Conference PaperPDF Available

Is problem posing about posing „problems“? A terminological framework for researching problem posing and problem solving.



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.
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. 2131). Mnster, Germany: WTM-Verlag.
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.
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 Plya’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
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.
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)
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”?
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.
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
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).
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.
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”?
Write a story to match the graph shown below.
Jiang & Cai (2014, p. 397)
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
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
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”?
(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)
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)
Write a problem based on the
following picture:
Christou et al. (2005, p. 152)
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
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.
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”?
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.
We would like to thank Zoltán Kovács, Ioannis Papadopoulos, and Ana Kuzle for
their constructive suggestions, remarks and comments within the review process.
Arıkan, E. E., & Ünal, H. (2015). An investigation of eighth grade studentsproblem
posing skills. International Journal of Research in Education and Science, 1(1), 23
Brown, S. I., & Walter, M. I. (1983). The art of problem posing. Mahwah, NJ: Erlbaum.
Bruder, R. (2000). Akzentuierte Aufgaben und heuristische Erfahrungen Wege zu
einem anspruchsvollen Mathematikunterricht fr alle. In L. Flade & W. Herget
(Eds.), Mathematik. Lehren und Lernen nach TIMSS. Anregungen fr die
Sekundarstufen (pp. 6978). Berlin, Germany: Volk und Wissen.
Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese
students’ mathematical problem solving and problem posing. Journal of
Mathematical Behavior, 21(4), 401421.
Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in
mathematics education: Some answered and unanswered questions. In F. M. Singer,
N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing. From research to
effective practice (pp. 334). New York, NY: Springer.
Cambridge Dictionary (2018). Pose. Retrieved, from
Cantor, G. (1966). Gesammelte Abhandlungen mathematischen und philosophischen
Inhalts. Mit erläuternden Anmerkungen sowie mit Ergänzungen aus dem
Briefwechsel Cantor-Dedekind. Hildesheim, Germany: Georg Olms. (Original work
published 1932)
Christou, C., Mousoulides, N., Pittalis, M., Pitta-Pantazi, D., & Sriraman, B. (2005). An
empirical taxonomy of problem posing processes. ZDM The International Journal
on Mathematics Education, 37(3), 149158.
Dörner, D. (1979). Problemlösen als Informationsverarbeitung (2nd ed.). Stuttgart,
Germany: Kohlhammer.
English, L. D. (1997). The development of fifth-grade children’s problem-posing
abilities. Educational Studies in Mathematics, 34(3), 183217.
Baumanns & Rott
Felmer, P., Pehkonen, E., & Kilpatrick, J. (Eds.). (2016). Posing and solving
mathematical problems. Advances and new perspectives. Basel, Switzerland:
Springer International. doi:10.1007/978-3-319-28023-3
Halmos, P. R. (1980). The heart of mathematics. The American Mathematical Monthly,
87(7), 519524.
Jiang, C., & Cai, J. (2014). Collective problem posing as an emergent phenomenon in
middle school mathematics group discourse. In P. Liljedahl, C. Nicol, S. Oesterle, &
D. Allan (Eds.), Proceedings of the 38th Conference of the International Group for
the Psychology of Mathematics Education and the 36th Conference of the North
American Chapter of the Psychology of Mathematics Education (Vol. 3, pp. 393
400). Vancouver, Canada: PME.
Kilpatrick, J. (1987). Problem formulating: Where do good problems come from? In A.
H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 123147).
Hillsdale, MI: Erlbaum.
Lang, S. (1989). Faszination Mathematik Ein Wissenschaftler stellt sich der
Öffentlichkeit. Braunschweig, Germany: Vieweg.
Plya, G. (1966). On teaching problem solving. In The Conference Board of the
Mathematical Sciences (Ed.), The role of axiomatics and problem solving in
mathematics (pp. 123129). Boston, MA: Ginn.
