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In A. Kuzle, I. Gebel, & B. Rott (Eds.), Proceedings of the 2018 Joint Conference of ProMath and the
GDM Working Group on Problem Solving (pp. 21–31). Mnster, Germany: WTM-Verlag.
IS PROBLEM POSING ABOUT POSING “PROBLEMS”?
A TERMINOLOGICAL FRAMEWORK FOR RESEARCHING
PROBLEM POSING AND PROBLEM SOLVING
Lukas Baumanns and Benjamin Rott
University of Cologne, Germany
In this literature review, we critically compare different problem-posing
situations used in research studies. This review reveals that the term “problem
posing” is used for many different situations that differ substantially from each
other. For some situations, it is debatable whether they provoke a posing activity
at all. For other situations, we propose a terminological differentiation between
posing routine tasks and posing non-routine problems. To reinforce our
terminological specification and to empirically verify our theoretical
considerations, we conducted some task-based interviews with students.
INTRODUCTION
In his article “The heart of mathematics”, the mathematician Paul Halmos
concluded: “I do believe that problems are the heart of mathematics, and I hope
that as teachers, in the classroom […] we will train our students to be better
problem-posers and problem-solvers than we are” (1980, p. 524). He emphasizes
two activities: The first activity, problem solving, received a lot of attention in the
last decades of mathematics education research, especially since Plya’s
(1945/1973) and Schoenfeld’s (1985) seminal works. The second activity,
problem posing, has also been emphasized as an important mathematical activity
by many mathematicians (Cantor, 1932/1966; Lang, 1989) and mathematics
educators (Brown & Walter, 1983; English, 1997; Kilpatrick, 1987; Silver, 1994).
As an important companion of problem solving, it can encourage flexible
thinking, improve problem-solving skills, and sharpen learners’ understanding of
mathematical content (English, 1997).
However, problem posing hast not been in the focus of mathematics education
research (Cai, Hwang, Jiang, & Silber, 2015). As a result, “the field of problem
posing is still very diverse and lacks definition and structure” (Singer, Ellerton, &
Cai, 2013, p. 4). Therefore, we conducted a literature review and critically
compared different problem-posing situations used in research studies and
theoretical papers that were labeled as problem posing.
This review revealed that one aspect in which this field of research lacks
definition and structure is the variety of situations in empirical studies that are
labeled as problem posing. This can be exemplified by two studies. Both Cai and
Hwang (2002) as well as Arıkan and Ünal (2015) show strong links between
students’ problem-posing and problem-solving performance. However, a
Baumanns & Rott
22
comparison of the problem-posing situations used in both studies reveals
significant differences. Cai and Hwang (2002) (see Table 1, Situation 1) invite
students to pose at least three tasks with varying degrees of difficulty for a given
dot pattern. This leads to a creative activity of exploring different paths to pose
numerous tasks, such as “How many white dots are there in the twentieth figure?”,
or “How many black dots are there in the first five figures?” (p. 413). Arıkan and
Ünal (2015, see Table 1, Situation 2) provide a defined calculation and three tasks
for which only one applies to the given calculation. Therefore, there is only one
correct answer and the situation does not encourage posing further tasks. Thus, if
both studies find a strong link between problem solving and problem posing, they
do not refer to the same relationship, because the situations they use differ
significantly from each other. This is like comparing a non-routine problem with
an algorithmic word problem and calling both mathematical problem solving.
Situation
1
Mr. Miller drew the following figures in a pattern, as shown below.
For his student’s homework, he wanted to make up three problems based on
the above situation: an easy problem, a moderate problem, and a difficult
problem. These problems can be solved using the information in the situation.
Help Mr. Miller make up three problems and write these problems in the
space below.
Cai & Hwang (2002, p. 405)
2
Which one of the below problems can be matched with the operation of
213 + 167 = 380?
A) Osman picked up 213 pieces of walnut. Recep picked up 167 more pieces
of nuts more than Osman. What is the total amount of the nuts that both
Osman and Recep picked up?
B) On Saturday, 213 and on Sunday 167 bottles of water were sold in a
market. What is the total number of bottles of water that were sold at this
market on these two days?
C) Erdem has 213 Turkish lira. His brother has 167 lira less than that. What
is the total amount of money that both Erdem and his brother have?
