11th International Congress on Mathematics Education
TG6: Activities and Programs for Gifted Students
Problem posing in the classroom and its relation to mathematical
creativity and giftedness
Ildikó Pelczer
Centre for Technical Development, UNAM, Mexico
[email protected]
“In the last decades, problem posing gained terrain in studies related to
mathematics education. There seem to be an agreement between teachers and educators
that problem posing is a creative process and, therefore, it can be employed to identify
gifted students.
However, it is not clear how classroom problem posing relates to an account of
mathematical creativity: what makes problem posing creative and what types of
creativity occur? In particular, it is not clear how the different assessments used by
researchers in their empirical evaluations count as creativity criteria. We propose a
meta-analysis of problem posing from the point of view of mathematical creativity as
defined and described by Ervynck (1991). We define what classroom problem posing is
and in which sense it is creative. There are several benefits in doing so. By highlighting
the connections at different levels between the creative process and problem posing we
argue for the creative nature of the classroom problem posing. We also define criteria for
assessing students’ creativity in problem posing tasks, both from process and results
point of view. Such general criteria can function as framework for the different
assessments found in the literature.
Research oriented;”
Introduction
As Silver (1994) stated problem posing, along with problem solving, is essential to
the nature of mathematical thinking. Professional mathematicians use to pose themselves
problems since their intellectual goal is the generation of novel conjectures or results
(Pollack, 1987). Pólya (1954) said that in “trying to solve a problem, we consider
different aspects of it in turn, we roll it over and over in our minds; variation of the
problem is essential to our work”. Students are not mathematicians, but still need to solve
and pose problems. So, one can ask: is classroom problem posing a creative act?
Hadamard observed (1954) "Between the work of the student who tries to solve a
problem in geometry and a work of invention ... there is only a difference of degree!"
There are several questions to answer. Where it is hidden the creativity in
classroom problem posing? Is it in the newly stated problem’s quality or rather in the
process (independently of the quality of that result)? Is it possible to define criteria such
to identify the levels of creativity involved in problem posing? Is it possible to teach how
to be creative in the problem posing task? Our interest lies in answering such questions
by situating classroom problem posing in the perspective of mathematical creativity.
Mathematical creativity
In 1991, Ervynck proposed a definition of mathematical creativity situated in a
framework of mathematical knowledge. He defines a formal theory of mathematics as a
framework consisting of concepts and relations and suggests that an act of creativity
requires the realization of at least one of the following objectives: create a useful new
concept, discover formerly not noted relations between two nodes or to construct a useful
ordering that puts forward the deductive order of a theory. Based on this specification, he
gives the following definition “Mathematical creativity is the ability to solve problems
and / or to develop thinking in structures, taking into account of the peculiar logicaldeductive nature of the discipline, and of the fitness of the generated concepts to integrate
into the core of what is important in mathematics.” It is interesting to mention that
Ervynck says that creativity is an ability that can be observed by its manifestations in
problem solving, structured thinking and, in novel and proper concepts. By such a
statement he is saying that creativity lies in the individual and not in the final product, a
thought that underlines the importance of the question: by which means this ability
produces the results? We interpret his definition as a stimulus to take into account, for the
assessment of the creativity, both the process of getting results and the quality of the
results.
For the classroom problem posing it is important to see whether such a view of
mathematical creativity allows giving an account of it, that is, to define what problem
posing is, why it would be creative and how it could be measured. We answer these
questions in the followings.
Classroom Problem posing as an instance of mathematical creativity
Classroom problem posing: a definition
For mathematicians problem posing refers to the process by which they obtain a
problem that has not been solved yet by anyone. In most empirical studies, though,
problem posing means the formulation of novel problems with the solution unknown at
least for its creator (Van den Heuval-Panhuizen et al. 1995). In other contexts it is
understood as reformulation of an existing problem (Cohen & Stover, 1981), mostly illdefined ones. We are interested in clarifying what it means to pose a problem in
classroom. Such a definition has to consider the nature of the mathematical knowledge,
but also the way in which this is structured for classroom activities. Ervynck (1991)
described the formal theory of mathematics as a “framework consisting of definitions of
concepts and relations between defined concepts, the latter being of a very particular
kind: the relations emerge from the implementation of very strictly prescribed (deductive)
rules”. Brinkmann (2005) claims that relations between mathematical objects around a
topic may be graphically represented by mind and concept maps in a way that
corresponds to the structure of mathematics. Therefore, to pose a problem in a topic it
means to formulate questions about 1) the existence of a mathematical object; 2) the
relation between different mathematical objects; 3) new properties of a given object
deduced or related to a set of specified properties. The result should have an intrinsic
difficulty (Pólya, 1967) in order to differentiate it from an exercise.
