The process of problem posing: development of a descriptive phase model of problem posing

Abstract

The aim of this study is to develop a descriptive phase model for problem-posing activities based on structured situations. For this purpose, 36 task-based interviews with pre-service primary and secondary mathematics teachers working in pairs who were given two structured problem-posing situations were conducted. Through an inductive-deductive category development, five types of activities (situation analysis, variation, generation, problem-solving, evaluation) were identified. These activities were coded in so-called episodes, allowing time-covering analyses of the observed processes. Recurring transitions between these episodes were observed, through which a descriptive phase model was derived. In addition, coding of the developed episode types was validated for its interrater agreement.

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https://doi.org/10.1007/s10649-021-10136-y
1 3
The process ofproblem posing: development ofadescriptive
phase model ofproblem posing
LukasBaumanns1 · BenjaminRott1
Accepted: 29 November 2021 /
© The Author(s) 2021
Abstract
The aim of this study is to develop a descriptive phase model for problem-posing activities
based on structured situations. For this purpose, 36 task-based interviews with pre-service
primary and secondary mathematics teachers working in pairs who were given two struc-
tured problem-posing situations were conducted. Through an inductive-deductive category
development, five types of activities (situation analysis, variation, generation, problem-
solving, evaluation) were identified. These activities were coded in so-called episodes,
allowing time-covering analyses of the observed processes. Recurring transitions between
these episodes were observed, through which a descriptive phase model was derived. In
addition, coding of the developed episode types was validated for its interrater agreement.
Keywords Problem posing· Phase model· Cognitive processes· Problem solving
1 Introduction
In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.
(Cantor, 1867, p. 26)
Transl.: In mathematics, the art of posing a question is of greater value than solving
it.
In his statement, Cantor emphasizes the importance of the ability to pose substantial
questions within mathematics. In fact, problem posing is considered a central activity of
mathematics (Hadamard, 1945; Halmos, 1980), and at the latest since the 1980s (Brown &
Walter, 1983; Butts, 1980; Kilpatrick, 1987), it is being investigated with growing interest
by mathematics education researchers. Since the 1990s, it has been widely used to identify
or assess mathematical creativity and abilities (Silver, 1994, 1997; Singer & Voica, 2015;
Van Harpen & Sriraman, 2013; Yuan & Sriraman, 2011). Silver (1997, p. 76) emphasizes
that to grasp such constructs, both products and processes of problem-posing activities can
be considered. However, a strong product orientation within research on problem posing
is noticeable (Bonotto, 2013; Singer etal., 2017; Van Harpen & Sriraman, 2013); that is,
* Lukas Baumanns
lukas.baumanns@uni-koeln.de
1 University ofCologne, Cologne, Germany
Published online: 27 December 2021
Educational Studies in Mathematics (2022) 110:251–269
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studies aiming to assess mathematical creativity, for example, often focus on the posed
problems rather than the processes that led to them. This is noteworthy, since processes are
central to educational research. As Freudenthal (1991) states:
[T]he use of and the emphasis on processes is a didactical principle. Indeed, didac-
tics itself is concerned with processes. Most educational research, however, and
almost all of it that is based on or related to empirical evidence, focuses on states (or
time sequences of states when education is to be viewed as development). States are
products of previous processes. As a matter of fact, products of learning are more
easily accessible to observation and analysis than are learning processes which, on
the one hand, explains why researchers prefer to deal with states (or sequences of
states), and on the other hand why much of this educational research is didactically
pointless. (p. 87, emphases in original)
Although there are studies considering problem-posing processes (Headrick etal., 2020;
Ponte & Henriques, 2013), general knowledge about learners’ problem-posing processes
remains limited (Cai & Leikin, 2020). Only a few studies are dedicated to the development
of a phase model for problem posing (Cruz, 2006; Pelczer & Gamboa, 2009). Those mod-
els still hold the potential for sufficient generalization and validation. This knowledge could
help to develop a more sophisticated process-oriented perspective on problem posing. The
few studies that examine the general process of problem posing (Koichu & Kontorovich,
2013; Patáková, 2014; Pelczer & Rodríguez, 2011) may benefit from a validated phase
model. Such a model may also be useful for the effective educational use of problem pos-
ing in the classroom. This study aims to develop a valid and reliable category system that
allows analyzing problem-posing processes. These kinds of conceptual frameworks play
a central role in mathematics education research as they enable a better understanding of
thinking processes (Lester, 2005; Schoenfeld, 2000).
2 Theoretical background
2.1 Problem posing
There are two widespread definitions of problem posing which are used or referred to in
most studies on the topic. As a first definition, Silver (1994, p. 19) describes problem pos-
ing as the generation of new problems and reformulation of given problems. Silver con-
tinues that both activities can occur before, during, or after a problem-solving process. As
a second definition, Stoyanova and Ellerton (1996, p. 218) 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.” In the following, we adopt the definition of Silver (1994) as the differentiation
between the activities of generation and reformulation is beneficial for identifying differ-
ent activities in problem-posing processes. However, both definitions are not disjunctive or
contradictory but describe equivalent activities.
