Content uploaded by Janina Krawitz
Author content
All content in this area was uploaded by Janina Krawitz on Jan 24, 2024
Content may be subject to copyright.
Journal of Mathematical Behavior 73 (2024) 101129
Available online 23 January 2024
0732-3123/© 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).
Do task variables of self-generated problems inuence interest?
Authenticity, openness, complexity, and students’ interest in
solving self-generated modelling problems
Janina Krawitz
a
,
*
, Luisa Hartmann
b
, Stanislaw Schukajlow
b
a
Institute of Mathematics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany
b
Institute of Mathematics Education and Computer Science Education, University of Münster, Henriette-Son-Str. 19, 48149 Münster, Germany
ARTICLE INFO
Keywords:
Problem posing
Mathematical modelling
Interest
Task variables
Motivation
Problem solving
ABSTRACT
Problem posing—generating one’s own problems—is considered a powerful teaching approach
for fostering students’ motivation such as their interest. However, research investigating the ef-
fects of task variables of self-generated problems on students’ interest is largely missing. In this
contribution, we present a study with 105 ninth- and tenth-graders to address the question of
whether the task variables modelling potential, assessed by openness and authenticity, or
complexity of self-generated problems have an impact on students’ interest in solving them.
Further, we investigated whether the effect of task variables of self-generated problems on stu-
dents’ interest differed among students with different levels of mathematical competence. High
modelling potential had a positive effect on interest in solving the problem for students with low
mathematical competence, whereas it had a negative effect for those with high mathematical
competence. However, complexity of self-generated problems did not affect students’ interest in
solving problems.
1. Introduction
Problem posing enables students to experience a feeling of competence by posing problems according to their individual learning
abilities and to develop learning autonomy by authoring their own problems (e.g., Silver, 1994). According to the self-determination
theory of motivation (Deci & Ryan, 2008), the experiences of competence and autonomy are essential for students’ intrinsic moti-
vation. As intrinsic motivation is closely related to interest (Reeve et al., 2003), problem posing could increase interest in solving
problems (e.g., Walkington, 2017). Enhancing learners’ interest via problem posing seems to be a fruitful goal, as studies have
demonstrated that students’ interest in a task or activity is associated with increased engagement, greater effort, more productive
learning behaviors, and better learning outcomes (Harackiewicz et al., 2008; Mitchell, 1993; Renninger & Hidi, 2017). In the present
contribution, we focus on problem posing that is based on descriptions of real-world situations and examine the role of task variables of
self-generated problems on students’ interest in solving the problems. An example of a real-world situation that we used as a
problem-posing stimulus is shown in Fig. 1.
To generate a problem based on a real-world situation, students can use their interpretation of the situation to construct a
mathematical problem according to their prior knowledge, problem-posing experience, and personal preferences. Thus, self-generated
* Corresponding author.
E-mail address: janina.krawitz@uni-paderborn.de (J. Krawitz).
Contents lists available at ScienceDirect
Journal of Mathematical Behavior
journal homepage: www.elsevier.com/locate/jmathb
https://doi.org/10.1016/j.jmathb.2024.101129
Received 24 March 2023; Received in revised form 12 December 2023; Accepted 14 January 2024
Journal of Mathematical Behavior 73 (2024) 101129
2
problems differ regarding task variables. Prior studies demonstrated that students can pose demanding and meaningful problems (Cai
& Hwang, 2002; Silver & Cai, 1996; Stoyanova & Ellerton, 1996), and engage in important modelling processes while posing own
problems (Hartmann et al., 2022, 2023). However, research also revealed that problem posing is demanding and self-generated
problems often have decits (English, 1998; Hansen & Hana, 2015; Hartmann et al., 2021). Learners’ experience with problem
posing might be an important factor in determining the task variables and thereby also the quality of self-generated problems (e.g.,
Chen et al., 2015). The task variables of self-generated problems in turn can affect learners’ motivation in solving these problems. If
students perceive a problem as new and complex and are condent that they can master it, they likely will nd it interesting (Silvia,
2005, 2008). In addition, individual factors, such as learners’ mathematical competence, can impact their perception of self-generated
problems and thus, determine whether or not they nd a particular problem interesting. For example, students with high levels of
mathematical competence are more likely to nd complex problems interesting because they are more condent in their ability to
solve such problems compared to students with lower levels of competence.
Although problem posing is presumed to have a positive effect on students’ interest, there is currently limited evidence regarding its
impact, especially regarding the task variables of the self-generated problems. Therefore, the overarching goal of the study was to
contribute to the theory of problem posing and motivation (Cai & Leikin, 2020) by investigating the conditions under which students
perceive self-generated problems interesting. To reach this goal we focused on task-specic factors, namely the modelling potential of
self-generated problems assessed by openness and authenticity, and the complexity of self-generated problems. We consider these
factors as important for students’ interest in solving self-generated problems that were developed based on real-world situations. In
addition, we focused on two individual factors, namely students’ mathematical competence and their problem-posing experience.
Mathematical competence is considered a signicant individual factor inuencing interest (Harackiewicz et al., 2008). It can
potentially shape how individuals perceive certain characteristics of a task and, consequently, impact their interest in solving it.
Furthermore, problem-posing experience is regarded as an important factor affecting the characteristics of self-generated problems
(Cai & Hwang, 2021). Problem-posing experience inuences how students formulate and structure problems based on their prior
experiences and knowledge.
The present study aims to investigate the impact of task variables, such as modelling potential and complexity, on students’ interest
in solving self-generated problems. Additionally, we examine whether the effect of these task variables on interest varies among
students with different levels of mathematical competence. Further, we investigate whether prior experience with problem-posing
predicts interest in solving self-generated problems via the task variables of self-generated problems.
Fig. 1. A real-world situation about the “salt mountain” together with a problem posing prompt adapted from Hartmann et al. (2021).
