Identifying Metacognitive Behavior in Problem-Posing Processes: Development of a Framework and a Proof of Concept

Abstract

Insights into the process of mathematical problem posing is a central concern in mathematics education research. However, little is known about regulative or metacognitive behaviors that are essential to understanding this process. In this study, we investigate metacognitive behavior in problem posing. We aim at (1) identifying problem-posing-specific metacognitive behaviors and (2) applying these identified metacognitive behaviors to illustrate differences in problem-posing processes. For these aims, we identified problem-posing-specific metacognitive behaviors of planning , monitoring & control , and evaluating in task-based interviews with primary and secondary pre-service teachers. As a proof of concept, the identified behaviors are applied on two selected transcript fragments to illustrate how a problem-posing-specific framework of metacognitive behavior reveals differences in problem-posing processes.

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Vol.:(0123456789)
International Journal of Science and Mathematics Education
https://doi.org/10.1007/s10763-022-10297-z
1 3
Identifying Metacognitive Behavior inProblem‑Posing
Processes
Development of a Framework and a Proof of Concept
LukasBaumanns1 · BenjaminRott1
Received: 17 February 2021 / Accepted: 12 June 2022
© The Author(s) 2022
Abstract
Insights into the process of mathematical problem posing is a central concern in
mathematics education research. However, little is known about regulative or meta-
cognitive behaviors that are essential to understanding this process. In this study,
we investigate metacognitive behavior in problem posing. We aim at (1) identifying
problem-posing-specific metacognitive behaviors and (2) applying these identified
metacognitive behaviors to illustrate differences in problem-posing processes. For
these aims, we identified problem-posing-specific metacognitive behaviors of plan-
ning, monitoring & control, and evaluating in task-based interviews with primary
and secondary pre-service teachers. As a proof of concept, the identified behaviors
are applied on two selected transcript fragments to illustrate how a problem-posing-
specific framework of metacognitive behavior reveals differences in problem-posing
processes.
Keywords Problem posing· Metacognition· Regulation
Introduction
In research on problem posing, analyzing the products, that is the posed problems,
plays an important role. One reason for this is that the ability to pose problems
is often assessed by the posed problems (Bonotto, 2013; Singeret al.,2017;Van
Harpen & Sriraman, 2013). However, in our analyses of problem-posing processes,
we found that the observable quality of processes can differ, even though the same
* Lukas Baumanns
lukas.baumanns@uni-koeln.de
Benjamin Rott
benjamin.rott@uni-koeln.de
1 Institute ofMathematics Education, University ofCologne, Gronewaldstr. 2, D-50931Köln,
Germany

L.Baumanns, B.Rott
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problems are posed, as will be illustrated with two exemplary processes. One con-
struct that has made this difference within processes tangible for us is metacognitive
behavior.
At least since Flavell’s (1979) seminal work, metacognition has been a central
construct of research in psychology and is an established topic of mathematics
education research (Schneider & Artelt, 2010). In particular, research on problem
solving has benefited from the consideration of metacognitive behavior in the past
decades. There are numerous characterizations of metacognitive behavior in prob-
lem solving (Artzt & Armour-Thomas, 1992;Garofalo & Lester, 1985; Schoenfeld,
1987; Yimer & Ellerton, 2010) as well as studies on the connection between meta-
cognitive behavior and successful problem solving (Desoete et al., 2001; Kuzle,
2013;Özsoy & Ataman, 2009; Rott, 2013).
Although problem-solving research has benefited greatly from this perspective,
it is noteworthy that far less conceptual or empirical research has been conducted
in the related field of problem posing. A systematic literature review in high-ranked
mathematics education journals has revealed that there are only a few studies that
explicitly investigate problem-posing-specific aspects of metacognitive behavior
(Baumanns & Rott, 2021a). For example, activities of reflection are observed where
problem posers consider solvability or the appropriateness of the posed problem for
a specific target group (Kontorovichet al., 2012; Pelczer & Gamboa, 2009). Fur-
thermore, there is a lack of a framework that explicitly addresses the analysis of
metacognitive behavior in problem-posing processes. That is in particular striking
as researchers often note that the field of problem posing lacks conceptual insights
into the activity. Such insights would enable a better analysis and interpretation of
the activity itself (Ellerton et al., 2015; Ruthven, 2020; Van Harpen & Sriraman,
2013). Similar to research on problem solving, it can be assumed that the considera-
tion and analysis of problem-posing-specific aspects of metacognitive behavior may
be a central enrichment to the field. This can contribute to a better understanding of
problem-posing processes.
Based on this research desideratum, this article aims at (1) identifying problem-
posing-specific aspects of metacognitive behaviors in problem-posing processes
and (2) applying these identified metacognitive behaviors to illustrate differences in
problem-posing processes.
Theoretical Background
Problem Posing
Similar to problem solving, problem posing is considered to be a central activity
of mathematicians’ (Hadamard, 1945; Halmos, 1980; Lang, 1989). Already (Pólya,
1945) mentioned problem posing as a partial activity in the context of problem solv-
ing. Despite this recognition by mathematicians, for a long time, problem posing
has received noticeably less attention in mathematics education research than prob-
lem solving. At the latest since the 1980s (Brown & Walter, 1983;Butts, 1980; Kil-
patrick, 1987), problem posing has been increasingly investigated. In recent years,

