Post on 21-Jan-2023
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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
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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
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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).
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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.
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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
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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
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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).
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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.
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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