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
3
2. Theoretical background
2.1. Problem posing based on real-world situations
Problem posing is considered to have great potential for the learning of mathematics (e.g., Cai et al., 2015; e.g., Liljedahl & Cai,
2021). In school curricula, problem posing has been described as an important learning goal and it has been suggested to be an
important tool for problem solving (National Council of Teachers of Mathematics, 2000). However, problem posing is still not used
much in mathematics classes, and research is just beginning to develop an understanding of the benets and challenges of this teaching
approach (Cai & Hwang, 2021). Problem posing is typically dened as the generation of new problems or the reformulation of given
problems (Silver, 1994). In the literature, the term problem posing is used for a wide range of activities, and several scholars (e.g., Cai
et al., 2015; Silver, 2013) suggested that there is a need to clarify different forms of problem posing. Problem-posing tasks can be
classied in various manners (Baumanns & Rott, 2021; Stoyanova & Ellerton, 1996). One such classication considers a task to consist
of two components: a problem situation and a problem-posing prompt (Cai & Hwang, 2023; Leikin et al., 2023). The problem situation
provides context and data that serve as sources for generating problems. For example, it can comprise intra-mathematical elements like
geometric gures, graphs or equations or elements that are connected to the real world such as textual descriptions or pictures of a
real-world situation. The problem-posing prompt instructs the learners on how to approach the problem situation. For example, it can
encourage learners to pose multiple problems, reformulate a given problem or pose a mathematical problem. In the present approach,
we look at problem posing from the perspective of modelling and applications in mathematics education. Consequently, we focus on
problem situations that are connected to the real world (here, textual descriptions of real-world situations) and problem-posing
prompts that stimulate the generation of real-world problems.
2.2. Task variables of self-generated problems based on real-world situations
The task variables of problems have an impact on what students learn, as different types of problems place varying cognitive
demands on learners. Consequently, the potential for problem posing to enhance problem-solving depends on the task variables of self-
generated problems.
In the eld of problem-solving research, multiple task variables have been identied to classify problem types. For instance, Goldin
& McClintock (1984) distinguished between syntax variables, content variables, structure variables, and heuristic behavior variables.
In the present study, we focus on structure variables of self-generated problems. Structure variables refer to factors that determine the
problem structure and strongly inuence the solution process, such as specic algorithms or strategies needed to solve it (Goldin &
McClintock, 1984). Task variables related to self-generated problems serve as indicators of learners’ problem posing abilities in
problem-posing research. Studies on modelling-related problem posing focused on several task variables including correctness,
authenticity, openness, mathematical reference, mathematical meaningfulness, and students’ involvement (Hansen & Hana, 2015;
Hartmann et al., 2021; Marco & Palatnik, 2022). The results highlight that students and pre-service teachers have problems to pose
open (Hartmann et al., 2021) and mathematical meaningful (Hansen & Hana, 2015) problems. However, when working over a longer
time period on their self-generated problems in a problem posing course pre-service teachers were found to improve their problems
regarding several task variables such as their correctness, authenticity, and students’ involvement (Marco & Palatnik, 2022). In our
study, we addressed modelling potential, assessed by openness and authenticity, as one structure variable from modelling research and
complexity as one structure variable from problem solving research.
2.2.1. Modelling potential
Modelling problems are problems, that require a demanding translation process between the real-world and mathematics (Niss &
Blum, 2020). To solve a modelling problem, students have to understand a given real-world situation, structure and simplify the
information to set up a mathematical model, work mathematically, interpret their results at the end of the solution process, and
validate their results with regard to the real-world situation. Modelling problems require specic solution procedures and strategies
(Niss & Blum, 2020). Thus, the modelling potential of a problem – the extent to which a problem conforms to modelling characteristics
– can be considered as a structure variable. Two key characteristics of modelling problems are their authentic connection to reality and
their openness (Maaß, 2010). Problems are considered authentic if they comprise questions that might be posed in a real-world sit-
uation (Palm, 2008). Openness can refer to the initial state of the problem, the goal state or the intermediate state (e.g., Silver, 1995;
Yeo, 2017). Here we focus on openness regarding the initial state of the problem, i.e. that the problem does not include all of the
information and that assumptions or data collection are necessary to solve it.
2.2.2. Complexity
A second important structure variable is complexity (Goldin & McClintock, 1984). The complexity of a problem depends on the
individual, but for the same student, one task can be judged as more or less complex than another (Yeo, 2017). In the literature, several
different frameworks are used to dene complexity (e.g., Stillman & Galbraith, 2003; Williams & Clarke, 1997). Prior research used the
number of solution steps in combination with other criteria such as the magnitude of the numbers or the level of required general-
ization as an indicator of the complexity of self-generated problems (Silber & Cai, 2017). Another factor that contributes to the
complexity of a problem is its’ mathematical structure (Williams & Clarke, 1997). In modelling problems, the required mathematical
model determines the mathematical structure. Non-complex problems can be characterized as problems that can be solved using
simple mathematical models, while complex problems require more advanced models. Similar to prior research in problem posing
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
4
(Silber & Cai, 2017), we use a combination of indicators to assess complexity. In the present study, we rely on the number of required
solution steps and the complexity of the mathematical model as indicators of the complexity of self-generated problems.
2.3. Problem posing and interest in solving self-generated problems
Interest is a person-object relationship that refers to (a) how much attention a person pays to and their effect regarding a particular
topic (situational interest) or (b) an enduring predisposition to reengage with a topic over time (individual interest) (Hidi & Renninger,
2006). Interest is specic to particular domains or content areas and is characterized by both cognitive and affective components
(Harackiewicz et al., 2016; Schiefele et al., 1992). Theoretical models of interest development suggest that students pass through
several stages as their interest in a topic evolves, starting from an unstable situational interest that is easily triggered to a more
enduring and personal interest that is well-developed (Hidi & Renninger, 2006). Repeated experiences of situational interest towards a
particular topic can lead to the development of individual interest over time. Therefore, the environment can play a role in fostering
the development of individual interest by creating opportunities to trigger situational interest and building on prior individual interest.
But what makes an object interesting? Research focusing on text-based interest identied several text characteristics as factors that can
enhance situational interest, including personal relevance, novelty, vividness, and comprehensibility (Schraw et al., 1995; Wade et al.,
1999). However, appraisal theories propose that not mere characteristics of an object of interest or event that humans are interested in,
but individuals’ subjective evaluations of objects or events are believed to arouse interest (Lazarus, 1991). Two specic appraisals, the
novelty-complexity appraisal and the coping-potential appraisal, are considered as contributors to interest (Silvia, 2005, 2008). The
appraisal of novelty-complexity refers to evaluating an event as new, unexpected, complex, hard to process, uncertain, or surprising.
The coping-potential appraisal refers to an individual’s assessment of their abilities and resources. If a person has the condence of
mastering a complex task, he or she will likely nd it interesting. Here we focus on situational, task-specic interest in solving
self-generated problems. Task variables of the self-generated problems and how they are perceived by the students might play an
important role for their situational interest in mathematical problems. The task variables of modelling potential and complexity may
contribute to interest through the novelty-complexity appraisal, while the fact that the problems are self-generated may contribute to
the coping-potential appraisal, as problem posing allows students to generate problems that match their abilities and prior knowledge.