Arıkan & Ünal (2015, p. 1410)
Table 1. Problem-posing situations from studies with similar research questions.
Is problem posing about posing “problems”?
23
This striking difference between two studies, which are based on a similar
research question, motivated a more in-depth investigation of problem-posing
situations used in research studies. Apparently, the term problem posing is used
for a variety of situations that seem to have characteristic differences. Therefore,
we propose a terminology to be able to differentiate between those situations and
to prevent misinterpretations of research results.
A similar approach – clarifying terms – has been made in research on problem
solving, decades ago. The term problem has been and still is used in multiple and
often contradictory meanings, which makes it difficult to interpret the literature.
In some cases, the term problem is used for any kind of mathematical task without
differentiating between routine tasks or textbook exercises and non-routine
problems, as researchers like Schoenfeld (1985, 1992) suggest.
THEORETICAL BACKGROUND
In this chapter, we want to present the current understanding of the term problem
posing. Afterwards, it will be further analyzed by breaking it down into its
etymological components, problem and posing.
What is problem posing?
There are two definitions of the term problem posing, at least one of which is used
or referred to in almost all mathematics education research papers on the topic.
The first definition was stated by Silver (1994, p. 19), who defines problem posing
as the activity of generating new problems and reformulating given problems
which, consequently, can occur before, during, or after a problem-solving
process. The second definition comes from Stoyanova and Ellerton (1996, p. 518),
who refer to problem posing as the “process by which, on the basis of
mathematical experience, students construct personal interpretations of concrete
situations and formulate them as meaningful mathematical problems”. Both
concepts of problem posing are not very restrictive and can be applied to a wide
spectrum of situations. In the following, we adopt the definition of Stoyanova and
Ellerton (1996) as the underlying understanding within this paper.
Stoyanova and Ellerton (1996) differentiate problem-posing situations between
free, semi-structured, and structured problem-posing situations, depending on
their degree of structure. Free situations provoke the activity of posing problems
out of a given, naturalistic or constructed situation without any restrictions. In a
semi-structured situation, the problem poser is invited to explore the structure of
an open situation by using mathematical knowledge, skills, and concepts of
previous mathematical experiences. It is noticeable that the differentiation
between free and semi-structured situations is difficult because there is no sharp
demarcation. We, therefore, plead to merge free and semi-structured situations,
resulting in unstructured situations which have varying degree of restrictions. In
structured situations, people are asked to pose further problems based on a
specific problem, e.g. by varying its conditions.
Baumanns & Rott
24
What is posing?
We now fragment the term problem posing into its two components, starting with
the definition of posing, for which we consulted dictionaries. The Cambridge
Dictionary (2018) defines “to pose” as “to ask a question”.
What is a problem?
In mathematics education research, the term problem has been (and still is) used
for any kind of mathematical task, leading to some difficulties and
misinterpretations in reading the research literature (Schoenfeld, 1992, p. 337).
Therefore, researchers like Schoenfeld (1985) suggest to differentiate between
mathematical tasks that are routine tasks or exercises and non-routine problems.
Following Schoenfeld (1985), we consider (mathematical) tasks to be the
overarching category that can be further differentiated into routine tasks or
exercises “if one has ready access to a solution schema” (p. 74) and non-routine
problems if the individual has no access to a solution schema.
In most cases, the decision whether a task is a routine task or a non-routine
problem is evident. Nevertheless, this attribution is specific to the individual; a
problem which is a non-routine problem for one person can be a routine task for
another person who knows a solution scheme (Dörner, 1979; Rott, 2012;
Schoenfeld, 1992). Thus, the demarcation between these categories may not be
sharp but the extreme cases are clearly recognizable (Pólya, 1966, p. 126).
METHODOLOGY
We systematically gathered problem-posing situations from 185 empirical studies
and theoretical papers about problem posing from the A*- and A-ranked journals
in mathematics education research (as classified by Törner & Azarello, 2012,
p. 53), the Web of Science, papers from the PME, and papers of the collection of
Singer, Ellerton, and Cai (2015) as well as the collection of Felmer, Pehkonen,
and Kilpatrick (2016). With the situations resulting from this review, we have
conducted a qualitative content analysis. An inductive category formation with
regard to the term posing lead to two categories, which will be presented in the
analysis. A deductive category assignment with regard to the term problem lead
to another two categories, which will be presented in the analysis.
