynamic geometry technology for teaching and learning. In: M.

Saraiva, J. Matos, I. Coelho, (Eds.) Ensinoe Aprendizagem de Geometria. Lisbon: SPCE. Retrieved

Sep. 2010 from: http://www.spce.org.pt/sem/JO.pdf.

Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19-28.

Sinclair, M. (2004). Working with accurate representations: the case of preconstructed

dynamic geometry sketches. Journal of Computers in Mathematics and Science Teaching,

23(2), 191-208.

ATARA SHRIKI, ILANA LAVY