In both definitions, the term problem is used for any kind of mathematical task,
whether it is a routine or a non-routine problem (Pólya, 1966). For the former, “one has
ready access to a solution schema” (Schoenfeld, 1985b, p. 74), and for the latter, one
has no access to a solution schema. Thus, problem posing can lead to any kind of task
252 L. Baumanns, B. Rott
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on the spectrum between routine and non-routine problems (Baumanns & Rott, 2019;
Baumanns & Rott, 2021a).
Stoyanova and Ellerton (1996) distinguish between free, semi-structured, and struc-
tured problem-posing situations depending on the degree of structure. A situation is an
ill-structured problem in the sense that its goal cannot be determined by all given ele-
ments and relationships (Stoyanova, 1997). Because this study focuses on structured
situations and Baumanns and Rott (2021a) encountered difficulties in distinguishing free
and semi-structured situations, in this article, we distinguish between unstructured and
structured situations. Unstructured situations form a spectrum of situations without an
initial problem. The given information of these situations reaches from nearly none (see
Table1, situation 1) to open situations with numerous given information, the structure
of which must be explored by using mathematical knowledge and mathematical con-
cepts (see Table1, situation 2). In structured situations, people are asked to pose fur-
ther problems based on a specific problem, for example, by varying its conditions (see
Table1, situation 3). The phase model developed in this article aims to describe prob-
lem-posing activities that are induced by situations like those in Table1. In particular,
the model is developed using structured situations.
Table 1 Unstructured and structured problem-posing situations
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2.2 Process ofproblem posing—state ofresearch
Because products may be more accessible by analysis than processes (Freudenthal, 1991),
most problem-posing studies focus on posed problems (Bicer etal., 2020; Van Harpen
& Presmeg, 2013; Yuan & Sriraman, 2011). However, consideration of the processes
increases in recent studies (Cai & Leikin, 2020; Crespo & Harper, 2020; Headrick etal.,
2020; Koichu & Kontorovich, 2013; Patáková, 2014; Pelczer & Rodríguez, 2011). Ponte
and Henriques (2013), for example, examine the problem-posing process in investiga-
tion tasks among university students and found that problem posing and problem-solving
complement each other in generalizing or specifying conjectures to obtain more general
knowledge about the mathematics contents. Christou etal. (2005) describe four thinking
processes that occur within problem posing, namely editing, selecting, comprehending/
organizing, and translating quantitative information. They found the most able students are
characterized through editing and selecting processes. However, compared to the present
study, these activities do not tend to describe problem-posing processes by phases. Instead,
Christou etal. (2005) intend to characterize thinking processes in problem posing. Cifarelli
and Cai (2005) include problem posing in their model to describe the structure of math-
ematical exploration in open-ended problem situations. They identify a recursive process in
which reflection on aproblem’s solutions serves as the source of new problems.
The studies cited above differ from the present study as follows: They describe and
analyze only individual processes, they describe the problem-posing process in terms
of thinking processes rather than phases, or they consider problem-posing processes as
a sub-phase of a superordinate process. However, there is a lack of studies that attempt
to derive a general, descriptive phase model of observed problem-posing processes
themselves from numerous processes. For problem-solving research, the analysis of pro-
cesses through phase models has been established at least since Pólya’s (1945) and Sch-
oenfeld’s (1985b) seminal works. While their models are normative, which means they
function as advice on how to solve problems, newer empirical studies on the process of
problem-solving develop and investigate descriptive models, which means they portray
how problems are actually solved by participants (Artzt & Armour-Thomas, 1992; Rott
etal., 2021; Yimer & Ellerton, 2009). This study also focuses on descriptive models.
Some researchers interpret problem posing as a problem-solving activity (Arıkan
& Ünal, 2015; Kontorovich etal., 2012; Silver, 1995), and there are several established
models of problem-solving processes (e.g., Mason etal., 1982; Pólya, 1945). Therefore,
it is a reasonable question whether a separate phase model for problem posing is needed.
From the observations of problem-solving and problem-posing activities within the present
study, we share the argument by Pelczer and Gamboa (2009) that the cognitive processes
involved in problem posing are of their own nature and cannot be adequately described by
the phase models of problem solving. For problem posing, Cai etal. (2015) state, “there
is not yet a general problem-posing analogue to well-established general frameworks for
problem solving such as Polya’s (1957) four steps” (p. 14).
To find existing research on this topic, we conducted a systematic literature review
(Baumanns & Rott, 2021a, 2021b). This review encompassed articles from high-ranked
journals of mathematics education, the Web of Science, PME proceedings, the 2013 and
2020 special issues in Educational Studies of Mathematics, the 2020 special issue in
International Journal of Educational Research, and two edited books on problem posing
(Felmer etal., 2016; Singer etal., 2015). From all reviewed articles, three were dedicated
to the development of general phases in problem posing similarly to the present study.