To the best of our knowledge, no scientic studies have been conducted to examine the effect of task variables of self-generated
problems on students’ interest in solving them. The ndings of research that focused on given problems are mixed. Enhancing the
modelling potential by adding meaningful real-world contexts to intra-mathematical problems was found to increase interest in
elementary school children (Cordova & Lepper, 1996). However, the studies conducted by Rellensmann and Schukajlow (2017) and
Schukajlow et al. (2012) revealed that students do not consider problems related to reality more interesting than intra-mathematical
problems, per se. In addition, students reported more positive feelings towards word problems (i.e., problems that do not require
making assumptions and have one solution) than modelling problems (Krawitz & Schukajlow, 2018; Parhizgar & Liljedahl, 2019). One
potential explanation for this unexpected nding is that the modelling problems were perceived as too difcult, leading to frustration.
However, it’s also possible that problems perceived as too easy may be less interesting to students. Krug and Schukajlow (2013) found
that interest tends to decrease after a task has been completed, suggesting that tasks for which students already know the answer may
be less interesting to them.
It is an open question whether these ndings also hold for self-generated problems. For given modelling problems learners do have
less possibilities to adapt the problems to their own needs and abilities. Thus, their appraisal for solving given modelling problems
might be high regarding the novelty-complexity facet but low regarding the coping potential facet. We expect that posing high-quality
problems, in terms of modelling potential and complexity, will enhance students’ interest in solving the self-generated problems,
because we expect these task variables to enhance novelty-complexity appraisal and the activity of problem posing to facilitate coping-
potential appraisal.
2.4. Mathematical competence as a moderator of the effect
Research has predominantly focused on achievement as the dependent variable of interest, but there is evidence indicating that
achievement is reciprocally related to interest (Harackiewicz et al., 2008). Therefore, students’ mathematical competence can be
expected to inuence their interest in solving the problem. Niss and Højgaard (2019) dened mathematical competence as the
readiness to act appropriately in response to all kinds of mathematical challenges pertaining to given situations. The task variables of
self-generated problems may interact with the individual’s mathematical competence to affect their interest. Depending on their level
of mathematical competence, students may experience varying levels of interest in tasks with certain characteristics. Mathematically
complex problems might be perceived as interesting by students with a high level of mathematical competence but might be perceived
as uninteresting by students with a low level of mathematical competence because of the difculty of such problems, which can lead to
lower self-efcacy in solving the problem (Carmichael et al., 2009). Furthermore, the modelling potential of a problem could be a
critical factor, as an authentic context may offer more opportunities for students to connect the problem to their own real-world in-
terests. This might be particularly benecial for students with a low level of mathematical competence because they might focus more
on non-mathematical aspects and might appreciate the chance to deal with these aspects more in mathematics class. The ndings from
the study conducted by Schukajlow et al. (2022) support these considerations. The study revealed that low-achieving students reported
greater interest in solving open modelling problems when compared to closed problems, whereas high-achieving students reported
similar interest in both open and closed modelling problems with a slight preference for closed problems.
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
5
2.5. Problem-posing experience and task variables of self-generated problems
Numerous studies have indicated that students are capable of posing interesting and important mathematical problems (e.g., Cai &
Hwang, 2002; Silver & Cai, 1996; Stoyanova & Ellerton, 1996). However, also decits in posing adequate problems have been reported
(Cai & Hwang, 2002; Silver & Cai, 1996). In particular, for modelling-related problem posing it was found that students tend to pose
over-simplistic and closed problems (English, 1998; Hartmann et al., 2021). One possible explanation for this nding is, that students
may lack experience in problem posing, as problem posing is rarely taught in math classes (Cai & Hwang, 2021). Problem posing can be
considered a competence that can be learned by engaging in problem posing activities (Chen et al., 2015). Therefore, it is reasonable to
expect a positive effect of problem-posing experience on the quality of self-generated problems in terms of various task variables. The
characteristics of self-generated problems, in turn, might affect students’ interest. When students pose problems with high modelling
potential and complexity, it is more likely that they will nd them interesting (as discussed in Section 2.3). Hence, we expect a positive
effect of problem-posing experience on interest via the task variables of self-generated problems. Intervention studies on problem
posing support this claim, as they have demonstrated that the quality of self-developed problems can be enhanced through
problem-posing activities (Abu-Elwan, 2002; Chen et al., 2015; English, 1998). However, in a long-term intervention with fth graders
Chen et al. (2015) showed that students’ difculties in posing high-quality problems are persistent and hard to change. In this study,
the problem posing intervention helped to improve the quality of self-generated problems for some task variables (e.g. originality) but
not for others (e.g. complexity). Hence, additional research is needed to determine whether experience with problem posing predicts
the quality of self-generated problems regarding different task variables.
3. Path model and hypotheses
The present study’s research questions were:
(1) Does posing problems with high modelling potential or high complexity positively affect interest in solving them?
(2) Does mathematical competence moderate the effect of posing problems with high modelling potential or high complexity on
interest in solving these problems?
(3) Does students’ experience with problem posing predict interest in solving self-generated problems via the task variables of
modelling potential and complexity?
Fig. 2. Hypothesized path models for testing direct, indirect, and moderating effects of problem-posing experience on interest in solving self-
generated problems. Model A illustrates Hypotheses 1a, 2a, and 3 and Model B illustrates Hypotheses 1b, 2b, and 3.
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
6
To answer these research questions, we set up and tested two path models (Fig. 2).
Based on theoretical considerations and research ndings, we posed the following hypotheses:
Hypothesis 1. (task variables on interest).
a) Posing problems with high modelling potential is benecial for students’ interest in solving the problems.
b) Posing complex problems is benecial for students’ interest in solving the problems.
Hypothesis 2. (moderation by mathematical competence).
The effect of posing problems with high modelling potential or complexity on interest in solving the problems depends on students’
mathematical competence.
a) Posing problems with high modelling potential is more benecial for increasing the interest of students with low levels of math-
ematical competence.
b) Posing mathematical complex problems is more benecial for increasing the interest of students with high levels of mathematical
competence.
Hypothesis 3. (Problem-posing experience on task variables and interest).
a) Students’ experience with problem posing will positively affect their interest in solving the problems. (total effects).
b) Students’ experience with problem posing will positively affect interest in solving the problems via posing problems with high
modelling potential or high complexity. (indirect effects).
