A descriptive model of problem-solving processes

Vol.:(0123456789)
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ZDM – Mathematics Education (2021) 53:737–752
https://doi.org/10.1007/s11858-021-01244-3
ORIGINAL PAPER
A descriptive phase model of problem‑solving processes
Benjamin Rott1
· Birte Specht2
· Christine Knipping3
Accepted: 18 February 2021 / Published online: 9 March 2021
© The Author(s) 2021
Abstract
Complementary to existing normative models, in this paper we suggest a descriptive phase model of problem solving. Real,
not ideal, problem-solving processes contain errors, detours, and cycles, and they do not follow a predetermined sequence, as
is presumed in normative models. To represent and emphasize the non-linearity of empirical processes, a descriptive model
seemed essential. The juxtaposition of models from the literature and our empirical analyses enabled us to generate such a
descriptive model of problem-solving processes. For the generation of our model, we reflected on the following questions:
(1) Which elements of existing models for problem-solving processes can be used for a descriptive model? (2) Can the model
be used to describe and discriminate different types of processes? Our descriptive model allows one not only to capture the
idiosyncratic sequencing of real problem-solving processes, but simultaneously to compare different processes, by means
of accumulation. In particular, our model allows discrimination between problem-solving and routine processes. Also,
successful and unsuccessful problem-solving processes as well as processes in paper-and-pencil versus dynamic-geometry
environments can be characterised and compared with our model.
Keywords Mathematical problem solving · Geometry · Descriptive process model
1 Introduction
Problem solving (PS)—in the sense of working on non-
routine tasks for which the solver knows no previously
learned scheme or algorithm designed to solve them (cf.
Schoenfeld, 1985, 1992b)—is an important aspect of doing
mathematics (Halmos, 1980) as well as learning and teach-
ing mathematics (Liljedahl et al. 2016). As one of several
reasons, PS is used as a means to help students learn how
to think mathematically (Schoenfeld, 1992b). Hence, PS is
part of mathematics curricula in almost all countries (e.g.,
KMK, 2004; NCTM, 2000, 2014). Accordingly, PS has been
a focus of interest of researchers for several decades, Pólya
(1945) being one of the most prominent scholars interested
in this activity.
Problem-solving processes (PS processes) can be charac-
terised by their inner or their outer structure (Philipp, 2013,
pp. 39–40). The inner structure refers to (meta)cognitive
processes such as heuristics, checks, or beliefs, whereas
the outer structure refers to observable actions that can be
characterised in phases like ‘understanding the problem’ or
‘devising a plan’, as well as the chronological sequence of
such phases in a PS process. Our focus in this paper is on
the outer structure, as it is directly accessible to teachers and
researchers via observation.
In the research literature, there are various characterisa-
tions of PS processes. However, almost all of the existing
models are normative, which means they represent idealised
processes. They characterise PS processes according to dis-
tinct phases, in a predetermined sequence, which is why they
are sometimes called ‘prescriptive’ instead of normative.
These phases and their sequencing have been formulated as a
norm for PS processes. Normative models are generally used
as a pedagogical tool to guide students’ PS processes and to
help them to become better problem solvers. The normative
models in current research have mostly been derived from
theoretical considerations. Nevertheless, real PS processes
look different; they contain errors, detours, and cycles, and
they do not follow a predetermined sequence. Actual pro-
cesses like these are not considered in normative models.
Accordingly, there are almost no models that guide teachers
* Benjamin Rott
brott@uni-koeln.de
1
Universität zu Köln, Köln, Germany
2
Universität Oldenburg, Oldenburg, Germany
3
Universität Bremen, Bremen, Germany

738 B. Rott et al.
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and researchers in observing, understanding, and analysing
PS processes in their ‘non-smooth’ occurrences (cf. Fernan-
dez et al. 1994; Rott, 2014). Our aim in this paper, therefore,
is to address this research gap by suggesting a descriptive
model.