In a small empirical study (n = 4 participants), we tested these theoretical
considerations by conducting task-based interviews with situations from the
review. The answers were analyzed with regard to the theoretical considerations
of the analysis presented in this article.
TERMINOLOGICAL ANALYSIS
Is it posing?
There are some situations for which it has to be discussed whether the term posing
fits to the activity it is supposed to induce (see Table 2).
Is problem posing about posing “problems”?
25
Situation
3
Write a story to match the graph shown below.
Jiang & Cai (2014, p. 397)
4
Write a question to the following story so that the answer to the problem is
‘385 pencils’. ‘Alex has 180 pencils while Chris has 25 pencils more than
Alex’.
Christou, Mousoulides, Pittalis, Pitta-Pantazi, & Sriraman (2005, p. 152)
Table 2. Problem-posing situations discussed regarding the term posing.
Situation 3 invites providing a context to a given data or calculation. A graph
showing the temperature of an unknown place from 6:00 a.m. to 10:00 p.m. is
given; the task is to write a story that matches with this graph. We conclude that
this task does not necessarily lead to a question that needs to be solved afterwards.
Our empirical study confirmed this theoretical conclusion: three out of four
participants did not ask a question when working on this situation. In our view,
this situation, therefore, is no posing activity. Nevertheless, it is an important
activity for students and, thus, of interest in mathematics education research – but
not in research on problem posing. We refer to these situations as context
providing tasks.
In situation 4, students are invited to search for the question to a given context
and its answer. The sought-after task, however, is predefined: “How many pencils
do they have in total?” Reacting to situation 4 in the expected way leads to posing
a task, but once the right question for the situation has been found, there is no task
that can be worked on because it has already been solved. Furthermore, because
the situation offers a defined goal, searching for the question that matches the
predetermined situation and answer makes working on this situation equivalent to
solving a reversed task. These are characterized as mathematical tasks with a
defined goal and an undefined question (e.g., Bruder, 2000). This applies to
situation 4 and therefore we consider describing these and similar situations with
a given goal and a basically unambiguous sought-after question not as problem
posing but rather as a reversed task.
Baumanns & Rott
26
Both context providing tasks and reversed tasks have in common that they have a
solution for which it can be decided whether it is correct or incorrect. Therefore,
we summarize them under the term answering tasks. Answering tasks are to be
distinguished from problem-posing situations. Whether a given setting is an
answering task or a problem-posing situation can be determined a priori by
analyzing whether the particular characteristics presented above apply.
Is it a problem?
In the following, we want to use the differentiation of mathematical tasks into
routine tasks and non-routine problems to further differentiate problem-posing
situations. In Table 3, there are four situations sorted by the presented and adapted
categories of Stoyanova and Ellerton (1996).
The shopping shelf in situation 5 comes along with three routine tasks for which
additional questions are to be posed. As the given tasks are routine tasks, the
situation provokes posing routine tasks like “You want to buy two badminton
rackets, a football and a basketball and you have $200 in your pocket. Do you
have enough money for these products? If not, what is the difference?” In fact,
six out of six tasks posed by the participants in the empirical investigation were
routine tasks.
Situation 6 is quite similar to situation 5 by also stating a structured situation with
a task to be solved. However, in contrast thereto, a non-routine problem is given.
Tasks should be posed by constraint or goal manipulation which, consequently,
leads to further non-routine problems like: “Which radius should the smallest
circle have, so that the area of the largest circle is π?” The empirical investigation
confirmed this evaluation: eight out of eight tasks posed by the participants were
non-routine problems.
Comparing these two situations, situation 5 provokes posing routine tasks
whereas situation 6 provokes posing non-routine problems. We want to apply this
established differentiation of mathematical tasks between routine tasks and non-
routine problems on the structured problem-posing situations 5 and 6. This paper
introduces the terms routine task posing and non-routine problem posing. The
former refers to the process of posing routine tasks, and the latter refers to the
process of posing non-routine problems. However, it is not sufficient to assess the
initial task of structured situations in terms of whether it is a non-routine problem
or a routine task. Even initial routine tasks can lead to a mathematically rich non-
routine problem. Therefore, when labeling a situation as routine task posing or
non-routine problem posing, it is necessary not only to assess a priori the initial
task of a structured situation but also to assess a posteriori the emerging tasks.