Pólya, G. (1973). How to solve it. Princeton, NJ: University Press. (Original work
published 1945)
Rott, B. (2012). Problem solving processes of fifth graders an analysis of problem
solving types. In S. J. Cho (Ed.), The Proceedings of the 12th International Congress
on Mathematical Education. Intellectual and attitudinal challenges (pp. 30113021).
Basel, Switzerland: Springer International. doi: 10.1007/978-3-319-12688-3
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
metacognition and sense-making in mathematics. In D. Grouws (Ed.), Handbook of
research on mathematics teaching and learning (pp. 334370). New York, NY:
Silver, E. A. (1994). On mathematical problem posing. For the Learning of
Mathematics, 14(1), 1928.
Singer, F. M., Ellerton, N. F., & Cai, J. (2013). Problem-posing research in mathematics
education: new questions and directions. Educational Studies in Mathematics, 83(1),
Singer, F. M., Ellerton, N. F., & Cai, J. (Eds.). (2015). Mathematical problem posing.
From research to effective practice, New York, NY: Springer.
Singer, F. M., Voica, C., & Pelczer, I. (2017). Cognitive styles in posing geometry
problems: implications for assessment of mathematical creativity. ZDM The
International Journal on Mathematics Education, 49(1), 3752.
Is problem posing about posing “problems”?
Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’
problem posing in school mathematics. In P. C. Clarkson (Ed.), Technology in
mathematics education (pp. 518525). Melbourne, Australia: Mathematics
Education Research Group of Australasia.
Trner, G., & Azarello, F. (2012). Grading mathematics education research journals.
EMS Newsletter, 5254.
Xie, J., & Masingila, J. O. (2017). Examining interactions between problem posing and
problem solving with prospective primary teachers: A case of using fractions.
Educational Studies in Mathematics, 96(1), 101118.
... Problem-solving skill is also significant in mathematics education (Baumanns & Rott, 2019;OECD, 2019;WEF, 2020). Studies show that before teaching students to solve mathematical problems, the relationship between mathematical problems and concepts in daily life should be taught. ...
... Problem-solving skill is not only about solving mathematical problems but also about the ability to find solutions to real-life problems (Burkholder, 2021). Therefore, a person should have problem-solving skills against all unusual situations that he may encounter (Aslan, 2021;Baumanns & Rott, 2019). ...
Full-text available
This study aims to examine the contribution of ArtiBos, which is designed as a Gamified Adaptive Intelligent Tutoring System for students' problem-solving skills. In the study, first of all, the system’s design features to improve problem-solving skills were examined, and then the effect of the system on problem-solving skills was evaluated. The study was carried out with 12 students studying in the ninth class of a High School in Türkiye and 6 mathematics teachers with different experiences working in the same school. A case study, one of the qualitative research methods were applied in this study through which ArtiBos system logs, student interviews, and teacher interviews were evaluated. Data pertaining to the number of solved problems, the number of problems created, the number of problems solved correctly, the duration of being online in the system, the rate of correct problem-solving, and the average solving time were examined to evaluate system logs. Interview questions have been prepared so that the contribution of system features to problem-solving skills can be evaluated. The data from the interview were analyzed and some codes for problem-solving skills were created. And then, sub-themes were created by combining the codes. The results show that ArtiBos affects students' problem-solving skills positively.
... Solving tasks is here conceptualised as an activity with varying degrees of routineness. The term problem in problem posing also refers to all types of tasks with differing degrees of routineness (Cai & Hwang, 2020;Baumanns & Rott, 2019). This understanding is used in this article for the analysis and, therefore, should be given concrete terms. ...
... In Analysis 3, a closer look is taken at all situations that had been identified as structured in Analysis 2. It is considered whether the initial problem of a structured problem-posing situations is routine or non-routine which may influence the emerging problems of the problemposing activity. A similar assessment can be made for unstructured situations: Some situations tend to cause routine problems, while others may lead to non-routine problems (Baumanns & Rott, 2019). The analysis of the emerged problems on the spectrum between routine and non-routine problems has been almost completely absent within the reviewed articles. ...