A descriptive model enables not only the representation
of real PS processes, but also reveals additional potential for
analyses. Our model allows one systematically to compare
several PS processes simultaneously by means of accumu-
lation, which is an approach that to our knowledge has not
been proposed before in the mathematics education com-
munity. In Sect. 6, we show how this approach can be used
to reveal ‘bumps and bruises’ of real students’ PS processes
to illustrate the practical value of our descriptive model
(Sect. 5.3, Fig. 5). We show how our model allows one to
discriminate problem-solving processes from routine pro-
cesses when students work on tasks. We illustrate how dif-
ferences between successful and unsuccessful processes can
be identified using our model. We also reveal how students’
PS processes, working in a paper-and-pencil environment
compared to working in a digital (dynamic geometry) envi-
ronment, can be characterised and compared by means of
our model.
Our descriptive model is based on intertwining theoreti-
cal considerations, in the form of a review of existing mod-
els, as well as on a video-study researching the processes
shown by mathematics pre-service teachers working on geo-
metrical problems.
2 Theoretical background
In this section, we first describe and compare aspects of
existing models of PS processes (which are mostly norma-
tive) to characterize their potential and their limitations for
analysing students’ PS processes (2.1). We then discuss why
looking specifically at students’ PS processes in geometry
and in dynamic geometry contexts is of particular value for
developing a descriptive model of PS processes (2.2).
2.1 Models of problem‑solving processes
Looking at models from mathematics, mathematics educa-
tion, and psychology that describe the progression of PS
processes, we find phase models, evolved by authors observ-
ing their own PS processes or those of people with whom
the authors are familiar. So, the vast majority of existing
PS process models are not based on ‘uninvolved’ empirical
data (e.g., videotaped PS processes of students); they were
actually not designed for the analysis of empirical data or to
describe externally observed processes, which emphasises
the need for a descriptive model.
2.1.1 Classic models of problem‑solving processes
Two ‘basic types’ of phase models for PS processes have
evolved in psychology and mathematics education. Any fur-
ther models can be assigned to one or the other of these basic
types: (1) the intuitive or creative type and (2) the logical
type (Neuhaus, 2002).
(1) Intuitive or creative models of PS processes originate
in Poincaré’s (1908) introspective reflection on his own
PS processes. Building on his thoughts, the mathema-
tician Hadamard (1945) and the psychologist Wallas
(1926) described PS processes with a particular focus
on subconscious activities. Their ideas are most often
summarised in a four-phase model: (i) After working
on a difficult problem for some time and not finding
a solution (preparation), (ii) the problem solver does
and thinks of different things (incubation). (iii) After
some more time—hours, or even weeks—suddenly, a
genius idea appears (illumination), providing a solution
or at least a significant step towards a solution of the
problem; (iv) this idea has to be checked for correctness
(verification).
(2) So-called logical models of PS processes were intro-
duced by Dewey (1910), describing five phases: (i)
encountering a problem (suggestions), (ii) specify-
ing the nature of the problem (intellectualization),
(iii) approaching possible solutions (the guiding idea
and hypothesis), (iv) developing logical consequences
of the approach (reasoning (in the narrower sense)),
and (v) accepting or rejecting the idea by experiments
(testing the hypothesis by action). Unlike in Wallas’
model, there are no subconscious activities described
in Dewey’s model. Pólya’s (1945) famous four-phase
model—(i) understanding the problem, (ii) devising
a plan, (iii) carrying out the plan, and (iv) looking
back—manifests, according to Neuhaus (2002), refer-
ences to Dewey’s work.
Research in mathematics education mainly focuses on
logical models for describing PS processes, following Pólya
or more recent variants of his model (see below). This is due
to the fact that PS processes of the intuitive or creative type
might take hours, days, or even weeks to allow for genuine
incubation phases, and PS activities in the context of school-
ing and university teaching are mostly shorter and more
contained. Therefore, we focus on logical models. In the
following, we compare prominent PS process phase models
that emerged in the last decades (see Fig. 1).