Is problem posing about posing “problems”?
27
Situation
structured
5
(1) If we want to buy 5 volleyballs,
how much do we need to pay?
(2) If we bought three footballs,
and paid the cashier 100 dollars,
how much can we get for change?
(3) If I want to buy one badminton racket and 10 badminton
shuttlecocks, how much do I need to pay?
(4) Please pose two more questions and answer them.
Jiang & Cai (2014, p. 396)
6
For the figure on the left, one mathematics
problem we could ask is: Given that the radius of
the smallest circle is one unit, what is the ratio of
the area of the largest circle to the area of the
smallest circle?
1. Think about how to solve this problem. […]
2. Pose problems using constraint manipulation or goal manipulation
strategy according to the given figure, or the problems you have posed,
or any other ideas you have. […]
Xie & Masingila (2017, p. 116)
unstructured
7
Write a problem based on the
following picture:
Christou et al. (2005, p. 152)
8
The figure contains: the square ABCD, the circle
inscribed in this square, and the circular arc of centre
A and radius AB. Pose as many problems as possible
related to this figure […]
Singer, Voica, & Pelczer (2017, p. 39)
Table 3. Problem-posing situations discussed regarding the term problem.
Baumanns & Rott
28
This qualitative difference between posing routine tasks and posing non-routine
problems also occurs in the two unstructured situations 7 and 8. The former
situation provokes routine tasks like: “For their kitchen, a family needs a stove
with an oven and a fridge. What do they have to pay in total?” Of course, also
non-routine problems can be posed, but they are less likely in this situation. This
is reinforced by the empirical investigation in which four out of four tasks posed
by the participants were routine tasks. Situation 8 induces posing tasks like:
“What is the ratio of the line segments AC to CE?” or “What is the area of the
rounded segment of the inscribed circle?” These are non-routine problems which
the figure provokes to pose. Nonetheless, there are also routine tasks that could
occur, though they are far less interesting with regard to the information the
situation provides. Actually, 10 out of 16 tasks posed by the participants in the
empirical investigation were non-routine problems.
We now apply the new terms: situation 7 provokes the activity of routine task
posing and situation 8 provokes the activity of non-routine problem posing.
However, for the unstructured situations, this distinction is less pronounced than
for the structured situations 5 and 6. Since no tasks are predefined, you are not
immediately urged in the direction of a specific type of task in the process of
posing. Depending on association, motivation, and mathematical experience, both
routine tasks and non-routine problems can be posed. Because of these difficulties
to predict whether the given situation provokes posing routine tasks or non-
routine problems, the differentiation between routine task posing and non-routine
problem posing is not dichotomous but rather a continuum.
CONCLUSION
The stated situations and their characteristics reveal two aspects. First of all, the
discussed answering tasks, which consist of context providing tasks and reversed
tasks, do not fit into the stated understanding of posing. We want to distinguish
them from problem-posing situations. Whether a situation or task is an answering
task or a problem-posing situation can be assigned a priori by analyzing the
characteristics of the situation or task. Secondly and similar to the terminological
differentiation of mathematical tasks into routine tasks and non-routine problems
(Pólya, 1966; Schoenfeld, 1985), we want to differentiate between routine task
posing and non-routine problem posing. An a priori assignment of this
differentiation is without complete certainty since it also depends on the problem
poser whether a situation provokes routine task posing or non-routine problem
posing. Instead, it is necessary to attribute a posteriori – and for each task
individually – whether the posed tasks are routine tasks or non-routine problems.
Since the problem-solving research has benefited from the differentiation between
routine tasks and non-routine problems (Schoenfeld, 1992), it is assumed that it
could also be beneficial for problem-posing research.
Is problem posing about posing “problems”?
29
The framework can now be used to determine the differences between the studies
from the introduction (see Table 1) more precisely. While Cai and Hwang (2002)
used problem-posing situations that supposedly induce non-routine problem
posing, Arıkan and Ünal (2015) used reversed tasks which – on the basis of this
framework – we do not consider to be a problem-posing activity. As mentioned
in the introduction, the field of problem posing lacks definition and structure. This
paper’s framework is a contributive attempt to close this gap.
Acknowledgements
We would like to thank Zoltán Kovács, Ioannis Papadopoulos, and Ana Kuzle for
their constructive suggestions, remarks and comments within the review process.
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