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Identifying Metacognitive Behavior inProblem‑Posing…
researchers from mathematics education have shown an increasing interest in inves-
tigating and understanding problem posing (Caiet al., 2015; Cai & Hwang, 2020;
Cai & Leikin, 2020; Lee, 2020; Silver, 2013). For example, problem posing has
been widely used to identify or assess mathematical creativity (Joklitschkeet al.,
2019;Silver, 1997;Singer & Voica, 2015;Van Harpen & Sriraman, 2013; Yuan &
Sriraman, 2011).
There are numerous definitions of problem posing that conceptualize more or
less equivalent activities. Silver (1994) defines problem posing as the generation of
new problems and reformulation of given problems which can occur before, during,
or after a problem-solving process. 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.” Cai and Hwang (2020, p. 2) subsume under
problem posing “several related types of activity that entail or support teachers and
students formulating (or reformulating) and expressing a problem or task based on a
particular context.”
Based on the categories by Stoyanova and Ellerton (1996), we distinguish
between unstructured and structured problem-posing situations depending on the
degree of given information (Baumanns & Rott, 2021b). Unstructured situations are
characterized by a given naturalistic or constructed situation in which tasks can be
posed without or with less restrictions, for example “Consider the following infi-
nite sequence of digits: 123456789101112131415…999100010011002… Note that
it is made by writing the base ten counting numbers in order. Ask some meaningful
questions. Put them in a suitable order” (Stoyanova, 1999, p. 32). To pose meaning-
ful questions, the structure of the situation has to be explored using mathematical
knowledge and concepts. In structured situations, people are asked to pose further
problems based on a specific problem, for example by varying its conditions. As
structured situations are used in this study, examples can be seen in Table2.
Metacognition
Going back all the way to Flavell (1979), who significantly influenced early research
on this topic, metacognition is described as “knowledge and cognition about cog-
nitive phenomena,” which roughly means thinking about thinking. Based on this
understanding, two facets of metacognition are distinguished: (1) knowledge about
cognition (Cross & Paris, 1988;Kuhn & Dean, 2004; Pintrich, 2002) and (2) regula-
tion of cognition (Schraw & Moshman, 1995; Whitebread etal., 2009).
(1) Knowledge about cognition includes declarative knowledge of strategy, task, and
person (Pintrich, 2002). Strategic knowledge refers to knowledge about strategies
(e.g. when solving problems) and when to apply them. Knowledge of tasks refers
to knowing about different degrees of difficulty of tasks and different strategies
required to solve them, for example. Person’s knowledge includes knowledge
about one’s own strengths and weaknesses (e.g. in problem solving)

L.Baumanns, B.Rott
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(2) Regulation of cognition refers to procedural knowledge with regard to processes
that coordinate cognition. This facet includes the activities of planning, moni-
toring, control, and evaluating (Pintrich, 2000). Planning refers to setting of
a target goal concerning the current endeavor, the activation of prior content
knowledge, and activation of metacognitive knowledge, for example knowledge
about specific tasks and how to solve them. Monitoring refers to metacognitive
awareness and monitoring of cognition, for example when verifying that one
has understood the current task. Control refers to the selection and adaption of
strategies, for example with the goal to solve a problem. Evaluating refers to
activities of reflecting and judging on one’s own performance and results in the
form of an outcome. These activities are not completely distinct from each other
and there are different perspectives on their overlap. We follow the perspective
of Pintrich (2000), who states that control is mostly conceptualized as dependent
or at least highly similar to monitoring (Pintrich, 2000). By that, we also follow
the approach of Cohors-Fresenborg and Kaune (2007) who summarize monitor-
ing and control under one category (see also Schraw & Moshman, 1995). This
is also consistent with past work by Pólya (1945), who indicated roughly these
three activities in problem solving even before the construct metacognition was
established (Cohors-Fresenborget al., 2010)
There are attempts to identify the behavior of planning, monitoring & control, and
evaluating in learning contexts (Kaune, 2006;Van der Stelet al., 2010). For exam-
ple, Cohors-Fresenborg and Kaune (2007) provide a category system for classify-
ing teachers’ and students’ metacognitive activities in class discussions. The main
categories of planning, monitoring & control (which they refer to as monitoring),
and evaluating (which they refer to as reflecting) are divided into several subcatego-
ries with different aspects. Table1 summarizes the codes for the main categories of
planning, monitoring, and evaluating.1 These codes are used to identify metacogni-
tive behavior by teachers and students through the analysis of verbal protocols of
classroom interactions. In addition to metacognitive behavior, Cohors-Fresenborg
and Kaune (2007) also consider (negative) discursivity in their coding scheme. Dis-
cursivity is understood as a culture in which the teacher and the students always
refer to each other’s expressions, work out differences in approaches, and regulate
their own understanding. Because it is an additional aspect besides metacognition,
and it is specific for classroom interactions which is not analyzed in the study at
hand, the aspect of discursivity will not be considered further. In the study at hand,
the focus is on the process of problem posing and, therefore, investigates regulation
of cognition. In particular, we want to differentiate the activities of planning, moni-
toring & control, and evaluating for problem posing by adapting the approach by
Cohors-Fresenborg and Kaune (2007).
Metacognition is often considered in conjunction with motivation and beliefs.
Zimmerman and Moylan (2009) state that proactive self-regulation depends on the
1 The current version of this coding manual can be found at https:// www. mathe matik. uni- osnab rueck. de/
filea dmin/ didak tik/ Proje kte_ KM/ Kateg orien system_ EN. pdf.

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Identifying Metacognitive Behavior inProblem‑Posing…
Table 1 Main categories of planning, monitoring, and evaluating in the category system for classifying teacher and students metacognitive activities in class discussions
by Cohors-Fresenborg and Kaune (2007)
Indication of a focus of attention, in particular
with regard to tools/methods to be used or
(intermediate) results or representations to be
achieved
Controlling of a subject-specific activity analysis of structure of a subject-specific expres-
sion
Controlling of terminology/vocabulary used for a
description/explanation of a concept
Reflection on concepts/analogies/metaphors
Controlling of notation/representation Result of reflection expressed by a wilful use of a
(subject-specific) representation
Controlling of the validity or adequacy of tools
and methods used, in particular with regard to
a planned approach or a modelling approach
Analysis of the effectiveness and application of
subject-specific tools or methods/indication of a
tool needed to achieve an intended result
Planning metacognitive activities Controlling of (consistency of an) argumenta-
tion/statement
Analysis of argumentation/reasoning with regard
to content-specific or structural aspects
Controlling if the results meet the question Reflection-based assessment or evaluation
Revealing a misconception Analysis of the interplay between representation
and conception
Self-monitoring