254 L. Baumanns, B. Rott
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Cruz (2006) postulates a phase model based on a training program for teachers (see
Fig.1). For this reason, this phase model is preceded by educative needs and goals. Once
a concrete teaching goal has been set (1), the episode type of problem formulating begins
(2). This episode has a problem as its output which is then solved (3). If it cannot be solved,
the problem may have to be reformulated (4). A solvable problem is further developed
in the episode type problem improving (5). The complexity of the problem is adapted to
the learning group and compared with the goal (6 and 7). If the comparison shows that
the problem is not suitable, either further changes are made to the task (8) or the task is
rejected as unsuitable.
Pelczer and Gamboa (2009) distinguish five phases—setup, transformation, formula-
tion, evaluation, and final assessment—based on the analysis of problem-posing processes
in unstructured situations. The setup includes the definition of the mathematical context
of a situation and the reflection on the knowledge needed to understand the situation. This
assessment serves as a starting point for the subsequent process. During the transforma-
tion, the conditions of a problem are analyzed, and possibilities for modification are identi-
fied, reflected, and executed. In the formulation, all activities related to the formulation of
a task are summarized. This includes the consideration of different possible formulations
of the problem as well as an evaluation of these formulations. In the evaluation, a posed
problem is assessed in terms of various aspects, for example, whether it fulfills the initial
conditions or further modifications are needed. In the final assessment, the process of pos-
ing a problem is reflected upon, and the problem itself is evaluated, for example, in terms
of difficulty and interest. In their study, Pelczer and Gamboa (2009) compare experts’ and
novices’ problem-posing processes, identifying different trajectories, that is, transitions
between the stages. While experts more often go through recursive processes, processes of
novices are more linear and often occur without transformation and final assessment.
Koichu and Kontorovich (2013) developed four stages, observed in the context of two
successful problem-posing activities: (1) In the warming-up phase, typical problems spon-
taneously associated with the given situation are posed that serve as a starting point. (2) In
the phase searching for an interesting mathematical phenomenon, participants concentrate
on selected aspects of the given task to identify interesting aspects that can be used for
forthcoming problems. (3) Since the intention is to develop interesting problem formula-
tions, in the phase hiding the problem-posing process in the problem formulation, the pos-
ers try to disguise to the potential solvers in which way the task was created. (4) Finally, in
the reviewing phase, the posers evaluate the problems based on individual criteria such as
the degree of difficulty or appropriateness for a specific target group.
In general, Cruz’ (2006) phase model does not allow for sufficient generalization to
processes of sample groups that do not pursue school learning goals such as students or
mathematicians. The model by Pelczer and Gamboa (2009) has the potential to verify the
validity by checking objective coding. The stages by Koichu and Kontorovich (2013) are
developed on a small sample of two people and therefore need to be tested for applicability
to larger sample groups. All these potentials will be addressed in this article. In addition,
Fig. 1 Phase model of problem
posing by Cruz (2006)
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1 3
although the models presented have certain similarities, they also show numerous charac-
teristic differences. In comparison, phase models for problem-solving (Artzt & Armour-
Thomas, 1992; Pólya, 1945; Rott etal., 2021; Schoenfeld, 1985b; Yimer & Ellerton, 2009)
share a very similar core structure. Thus, there is a conceptual and empirical need for a
generally applicable model for problem-posing research.
The need for developing a phase model for problem posing is, furthermore, based on
our general observation that the quality of the posed problems did not always match the
quality of the observed activity. In our opinion, it is therefore not enough to consider only
the products when, for example, problem posing is used to assess mathematical creativity.
Furthermore, developing a process-oriented framework serves as research for discussing
and analyzing these processes (Fernandez etal., 1994, p. 196).
2.3 Research questions
The research goal of this study is to develop a descriptive phase model for problem-posing
activities based on structured situations. The lack of phase models constitutes a desidera-
tum from which the following research questions emerge:
(1) Which recurring and distinguishable activities can be identified when dealing with
structured problem-posing situations?
(2) What is the general structure (i.e., sequence of distinguishable activities) of the
observed processes from which a descriptive phase model may be derived?
The goal of these research questions is to develop a descriptive phase model that allows
analyzing problem-posing processes. To evaluate the quality of this model, we draw on the
criteria by Schoenfeld (2000) that can be used for evaluating models in mathematics edu-
cation. As this type of coding is highly inferential (Rott etal., 2021; Schoenfeld, 1985b),
special emphasis is given to interrater agreement.
3 The study
3.1 Data collection
The present study is a generative study that aims to “generate new observation categories
and new elements of a theoretical model in the form of descriptions of mental structures
or processes that explain the data” (Clement, 2000, p. 557). For such studies, a less struc-
tured, qualitative approach is appropriate that is open to unexpected findings (Döring &
Bortz, 2016, p. 192), such as task-based interviews. Task-based interviews have particu-
larly been used in problem-solving research to gain insights into the cognitive processes
of participants (Konrad, 2010, p. 482). The interviews were conducted in pairs to create a
more natural communication situation and eliminate the constructed pressure to produce
something mathematical for the researcher (Schoenfeld, 1985a, p. 178). Johnson and John-
son (1999) also underline that cooperative learning groups such as pairs are “windows into
students’ minds” (p. 213). For this reason, the interviewer avoided intervening in the inter-
action process.