With this view of the problem posing, we can ask how different parts of this
knowledge network are drawn together in the process of problem posing and, regarding
the result, how novel and interesting (or unusual for that topic) are the questions about the
objects or relations?
Problem posing: why it is creative?
Silver (1997) asserts “…It is in this interplay of formulating, attempting to solve,
and eventually solving a problem that one sees creative activity. Both the process and the
products of this activity can be evaluated in order to determine the extent to which
creativity is evident.” He proposes aspects to be examined in order to assess the
creativity involved in the problem posing tasks: novelty of problem formulation or
problem solution, shifts in direction or focus during the process of reformulation and
number of paths explored as facets. Silver’s view is interesting because it insists on the
necessity to consider the process by which we get new problems, that is, it is the process
that makes problem posing an instance of creativity. It remains, still, a question on what
level is the similarity of the processes involved in problem posing and creative thinking.
Is it at a meta-level (like formulating, attempt to solve) or at the level of cognitive
mechanisms? Or there should be analogy at all possible levels (cognitive mechanism,
meta-level and results)?
The revision of hundreds of textbook problems and also students’ generated
problems (Pelczer & Gamboa, 2008) lead us to the idea that the creativity of the problem
posing process relies on the relational, multi-articulated nature of the mathematical
knowledge. We hypothesize that the knowledge available to the student is under a
continuous reordering during the problem posing process as the relevance of a piece of
knowledge is under change. Such a “reordering” allows cognitive change to occur and it
is the base for the “shift from association-based to causation-based thinking, which
facilitates the fine-tuning and manifestation of the creative work” (Gabora, 2002).
Therefore, we consider that classroom problem posing is creative because it involves
cognitive mechanisms that are typical for creative thought. Based on this, we formulate
the following criteria for the assessment of the problem posing process from creativity
point of view. The examples used below are from the study conducted by Pelczer &
Gamboa (2008) on gifted student’s problem generation. The problems were transcribed
exactly in the form proposed by the students.
Criteria for the assessment of the classroom problem posing
Criteria for creativity assessment of the problem posing process
We define as first level of creativity (algorithmic) one that it is characterized by
the employment of domain-specific algorithm in the problem posing. The typical
example for this case would be problem generation based on a rule. Example: Consider
(an ) n 0 a real sequence such that a1  2 , an 1  an  2 n . Find a closed expression for a n .
A second level of creativity is defined as the application of some domain-specific
rule along with some other type of knowledge. We shall use the term combined creativity
for this case. The “other knowledge” would be from another domain and its application
not straightforward for the most. It can also be seen in the form of the problem’s
question, that is, the question refers to something that is not typically from the domain.
Example: Consider
(an ) nN such
that
a0  1 and an 
na n 1
a n 2
. Prove that (2n  1)!
an an 1

2 2n 1
.
2n  1
A third level of creativity is identified as innovative creativity and it is defined as
the process of using knowledge from outside of the domain for which the problem is
generated. In many cases, even the question (like an important constitutive part of the
problem) falls outside of range of the typical questions. Example: Consider the following
sequence: a1  3 , a n 1  a n 2  a n . Decide whether 396,138,794,300,000 is term of the
sequence.
We gave the last example in order to underline that assessing the process’
creativity need not to give us the same result as the assessment of the result. The above
problem is a very easy one, of low level of interest, however it is creative in terms that
appeals to knowledge from another domain in order to define the problem.