4. Method
4.1. Participants and procedure
The sample involved 105 ninth- and tenth-graders from 12 classes (52.88% female; mean age M =15.30 years, SD =0.66).
Students were given a paper-pencil test with six problem situations and were asked to create problems that were based on the
situations. After posing each of the self-generated problems, students were asked to answer a questionnaire (as part of the test) about
their situational interest in solving the self-generated problem. Then they were asked to solve the self-generated problem. At the end of
the test-booklet students answered a questionnaire about their gender, their experience with problem-posing and their mathematics
grades on their last school report.
The problem situations were descriptions of real-world situations. An example for one of the situations is provided in Fig. 1. The
other situations were taken from Hartmann et al. (2021). The problem-posing prompt in the test booklet was:
“In this booklet, you will nd a number of different situations from the real world. Unlike most of the tasks you are familiar with,
there is no mathematical problem for you to solve for these situations because today you will develop the problem yourself.
First, read the description of the situation. Then think about a mathematical problem you can pose based on the given situation
that can be solved by using information from these situations and write this problem down. Then you should solve your self-
generated problem.”
We included the phrase “that could be solved using information from the situation” to prompt students to pose mathematical
problems and to minimize the number of non-mathematical questions. The prompt was given to the students at the beginning of the
test. After each real-world situation they were asked to write down a self-generated problem (see Fig. 1).
Table 1 provides examples of problems posed by the students. The descriptions of real-world situations were adapted from
modelling problems that were used in prior studies (e.g., Krawitz & Schukajlow, 2018). The descriptions were enriched with additional
information to provide students with the opportunity to pose different problems that are related to various situational and
Table 1
Sample problems with codes for modelling potential.
Problem Modelling potential
Calculate the surface of the salt mountain. Non-authentic and closed (scored 0)
How long is side b? Non-authentic and closed (scored 0)
How many trucks are needed to transport the salt mountain? Authentic and closed (scored 0.5)
How many times does the truck have to go back and forth until it has transported all the salt? Authentic and closed (scored 0.5)
How long does it take to transport the salt mountain? Authentic and open (scored 1)
How much does one grain of salt weigh? Authentic and open (scored 1)
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
7
mathematical aspects. The test was administrated by instructed master students who followed a standardized instruction manual. A
pilot study (Hartmann et al., 2021) was conducted and ensured that the time given (two school lessons, 90 min) was sufcient to nish
the test.
4.2. Measures
4.2.1. Scales for problem-posing experience, mathematical competence and interest
Students’ experience with problem posing was measured with a questionnaire. Students were asked to indicate the extent to which
they agreed with three given statements on a scale ranging from 1 =not at all true to 5 =completely true (three items, e.g. “In math
lessons, we pose our own word problems that we can solve by using math.”) The reliability (Cronbach’s
α
) of the scale was.714.
As an indicator of students’ mathematical competence, students’ mathematics grades on their last school report card were used
(inversely scored; ranging from 1 =“low mathematical competence” to 5 =“high mathematical competence”). Students’ mathematics
grades are considered a good measure of mathematics achievement as they take into account a substantial timeframe and draw on
different sources (Roth et al., 2015). They were found to reect students’ achievement when different external criteria, such as a
general mathematics achievement scale and an understanding proofs scale were used (Rakoczy et al., 2008).
Students’ interest in solving the self-generated problems was measured by using a questionnaire. For each of the six problem
situations students were asked after generating a problem and before solving the self-generated problem, whether they were interested
in solving their self-generated problem (5-point Likert scale: 1 =not at all true, 5 =completely true). Students’ responses indicated the
extent to which they agreed with the following statement: “It would be interesting to solve my problem.” Interest in solving the
problems was aggregated for the six problem situations. Leading to a scale consisting of 6 items with an internal consistency (Cron-
bach’s
α
) of.859.
4.2.2. Scales for the task variables modelling potential and complexity of self-generated problems
Students’ self-generated problems to given six real-world situations were analyzed with regard to their modelling potential.
Thereby we focused on openness and authenticity as typical characteristics of modelling problems (Table 1). Each problem was
assigned to one of the following categories: non-authentic and closed (scored 0), either authentic or open (scored 0.5), or authentic and
open (scored 1). Problems were coded as authentic if the question might actually be posed in a real-world situation (Palm, 2008). A
sample problem that was coded as authentic is: “How many trucks are needed to transport the salt mountain?” Problems were coded as
non-authentic if they were mainly dressed-up mathematical questions that were not meant to contribute to an understanding or
exploration of the real-world situation (e.g., “Calculate the surface of the salt mountain”). Regarding the openness of the problems, we
focused on the initial problem state and dened a problem as open if it required the problem solver to make assumptions. As an
example, the problem “How much does one grain of salt weigh?” was scored as an open problem because additional information or
assumptions were necessary to solve the problem. Also, the problem “How long does it take to transport the salt mountain?” was scored
as open, because it requires assumptions about the time needed to load the truck, the number of trucks, driving time, and so on. We
used students’ solutions as an additional source for coding the task variables. For instance, if a solution to a problem that could be
interpreted either as open or closed included realistic considerations or assumptions, we coded it as an open problem. Conversely, the
problem “How many trucks are needed to transport the salt mountain?” was scored as closed because the information that was given
was sufcient to solve the problem. Table 1 presents sample problems to illustrate how the scoring works. The scale reliability
(Cronbach’s
α
) was.676. Two trained raters scored the self-generated problems following a coding manual that was developed in a
prior study (Hartmann et al., 2021). To assess interrater agreement, 20% of the self-generated problems were independently coded by
both raters. The interrater agreement (Cohen’s κ) for modelling potential ranged between 0.611 and 1.00 for the six problem posing
tasks (M =0.860). The double coding was analyzed to identify systematic disagreements, which were subsequently discussed and
resolved.
In addition, the self-generated problems were also scored concerning their mathematical complexity. We distinguished between
non-complex problems (problems that can be solved with simple mathematical models such as one basic mathematical operations) and
complex problems (advanced mathematical models, e.g., models that are based on the Pythagorean theorem or require multiple non-
routine steps). For example, the problem “How many trucks are needed to transport the salt mountain?” was scored non-complex,
because it can be solved with a simple mathematical model (the weight of the salt mountain divided by the maximum load capac-
ity of the trucks). Problems that could be solved directly by taking information from the text were also scored non-complex. For
example, the problem “How much does the salt mountain weigh?” was scored non-complex, because the answer 3740 tons is given in
the text. Conversely, the problem “What is the height of the salt-mountain was scored as complex, because Pythagorean theorem needs
to be applied to solve the problem. The reliability (Cronbach’s
α
) of the scale was .563. The intercoder reliability (Cohen’s κ) ranged
between 0.444 and 0.673 (M =0.573), indicating at least a moderate agreement.