The interviews were conducted with 64 pre-service primary and secondary mathemat-
ics teachers (PST). The PSTs worked in pairs on one of two structured problem-posing
256 L. Baumanns, B. Rott
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1 3
situations, either (A) Nim game or (B) Number pyramid, which are presented in Table2.
The participants were informed that both problemsolving and problem posing were cen-
tral. After the initial problemsolving, both situations stated: “Based on this task, pose as
many mathematical tasks as possible.” This open and restriction-free question should stim-
ulate a creative process. A common question of understanding from participants was, using
the example of situation (A), whether they should now pose further Nim games or were
also allowed to depart from them. This decision was left to the PSTs’ creativity.
In total, 15 processes of situation (A) and 17 processes of situation (B), ranging from
9 to 25min, have been recorded and analyzed. The processes ended when no ideas for
further problems emerged from the participants. In total, 7h and 46min of video material
were recorded and analyzed. Thus, the processes had an average length of 14.5min. Four
pairs of PSTs each were in the same room under authentic university seminar conditions. A
camera was positioned opposite the pairs capturing all the participants’ actions. To accus-
tom them to natural communication in front of the camera, short puzzles were performed
before problem posing.
3.2 Data analysis
For data analysis, we adapted Schoenfeld’s (1985b) verbal protocol analysis, originally
used to analyze problem-solving processes. This method is an event-based sampling. Com-
pared to time-based sampling, the processes are not divided into fixed time segments (e.g.,
30s), which are then coded. Instead, new codes are set when the participants’ behavior
changes. This method has two steps: At first, the recorded interviews are segmented into
“macroscopic chunks of consistent behavior” (Schoenfeld, 1985b, p. 292) that are called
episodes in which “an individual or a problem-solving group is engaged in one large task
[...] or closely related body of tasks in the service of the same goal” (Schoenfeld, 1985b, p.
292). In a second step, the episodes are then characterized in terms of content.
Table 2 Structured problem-posing situations used in this study
Situation
(
A)
Nim game
There are 20 stones on the table. Two players A and B ma y
alternately remove one or two stones from the table.
Whoever makes the last move wins. Can player A, who
starts, win safely? Based on this task, pose as many
mathematical tasks as possible.
(cf. Schupp, 2002, p. 92)
(
B)
Number pyramid
In the following numb er pyramid, which number is in 8th place from the right in the
67th line?
Based on this task, pose as many mathematical tasks as possible.
(cf. Stoyanova, 1997, p. 70)
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To answer the first research question, verbal protocol analyses were employed in terms
of inductive category development (Mayring, 2014, pp. 79–87), meaning that the episode
types were developed data-derived. The descriptions of the episode types were addition-
ally concretized in a theory-based manner. For that, the above-mentioned conceptual and
empirical findings of problem-posing research (Cruz, 2006; Pelczer & Gamboa, 2009; Sil-
ver, 1994), as well as findings of research on phase models in problemsolving (Pólya,
1945; Schoenfeld, 1985b), were used. This procedure aims to develop exclusive and
exhaustive codes (Cohen, 1960), that is, episode types, that can be assigned to the observed
problem-posing processes.
To answer the second research question, recurring sequences of the episode types were
identified to develop a general phase model. Both general sequences in the observed pro-
cesses, as well as conceptual insights about problem-posing activities in general, were con-
sidered. To analyze the interrater agreement, an independent second coder was trained. At
first, the second coder was given the coding manual and a process to code without further
comment. For this first coding, cases of doubt were discussed within 2h of training. After
this training, the second coder analyzed about 2h and 23 min of the total video material
of 7h and 46min which means 10 randomly chosen processes out of 32. Thus, the second
coder analyzed about 30.7% of the total video material. Finally, cases of doubt of coding
were discussed via consensual validation. These codings were used to calculate the inter-
rater agreement to the author’s coding.
The interrater agreement was calculated with the EasyDIAg algorithm by Holle and
Rein (2015). EasyDIAg provides an algorithm that converts two codes of an event-based
sampling data set into an agreement table from which Cohen’s kappa (Cohen, 1960) is cal-
culated through an iterative proportional fitting algorithm. Furthermore, in contrast to the
classical Cohen’s kappa, EasyDIAg provides an interrater agreement score for each value
of a category. EasyDIAg considers raters’ agreement on segmentation and categorization
as well as the temporal overlap of the raters’ annotations. This makes this algorithm par-
ticularly suitable for assessing the interrater agreement of the event-based sampling data
set at hand. For the agreement, we used an overlap criterion of 60% as suggested by Holle
and Rein (2015). In the online supplement, we provide an example analysis of a process
that was coded by the authors and the second rater followed by the calculation of the inter-
rater agreement in this manner.
4 Results
First, to retrace the inductive-deductive category development, the problem-posing process
of the Nim game by Theresa and Ugur will be described in order to refer back to it when
describing the developed episode types. The individual episodes are described without
labelling them. The given periods indicate the minutes and seconds (mm:ss) of the respec-
tive episodes. The recorded time starts with the first attempt at posing problems after the
initial problem has been solved. Compared to other participants, Theresa and Ugur get the
solution of the Nim game quickly and without assistance.