Criteria for creativity assessment of problem posing result
The dominant view of creativity in the literature is like the process that leads to
products that will be considered as novel and interesting (valuable, adequate, surprising,
unusual or aesthetic). Since novelty is assessed by comparison, it is necessary to specify
the aspects of the problem that will be considered in such process. Generally speaking,
we treat a problem as having a given part, requested part, form of the question,
restrictions (when asking to apply some particular method, for example) and solutions. In
some particular domain (sequences, for example), we can speak about problem types as
determined by the expressions involved in the given part. Based on these parts we define
the following levels of novelty.
At the lowest level we define the algebraic novelty which consists of small
differences in the expressions in the given or requested part, meanwhile all the rest
remains unchanged.
The second level of novelty occurs when there is a significant change in the given
or requested or form of the question part, but the rest stays equal. Such change it is
reflected at the level of the nature of the element, therefore we shall use the term of
conceptual novelty.
A third level of novelty is the methodological one. In this case, a modification in
the expression causes modifications in other parts of the problem, especially in the
possible solutions.
From the list of second attributes we chose interestingness. The level of a
problem’s interestingness is defined in parallel to the level of creativity in the process of
problem posing. Therefore, a problem presents low level of interest if it is a simple
application of a known (domain-specific) algorithm.
At the next level (average) are the problems that request to connect different types of
knowledge. In a network view of the mathematical knowledge this means that concepts
from different parts of the network are drawn together by the problem. Again, the main
aspect of the problem remains domain-specific, but involves other type of knowledge.
At the third level of interestingness, high, we consider those problems that move
us to another domain, that is, in order to solve it we have to have knowledge from another
domain. In this case, the domain-specific algorithms are not of great usefulness, since the
question of the problem can ask for concepts from another domain. Many of existence
problems (of the type: is there any sequence such that property P is true) can have this
level of interestingness. To question the existence of a mathematical object, as activity,
falls into the description of the mathematical creativity (in sense of Ervynck, 1991),
therefore we consider that has a high level of interestingness.
However, in classroom problem posing adequacy can be another useful aspect to
assess the creativity. The term can be interpreted as correspondence to a specific goal, for
example, in case of teachers the problem can be judged as adequate if it complies with
some specific pedagogical aim or, in case of students, if it has a desired difficulty level or
satisfies some restriction of the task.
The above criteria being topic-independent allow specifying more concrete forms for
them for specific domains or topics and, on other hand, to compare performance on
problem posing tasks between different domains.
Fostering problem posing skills
What should be done in order to enhance student’s problem posing skills? On
which aspects of problem posing should we focus? In the past, several research results
were reported about enhancing student’s problem posing skills in different domains (for
example, Yevdokimov, 2005; Gonzales, 1998). Another set of investigations highlighted
factors that affect problem posing (for example, Nicolau & Phillipou, 2002; English,
1998). We want to treat the subject in a more general note and in order to do that we draw
in the elements enumerated by Ervynck as being essential for mathematical creativity.
We give a reinterpretation them for the case of classroom problem posing and build on it
to suggest activities that can enhance problem posing skills.
Ervynck mentioned four elements as being the motive power for mathematical
creativity.
Understanding: Ervynck underlines that understanding has to be seen in both
terms of instrumental and relational understanding, in the definition of Skemp (1976), but
even more important is the “meaningful relationship between the concepts in the
context”. Therefore, for teaching it would be important to explicitly show the connection
between different concepts in the specific context in which they are used. For example,
for the topic of sequences we can draw the attention on how sequences relate to definite
integrals or to matrixes.
Intuition: the interpretation that Ervynck gives to the notion of intuition (see
above), leads us to examples as typical illustrations of concepts. Examples have several
uses for mathematicians, as synthesized by Alcock (2004): in understanding the
statement, generating an argument and in checking an argument. By giving good
examples to the introduced theorems, criteria and concepts, students can grasp the
meaning of it: not only has what made a mathematical object to belong to a class, but also
the limitations of such relationship.