L.Baumanns, B.Rott
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presence of motivational beliefs. For example, the metacognitive activity of plan-
ning depends highly on aspects like the intrinsic interest into the current endeavor,
self-efficacy perceptions, or learning goal orientation. These aspects are deeply
interwoven with motivation. It follows that for a holistic view of learners’ effort, for
example in problem solving, metacognitive processes should be considered in addi-
tion to motivational beliefs (Zimmerman & Moylan, 2009).
Research onMetacognition inMathematics Education
In mathematics education research, metacognition is of immense importance (for an
overview, see Schneider & Artelt, 2010). Assessing metacognitive behavior is often
used to investigate the mathematical skills of participants (Mevarech & Fridkin, 2006;
Van der Stel etal., 2010). For example, Van der Stel etal. (2010) found out that the
quality of metacognitive behavior seems to be a predictor of the mathematical perfor-
mance in the future by analyzing the regulation of cognition in thinking-aloud proto-
cols of second- and third-year students. Most prominently, metacognition is consid-
ered in problem-solving research. Considerations on this are mainly based on the ideas
of Pólya (1945). Especially Schoenfeld(1985b, 1987, 1992) emphasizes the impor-
tance of control or self-regulation in problem solving. He observed that even when
students have the content knowledge necessary to solve a problem, a lack of ability to
keep track of what they are doing, that is metacognitive behavior, might lead to failure
in solving a problem. His theoretical and empirical analyses initiated numerous studies
on metacognition in mathematics education — including the present study.
Extensive discussion has been done on characterizing metacognitive behavior in
problem solving (Artzt & Armour-Thomas, 1992; Garofalo & Lester, 1985; Sch-
oenfeld, 1985b; Yimer & Ellerton, 2010). For example, Artzt and Armour-Thomas
(1992) and Schoenfeld (1985b) both identify the analysis of the problem and plan-
ning the solution as predominantly metacognitive behavior. In the analysis, select-
ing an appropriate way to reformulate the given problem, for example in order to
make it simpler, is referred to as metacognitive behavior. The phase of planning is
also metacognitive by nature as it involves controlling and self-regulating the solv-
ing process (Garofalo & Lester, 1985). Building on that, several studies investigate
the relationship between metacognitive activities and successful problem solving
and come to the result that successful problem solving is mostly associated with the
presence of metacognitive activities (Desoete etal., 2001; Kimet al., 2013; Kuzle,
2013; Özsoy & Ataman, 2009).
Research onMetacognition inProblem Posing
Due to the relevance of considering metacognitive behavior in problem solving,
it can be assumed that the analysis of metacognitive behavior could be similarly
relevant in problem posing. However, the potential in this area has not been suf-
ficiently exploited to date. As some researchers interpret problem posing as a
problem-solving activity (Kontorovich etal., 2012;Silver, 1995) and since there
are several established frameworks of metacognitive behavior in problem solving

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Identifying Metacognitive Behavior inProblem‑Posing…
as described above, it is a reasonable question whether there is even a need for a
separate framework for metacognitive behavior in problem posing. We follow the
perspective of previous studies (Baumanns & Rott, 2022;Cruz, 2006;Pelczer &
Gamboa, 2009) that argue, based on observations of problem-posing processes,
that there are characteristic differences between problem-posing processes and
problem-solving processes in general. Therefore, also metacognitive processes
involved in problem posing differ from those in problem solving. Cognitive and
metacognitive processes involved in problem posing seem to be of their own
nature for which metacognitive frameworks for problem solving can only serve
as a stimulus. From this derives the interest addressed in this study to investigate
these problem-posing-specific aspects of metacognitive behavior.
Problem posing includes different cognitive and metacognitive processes (Bau-
manns & Rott, 2022;Christou et al., 2005; Koichu & Kontorovich, 2013;Pelczer
& Gamboa, 2009) which are indicated in the following informally: Problem pos-
ers analyze the given situation, examine which mathematical knowledge could be
relevant for this, and possibly look for structures in the situation that may lead to
an interesting problem. Then, problems are posed and suitable representations of
them are sought. The task may then be solved and while solving, the posers may
reflect on the difficulty of the task, its appropriateness for an intended target group,
or the general interest in the solution. These activities are not limited to cogni-
tive processes, but may also contain metacognitive behavior. Cognitive behavior
in problem posing is, for example, applying knowledge of previous mathematical
experiences to a given problem-posing situation to pose a problem. Metacognitive
behavior would be, for example, to attack a problem in order to assess whether the
problem is well-defined or solvable at all.
Theoretical considerations on problem posing implicitly contain some aspects of
metacognition and metacognitive regulation in particular (Carrillo & Cruz, 2016;
Ghasempouret al., 2013;Kontorovich etal., 2012; Pelczer & Gamboa, 2009;Singer
& Voica, 2015), yet metacognition is rarely explicitly addressed as the systematic
literature review on problem-posing studies by Baumanns and Rott (2022) has
revealed. The following sections address the few studies that review has identified
that implicitly or explicitly address metacognitive behavior.
Voicaet al. (2020) mention that they found metacognitive behavior in their study
with students as they were able to analyze and reflect on their own posed problems
and thinking processes which helped them to become aware of their strengths. Other
studies have also pointed to the lack of metacognitive activities from the participants
(Crespo, 2003; Tichá & Hošpesová, 2013). For example, Crespo (2003) writes that
four of her thirteen participants “posed [problems] without solving beforehand or
deeply understanding the mathematics” and they “indicate unawareness of the math-
ematical potential and scope of [their] problem[s]” (p. 251). Crespo (2003) identifies
unawareness in her participants in the sense of a lack of reflection. This can also be
interpreted as a lack of metacognitive behavior. Erkan and Kar (2022) were able to
observe metacognitive factors in pre-service mathematics teachers, but these factors
varied depending on the problem-posing situation. Their participants considered the
strengths and weaknesses of their mathematical knowledge in order to write math-
ematically valid problems.

L.Baumanns, B.Rott
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Pelczer and Gamboa (2009) as well as Kontorovich etal. (2012) specify thoughts
on reflection on the posed problems. In their descriptive process model, Pelczer and
Gamboa (2009) state that in the phase of evaluation, expert problem posers assess their
posed problems in terms of various aspects, for example, whether further modifica-
tions are needed. In the phase of final assessment, the process of posing a problem is
reflected upon and the problem itself is evaluated, for example in terms of difficulty and
one’s own interest in solving it. Similarly, the framework for handling the complexity
of problem-posing processes in small groups by Kontorovich etal. (2012) integrates
the facet of individual considerations of aptness. Consideration of aptness includes, for
example, to what extent the problem poser is satisfied with the quality of the posed
problem or whether the posed problem is appropriate for a specific group of solvers.
Research Objectives
As the state of research has shown, metacognition has not been an important factor
in problem-posing research. Therefore, the aim of this study is to offer a focused
perspective on metacognitive behavior in problem posing. This lack of conceptual
and empirical insight constitutes a desideratum which leads us to the following
research objective:
(1) Development of a framework for identifying problem-posing-specific aspects of
metacognitive behavior (i.e. planning, monitoring & control, and evaluating) in
pre-service teachers’ problem-posing processes
As a second research objective, we pursue a proof of concept to identify differ-
ences in problem-posing processes based on metacognition. The problem-posing-
specific metacognitive behaviors developed in research objective (1) will be applied
to selected transcript excerpts of two problem-posing processes. This proof of con-
cept is intended to show to what extent the analysis of metacognitive behaviors in
problem-posing processes with the framework developed in (1) makes differences
regarding metacognitive behavior in problem-posing processes visible.
(2) Application of the framework developed in (1) as a proof of concept to make dif-
ferences regarding metacognitive behavior in problem-posing processes visible
Methods
Research Design forData Collection
There are several ways to assess metacognitive behavior, for example through
interviews, stimulated recall, or eye-movement registration (for an overview,
see Veenmanet al., 2006). A clear distinction exists between off-line methods