4.3. Data analysis
We used the MPlus software to test the hypotheses with two path models, one model for modelling potential and one model for
complexity. The reported p-values for the effects that referred to our hypotheses were one-tailed because our expectations were
directional. Each model included 9 free parameters and 105 participants. The ratio (participants/free parameters) was above the
critical value of 5, which indicates that the sample size was sufcient for obtaining solid results (Kline, 2005).
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
8
5. Results
5.1. Analysis of model t for the path models and descriptive statistics
The goodness-of-t statistics indicated that the path models t well. For the model examining the construction of problems with
high modelling potential, the comparative t index (CFI) was.923, the standardized root mean squared residual (SRMR) was.031,
which is below the critical value of.05, and the Chi-square test was nonsignicant, Х
2
=3.349, p=.187. For the model examining the
construction of complex problems, the t indices indicated a good model t (CFI =1.0; SRMR =.013; Х
2
=0.155, p =.926).
The estimates of the path models were based on the correlation matrix presented in Table 2. The results for the path model are
summarized in Fig. 3.
5.2. Effects of posing problems with high modelling potential or complexity on interest
Posing problems with high modelling potential was expected to positively affect students’ interest in self-generated problems
(Hypothesis 1). Contrary to our expectations neither posing problems with high modelling potential nor posing complex problems did
affect interest in solving self-generated problems (modelling potential (Model A): β= − 0.043, p=0.643; complexity (Model B):
β=0.040, p=0.643).
5.3. Mathematical competence as a moderator
We further expected that the effect of posing problems with high modelling potential on interest in solving the problems would
depend on students’ mathematical competence. In particular, we expected that posing problems with high modelling potential would
enhance interest more in students with a low level of mathematical competence (Hypothesis 2a), whereas posing complex problems
would enhance interest more in students with a high level of mathematical competence (Hypothesis 2b). In line with our expectations,
we found that students’ mathematical competence moderated the effect of posing problems with high modelling potential on their
interest in solving the problems (β= − 1.136, p=.009), indicating that posing problems with high modelling potential is more
benecial for interest in solving the problems for students with a low level of mathematical competence than for students with a high
level of mathematical competence. We took a closer look at the potential differences and analyzed the effect at different levels of
mathematical competence. One standard deviation of mathematical competence corresponds to about one grade (SD =0.95). If the
grade was one standard deviation below the arithmetic mean (low mathematical competence), posing problems with high modelling
potential was positively related to students’ interest (β=0.885, p=.037), whereas, if the grade was one standard deviation above the
arithmetic mean (high mathematical competence), posing problems with high modelling potential was negatively related to students’
interest (β= − 1.240, p=.039).
Regarding the effect of posing complex problems on students’ interest in solving the problems, contrary to our expectations, we
found no signicant moderating effect of mathematical competence (β=0.165, p=.263). Hence, our hypothesis that students with a
higher level of mathematical competence would be more interested in solving mathematically complex self-generated problems than
students with a lower level of mathematical competence was not supported by the data.
5.4. Effects of problem-posing experience on interest
We further analyzed the total effects in the two path models. The total effects included all direct and indirect effects of problem-
posing experience on interest. We expected that experience with problem posing would positively affect interest (Hypothesis 3a). This
expectation was conrmed by the results of both models (Model A: β=0.239, p=.002; Model B: β=0.205, p=.011). Further, we
expected an indirect effect of problem-posing experience on interest via posing problems with high modelling potential or via posing
complex problems as intervening variables (Hypothesis 3b). Contrary to our expectations, no indirect effects were found in the models
(indirect effects, Model A: β= − 0.012, p=.331, Model B: β=0.001, p=.454). Thus, the effect of problem-posing experience on
interest in solving the self-generated problems was not mediated by posing problems with these task variables. Problem-posing
experience had a positive effect on the posing of problems with high modelling potential (direct effect, β=0.285, p<.001), but
posing problems with high modelling potential did not affect interest in the self-generated problems (direct effect, β= − 0.043,
p=.327). For Model B, problem-posing experience did not enhance the posing of complex problems (direct effect, β=0.010,
p=.453), and in turn, posing complex problems did not increase interest in the problems (direct effect, β=0.040, p=0.322).
Table 2
Means, standard deviations, and correlations of all variables (**p<.01, *p<.05 two-tailed).
Variable (score range) M SD 1 2 3 4 5
1. Problem-posing experience (1-5) 1.65 0.77 -
2. Mathematical competence (1-5) 3.20 0.95 .05 -
3. Modelling potential (0-1) 0.51 0.13 .28** .11 -
4. Complexity (0-1) 0.43 0.23 −.01 −.05 .30** -
5. Interest (1-5) 3.14 0.89 .23* .22* .04 .06 -
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
9
6. Discussion
The goal of the present study was to investigate the interplay between different factors that help to explain how problem posing can
contribute to situational interest in mathematics. Thereby, we included task-specic factors (the task variables of self-generated
problems) and two individual factors (prior experience with problem posing and mathematical competence) in our analyses.
Our study makes a novel empirical contribution by assessing, for the rst time, the effect of task variables of self-generated
problems on students’ interest in solving these problems. A path analysis based on theories of problem posing, modelling, and in-
terest revealed that contrary to our expectations posing problems with high modelling potential or complexity did not affect students’
interest in self-generated problems across the whole sample. However, an important nding of our study was that posing problems
with high modelling potential had a positive effect on students with low mathematical competence, but a negative effect on those with
high mathematical competence. For students with low mathematical competence, characteristics of modelling problems such as
authenticity and openness may be more important as they allow them to connect their real-world interests to the problems. Students
with high mathematical competence may be more interested in solving problems that are more obviously mathematical. A theoretical
implication of this nding is that mathematical competence is an important factor that has to be considered in the context of modelling
problems and interest in solving the problems. This result sheds light on the ndings of prior studies in which modelling problems were
perceived as similar or even less interesting than dressed-up word problems and intra-mathematical problems (Parhizgar & Liljedahl,
2019; Rellensmann & Schukajlow, 2017; Schukajlow et al., 2012). Future studies should investigate whether the positive effect of
generating problems with high modelling potential on situational interest among low-achieving students can be extended to their
interest in solving given modelling problems.