Full-text available
In research on mathematical problem posing, a broad spectrum of different situations is used to induce the activity of posing problems. This review aims at characterizing these so-called problem-posing situations by conducting three consecutive analyses: (1) By analyzing the openness of potential problem-posing situations, the concept of ‚mathematical posing' is concretized. (2) The problem-posing situations are assigned to the categories free, semi-structured, and structured by Stoyanova and Ellerton to illustrate the distribution of situations used in research. (3) Finally, the initial problems of the structured problem-posing situations are analyzed with regard to whether they are routine or non-routine problems. These analyses are conducted on 271 potential problem-posing situations from 241 systematically gathered articles on problem posing. The purpose of this review is to provide a framework for the identification of differences between problem-posing situations.
... Las investigaciones que examinan la relación entre la resolución de problemas y la invención de problemas coinciden en que la habilidad o capacidad que el alumnado muestra al resolver un problema está directamente relacionada con la habilidad o capacidad de inventar un nuevo problema (Baumanns y Rott, 2018;Cai y Hwang, 2002;Silver y Cai, 2005). En la presente investigación seguiremos el estudio de Cai y Hwang (2002) en el que examinaron la relación que existe entre la estrategia que usa el alumnado al resolver un problema de patrones matemáticos y el tipo de problemas que inventan. ...
Conference Paper
Full-text available
Resumen El presente estudio muestra una experiencia que incluye resolución e invención de problemas llevada a cabo con dos grupos de alumnos de sexto de educación primaria. La investigación toma como objetivo categorizar los problemas que el alumnado inventa con relación a sus estrategias como resolutores. Los resultados enfatizan que aquellos alumnos que generan buenos problemas de patrones son aquellos alumnos que desarrollan el patrón aritmético como resolutores. En contraposición, aquellos alumnos que no generan problemas de patrones usan como estrategia de resolución la ampliación del dibujo presente en el enunciado del problema que se les propone.
... The following examples illustrate that the emerging tasks of a problem-posing situation can result on a spectrum between routine and non-routine problems (cf. Baumanns & Rott, 2019). In Table 7 on page 13, three problemposing situations are shown to motivate this differentiation. ...
Full-text available
This article aims to develop a framework for the characterisation of problem-posing activities. The framework links three theoretical constructs from research on problem posing, problem solving, and psychology: (1) problem posing as an activity of generating new or reformulating given problems, (2) emerging tasks on the spectrum between routine and non-routine problems, and (3) metacognitive behaviour in problem-posing processes. These dimensions are first conceptualised theoretically. Afterward, the application of these conceptualised dimensions is demonstrated qualitatively using empirical studies on problem posing. Finally, the framework is applied to characterise problem-posing activities within systematically gathered articles from high-ranked journals on mathematics education to identify focal points and under-represented activities in research on problem posing.
... Die entstehenden Aufgaben einer Problem-Posing-Tätigkeit resultieren im Spektrum zwischen Routineaufgaben und Problemen (vgl. Baumanns & Rott, 2019). ...
Conference Paper
Full-text available
Schwerpunkt der vorliegenden Studie ist die Entwicklung eines deskriptiven Prozessmodells von Problem-Posing-Prozessen. Dazu wurden 17 Prozesse erhoben, in denen Lehramtsstudierende in Paaren neue Probleme zum sogenannten NIM-Spiel aufwerfen sollten. Aus der Analyse dieser Prozesse wurden fünf inhaltstragende Episodentypen abgeleitet, mit denen sich die Prozesse zeitdeckend beschreiben lassen. Diese Episodentypen wurden sowohl induktiv durch die beobachteten Prozesse als auch deduktiv aus der Theorie zum Problem Posing gewonnen. In einem Ausblick werden Verwendungsmöglichkeiten des deskriptiven Prozessmodells skizziert.