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Identifying Metacognitive Behavior inProblem‑Posing…
which are carried out before or after the current endeavor (often via self-report
questionnaires), and on-line methods which are carried out during the current
endeavor. As on-line methods seem to be more predictive with regard to the
learning performance (Veenman etal., 2006), we chose the approach of video-
based content analysis on problem-posing processes of pairs of pre-service
teacher students. This method is based on the commonly used assumption that
we can make interpretive conclusions about participants’ cognition and meta-
cognition from their verbal expressions working in small groups (Artzt &
Armour-Thomas, 1997; Gooset al., 2002).
The interviews were conducted in pairs in the same room to create a natu-
ral communication situation and eliminate the constructed pressure to produce
something mathematical for the researcher (Schoenfeld, 1985a, p. 178). John-
son and Johnson (1999) underline that cooperative learning groups such as pairs
are “windows into students’ minds” (p. 213). For this reason, the interviewer
avoided intervening in the interaction process. The participants were asked to
speak aloud at any time during interview while posing new problems. The inter-
views were conducted with 64 pre-service primary and secondary mathemat-
ics teachers. Sixteen students were in the first bachelor semester, 22 in the fifth
bachelor semester, and 26 in the third master semester. The students worked in
pairs on one of two structured problem-posing situations, either (A) Nim game
or (B) Number pyramid (see Table2). In total, 15 processes of situation (A) and
17 processes of situation (B) that range from 9 to 25 min with an average length
of 14.5 min have been recorded. The processes ended when no ideas for further
problems emerged from the participants. In total, 7h 46min of video material
was recorded. Four pairs of students each were in the same room under authen-
tic university seminar conditions. A camera was positioned opposite the pairs
in order to capture all of the participants’ actions. In order to accustom them to
a natural communication in front of the camera, short puzzles were performed
before solving the initial task and the consecutive problem posing.
Table 2 Structured problem-posing situations used in the study
Situations
(A) Nim game
There are 20 stones on the table. Two players A and B may
alternately remove one or two stones from the table. Who-
ever 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 number 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)

L.Baumanns, B.Rott
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Data Analysis — Assessment ofMetacognitive Behavior
To answer research question (1), we conducted a qualitative content analysis
(Mayring, 2000). The main categories of the metacognitive behavior of plan-
ning (P), monitoring & control (MC), and evaluating (E) stem from theoretical
considerations on regulation of cognition presented above (Schraw & Mosh-
man, 1995; Whitebread et al., 2009), and more specifically from Cohors-
Fresenborg’s and Kaune’s (2007; see Table 1) considerations. Although this
framework is developed for analyzing classroom interaction, it has been used
successfully in paired problem-solving processes (Rott, 2014). However,
because problem posing is a mathematical activity of its own kind in our obser-
vations (Baumanns & Rott, 2022), a problem-posing-specific approach was
chosen for the present proof of concept. The individual problem-posing-spe-
cific characteristics were obtained data-driven through an inductive category
development (Mayring, 2000). The category development had four steps: (1)
As the unit of analysis, the statements of the participants on the videotaped
problem-posing processes were used for the category development. To iden-
tify the statements of planning, monitoring & control, and evaluating, the 32
recorded problem-posing processes were analyzed as follows: For the category
of planning, we identified participants’ statements of setting a target goal for
the current problem-posing situation, activating prior mathematical knowledge
that helped posing new problems, or activating metacognitive knowledge in the
form of knowledge on how to pose new problems in general. For the category
of monitoring & control, we identified participants’ statements of awareness
and monitoring of cognition as well as selecting and adapting problem-posing
strategies. For the category of evaluating, we identified participants’ statements
of assessing their problem-posing process or their products, that is their posed
problems (e.g. “This is a good problem because it is not too difficult and it
is novel compared to the initial problem.”). (2) For each identified statement
in step 1, a short description was obtained (e.g. evaluation of the posed prob-
lem based on specific characteristics). Similar descriptions within the main
categories planning, monitoring & control, and evaluating were then clustered
into subcategories. (3) Afterwards, the developed subcategories were revised
by reanalyzing seven problem-posing processes in which particularly many
and different metacognitive behaviors were observed and in which participants
expressed particularly many verbalizations regarding the interpretation of meta-
cognitive behavior. This reanalysis was used to further refine the descriptions
of the categories. (4) After the categories were specified, all the video material
was reviewed again in order to draw attention to possible additional categories.
No new categories were found in this last step. The quality of the coding was
ensured through consensual validation in team discussions which is a common
method of ensuring scientific quality in qualitative research (Flick, 2007).
Only for answering research question (2), selected sections of the videotaped
problem-posing processes were transcribed. The framework developed in (1) is
applied as a proof on concept in the second part of the paper onto the transcripts of
these selected sections of two problem-posing processes. This proof of concept is