Further, our results showed that problem-posing experience positively affected students’ interest in the problems. However,
contrary to our expectations, problem-posing experience did not affect students’ interest indirectly via the task variables of the self-
generated problems. In line with our expectations, problem-posing experience did enhance the posing of problems with high modelling
potential, and posing problems with high modelling potential in turn had benecial effects on the interest of students with a low level
Fig. 3. Path models for testing the hypotheses. Note. Signicant paths (p<.05) are presented as solid lines. The estimates are standardized
(STDYX). Thus, they can be interpreted as predicted changes in the criterion measures in standard deviation units when the covariate changes by
one standard deviation.
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
10
of mathematical competence. Consistent with theories that explain how the appraisal of events contributes to interest (Silvia, 2005,
2008), it is important to consider not only the characteristics of self-generated problems but also how they interact with personal
factors that inuence how a person evaluates a task. Regarding the complexity of self-generated problems and contrary to our ex-
pectations, we found no effect of problem-posing experience on the posing of complex problems. Similar to previous research, students
showed a tendency to pose simple problems (English, 1998; Hartmann et al., 2021). In addition, there was no effect of posing complex
problems on students’ interest in the problems, and students’ mathematical competence did not moderate the effect of posing complex
problems on their interest. One possible explanation for the absence of signicant effects related to the focused task variables
modelling potential and mathematical complexity on students’ interest in solving the problems could be that other task variables are
more important for students. For instance, they may prioritize the mathematical correctness of problems, which was identied as
important task variable in prior research on modelling-related problem posing (Hansen & Hana, 2015; Marco & Palatnik, 2022).
Therefore, future studies should explore in addition to modelling potential and mathematical complexity other task variables to un-
cover factors that might play an important role in stimulating students’ interest in solving self-generated problems.
Our expectation that problem-posing experience would have an indirect effect on interest in solving self-generated problems via the
task variables of the problem was not supported. However, problem-posing experience had a signicant direct effect on students’
interest in solving self-generated problems. Thus, teaching problem posing in class nevertheless seems to be important for students’
interest in solving the problems. Future studies should clarify what task-related or personal factors mediate the effects of experience on
students’ interest in solving problems. Perceived competence, autonomy, and enjoyment have been revealed to be important for
students’ interest in mathematical problem solving in the past (Schukajlow & Krug, 2014; Schukajlow & Rakoczy, 2016), and they
might also be important in the context of problem posing (Cai & Leikin, 2020; Liu et al., 2020).
In addition, our ndings indicate that modelling problems have the potential to boost interest in solving self-generated problems for
students with low levels of mathematical competence. This needs to be conrmed in future studies. A practical implication would be
that modelling problems are an important tool for supporting students’ interest via problem posing, and important characteristics of
modelling problems (e.g., openness) might support the development of interest in particular in students with low levels of mathe-
matical competence. This aspect should be discussed further in modelling research, and future studies should focus on how interest,
modelling, and mathematical competence are interrelated.
7. Strength and limitations
Among some obvious limitations, such as the limited number of tasks and participants as well as limitations that hail from the
choice of concrete real-world situations, the following points should also be considered. In our study, rich real-world situations were
carefully selected and offered to students. Thus, students’ choices for constructing problems were limited to specic situations. The
limited choices may have negatively impacted the effects of problem posing on students’ interest in solving the problems and may have
contributed to the absence of indirect effects on interest via the task variables of the self-generated problems.
Our framework for analyzing self-generated problems primarily focused on modelling problems. However, we acknowledge that
different results might arise if other frameworks such as those centered on word problems (Verschaffel et al., 2020), had been utilized.
Furthermore, students are generally more familiar with closed problems (Verschaffel et al., 2000). Hence, they may have assumed that
they should generate closed problems. This tendency might have been reinforced by our prompt, which asked students to pose
problems that can be solved using information from the situations. The reason for the prompt was that we did not want to force
students to pose open problems. To address this issue, we performed a statistical analysis that excluded openness from the score of
modelling potential to check how these changes would affect the results of the statistical analysis. The analysis with the authenticity
(and without openness) as a score for modelling potential, revealed that most regressions did not change and remained signicant. The
only important change was in the signicance of the path from ‘problem posing experience’ to ‘posing problems with high modelling
potential’, which became not signicant after excluding openness (p=.25). To learn more about the effects of problem-posing
prompts on self-generated problems, future studies need to systematically examine different types of prompts (Cai & Hwang, 2023).
Further, the internal consistency of the mathematical complexity scale was rather low and the interrater reliability for mathe-
matical complexity had borderline values for some of the tasks. Thus, results related to this task variable should be interpreted with
caution. The low reliability indicates that the extent to which problems with different levels of complexity can be posed depends on the
real-world situation that a student was offered. Future investigations should uncover the factors that inuence students’ attempts to
pose problems, and different real-world situations should be explored.
In addition, the problem situations in the test were presented for all students in the same order. Hence, it is possible that differences
between interest in solving the tasks at the beginning or at the end occurred because of for example tiredness of the students.
Funding
This research did not receive any specic grant from funding agencies in the public, commercial, or not-for-prot sectors.
CRediT authorship contribution statement
Krawitz Janina: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Writing – original draft, Writing –
review & editing. Hartmann Luisa: Data curation, Writing – review & editing. Schukajlow Stanislaw: Conceptualization, Writing –
review & editing.
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
11
Declaration of Generative AI and AI-assisted technologies in the writing process
During the preparation of this work the authors used ChatGPT-3.5 to conduct a language. After using this tool, the authors reviewed
and edited the content as needed and take full responsibility for the content of the publication.
Declaration of Competing Interest
The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to
inuence the work reported in this paper.
References
Abu-Elwan, R. (2002). Effectiveness of problem posing strategies on prospective mathematics teachers’ problem solving performance. Journal of Science and
Mathematics Education in Southeast Asia, 25(1), 56–69.
Baumanns, L., & Rott, B. (2021). Developing a framework for characterising problem-posing activities: A review. Research in Mathematics Education, 23(1). https://doi.
org/10.1080/14794802.2021.1897036
Cai, J., & Hwang, S. (2002). Generalized and generative thinking in US and Chinese students’ mathematical problem solving and problem posing. The Journal of
Mathematical Behavior, 21(4), 401–421. https://doi.org/10.1016/S0732-3123(02)00142-6
Cai, J., & Hwang, S. (2021). Teachers as redesigners of curriculum to teach mathematics through problem posing: Conceptualization and initial ndings of a
problem‑posing project. ZDM Mathematics Education, 53(6), 1403–1416. https://doi.org/10.1007/s11858-021-01252-3
Cai, J., & Hwang, S. (2023). Making mathematics challenging through problem posing in the classroom. In R. Leikin, C. Christou, A. Karp, D. Pitta-Pantazi, & R. Zazkis
(Eds.), Mathematical challenges for all. Springer.