... • Unterscheidung zwischen strukturierten und unstrukturierten Situationen (Baumanns & Rott, 2019;Stoyanova & Ellerton, 1996) ▹ Hier im Fokus: Strukturierte Problem-Posing-Situationen → Aufwerfen neuer Probleme auf der Grundlage eines vorhandenen und zu lösenden Problems (vgl. rechts Beispielaufgabe: NIM-Spiel) ▹ Eine in der deutschsprachigen Literatur bekannte Art des strukturierten problem posings sind Aufgabenvariationen (Schupp, 2002) • Problem Posing fehlt es an Aufmerksamkeit innerhalb mathematikdidaktischer Forschung: "[T]he field of problem posing is still very diverse and lacks definition and structure" (Singer, Ellerton und Cai, 2013) ...
Full-text available
Probleme sind das Herz der Mathematik (Halmos, 1980, S. 524). Dabei steht bei der Beschäftigung mit mathematischen Problemen nicht alleine das Problemlösen im Zentrum, sondern auch das Aufwerfens mathematischer Probleme, das sogenannte Problem Posing. Dieser Tätigkeit schreibt die Mathematik eine zentrale Bedeutung zu, und doch liegt auf ihr kein Fokus innerhalb der mathematikdidaktischen Forschung. Insbesondere fehlen gesicherte Erkenntnisse, ob bei dieser Tätigkeit bestimmte, immer wiederkehrende Phasen durchlaufen werden (Cai u. a., 2015, S. 14). Entsprechende normative bzw. deskriptive Prozessmodelle sind für das Problemlösen bereits etabliert (z. B. Pólya, 1949; Schoenfeld, 1985; Rott, 2013); für das Problem Posing gibt es bislang kein allgemein akzeptiertes Modell. Innerhalb des vorliegenden Forschungsvorhabens zum Problem Posing wird untersucht, inwieweit sich die Tätigkeit des Aufwerfens mathematischer Probleme durch ein Modell phasisch beschreiben lässt. Können in Problem-Posing-Prozessen Phasen identifiziert und inhaltlich beschrieben werden? Treten bei wiederkehrende Muster auf und lässt sich ein phasischer Aufbau erkennen?
Conference Paper
Full-text available
In der Problemlöseforschung ist die Unterscheidung mathematischer Aufgaben in Routineaufgaben und Probleme weitgehend etabliert. Ein Review zur ergänzenden Tätigkeit des Problem Posings hat ergeben, dass die Unterscheidung zwischen dem Aufwerfen von Routineaufgaben bzw. Problemen auch beim Problem Posing fruchtbar ist (Baumanns & Rott, im Review). Die insb. durch Schupp (2002) bekannt gemachte Aufgabenvariation ist eine Form des Problem Posings. In diesem Beitrag wird Schupps Buch „Thema mit Variation“ vor dem Hintergrund der Unterscheidung zwischen Routineaufgaben und Problemen analysiert.
Conference Paper
Full-text available
In this article, we take an in-depth look at research on the intersection of problem posing and creativity in order to present its current state of research in a systematic review. A full search in top journals from mathematics education and the Web of Science revealed only 15 articles from different genres, of which 11 were included in the analysis. Those articles were sorted into two clusters, depending on whether the articles focus on the identification or the fostering of creativity.
Full-text available
Existing studies have quantitatively evidenced the relatedness between problem posing and problem solving, as well as the magnitude of this relationship. However, the nature and features of this relationship need further qualitative exploration. This paper focuses on exploring the interactions, i.e., mutual effects and supports, between problem posing and problem solving. More specifically, this paper analyzes the forms of interactions that happened between these two activities, the ways that those interactions supported prospective primary teachers’ conceptual understanding, and the difficulties that prospective teachers encountered while engaged in alternating problem-posing and problem-solving activities. The results indicate that problem posing contributes to problem-solving effectiveness while problem solving supports participants in posing more reasonable problems. Finally, multiple difficulties that demonstrate prospective primary teachers’ misunderstanding with fractions and their operations provide insight for teacher educators to design problem-posing tasks involving fractions.