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2.1.2 Recent models of problem‑solving processes
In Fig. 1, different models are presented (for more details
see the appendix). These build on and alter distinct aspects
of Pólya’s model, especially envisioned phases and possible
transitions between these phases. They mark this distinction
by using different terminology for these nuanced differences
in the phases. The models by Mason et al. (1982), Schoen-
feld (1985, Chapter 4), and Wilson et al. (1993; Fernandez
et al. 1994) are normative; they are mostly used for teach-
ing purposes, that is, to instruct students in becoming better
problem solvers. Compared to actual PS processes, these
models comprise simplifications; looking at and analysing
students’ PS processes requires models which are suited to
portray these uneven and cragged processes.
In several studies, actual PS processes are analysed; how-
ever, only a few of these studies use any of these normative
models that describe the outer structure of PS processes.
Even fewer studies present a descriptive model as part of
their results. Some of the rare studies that attempt to derive
such a model are presented in more detail in the appendix;
their essential ideas are presented below (Artzt & Armour-
Thomas, 1992; Jacinto & Carreira, 2017; Yimer & Ellerton,
2010).
2.1.3 Comparing models of problem‑solving processes
In this section, we compare the previously mentioned as
well as additional phase models with foci on (a) the differ-
ent types of phases and (b) linearity or non-linearity of the
portrayed PS processes. Figure 1 illustrates similarities and
differences in these models, starting with those of Dewey
(1910) and Pólya (1945) as these authors were the first to
suggest such models. Schoenfeld (1985) and Mason et al.
(1982) introduced this discussion to the mathematics educa-
tion community, referring back to ideas of Pólya. Then, we
discuss those of Wilson et al. (1993), and Yimer and Ellerton
(2010), as examples of more recent models in mathematics
education.
(a) Different types of phases
The presented models comprise three, four, or more dif-
ferent phases. However, we do not think that this number is
important per se; instead, it is interesting to see which activi-
ties are encompassed in these phases of the different models
and in the extent and manner in which they follow Pólya’s
formulation, adopt it, or go beyond his ideas. In Fig. 1, we
indicated Pólya’s phases with differently patterned layers in
the background.
Dewey’s (1910) model starts with a phase (named “sug-
gestions”) in which the problem solvers come into contact
with a problem without already analysing or working on it.
Such a phase is seldom found in phase models in the con-
text of mathematics education. In mathematics though this
phase at the beginning is typical and important, as Dewey
already pointed out. In the context of teaching, on the other
hand, PS mostly starts with a task handed to the students
by their teachers. Analysing and working on the problem is
expected right from the beginning; this is part of the nature
of the provided task. So, in educational research the phase
of “suggestions” is rarely mentioned, as it normally does not
occur in students’ PS processes.
“Understanding the problem”, Pólya’s (1945) first phase,
is comparable to the second phase (“intellectualization”) of
Dewey’s model. In this phase, problem solvers are meant to
make sense of the given problem and its conditions. Such a
phase is used in all models, though often labelled slightly
differently (see Fig. 1 for a juxtaposition). Artzt and Armour-
Thomas, (1992) facing the empirical data of their study, dif-
ferentiated this phase of “understanding the problem” into
a first step, where students are meant to apprehend the task
(“understanding”), and a second step, where students are
actually expected to comprehend the problem (“analys-
ing”); a similar differentiation is presented by Jacinto and
Fig. 1 Different phase models of problem-solving processes

740 B. Rott et al.
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Carreira (2017) into “grasping, noticing” and “interpreting”
a problem.
The next two phases incorporate the actual work on the
problem. Pólya describes these phases as “devising” and
“carrying out a plan”. Especially the planning phase encom-
passes many different activities, such as looking for simi-
lar problems or generalizations. These two phases are also
integral parts of the models by Wilson et al. (1993), and
Yimer and Ellerton (2010) (see Fig. 1), or Jacinto and Car-
reira (2017, there called “plan” and “create”). Mason et al.
(1982) chose to combine both phases, calling this combined
phase “attack”. According to their educational and research
experience, they noted that both phases cannot be distin-
guished in most cases; therefore, a differentiation would
not be helpful for learning PS and describing PS processes.