Cai, J., Hwang, S., Jiang, C., & Silber, S. (2015). Problem-posing research in mathematics education: Some answered and unanswered questions. In F. M. Singer,
N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 3–34). Springer. https://doi.org/10.1007/978-1-4614-6258-3_1.
Cai, J., & Leikin, R. (2020). Affect in mathematical problem posing: Conceptualization, advances, and future directions for research. Educational Studies in Mathematics,
105(3), 287–301. https://doi.org/10.1007/s10649-020-10008-x
Carmichael, C., Callingham, R., Watson, J., & Hay, I. (2009). Factors inuencing the development of middle school students’ interest in statistical literacy. Statistics
Education Research Journal, 8(1), 62–81.
Chen, L., van Dooren, W., & Verschaffel, L. (2015). Enhancing the development of Chinese fth-graders’ problem-posing and problem-solving abilities, beliefs, and
attitudes: A design experiment. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From research to effective practice (pp. 309–329).
Springer. https://doi.org/10.1007/978-1-4614-6258-3_15.
Cordova, D. I., & Lepper, M. R. (1996). Intrinsic motivation and the process of learning: Benecial effects of contextualization, personalization, and choice. Journal of
Educational Psychology, 88(4), 715–730. https://doi.org/10.1037/0022-0663.88.4.715
Deci, E. L., & Ryan, R. M. (2008). Self-determination theory: A macrotheory of human motivation, development, and health. Canadian Psychology/Psychologie
Canadienne, 49(3), 182–185. https://doi.org/10.1037/a0012801
English, L. D. (1998). Children’s problem posing within formal and informal contexts. Journal for Research in Mathematics Education, 29(1), 83–106. https://doi.org/
10.2307/749719
Goldin, G. A., & McClintock, C. E. (1984). Task variables in mathematical problem. Franklin Institute Press.
Hansen, R., & Hana, G. M. (2015). Problem posing from a modelling perspective. In F. M. Singer, N. F. Ellerton, & J. Cai (Eds.), Mathematical problem posing: From
research to effective practice (pp. 36–46). Springer.
Harackiewicz, J. M., Durik, A. M., Barron, K. E., Linnenbrink-Garcia, L., & Tauer, J. M. (2008). The role of achievement goals in the development of interest:
Reciprocal relations between achievement goals, interest, and performance. Journal of Educational Psychology, 100(1), 105–122. https://doi.org/10.1037/0022-
0663.100.1.105
Harackiewicz, J. M., Smith, J. L., & Priniski, S. J. (2016). Interest matters: The importance of promoting interest in education. Policy Insights from the Behavioral and
Brain Sciences, 3(2), 220–227. https://doi.org/10.1177/2372732216655542
Hartmann, L., Krawitz, J., & Schukajlow, S. (2021). Create your own problem! Do students pose modelling problems that are based on given descriptions of real-world
situations? ZDM Mathematics Education, 53(4), 919–935. https://doi.org/10.1007/s11858-021-01224-7
Hartmann, L., Krawitz, J., & Schukajlow, S. (2022). The process of modelling-related problem posing - a case study with preservice teachers. In C. Fernandez,
S. Llinares, A. Gutierrez, & N. Planas (Eds.), In Proceedings of the forty fth conference of the international group for the psychology of mathematics education, 2 pp.
355–362). PME.
Hartmann, L., Krawitz, J., & Schukajlow, S. (2023). Posing and solving modelling problems—extending the modelling process from a problem posing perspective.
Journal Für Mathematikdidaktik, 44, 533–561.
Hidi, S., & Renninger, K. A. (2006). The four phase model of interest development. Educational Psychologist, 41(2), 111–127. https://doi.org/10.1207/
s15326985ep4102_4
Kline, R. B. (2005). Principles and practice of structural equation modeling. Guilford Press.
Krawitz, J., & Schukajlow, S. (2018). Do students value modelling problems, and are they condent they can solve such problems? Value and self-efcacy for
modelling, word, and intra-mathematical problems. ZDM Mathematics Education, 50(1), 143–157. https://doi.org/10.1007/s11858-017-0893-1
Krug, A., & Schukajlow, S. (2013). Problems with and without connection to reality and students’ task-specic interest. In A. M. Lindmeier & A. Heinze (Eds.), In
Proceedings of the thirty seventh conference of the international group for the psychology of mathematics education (Vol. 3, pp. 209–216).
Lazarus, R. S. (1991). Emotion and adaptation. Oxford.
Leikin, R., Christou, C., Karp, A., Pitta-Pantazi, D., & Zazkis, R. (2023). Mathematical challenges for all. Springer.
Liljedahl, P., & Cai, J. (2021). Empirical research on problem solving and problem posing: A Look at the state of the art. ZDM Mathematics Education, 53(4), 723–735.
https://doi.org/10.1007/s11858-021-01291-w
Liu, Q., Liu, J., Cai, J., & Zhang, Z. (2020). The relationship between domain- and task-specic self-efcacy and mathematical problem posing: A large-scale study of
eighth-grade students in China. Educational Studies in Mathematics, 105(3), 407–431. https://doi.org/10.1007/s10649-020-09977-w
Maaß, K. (2010). Classication scheme for modelling tasks. Journal Für Mathematik-Didaktik, 31(2), 285–311. https://doi.org/10.1007/s13138-010-0010-2
Marco, N., & Palatnik, A. (2022). Dimensions of variation in teachers’ applied mathematics problem posing. In C. Fernandez, S. Llinares, A. Gutierrez, & N. Planas
(Eds.), Proceedings of the forty fth Conference of the International Group for the Psychology of Mathematics Education (pp. 163–170). PME.
Mitchell, M. (1993). Situational interest: Its multifaceted structure in the secondary school mathematics classroom. Journal of Educational Psychology, 85(3), 424–436.
https://doi.org/10.1037/0022-0663.85.3.424
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
Niss, M., & Blum, W. (2020). The learning and teaching of mathematical modelling. Taylor & Francis Group.