Episode 1 (00:00–00:49): Theresa and Ugur first read the task that should initiate the
problem posing. Ugur considers whether new tasks should now be posed in relation to the
solution strategy of working backwards. Theresa considers whether the stones should be
the focus of new tasks. Afterward, both reflect again on their solution strategy and consider
to what extent they can use it for new tasks.
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Episode 2 (00:49–02:14): Then other games like Connect Four or Tic-tac-toe, which
may have a winning strategy similar to the Nim game, are collected.
Episode 3 (02:14–05:50): Both participants want to figure out whether there is a
winning strategy for Tic-tac-toe. After about 3min, they assume that an optimal game
always results in a draw. They return to the Nim game and ponder whether player B also
has a chance to win safely. They conclude that player B can only win if player A does
not make the first move according to the winning strategy.
Episode 4 (05:50–07:43): They pose the task of how many stones are necessary for
player B to win safely. Afterward, the text of the task is formulated. They also ask how
many moves player A needs in order to win.
Episode 5 (07:43–09:03): The last-mentioned question of episode 4 is solved and
also generalized. Ugur says, you find the number of moves of player A by going from
the number of stones to the next higher number divisible by three, and then dividing this
number by three.
Episode 6 (09:03–09:44): Ugur suggests increasing the number of stones that can
be removed from the table. Specifically, he suggests that one to three stones can be
removed. Meanwhile, Theresa writes down these ideas.
Episode 7 (09:44–10:32): Theresa writes down the previously posed problems with-
out working on the content of the formulations.
Episode 8 (10:32–13:48): Both play the variation of the Nim game raised in episode
6. They express that they want to develop a winning strategy for this variation. They
quickly realize that player B can safely win the game since multiples of four are now
winning numbers and the 20 stones that are on the table at the beginning are already
divisible by four. They validate this strategy afterward. At the last minute, the newly
posed variation is also evaluated as exciting.
Episode 9 (13:48–14:13): Ugur wants to generalize the game further and poses the
task of how to win when the players can remove one to n stones. Theresa asks Ugur if
his goal is a general formula.
Episode 10 (14:13–15:48): This task is then solved by Ugur by transferring the struc-
ture of the solution of the initial problem to the generalization. Ugur formulates that if
you are allowed to remove one to n −
1 stones, the player who has the turn must bring
the number of stones to n by his turn to win safely.
Episode 11 (15:48–16:42): Subsequently, both work on a suitable formulation for
this generalized task.
Episode 12 (16:42–18:28): Theresa notes that solving the initial problem is challeng-
ing and therefore suggests providing help for pupils. Theresa suggests that it might help
when the pupils first develop a winning strategy for the simple case that the players can
only remove one stone. Ugur suggests further help cards which can be requested by the
pupils themselves if they get stuck.
Episode 13 (18:28–19:50): Theresa wants to focus on new tasks again. They move
away from the initial problem and use the stones to create an iconic representation of
the triangular numbers (1, 3, 6, ...). They formulate the task to find a general formula to
calculate the n-th triangular number.
Episode 14 (19:50–21:33): Theresa puts the stones in rows of three so that the struc-
ture that leads to the winning strategy is more visible. She evaluates this presentation by
emphasizing the usefulness of this method for extensions of the Nim game with more
than 20 stones on the table. The process comes to an end as Theresa and Ugur, when
asked by the interviewer, agree not to generate any more ideas.
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4.1 Category development ofepisode types inproblem posing
Using the described evaluation method, five episode categories were developed which
allow the observed processes to be described in a time-covering manner. These episode
categories are situation analysis, variation, generation, problem-solving, and evaluation. In
the following, the developed categories of episode types are described. The episodes of the
process by Theresa and Ugur (T&U) described above are assigned to these episode types
for a better comprehension of the episode types. In addition, we provide further anchor
examples in the online supplement. Subsequently, indications are given for coding the indi-
vidual categories. Finally, the categories are discussed regarding the state of research.
4.1.1 Situation analysis
Description During the situation analysis, the posers capture single or multiple conditions
of the initial task. They usually recognize which conditions are suitable and to what extent,
to create a new task by variation (changing or omitting single or multiple conditions) or
generation (constructing single or multiple new conditions). In addition, the subsequent
investigation of the initial task’s solution is summarized in this episode. This also includes
the creation of clues or supporting tasks that lead to the solution of the initial task.
In the process of T&U, episode 1 is coded as situation analysis as the participants still
reflect on their solution strategy. Also, episode 12 is coded as situation analysis because
both PSTs try to come up with ideas on how to support students with solving the initial
problem. A further example of other participants who capture the conditions of the initial
problem can be found in the online supplement.
Coding instructions It is not always clear when the posers are engaged in reading (see
non-content-related episodes below) or have already moved on to situation analysis.
Simultaneous coding is possible here. The creation of supporting tasks, which are sup-
posed to assist in solving the initial problem, is interpreted as an analytical examination of
the situation and is therefore coded as situation analysis.
4.1.2 Variation
Description During variation, single or multiple conditions of the initial task or a task
previously posed in the process are changed or omitted. No additional conditions are con-
structed. In addition, writing down and formulating the respective task is included under
this episode.