Insight: About insight, Ervynck says that it involves “refocusing of interest and a
reorientation to consolidate what is important, and even more, to envision will be
important in the future”. When posing problems, often the start point is some type of
requirement, theorem or something that the persons considers as fitting with the very
general idea of what he wants to build (Pelczer & Gamboa, 2008; Cruz Ramirez, 2006).
Such an idea acts as a guide trough the problem posing process, but it is necessary for the
person to see the “essence” of that idea. For teaching skills it would be important to show
students how restrictive are the terms of a theorem or the limitations of a criteria, etc. For
example, in the topic of sequences it should be underlined (as it can be easily biased by
the repetitive use of some theorems) that monotony is not a necessary condition for
convergence. Another aspect of this issue refers to solutions. One learns problem posing
methods from reading solutions to given problems. It is necessary to identify those
features that can be extracted from the solution (as they are not problem-dependent, for
example) and kept as methods for their own. For example, for sequences, such case can
be seen in problems where in order to solve the problem given by a recurrent relation,
part of this recurrence is replaced by a new sequence (like an  x n  x n1 ) obtaining a new
problem in terms of a n instead of x n . On a second term, multiple solutions underline the
multifaceted mature of a problem and can give new ideas on how to build new problems.
Generally speaking, these two issues belong to the “look-back” phase in the Pólya
proposed problem solving model, but often is ignored by students. Teachers should
explicitly underline its importance and show in which ways it could be employed for
problem posing.
Generalization: Ervynck mentions two types of generalizations: expansive and
reconstructive ones. The first one “broadens the applicability of the theory without
changing the nature of the cognitive structure”, meanwhile the second one “requires the
knowledge structure to be reorganized”. We concentrate on the first type of
generalization mentioned by Ervynck, defined similarly by Pólya (1954), and propose
two subtypes of such generalization.
The first subtype is based on the concept map related to the topic. For example,
from the expression an

1
n
we can easily get a new one by thinking of 1 and n as
particular polynomials, so we could get for example an

n2  1
n3
. Furthermore, we can see
the polynomial (polynomial function) as a particular case of a function, and a sequence as
another particular functions, therefore by combination we could get.
an 
an 1  1
an 2  an 1
.
Obviously, getting new expressions is not a guarantee for getting quality sequence
problems, but here we concentrate only eon the generalization as method. A second type
of generalization is possible along the function or property of an expression. In this case,
we should think of 1 as a bounded entity and we could get
an 
sin(n )
easily
n
by replacing
it with any other bounded entity. These two categories of generalization comprise the
“what-is-not” method proposed by Brown and Walter (1990).
In the classroom, the accent should be put on these two types of generalizations.
The first one also relates to the element of understanding, meanwhile the second one to
the element of insight, since the function or property of interest depends on the given
context. Learning how to apply these generalizations, sometimes also subject to
restrictions (for example, to be applicable the same theorem), is the key for successful
problem posing.
Problem posing and the mathematically gifted students
Many authors underline the necessity of creating a learning environment that
would help the gifted to grow into a creative person; however there are no official
guidelines on how this should be done. The term “mathematically gifted” is employed as
to designate a person with strong math skills, such as spatial or logical reasoning and
understanding mathematical ideas. It is our standpoint that mathematical giftedness itself
expresses just a potential and this potential needs to be nurtured by an adequate
education. These children need a special attention that could come, on one hand, through
situations that allow them to generate and explore ideas, to supervise their thinking
processes and to monitor their own advance and, on other hand, by mathematically
challenging situations that would require or promote creative thought.
A way to start is to think about the nature of the mathematical knowledge.
Mathematics is a highly structured network of ideas (Fisher, 1990); therefore, to think
mathematically means to derive information from the connections in the network and,
also, to form connections. As defined above, problem posing is seen as the process of
formulating questions about the objects in this network and the relations between them. It
can be said, then, that the very nature of the problem posing tasks invite students to
explore their own knowledge structures, to reformulate existing relations, generalize,
propose new relations or objects, link seemingly unrelated domains, etc. All these
activities are creative expressions. In the same time, it can be argued that (at least some
of) these aspects can be derived from problem solving tasks, too. However, there is a
particular aspect of problem posing that differentiates it from problem solving.