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Identifying Metacognitive Behavior inProblem‑Posing…
intended to make differences regarding metacognitive behavior in problem-posing
processes visible. In the transcripts, the participants’ statements are reproduced ver-
batim. Important actions of the participants are noted in parentheses for understand-
ing the scene. Filler words that do not affect the content have been removed for read-
ability. For the analysis, the transcripts were first read iteratively in order to obtain a
rough understanding of the text and to be able to better integrate finer sections of the
text into the overall context of the text. The codes developed in research question (1)
are then applied to the transcript. The coding of metacognitive behavior of ,
, and is color-coded in , , and in the
style of Kaune (2006; Cohors-Fresenborg & Kaune, 2007) to illustrate the distribu-
tion of the main categories. We want to emphasize that the analyses of metacogni-
tive behavior were not linked to the correctness of the (mathematical) content. With
a wrong argumentation, metacognitive behavior can be just as visible and evaluated
as with a correct argumentation.
Results
Development ofaFramework forIdentifying Problem‑Posing‑Specific Aspects
ofMetacognitive Behavior
We first want to establish the identified problem-posing-specific aspects of metacog-
nitive behavior of planning, monitoring & control, and evaluating. To do that, the
developed types of metacognitive behavior are described and anchor examples of
observed processes with regard to the Nim game and Number pyramid (see Table2)
are presented.
Planning
In Table3, the categories of regulation of cognition in terms of planning are pre-
sented. T1 and T2 each represent any participants in the study. In Table3, refers
to focussing on a starting point of a given situation from which a new problem can
be posed. This can be, for example, a certain condition, context, or solution structure
of the given initial problem. Behavior is reminiscent of the well-known “What-
if-not”-strategy (Brown & Walter, 2005), in which a similar activity is suggested
before the actual problem posing. Behavior ( ) refers to activities in which partici-
pants have partly considered what knowledge they or the potential solvers of a posed
problem need to have in order to be able to solve it. This behavior could have a
greater significance when confronted with unstructured situations in which the nec-
essary knowledge may not be obvious because no concrete initial problem is given.
In fact, the phase of setup in the mentioned framework by Pelczer and Gamboa
(2009) refers to the reflection on the knowledge needed to understand the situation.
Their framework is based on unstructured situations. Finally, participants named a
general procedure for the upcoming problem-posing process, e.g., first vary the ini-
tial task in multiple ways, then solving the varied tasks ( ).

L.Baumanns, B.Rott
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Table 3 Planning activities in problem posing
Code Description Anchor example
Focus on a starting point of the problem-posing
situation to generate new problems
(The participants chose one variation of the Nim game at the very beginning of their problem-posing
process.)
T1 When you think about a new game, I’m thinking about when you’re allowed to remove one to three
stones.
T2 Yes or, if you, exactly, if you increase the number of stones you are allowed to remove.
Capturing the conditions and identifying the
restrictions of the given problem-posing situation
(The participants are at the end of a longer period of varying the Nim game and are now thinking about
what other conditions there are that could be varied.)
T1 Are there any other possibilities, I mean variables that can be influenced in this? This would really be
the number of players, number of stones, how many do you remove?
T2 Amount of steps.
T1 Exactly.
Reflect necessary knowledge (The participants posed a variation of the Nim game in which you are allowed to remove one to three
stones. Afterwards, they pose another variation in which you can remove one to four stones. Both
recognize a pattern and reflect the mathematical content.)
T1 Actually, [...] that would be a cool introductory assignment to me introducing modular arithmetics.
T2 Yeah.
T1 A little bit at least, right? Or at least a more advanced task.
Express general procedure for problem posing T1 Yes, now I can try to vary all sorts of things. For example, I could vary the number of stones at the
beginning. [...] Change the number of removable stones per turn or something. Change the number of
players. I have no idea yet which of these things will be difficult to say simply trivial.

1 3
Identifying Metacognitive Behavior inProblem‑Posing…
Monitoring
In Table 4, the categories of regulation of cognition in terms of monitoring
are presented. follows on from and characterizes that metacognitive
behavior in which participants control the problem-posing process. This occurs,
for example, by suggesting to solve a posed problem first before focussing on
another one. Controlling the notation or representation of the posed problems
() refers to, for example, figures drawn to illustrate a problem or to the
formulation of the specific question so that it becomes understandable and pre-
cise. We observed that participants made a modification to the initial problem
and analyzed the consequences that this modification had on the newly created
problem ( ), for example for the solution structure or its difficulty. The code
was identified when participants analyzed the mathematical structure of
the given situation in order to get to a new problem or analyzed the structure of
a posed problem in order to be able the characterize it, for example with regard
to its solvability or appropriateness.
Evaluating
In Table5, the categories of regulation of cognition in terms of evaluating are pre-
sented. Assessing and reflecting on the characteristics of a posed problem ( ) was
seen when participants posed a problem and got an idea about how to solve it. Then,
they often evaluated whether their posed problem is interesting, solvable, or appro-
priate for a specific target group. This behavior was already mentioned in previous
studies on the problem-posing process (Baumanns & Rott, 2022;Kontorovich etal.,
2012;Pelczer & Gamboa, 2009). A reflection on modifications of the posed prob-
lems ( ) was observed when the posed problem lacks a specific characteristic, for
example it is too easy or too difficult, it is not very interesting, or it is too similar to
the initial problem.
Proof ofConcept: theCases ofTino & Ulrich andValerie & Wenke
In this section, we will analyze two cases — one by Tino & Ulrich and the other by
Valerie & Wenke — with regard to the aspects of metacognitive behavior that have
been developed in the previous section. The aim of this section is to illustrate differ-
ent degrees of metacognitive behavior in two problem-posing processes that have a
similar product. The two selected fragments are identical in numerous features of
the outer structure: All four participants are pre-service high-school teachers in their
third semester of their master degree. In the analyzed fragments, they work on pos-
ing new problems for the Nim game (see Table2). In the shown fragments, both
pairs of participants pose the same problem, namely the variation of the Nim game
in which the players are allowed to remove 2 or 3 stones from the table. The tran-
script analysis is conducted to illustrate the different degree of metacognitive behav-
ior between those two processes in which a largely identical product is produced.

L.Baumanns, B.Rott
1 3
Table 4 Monitoring & control activities in problem posing
Code Description Anchor example
Controlling the general procedure for problem posing (The participants are relatively at the beginning of their problem-posing process with regard to
the Number pyramid and initially pose several variations. T1 asks whether the questions of the
initial task should be retained. T2 suggests as a procedure first tasks with the same question are
posed and then further changes can be made.)
T1 Are we going to use the same ... question?
T2 Yes, then we can pose another task with a different one afterwards.
Controlling the notation or representation of the posed problems T1 You could also make a pyramid like this and then start with 5 in the next row. That this is not
always the row whose square number it is, but that it is done in steps of 5, so that you first have
5 numbers, then 10.
T2 This is not a pyramid, this is something like this (shows the shape of the resulting figure by
sharp movements).
Assessing consequences for the problem’s structure through the
modified or new constructed conditions
(The participants play the variation of the Nim game in which the players are allowed to remove
1 or 3 stones from the table.)
T1 So, here I always win.
T2 Yes, you always win in this situation.
T1 Just in a different system.
T2 Yes, but the other way round, because now the one who does not start always wins.
Controlling mathematical activities related to a posed problem T1 In which row is the second triplet of prime numbers, or so?
T2 Well, there aren’t that many prime triplets, are there?
T1 Aren’t there two? There are those right at the beginning. 3, 5 and 7 and then there’s another
one somewhere. Wasn’t that 3... No, it wasn’t.
T2 19, 21, 23. Oh no, 21 is not a prime number.