In the process of T&U, episodes 4, 6, 9, and 11 are coded as variation. In episode 6, for
example, Ugur varies one specific rule of the Nim game and states that the players are now
allowed to remove one to three stones from the table. In episode 9, this is further general-
ized by variation.
Coding instructions For the identification of variation, the What-If-Not-strategy by
Brown and Walter (2005) should be used. The first step of this strategy is intended to
extract the conditions of a problem. The Nim game, for example, has at least the following
five conditions: (1) 20 stones, (2) two players, (3) alternating moves, (4) one or two stones
260 L. Baumanns, B. Rott
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are removed, and (5) whoever empties the table wins. This analysis should be done before
coding. Omitting or varying these analyzed conditions will be coded as variation. Also,
omitting or varying conditions of a previously posed problem is coded as variation.
4.1.3 Generation
Description During generation, tasks are raised by constructing new conditions to the
given initial task or a task previously posed in the process. Due to the possible change in
the task structure, posers sometimes explain the new task. In addition, writing down and
formulating the respective task is summarized under this episode type. Also, free associa-
tions, in which tasks similar to the initial task are reminded, are coded as generation.
In the process of T&U, episodes 2 and 13 are coded as generation. In episode 13, for
example, they move further away from the Nim game and use the stones to ask questions
about dot patterns.
Coding instructions The episode types variation and generation are not always clearly
distinguishable from each other. Although the coding focuses on the activity of the poser
and not on the emerged task, it can help to examine the characteristics of a task resulting
from variation or generation. In the case of a varied task, the question or the solution struc-
ture often remains unchanged. In the case of a generated task, there is usually a fundamen-
tally different task whose solution often requires different strategies.
4.1.4 Problem solving
Description Problemsolving describes the activity in which the posers solve a task that
they have previously posed. If a non-routine problem has been posed, the respondents go
through a shortened problem-solving process in which the phases of devising and carrying
out the plan (Pólya, 1945) are the main focus. In some cases, the posers omit to carry out
the plan if the plan already provides sufficient information on the solvability and complex-
ity of the posed problem. If a routine problem has been posed, the solution is usually not
explained, since the method of solution is known. However, longer phases of solving rou-
tine tasks are also coded as problemsolving.
In the process of T&U, episodes 3, 5, 8, and 10 are coded as problemsolving as the par-
ticipants are engaged in solving their posed problems.
Coding instructions Although solving a routine problem should be differentiated from
solving a non-routine problem, both activities are labelled with the same code. However,
the commentary of the coding should specify whether an episode is an activity of solving a
routine or a non-routine problem.
4.1.5 Evaluation
Description In the evaluation, the posers assess the posed tasks based on individually
defined criteria. In the processes observed, posers asked whether the posed problem is
solvable, well-defined, similar to the initial task, appropriate for a specific target group, or
interesting for themselves to solve. On the basis of this evaluation, the posed task is then
accepted or rejected.
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In the process of T&U, episode 14 is coded as evaluation, and in episode 8, there is
a simultaneous coding of problem-solving and evaluation. In episode 8, for example, the
participants are initially engaged in problem-solving. Towards the end of this episode, they
both assess their posed problem based on their interest in solving it.
Coding instructions Often, evaluative statements are made about the course of an episode
of problem-solving, since the criteria for the evaluation of a posed problem (e.g., solvabil-
ity or interest) are based on sufficient knowledge about the solution of the posed prob-
lem. In such cases, the episode types of problem-solving and evaluation cannot be sepa-
rated empirically, which is why simultaneous coding is permitted. The criterion for this
simultaneous coding is that during an episode of problem-solving, an evaluative statement
must come within a 30-s window for a simultaneous coding to be made. For example, if
at least one evaluative statement falls during the first 30s of a problem-solving episode,
both types of episodes are coded simultaneously. If at least one evaluative statement also
falls within the following 30s of problem-solving, both episode types are again encoded
simultaneously.
4.1.6 Non‑content‑related episode types
When participants, for example, ran out of ideas or became distracted during the inter-
view, they engage in the following non-content-related activities. Such activities were
also identified in descriptive models of problemsolving (Rott etal., 2021). In the pro-
cess of T&U, episode 7 was coded as non-content-related episode.
Reading The episode of reading consists of reading the situation text as well as a shorter
exchange about what has been read to make sure that the text is understood. Since the par-
ticipants have usually already solved the initial task of the situation, the reading takes place
rather in between.
Writing In the episode of writing, posers write down the text of a problem they have
already worked out orally. Also, the posers write down the solution of a previously posed
problem. Writing is only coded if no solution or problem formulation is being worked on in
terms of content (e.g., specify the problem text).
Organization Organization includes all activities in which the poser is working on the
situation, but where no content-related work is apparent. This includes, for example, the
lengthy production of drawings.
Digression The episode digression is encoded when the posers are not engaged with the
situation. This may include informal conversations with the other person about topics that
are not related to the task (e.g., weekend activities) or looking out of the window for a long
time.