Problem posing asks for control: controlling ones own creation, supervising the
process of subsequent transformations. This aspect relates to metacognition. Smith (1994)
gives the following definition “Metacognitive processes are presumed to take place when
we think about our own thinking, for example, when we reflect upon whether we know
something, whether we are learning, or whether we have made a mistake.” During
problem solving, it is necessary for the student to monitor his advance, that is, to have
good metacognition skills. However, in problem solving the final state (the goal state to
reach) is often clear and the student has some (learned) strategies by which he can
evaluate his progress toward the goal. On contrary, problem posing tasks require from the
student to formulate and continuously reformulate his goal state, the criteria used to
assess advance and the termination of the process. Many studies, between them the ones
mentioned in the section regarding the definition of creativity assessment criteria,
concluded that students dispose of techniques to invent problems, but they often fail to
get to a correctly formulated problem. This failure is due to the lack of evaluation criteria
of the proposed intermediate solutions. Problem posing tasks can draw the attention to the
importance of developing some particular skills, like inventing evaluation criteria and
decision taking to change direction as consequence of such evaluation. These particular
skills are not easily reachable by problem solving activities.
Another aspect of problem posing that could highly benefit mathematically gifted
students is the exploration of knowledge. When asked to pose a problem (in free or semistructured situations, as defined by Stoyanova, 1998) students need to identify knowledge
that could be relevant for the task. When solving a problem the topic is fixed and students
tend to look for solution inside of the topic. In case when the problem to be solved is
reformulated in another topic or domain and solved there, we have a creative problem
solving. Still, we claim that such situation, although desirable, is not so frequent. On
other hand, when students are asked to pose a problem, as the very first step they do such
exploration. It might be in the starting point they define for the posing process or in the
final formulation of the problem or in the type of question they pose, but they have an
open situation in which they can move freely between topics and domains and also
express personal preferences (by the type of problem they generate). We shall illustrate
this latest aspect, since it is the least studied one in problem posing. Also, we consider
that by involving something personal, the student will feel more motivated and personally
compromised with the task. Ultimately, this can influence the emotional experience of
completing it and the desire to engage again in similar tasks.
The first example we give comes from a high school student and illustrates how
his conception of a problem lead to the final formulation of a sequence problem. In the
post-experiment questionnaire he said about the evaluation of problem difficulty: “I took
into consideration the problem’s complexity and the measure until which different
domains of mathematics are mixed in it. The problem should be interesting, thought
provoking and ask for the use of knowledge beyond of that about sequences.” As easy
problem he proposed the following one:
“We have a cylindrical tube of 10.10cm radius to which a rope is fixed. The rope
has 1 cm diameter and is of 100m length. How many times can we go around the cylinder
if we have to put the rope over the previously rolled rope?”
The second example come from an Olympiad participant and it is interesting,
because speaks about his (not mathematical) concerns. Although the problems are not of
special interest from mathematical point of view, they definitively are from personal
point of view: mathematics is seen as a way to speak up, to let to know about problems of
other types. It is not the place to discuss whether such problems are needed or not nor if
they are interesting/well formulated or not (from mathematical point of view), here we
are interested only of what personally means for this student the experience.
For the average difficulty problem he proposed the following one: “Consider a n
being the number of smokers in DF (Mexico City, note of the authors) in the year
2000+n. We know that a n1  a n  10%a n . Compute the term a8 if we know that a3  1000000 . ”
For the difficult problem he gave the following one: “The level of CO2 in our
country is growing with 2% by year and our planet is getting warmer. Demonstrate that
if the temperature continues to grow, then one day the humanity will cease to exist. Note:
consider that humans can stand a temperature higher to 100 o C.”
Therefore, we claim that problem posing allows reinforcing not only mathematical
knowledge, but also meta-knowledge and by allowing the expression of personal interests
or preferences can be more motivating and emotionally engaging mathematical task than
problem solving.
As conclusion, we consider problem posing tasks as being an invitation to perform
creative acts and the challenge inherent to such an act is the particular aspect that can
benefit mathematically gifted students.