1 3
Identifying Metacognitive Behavior inProblem‑Posing…
Table 5 Evaluating activities in problem posing
Code Description Anchor example
Assessing and reflecting on the characteristics of the
posed problems (e.g. if it is appropriate for a specific
target group, solvable, interesting, well-defined)
(The participants posed the variation where the stones are built up into a square pyramid with 42 stones
at the bottom, then 32, then 22 and 11 at the top. You can remove 1 or 2 pieces per turn but only from
one level.)
T1 But now you have to think, ok do you want ... at the end there are 16 stones, do you want to start in
this layer or not? Then there are 9 stones above that, ok. Then you have to decide, ok, if there are 9
stones, do you want to start in this layer or not? But I think that’s actually cool. Because then you still
have to think about the number of stones in each layer. That’s actually a cool game, because you have
to apply the rules differently again.
T2 Yes. It’s really good.
T1 It’s good, right?
Reflect on possible modifications of the posed problems (T1 raises the modification that player A may remove 1 to 3 and player B 1 to 2 pieces from the table.
They then play through this idea and realize that player A can of course use the same winning strat-
egy. Oskar reflects this as player A and proposes a modification on this basis.)
T2 (laughs) It’s cheeky to put them down like that (and points to the triple packs that Oskar divided the
pieces into)
T1 (laughs) Yes, that’s why I mean that. It would be more interesting if I could start and only take two
away. Or three or two, for example.

L.Baumanns, B.Rott
1 3
Tino & Ulrich — Analysis oftheMetacognitive Behavior
The following excerpt from the process of Tino and Ulrich takes place in the first half of
their process. Beforehand, they already posed, solved, and analyzed several new varia-
tions of the Nim game such as what if there are 21 stones on the table in the beginning?
What if you could remove 1, 2, or 3 stones from the table? What if you also get a win-
ning point if you have removed more stones from the table than your opponent? Then,
they pose the problem that you are only allowed to remove 2 or 3 stones from the table.
The development of this problem is shown in the following transcribed excerpt that takes
3m 12s. For a better visual assessment of the density of metacognitive behavior, in this
transcript, the codes for the metacognitive behavior are already set and marked in color.
In turn 1, Ulrich poses a new variation, in which only 2 or 3 stones may be
removed from the table, as starting point. This new starting point is derived from a
previous task (1 to 3 pieces may be removed). Since Ulrich sets a new focus for the
upcoming problem-posing activity, this statement is coded as planning ( ). After
Tino has thrown in what happened to one of the previous ideas, Ulrich refocuses
on the problem he just posed and says that the task Tino mentions can be dealt with
later. Therefore, this statement is coded as planning ( ).
In turn 10, Tino tries to find a formulation for the problem that was posed in turn 1. He
writes down this task as a negation that one may not just remove one stone from the table.

1 3
Identifying Metacognitive Behavior inProblem‑Posing…
His thinking about the formulation of the question represents a control of the notation or
representation of the problem and is therefore coded as monitoring & control ( ).
Ulrich says that this change results in a “new game.” He probably means a new
kind of outcome of the game, where nobody wins. This assessment of the conse-
quences that their variation has for the Nim game was coded as monitoring & control
(). Ulrich states that he likes the consequences that follow from their variation
since they are different from the initial task. Therefore, this is coded as evaluation
( ). In turn 18, Tino agrees with Ulrich’s positive evaluation of the game.
Tino interjects in turn 20 whether they should modify the new game due to this
situation by adding that a player loses even if s/he can no longer remove stones from
the table. Ulrich says that he would not make this change. In both statements, the par-
ticipants consider to modify the posed problem so that the game has a definite win-
ner. Therefore, statements related to that consideration are coded as evaluation (
E2
).
Tino states in turn 24 that this change would restore the original winning strategy of the
initial task. By that, he assesses the consequences of his slight modification and compares
it to the initial task. Therefore, this statement is coded as monitoring & control (
MC3
).
Ulrich does not seem to like this change, perhaps because it would bring him too
close to the initial task. In turn 13, he seemed to like this new element very much.
This statement is coded as evaluation (
E1
).
Tino reflects in turn 26 that one could modify the game with his suggestion in
order to maintain the original winning strategy of the initial task. This is a reflection
on their modification and, thus, is coded as evaluation (
E2
).
Ulrich initially agrees with Tino’s previous assessment (
E2
). Then, he focuses
on a solution strategy of the modified game again and thinks about the situation in
which five stones lie on the table. With his statement, he is controlling the process
which is why this statement is coded as monitoring & control (
MC1
).
30 T: Right.IfItake3...
31
U: Your goal is 5.
32
T: ...you have 2tochoose from.IfItake2,you have 3tochoosefrom.
33
U: Your goal is 5and then it is exactly thesame.
34
T: Yes.
35
U: Then yougo..itisamultiple of 5.
36
T: That means, what is thechange now? Before it wasamultiple of 4, whereyou
hadlost. (referring back to apreviously posedproblem whereyou can remove
1to3stones)
MC
3
37
U: Exactly,because because youalwaysadd thelargest plus thesmallest. Yo u
couldhavemaximum 3and minimum1andherethe largestis3plus the
smallestis2.
MC
3
38 T: 2, yes. 1P?4ro3evomerdluocenofitahW
In turn 36, Tino now assesses the changes in the solution strategy of the new game
against the background of the previously posed tasks. In the next turn, Ulrich contrasts
the solution strategy of the new game against the background of the solution strategy
of previous game and tries to bring both together under one mathematical thought. As
both statements are reflections on the winning strategy of the new game and how it is
related to the previous game, they are coded as monitoring & control (
MC3
).