Other All episodes that cannot be assigned to any other episode type are coded as other.
262 L. Baumanns, B. Rott
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1 3
4.1.7 Discussion
To provide a theoretical justification of the data-driven episode types of problem posing,
we want to connect the five episode types with the presented state of research on problem-
posing phase models.
Situation analysis In Pelczer’s and Gamboa’s (2009) phase model, we find aspects of situ-
ation analysis in their transformation stage. One sub-process of this transformation stage is
the analysis of the problem’s characteristics. Terminologically, the episode name is based
on Schoenfeld’s (1985a, 1985b) analysis, because we observed that, similar to problem-
solving, posers identify what possibilities for problem posing the given situations provide
through their conditions.
Variation Pelczer and Gamboa (2009) have aspects of variation in the stage of formula-
tion in which a problem is written down and the formulation is evaluated. Problem formu-
lating can also be found in the model by Cruz (2006). The principle of variation also plays
a central role in problemsolving. Schoenfeld (1985b), for example, suggests posing modi-
fied problems by replacing or varying the conditions of a particular problem that is difficult
to solve.
Generation Koichu and Kontorovich (2013) consider spontaneously associated problems
related to a given problem-posing situation in their model, yet this is only one aspect of
the generation described above. The distinction between variation and generation is theo-
retically already conceptualized by Silver (1994). In empirical studies on problem posing,
there are so far no objective criteria that enable distinct identification of both activities. The
phase model at hand proposes criteria for this distinction.
Problem‑solving Cruz (2006) explicitly mentions problemsolving as a stage in his prob-
lem-posing phase model. In the model by Pelczer and Gamboa (2009), problemsolving is
implicit in the evaluation phase, in which the posed problem is assessed and modified. This
is presumably done based on the solution of it.
Evaluation The stage of evaluation in the phase model by Pelczer and Gamboa (2009)
shares the same name and has similar characteristics. Cruz (2006) implicitly considers
evaluation when the posers improve the posed problem when they deem it not suitable
for a specific learning group. The activity of evaluation is closely related to the metacog-
nitive activity of the regulation of cognition (Flavell, 1979; Schraw & Moshman, 1995).
In research on problem posing, there are hardly any studies that investigate metacognitive
behavior, yet some frameworks implicitly include aspects of it. Kontorovich etal. (2012),
for example, consider aptness by means of fitness, suitableness, and appropriateness of a
posed problem.
4.2 Derivation ofadescriptive phase model forproblem posing
There is no predetermined order of episode types which means there can be transitions
from any episode type to any other. However, there is a kind of “natural order” in which
episode types appear in most processes and in which transitions often occur. This has
263The process of problem posing: development of a descriptive…
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1 3
been indicated by the order in which the episode types were presented in Sect.4.1. It was
observed that first the conditions of a situation are grasped (situation analysis) and then
new tasks are posed through variation or generation; these tasks are solved in order to
evaluate them based on the solution. Of course, we did not observe exactly this order in
every process, but across the participants and the different problem-posing situations, parts
of this superordinate pattern were identified. Often the situation analysis was observed at
the beginning of the process and at the end of a longer phase of variation. Also typical
were frequent changes between variation or generation and problemsolving (sometimes
in combination with evaluation). Furthermore, problem posing was identified as a cycli-
cal activity. Several participants were observed to revise or to further vary their previously
posed problems. Figure2 shows the T&U’S process following Schoenfeld’s (1985b) illus-
trations of problem-solving processes. Several characteristic transitions can be observed in
this process. The vertical lines shown in this figure indicate points in time when a new task
(either by variation or by generation) was posed.
From these theoretically justifiable as well as empirically observable patterns in the
sequence of episodes, the descriptive phase model shown in Fig.3 was derived. It contains
Fig. 2 Example of a timeline chart of the problem-posing process by Theresa and Ugur as described in
Sect.4 following the illustrations by Schoenfeld (1985b)
Fig. 3 Descriptive phase model for problem posing based on structured situations
264 L. Baumanns, B. Rott
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1 3
all five content-related episodes as a complete graph. All transitions indicated by arrows
can occur and have been observed empirically in the study. However, not all episode types
need to occur in a process. Several participants were observed to revise or to further vary
their previously posed problems. In addition, in most cases, not only one but several prob-
lems are posed in numerous cycles. The model reflects this observation through its cyclic
structure. The model is used to represent all these possible paths within the problem-pos-
ing process.
To check the interrater agreement, 30.7% of the total video material of 7h and 46min
was coded by a second independent rater and combined into an agreement table (see
Table3) using the EasyDIAg algorithm (Holle & Rein, 2015). As explained in Sect.4.1.5,
the episode types of problemsolving and evaluation have empirically often been observed
simultaneously, which is why simultaneous coding was allowed. We have, therefore, con-
sidered this simultaneous coding as a separate category for the verification of interrater
agreement. If the start or end of a process was coded differently in time by the two raters,
there are unlinked events in the agreement which are coded as X. The entry X–X in Table3
can, therefore, not occur empirically.