Full-text available
While a wide range of approaches and tools have been used to study children’s creativity in school contexts, less emphasis has been placed on revealing students’ creativity at university level. The present paper is focused on defining a tool that provides information about mathematical creativity of prospective mathematics teachers in problem-posing situations. To characterize individual differences, a method to determine the geometry-problem-posing cognitive style of a student was developed. This method consists of analyzing the student’s products (i.e. the posed problems) based on three criteria. The first of these is concerned with the validity of the student’s proposals, and two bi-polar criteria detect the student’s personal manner in the heuristics of addressing the task: Geometric Nature (GN) of the posed problems (characterized by two opposite features: qualitative versus metric), and Conceptual Dispersion (CD) of the posed problems (characterized by two opposite features: structured versus entropic). Our data converge on the fact that cognitive flexibility—a basic indicator of creativity—inversely correlates with a style that has dominance in metric GN and structured CD, showing that the detected cognitive style may be a good predictor of students’ mathematical creativity.
Full-text available
Full-text available
This chapter synthesizes the current state of knowledge in problemposing research and suggests questions and directions for future study. We discuss ten questions representing rich areas for problem-posing research: (a) Why is problem posing important in school mathematics? (b) Are teachers and students capable of posing important mathematical problems? (c) Can students and teachers be effectively trained to pose high-quality problems? (d) What do we know about the cognitive processes of problem posing? (e) How are problem- posing skills related to problem-solving skills? (f) Is it feasible to use problem posing as a measure of creativity and mathematical learning outcomes? (g) How are problem-posing activities included in mathematics curricula? (h) What does a classroom look like when students engage in problem-posing activities? (i) How can technology be used in problem-posing activities? (j) What do we know about the impact of engaging in problem-posing activities on student outcomes?.
Updated and expanded, this second edition satisfies the same philosophical objective as the first -- to show the importance of problem posing. Although interest in mathematical problem solving increased during the past decade, problem posing remained relatively ignored. The Art of Problem Posing draws attention to this equally important act and is the innovator in the field. Special features include: •an exploration ofthe logical relationship between problem posing and problem solving •a special chapter devoted to teaching problem posing as a separate course •sketches, drawings, diagrams, and cartoons that illustrate the schemes proposed a special section on writing in mathematics. © 1990 by Stephen I. Brown and Marion I. Walter. All rights reserved.
This book comprises the Proceedings of the 12th International Congress on Mathematical Education (ICME-12), which was held at COEX in Seoul, Korea, from July 8th to 15th, 2012. ICME-12 brought together 4700 experts from 100 countries, working to understand all of the intellectual and attitudinal challenges in the subject of mathematics education as a multidisciplinary research and practice. This work aims to serve as a platform for deeper, more sensitive and more collaborative involvement of all major contributors towards educational improvement and in research on the nature of teaching and learning in mathematics education. It introduces the major activities at ICME-12 which has successfully contributed to the sustainable development of mathematics education across the world. The program provides food for thought and inspiration for practice for everyone with an interest in mathematics education and makes an essential reference for teacher educators, curriculum developers and researchers in mathematics education. The work includes the texts of the four plenary lectures and three plenary panels and reports of three survey groups, five National presentations, the abstracts of sixty one Regular lectures, reports of thirty seven Topic Study Groups and seventeen Discussion Groups.
To pose a problem refers to the creative activity for mathematics education. The purpose of the study was to explore the eighth grade students’ problem posing ability. Three learning domains such as requiring four operations, fractions and geometry were chosen for this reason. There were two classes which were coded as class A and class B. Class A was consisted of successful students in comparison to class B in terms of mathematical acquisition. The study has been carried out by means of qualitative research. On the other hand, independent samples T test was used for obtaining statically inference. Moreover, chi-square test was used whether this students’ problem posing ability is independent of mathematics topics.
Conference Paper
This paper reports on an exploratory study which investigates (mathematical) problem solving processes of fifth graders (ages 10 to 12) from German secondary schools. An overview about the problem solving processes of 32 students working in pairs on a geometry task shows a significant correlation between the students' problem solving behavior and their success (or failure). The videotapes which supplied the raw data were coded using an adapted version of the protocol analysis framework from Schoenfeld (1985).