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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

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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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