With a Cohen’s kappa of κ = 0.81, the interrater agreement is almost perfect (Landis &
Koch, 1977, p. 165). This high level of agreement is particularly gratifying as the evalua-
tion method is a highly subjective and interpretative procedure, yet the developed catego-
ries are capable of consistent coding. As anticipated, the biggest coding differences are
observed for the categories variation and generation as well as the distinction between the
categories of problemsolving, problemsolving and evaluation, and evaluation. The kappa
calculated for the separate categories are (with the abbreviations from Table3 as indices)
κSA = 0.87, κV = 0.83, κG = 0.72, κPS = 0.87, κPS/E = 0.73, κE = 0.49, and κO = 97.
5 Discussion
This study aimed to develop a valid and reliable model to describe and analyze problem-
posing processes. Schoenfeld (2000) provides eight criteria for evaluating models in math-
ematics education: (i) descriptive power, (ii) explanatory power, (iii) scope, (iv) predictive
Table 3 Agreement table for all seven categories of episodes as determined by EasyDIAg. The %overlap
parameter was set to 60%. Abbreviations: SA situation analysis, V variation, G generation, PS problem-
solving, PS/E problemsolving and evaluation (simultaneous coding), E evaluation, O others, X no match)
Rater 1
SA V G PS PS/E E OX Totalp
Rater 2
SA 22 0400 0002
60
.10
V2680 00 0227
40
.28
G0630 40 0004
00
.15
PS 000322 0003
40
.13
PS/E 044226 2003
80
.14
E00002200
40
.02
O00000042 04
20
.16
X0310000–
40
.02
Total24 81 39 38 30 4442 2621.00
p
0.09 0.31 0.15 0.14 0.11 0.02 0.17 0.01 1.00 0.81
265The process of problem posing: development of a descriptive…
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1 3
power, (v) rigor and specificity, (vi) falsifiability, (vii) replicability, and (viii) multiple
sources of evidence. Criteria (i), (iii), (v), and (vii) will be outlined to discuss the potential
and limitations of the presented framework.
Regarding research question (1), five content-related episode types—situation analysis,
variation, generation, problemsolving, and evaluation—were identified inductively which
enable objective coding through their operationalization. The episode types of the devel-
oped phase model enable a specific descriptive perspective on all observed problem-posing
processes in the study in a time-covering manner. This description, we argue, provides a
better understanding of problem-posing processes in general (i). Furthermore and with
regard to research question (2), from the observed processes, a general structure in terms of
the sequence of the episodes was identified from which we were able to derive a descrip-
tive process model for problem posing. The high interrater agreement attests to the replica-
bility of the model (vii). The participants of the study were heterogeneous and ranged from
PSTs in the first bachelor’s semester for primary school to PSTs in the 3rd master’s semes-
ter for high school. Equally heterogeneous were the processes that could nevertheless be
analyzed by the developed model (iii). The detailed descriptions, coding instructions, and
theoretical classifications provide specificity to the terms. In the online supplement, anchor
examples serve for additional specification (v).
The model developed here provides additional insights compared to existing models
(e.g., Cruz, 2006; Pelczer & Gamboa, 2009): It distinguishes the episode types variation
and generation empirically which Silver (1994) already conceptualized theoretically. Addi-
tionally, the model encompasses non-content-related episodes for the description that have
also been identified in descriptive models of problem-solving (Rott etal., 2021).
The phase model can now be used to characterize, for example, different degrees of
quality of the problem-posing process which is still a recent topic in problem-posing
research and for which considering the products and processes seems advisable (Kon-
torovich & Koichu, 2016; Patáková, 2014; Rosli etal., 2013; Singer & Voica, 2017). Thus,
as in problem-solving research (cf. Schoenfeld, 1985b), a comparison between experts and
novices might be a fruitful approach to identify different types of problem posers. Further-
more and following the process-oriented research on problem-solving (Rott etal., 2021), it
would be conceivable that the process of posing routine tasks proceeds differently than the
process of posing non-routine problems.
Finally, possible limitations to the generalizability of the developed model will be
addressed. In general, the model offers one possible perspective on problem-posing pro-
cesses. Depending on the selected problem-posing situation, sample, or study design, it
cannot be ruled out that slightly different or even additional episode types may also occur.
We also find other perspectives on problem-posing processes in research (e.g., Headrick
etal., 2020). This study considers two specific structured situations with a non-routine ini-
tial problem. However, the developed phase model has also been successfully applied to
situations with routine initial problems and other mathematical contents within bachelor
and master theses. With small changes, the model was also successfully applied to pro-
cesses based on unstructured situations in several master theses. Moreover, this study has
PSTs as a sample. The phase model was successfully applied in bachelor and master the-
ses to other sample groups such as school students and teachers (iii). Therefore, there are
strong indications that support the generalizability of the phase model, which could still be
clarified in follow-up studies.
Supplementary Information The online version contains supplementary material available at https:// doi.
org/ 10. 1007/ s10649- 021- 10136-y.
266 L. Baumanns, B. Rott
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1 3
Funding Open Access funding enabled and organized by Projekt DEAL.
Declarations
We declare that this work has not been submitted for publication elsewhere.
Competing interests The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-
mons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is not
permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.
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