Schoenfeld (1985), on the contrary, further differentiated
those phases by splitting Pólya’s second phase into a struc-
tured “planning” (or “design”) phase and an unstructured
“exploration” phase. When “planning”, one might adopt a
known procedure or try a combination of known procedures
in a new problem context. However, when known procedures
do not help, working heuristically (e.g., looking at exam-
ples, counter-examples, or extreme cases) might be a way to
approach the given problem in “exploration” (Schoenfeld,
1985, p. 106). According to Schoenfeld, exploration is the
“heuristic heart” of PS processes.
The last phase in Pólya’s model is “looking back”,
the moment when a solution should be checked, other
approaches should be explored, and methods used should
be reflected upon. This phase is also present in other mod-
els (see Fig. 1). In their empirical approach, Yimer and
Ellerton (2010), for example, differentiated this phase into
two steps, namely, “evaluation” (i.e., checking the results),
which refers to looking back on the recently solved problem,
and “internalization” (i.e., reflecting the solution and the
methods used), which focuses on what has been learnt by
solving this problem and looks forward to using this recent
experience for solving future problems. Jacinto and Carreira
(2017) used the same “verifying” phase as Pólya, but added
a “disseminating” phase for presenting solutions, as their
final phase.
Other researchers (see the appendix) came to insights
similar to those of these researchers, using slightly different
terminologies when describing these phases or combinations
of these phases.
(b) Sequence of phases: linear or non-linear problem-
solving processes
Other important aspects are transitions from one phase
to another, and how such transitions occur. The graphical
representations of different models in Fig. 1 not only indi-
cate slightly different phases (and distinct labels for these
phases), but also illustrate different understandings of how
these phases are related and sequenced.
There are strictly linear models like Pólya’s (1945), which
outline four phases that should be passed through when solv-
ing a problem, in the given order. Of course, Pólya as a math-
ematician knew that PS processes are not always linear; in
his normative model, however, he proposed such a stepwise
procedure, which has often been criticised (cf. Wilson et al.
1993). Mason et al. (1982) and Schoenfeld (1985) discarded
this strict linearity, including forward and backward steps
between analysing, planning, and exploring (or attacking,
respectively) a problem. Thereafter, PS processes linearly
proceed towards the looking back equivalents of their mod-
els. Wilson et al. (1993) presented a fully “dynamic, cyclic
interpretation of Polya’s stages” (p. 60) and included forward
and backward steps between all phases, even after “looking
back”. The same is true for Yimer and Ellerton (2010), who
included transitions between all phases in their model.
As we illustrate later, transitions from one phase to
another reflect also characteristic features of routine and
non-routine processes in general, and can be also distinctive
for students’ PS processes in traditional paper-and-pencil
environments compared to Dynamic Geometry Software
(DGS) contexts. Our descriptive model of PS processes,
which we present in Sect. 5, also evolved by comparative
analyses of students’ PS processes in both learning contexts.
Thus, we comment briefly in 2.2 on what existing research
has found in this respect so far.
2.2 Problem solving in geometry and dynamic
geometry software
Overall, geometry is especially suited for learning math-
ematical PS in general and PS strategies or heuristics in
particular (see Schoenfeld, 1985). Notably, many geometric
problems can be illustrated in models, sketches, and draw-
ings, or can be solved looking at special cases or working
backwards (ibid.). Additionally, the objects of action (at least
in Euclidean geometry) and the permitted actions (e.g., con-
structions with compasses and ruler) are easy to understand.
Therefore, in our empirical study (see Sect. 4), we opted
for PS processes in geometry contexts, knowing that other
contexts could be equally fruitful.