L.Baumanns, B.Rott
1 3
Tino focuses on another variation that seems to result from the above consid-
erations. The background could be that Ulrich’s consideration in Turn 37 is to be
checked on a similar task. In that way, Tino sets a new focus for the process which
is why this statement is coded as planning (
P1
).
Overall, it can be seen that Ulrich has a controlling influence on their process
(
MC1
). In both Turn 5 and Turn 27, he determines the direction in which the
process should continue. First, by specifying the focus of the next considerations
and then by wanting to better understand the problem that they have posed, Tino’s
behavior is characterized by the fact that he wants to further modify the game that
has been posed (
E2
) so that the inevitable situation that has arisen can be avoided.
Valerie & Wenke — Analysis oftheMetacognitive Behavior
The following excerpt from the process of Valerie and Wenke takes place quite
early in the process. Beforehand, they both analyzed the solution to the initial
problem and then asked themselves how they could now come to new problems
or what they could specifically vary about the game in order to come to new prob-
lems. Afterwards, they pose the variation that you may only remove 2 or 3 stones
from the table. The following excerpt takes 1m 28s and shows the creation of this
problem. As in the section before, the codes for the metacognitive behavior are
already set and marked in color. The coding is commented on after the transcript.
1W:We can take2or 3stonesfrom it. P1
2V
:2or 3?
3W
:Mhmm.
4V
:And howmanyare there? (pointstothe pile of stones)
5W
:(laughs)Ihavenot counted.Ijust took it. (Wenke counts off20stones.)20
arehere already.
6V
:(countsdow n12stones.) 12.
7W
:(countsoff 10 stones.) 32.42.
8V
:Yes.
9W
:Howmanydowetake? 30? MC
2
10
V: Yes. (5 seconds)Doyou want to be Player Athistime?
11
W: Yeah!(bothstartplaying.)It wouldbeterrificifthere wa sonlyone left now
(laughs). Unfortunately Ican’t make amovethen(laughs).
MC
3
12
V: (Giggles.Afterwardsbothcontinue playing).
13
W: That certainly doesnot work.(3seconds)Itwillnot work (giggles).
14
V: (giggles)
15
W: Yes, butnow youhavetaken especiallysothatitworks out, right?(laughs)
I’mnot stupid!
16
V: (laughs)
17
W: 1E.3ro2htiwesneshcumekamt’nseodtaht,seY
18
V: No.
19 W: 1CM.gnihtemosfoknihtnacuoywoN.aediwen,yakO
Wenke brings a new problem into the focus in turn 1. This set of a new focus
is coded as planning (
P1
). After counting how many stones they have on the
table, Wenke asks in turn 9 how many stones to use for the new game. This

1 3
Identifying Metacognitive Behavior inProblem‑Posing…
statement is interpreted as controlling the representation of the posed problem,
as the focus is already set in the form of a new game and they now have to agree
on a number of stones at the beginning in order to be able to play the game
(
MC2
). Wenke notices in turn 11 that the change they have posed can cause
a situation in which one does not know who will win. This assessment of the
consequences of the new game through their modification is coded as monitor-
ing & control (
MC3
). In turn 17, Wenke says that their new problem does not
make sense because of this new situation in which nobody wins. This statement
is coded as evaluation since Wenke does not see much meaning in the new game
where an undecidable situation can occur (
E1
). Wenke obviously does not intend
to pursue this problem further in turn 19 and suggests that a new problem should
be posed. She has a steering effect on the process which is why this statement is
coded as monitoring & control (
MC1
).
Overall, in this process, Valerie and Wenke show noticeably insecurities. This
can be seen through the numerous occurrences of laughter. Nevertheless, they
also show metacognitive behavior — although less frequently than Tino and
Ulrich. Like Tino and Ulrich, they realize during the game that an inevitable
situation could occur. In Turn 17, Wenke also evaluates this, but they see this
situation as a reason to reject the posed game.
Discussion
Research Objective (1): Development of a framework for identifying problem‑pos‑
ing‑specific aspects of metacognitive behavior (i.e. planning, monitoring & control,
and evaluating) in pre‑service teachers’ problem‑posing processes Tables3,4, and5
summarize the identified metacognitive activities. In total, four planning activities,
four monitoring & control activities, and two evaluating activities were identified.
Some of these activities may be considered as cognitive, but being able to intention-
ally use these kinds of cognitive behavior is a sign for metacognitive abilities. How-
ever, when metacognitive behavior is mentioned here, it always means the primarily
metacognitive behavior in interaction with cognitive behavior. For example, search-
ing for a solution can be seen as cognitive behavior, but considering the solution
in order to get a better idea whether the posed problem is, for example, solvable or
appropriate for a specific target group can be seen as metacognitive behavior. Most of
the identified activities are indeed problem-posing specific. However, there are also
activities (e.g.
P3
: Reflect necessary knowledge) that are not problem-posing specific.
Moreover, not all codes (i.e. subcategories) within the superordinate categories of
planning, monitoring & control, and evaluating are separable from each other. How-
ever, a clear separation between the superordinate categories should be recognizable.
We want to highlight parallels and differences between the category system
for classifying teacher and students metacognitive activities in class discussions
by Cohors-Fresenborg and Kaune (2007) (see Table 1) and our developed frame-
work. The category
P1
(indication of a focus of attention, in particular with regard
to tools/methods to be used or (intermediate) results or representations to be
achieved) in their system can be found in the categories
P1
and
P2
(see Table3) in the