Niss, M., & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102(1), 9–28. https://doi.org/10.1007/s10649-019-09903-
9
Palm, T. (2008). Impact of authenticity on sense making in word problem solving. Educational Studies in Mathematics, 67(1), 37–58. https://doi.org/10.1007/s10649-
007-9083-3
J. Krawitz et al.
Journal of Mathematical Behavior 73 (2024) 101129
12
Parhizgar, Z., & Liljedahl, P. (2019). In S. A. Chamberlin, & B. Sriraman (Eds.), Affect in mathematical modelingTeaching modelling problems and its effects on students’
engagement and attitude toward mathematics (pp. 235–256). Springer International Publishing. https://doi.org/10.1007/978-3-030-04432-9_15.
Rakoczy, K., Klieme, E., Bürgermeister, A., & Harks, B. (2008). The interplay between student evaluation and instruction. Zeitschrift Für Psychologie Journal of
Psychology, 216(2), 111–124. https://doi.org/10.1027/0044-3409.216.2.111
Reeve, J., Nix, G., & Hamm, D. (2003). Testing models of the experience of self-determination in intrinsic motivation and the conundrum of choice. Journal of
Educational Psychology, 95(2), 375–392. https://doi.org/10.1037/0022-0663.95.2.375
Rellensmann, J., & Schukajlow, S. (2017). Does students’ interest in a mathematical problem depend on the problem’s connection to reality? An analysis of students’
interest and pre-service teachers’ judgments of students’ interest in problems with and without a connection to reality. ZDM, 49(3), 367–378. https://doi.org/
10.1007/s11858-016-0819-3
Renninger, K. A., & Hidi, S. (2017). The power of interest for motivation and engagement. Routledge. https://doi.org/10.4324/9781315771045
Roth, B., Becker, N., Romeyke, S., Sch¨
afer, S., Domnick, F., & Spinath, F. M. (2015). Intelligence and school grades: A meta-analysis. Intelligence, 53, 118–137. https://
doi.org/10.1016/j.intell.2015.09.002
Schiefele, U., Krapp, A., & Winteler, A. (1992). Interest as a predictor of academic achievement: A meta-analysis of research. In K. A. Renninger, S. Hidi, & A. Krapp
(Eds.), The role of interest in learning and developement (pp. 183–196). Erlbaum.
Schraw, G., Bruning, R., & Svoboda, C. (1995). Sources of Situational Interest. Journal of Reading Behavior, 27(1), 1–17. https://doi.org/10.1080/
10862969509547866
Schukajlow, S., Krawitz, J., Kanefke, J., & Rakoczy, K. (2022). Interest and performance in solving open modelling problems and closed real-world problems. In
C. Fern´
andez, S. Llinares, & A. Guti´
errez (Eds.), Proceedings of theforty fth conference of the international group for the psychology of mathematics education, 3 pp.
403–410). PME.
Schukajlow, S., & Krug, A. (2014). Do multiple solutions matter? Prompting multiple solutions, interest, competence, and autonomy. Journal for Research in
Mathematics Education, 45(4), 497–533. https://doi.org/10.5951/jresematheduc.45.4.0497
Schukajlow, S., Leiss, D., Pekrun, R., Blum, W., Müller, M., & Messner, R. (2012). Teaching methods for modelling problems and students’ task-specic enjoyment,
value, interest and self-efcacy expectations. Educational Studies in Mathematics, 79(2), 215–237. https://doi.org/10.1007/s10649-011-9341-2
Schukajlow, S., & Rakoczy, K. (2016). The power of emotions: Can enjoyment and boredom explain the impact of individual preconditions and teaching methods on
interest and performance in mathematics. Learning and Instruction, 44, 117–127. https://doi.org/10.1016/j.learninstruc.2016.05.001
Silber, S., & Cai, J. (2017). Pre-service teachers’ free and structured mathematical problem posing. International Journal of Mathematical Education in Science and
Technology, 48(2), 163–184. https://doi.org/10.1080/0020739X.2016.1232843
Silver, E. A. (1994). On mathematical problem posing. For the Learning of Mathematics, 14(1), 19–28.
Silver, E. A. (1995). The nature and use of open problems in mathematics education: Mathematical and pedagogical perspectives. ZDM The International Journal on
Mathematics Education, 27(2), 67–72.
Silver, E. A. (2013). Problem-posing research in mathematics education: Looking back, looking around, and looking ahead. Educational Studies in Mathematics, 83(1),
157–162.
Silver, E. A., & Cai, J. (1996). An analysis of arithmetic problem posing by middle school students. Journal for Research in Mathematics Education, 27. https://doi.org/
10.2307/749846
Silvia, P. J. (2005). What is interesting? Exploring the appraisal structure of interest. Emotion, 5(1), 89–102. https://doi.org/10.1037/1528-3542.5.1.89
Silvia, P. J. (2008). Interest: The curious emotion. Current Directions in Psychological Science, 17(1), 57–60. 〈http://www.jstor.org/stable/20183249〉.
Stillman, G., & Galbraith, P. (2003). Towards constructing a measure of the complexity of application tasks. In S. J. Lamon, W. A. Parker, & K. Houston (Eds.),
Mathematical modelling (pp. 179–188). Woodhead Publishing. https://doi.org/10.1533/9780857099549.4.179.
Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’ problem posing. In C. P (Ed.), Technology in mathematics education (pp. 518–525).
Mathematics Education Research Group of Australasia.
Verschaffel, L., Greer, B., & Corte, E. de (2000). Contexts of learning. Making sense of word problems. Lisse, The Netherlands: Swets & Zeitlinger.
Verschaffel, L., Schukajlow, S., Star, J., & van Dooren, W. (2020). Word problems in mathematics education: A survey. ZDM, 52(1), 1–16. https://doi.org/10.1007/
s11858-020-01130-4
Wade, S. E., Buxton, W. M., & Kelly, M. (1999). Using Think-Alouds to Examine Reader-Text Interest. Reading Research Quarterly, 34(2), 194–216.
Walkington, C. (2017). Design research on personalized problem posing in algebra. In E. Galindo, & J. Newton (Eds.), Proceedings of the thirty ninth annual meeting of
the North American Chapter of the International Group for the Psychology of Mathematics Education. Hoosier Association of Mathematics Teacher Educators.
Williams, G., & Clarke, D. J. (1997). The complexity of mathematics tasks. In N. Scott, & H. Hollingsworth (Eds.), Mathematics: Creating the future (pp. 451–457).
AAMT.
Yeo, J. B. W. (2017). Development of a framework to characterise the openness of mathematical tasks. International Journal of Science and Mathematics Education, 15
(1), 175–191. https://doi.org/10.1007/s10763-015-9675-9
J. Krawitz et al.