Conclusions
We gave an account of mathematical classroom problem posing from the
perspective of mathematical creativity. We based our analysis on Ervynck’s view on
mathematical creativity and build up a definition for classroom problem posing along
with an argument on why it is creative. We proposed three creativity levels for the
process of problem posing and also criteria to evaluate the novelty and interestingness of
the resulted problems. The meta-level of our analysis allow to integrate various, previous
finding and proposals into a unitary treatment of classroom problem posing. The problem
of fostering problem posing skills was treated also in the context of the Ervynck’s view
of mathematical creativity. Finally, we briefly explored the relation between problem
posing tasks and the personal development of mathematically gifted students. A main
conclusion is that we need to redefine mathematical creativity when speaking about the
classroom and in the same time we need to develop a working framework that would
allow disposing of shared assessment criteria. Such a framework would help researchers
to compare their empirical findings and make progress toward a theory of problem
posing.
References
Alcock, L. (2004), Uses of example objects in proving, International Group for the
Psychology of Mathematics Education 28th, , July, Bergen, Norway.
Brinkmann, A. (2005), Knowledge maps – Tools for building structure in
mathematics, International Journal for Mathematics Teaching and Learning.
Brown, S. I. & Walter, M. I. (1990) The art of problem posing (2nd ed.), Hillsdale,
NJ: Erlbaum.
Cohen, S.A. & Stover, G. (1981). Effects of teaching sixth-grade students to
modify format variables of math word problems. Reading Research Quaterly 16(2), 175200.
Cruz Ramirez, M. (2006), A Mathematical Problem–Formulating Strategy,
International Journal for Mathematics Teaching and Learning, ISSN 1473 – 0111,
December.
Ervynck, G. (1991), Mathematical Creativity. In D. Tall (Ed.), Advanced
Mathematical Thinking. 42-53. Dordrecht, the Netherlands: Kluwer Academic
Publishers.
Fisher, R. (1990). Teaching children to think, Oxford: Basil Blackwell Publishing
House.
Gabora, L. (2002), Cognitive mechanisms underlying the creative process.
Proceedings of the Fourth International Conference on Creativity and Cognition,
Loughborough Univ., UK, 126-133.
Hadamard, J. (1954). Psychology of invention in the mathematical field. Princeton,
NJ: Princeton University Press.
Pelczer, I. & Gamboa, F. (2008), Problem posing strategies of mathematically
gifted students, The 5th International Conference on Creativity in Mathematics and the
Education of Gifted Students, February, Haifa, Israel.
Pollack, H.O. (1987), Cognitive science and mathematics education: a
mathematician’s perspective – In . Schoenfeld, A.H. (ed.), Cognitive science and
mathematics education. Hillsdale, NJ: Lawrence Erlbaum Associates.
Pólya, G. (1954), Mathematics and plausible reasoning, Princeton, NJ: Princeton
University Press.
Pólya, G. (1967), Mathematical discovery: On understanding, learning, and
teaching problem solving, John Wiley & Sons, INC.
Silver, E.A. (1994), On mathematical problem posing, In: For the learning of
mathematics, 14(1), 19-28.
Silver, E.A. (1997), Fostering creativity through instruction rich in mathematical
problem solving and problem posing, ZDM, 29(3).
Skemp, R. (1976), Relational understanding and instrumental understanding,
Mathematics Teaching, 77, 20–26.
Smith, F. (1994). Understanding reading (5th ed.). Hillsdale, New Jersey:
Lawrence Erlbaum Associates.
Stoyanova, E. (1998). Problem posing in mathematics classrooms. In A. McIntosh
& N. Elerton (Eds). Research in Mathematics Education: a contemporary perspective
(pp. 164-185) Edith Cowan University: MASTEC.
Yevdokimov, O. (2005), On development of students’ abilities in problema
posing: a case of plane geometry, Proceedings of the 4th Mediterranean Conference on
Mathematics Education, Italy.
Van den Heuval-Panhuizen et al. (1995). Students-generated problems: easy and
difficult problems on percentage. For the learning of mathematics, 15(3), 21-27.