L.Baumanns, B.Rott
1 3
problem-posing-specific framework developed here.
P2
is considered separately as it
is a central component of problem posing (Baumanns & Rott, 2022; Brown & Wal-
ter, 2005).
P4
(express general procedure for problem posing) corresponds to a spec-
ification for problem posing of the category
P2
in Table1. For monitoring & con-
trol,
MC1
represents a specification for problem posing of the category
M8
and
MC4
represents a specification for problem posing of the category
M1
. Categories
M2
and
M3
were merged into the problem-posing-specific category
M2
and categories
M4
–
M7
were merged into the problem-posing-specific category
M3
. Finally, for evalu-
ation, rather rough parallels can be drawn between categories
R1
–
R4
and
E1
as well
as
R5
–
R7
and
E2
.
Research Objective (2): Application of the framework developed in (1) as a proof of
concept to make differences regarding metacognitive behavior in problem‑posing
processes visible The analysis of metacognitive behavior in the problem-posing pro-
cesses of Tino and Ulrich as well as Valerie and Wenke revealed differences. Tino
and Ulrich’s process had a greater frequency and density of metacognitive behav-
ior compared to Valerie and Wenke’s process. However, the coding does not allow
for a statement about the depth of the metacognitive behavior, that is how sophisti-
cated the specific metacognitive behaviors are. For example, in both processes, the
posed problem was evaluated (
E1
and/or
E2
). In the process by Valerie and Wenke,
the evaluation remains a single mention of futility in Turn 17 from which they dis-
card the problem. Tino and Ulrich take their thoughts further and relate their evalu-
ation to the new possible outcome of the game, which differs from the initial task
(Turns 13, 17, and 18). They even reflect on whether they want to restore similarity
to the initial task by modifying it further (Turns 20, 21, 26, and 27). In contrast to
Valerie and Wenke, they do not discard the problem. From a quantitative perspec-
tive, the observed processes do not claim to be representative. Therefore, a quantita-
tive counting on the frequency of the individually occurring metacognitive behav-
iors would yield only insufficiently helpful new insights.
The perspective on the metacognitive behavior of both pairs reveals differences
between the processes. Since in the selected excerpts the posed problems are rather
identical, no differences would have been attested by looking only at the products. It
also matters that Tino and Ulrich have a better understanding of the posed problem
by suggesting a solution strategy as well as their motivational beliefs. However, the
task did not explicitly ask for solving the posed problems. Tino’s and Ulrich’s impe-
tus to do so independently may attest motivational beliefs that are certainly relevant
in the context of metacognitive behavior.
Conclusion
The aim of the present explorative study was to investigate metacognitive behavior
in problem-posing processes, which has been widely disregarded in problem-posing
research. Analyses of 32 problem-posing processes of pre-service teacher students

1 3
Identifying Metacognitive Behavior inProblem‑Posing…
were conducted to identify metacognitive behaviors of planning, monitoring & con-
trol, and evaluating. Tables3,4, and5 summarize these inductively developed cat-
egories of problem-posing-specific aspects of metacognitive behavior. In addition,
two transcript excerpts were analyzed using the previously developed codes as a
proof of concept to make differences regarding metacognitive behavior in problem-
posing processes visible. Although, in both transcript excerpts, the product in terms
of the posed problem is almost identical, the metacognitive behavior, as the analysis
has shown, differs. The identified problem-posing-specific aspects of metacognitive
behaviors enabled the disclosure of these differences. In addition, the analyses have
shown that the consideration of metacognitive behavior allows a tentative assess-
ment of the quality of the activity in general. This assessment is a new perspective
on problem-posing processes.
Limitations of this study lie especially in the method of analyzing statements of
pairs of students in a video-based content analysis to assess metacognitive behavior.
As Goos etal. (2002) pointed out “student-student interactions could either help or
hinder metacognitive decision making during paired problem solving, depending on
students’ flexibility in sharing metacognitive roles” (p. 197). This approach can be
enriched in future studies by using stimulated recall interviews. Eye-tracking can
serve as a potential stimulus of such interviews. Individual interviews with a think-
ing-aloud approach were not used because thinking aloud, unpracticed, can interfere
with the natural flow of such processes and, in particular, affect metacognitive activ-
ities (McKeown & Gentilucci, 2007), which could distort the analyses.
The framework developed in this study provides numerous opportunities for follow-
up research on problem posing. As this study is based on problem-posing processes
of student teachers, it would be of interest to see if additional problem-posing-specific
aspects of metacognitive behaviors can be identified in a sample of, for example, pupils
or expert problem posers. A larger analysis could also address the question of which
metacognitive behaviors of planning, monitoring & control, and evaluating are particu-
larly prevalent. As in research on problem solving, a comparison between metacognitive
behaviors of experts and novices could reveal whether metacognitive behavior related to
successful problem posing or whether there are substantial differences in metacognitive
behavior in general. The data of this study was collected through structured problem-
posing situations. Future studies could address the question of whether there are differ-
ent metacognitive behaviors in unstructured situations or whether there are significant
differences in the frequency of the different metacognitive behaviors between structured
and unstructured problem-posing situations. Often, the ability to pose problems is meas-
ured by analyzing the products of a problem-posing process (Bonotto, 2013; Singer
etal., 2017; Van Harpen & Sriraman, 2013). The analysis of metacognitive behavior
could be used to assess the quality of problem posing on a process-oriented level. It is
likely that there is a strong correlation between the products and the processes of prob-
lem posing. However, there may also be high-quality processes by means of metacog-
nitive behavior, but whose products attest lower quality because few and non-original
problems were posed. Neglected in this study was the metacognitive facet knowledge
about cognition. The importance of this facet of metacognition could also be the focus
of future studies. Furthermore, metacognitive processes should be considered in addition
to motivational beliefs (Zimmerman & Moylan, 2009). Those beliefs could certainly

L.Baumanns, B.Rott
1 3
enrich the comparison of the two processes. Tino and Ulrich exhibit numerous behaviors
indicative of their motivational beliefs. Even before the analyzed extract, Tino and Ulrich
have numerous ideas about the Nim game. In turn 5, Ulrich shows that he would like
to deal with the different ideas one after the other, presumably in order to do sufficient
justice to all of them. Also, their numerous evaluative statements regarding their ideas
(Turn13; 17; 18; 25) as well as thinking their ideas further (Turn20; 21; 26; 27) are an
expression of intrinsic interest in posing new problems. In Valerie and Wenke’s process,
such behaviors and especially the positive reference to the activity of problem posing
are largely absent. Their laughter rather speaks for a general insecurity and a tendency
towards low interest in problem posing. Future studies could focus more on this interplay
between metacognitive behavior and motivational beliefs.
For teaching in school and university settings, these study’s findings can be used
to plan problem-posing activities in the classroom (cf. Kontorovich etal., 2012). For
example, students could be encouraged to engage in metacognitive behavior during
problem posing through appropriate construction of the problem-posing situation by
the teacher. The teacher could ask the students to solve their posed problems or they
could encourage reflection on the tasks by requiring students to pose tasks of vary-
ing difficulty for their classmates.
This study provides a first, qualitative insight into metacognitive behavior in
problem posing. We hope that the perspective of metacognition will stimulate fur-
ther studies in the field of problem posing research to gain further insights.
Funding Open Access funding enabled and organized by Projekt DEAL.
Declarations
Ethical approval Informed consent was obtained from all participants included in the study.
Data handling and analysis followed the rights of the General Data Protection Regulation (Datenschutz-
Grundverordnung - DSGVO).
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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