One particular tool to support learning and working in
the context of geometry, since the 1980s, is DGS, which is
characterised by three features, namely, dragmode, macro-
constructions, and locus of points (Sträßer, 2002). With
these features, DGS can be used not only for verification
purposes, but also for guided discoveries as well as working
heuristically (e.g., Jacinto & Carreira, 2017). However, as
Gawlick (2002) pointed out, to profit from such an environ-
ment, students—especially low achievers—need some time
to get accustomed to handling the software. Comparing DGS

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and paper-and-pencil environments, Koyuncu et al. (2015)
observed that in a study with two pre-service teachers, “[b]
oth participants had a tendency toward using algebraic solu-
tions in the [paper-and-pencil based] environment, whereas
they used geometric solutions in the [DGS based] environ-
ment.” (p. 857 f.). These potential differences between PS
processes in paper-and-pencil versus DGS environments are
interesting for research and practice. Therefore, we com-
pared students’ PS processes in these two environments in
our empirical study.
3 Research questions
With regard to research on PS processes, it is striking that
there is only a small number of studies, often with a low
number of participants, that present and apply a descriptive
model of PS processes. Further, the identified models are
not suited for comparing PS processes across groups of stu-
dents, but can only describe cases. Last but not least, in most
empirical studies, the selection of phases that are included,
and the assumption of (non-)linearity, are not discussed
and/or justified. In all these respects, we see a research gap.
Contributing to filling this gap was one of the motivations
for the study presented here. Based on the existing research
literature, we formulated two main research questions:
(1) What elements of the already discussed PS process
models can be used for a descriptive model? In particu-
lar, what is necessary so that such a descriptive model
enables
• a recognition of types of phases and an identification
of phases in actual PS processes as well as
• an identification of the sequence (i.e., the order, lin-
ear or non-linear) of phases and transitions between
phases?
(2) Can the model be used to describe and discriminate
among different types of PS processes, for example
• routine and non-routine processes,
• successful and not successful processes, or
• paper-and-pencil vs. DGS processes?
These questions guided our study and the motivation for
developing a descriptive model of PS processes. Next, we
present the methodology, before we discuss results of our
empirical study and present our model.
4 Methodology
In a previous empirical study, we looked at PS processes
of pre-service teacher students in geometry contexts. The
data in this study were enormously rich and challenged us
in their analyses in many ways. Existing PS models did
not allow us to harvest fully this rich data corpus and we
realised that with respect to our empirical data, we needed
a descriptive model. So we formulated the research ques-
tions listed above in order to explore the potential and
necessary extensions of the existing normative PS pro-
cess models. We changed our perspective and focused on
the development of an empirically grounded theoretical
model. We required an approach that would allow us to
mine the data of our empirical study and to provide a con-
ceptualisation that could be helpful for further research on
students’ problem-solving processes. The methodological
approach we used is described in the following.
4.1 Our empirical study
About 250 pre-service teacher students attended a course
on Elementary Geometry, which was conceived and con-
ducted by the third author at a university in Northern
Germany. The course lasted for one semester (14 weeks);
each week, a two-hour lecture for all students as well as
eight 2-h tutorials for up to 30 students each, supervised
by tutors (advanced students), took place. Four tutorials
(U1, Ulap2, U3, and Ulap4) were involved in this study:
in U1 and U3 the students worked in a paper-and-pencil
environment, in Ulap2 and Ulap4 the students used laptop
computers to work in a DGS environment. (The abbrevia-
tions consist of U, the first letter ‘Uebung’, German for
tutorial, with an added ‘lap’ for groups which used lap-
top computers as well as an individual number.) Students
worked on weekly exercises, which were discussed in the
tutorials. In addition, over the course of the semester, in
groups of three or four, the students worked on five geo-
metric problems in the tutorials (approx. 45 min for each
problem), accompanied by as little tutor help as possible.
In this paper, we focus on these five problems. See the
appendix for additional information regarding the organi-
sation of our study.
The five problems were chosen so that students had the
opportunity to solve a variety of non-routine tasks, which
at the same time did not require too much advanced knowl-
edge that students might not have.
For each of the five problems, two groups from each
of the four tutorials were observed. Each problem was
therefore worked on by four groups with and four groups
without DGS (minus some data loss because of students

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missing tutorials or technical difficulties). The collected
data were videos of the groups working on these problems
(processes), notes by the students (products), as well as
observers’ notes. Overall, 33 processes (15 from paper-
and-pencil as well as 18 from DGS groups) from all five
problems, with a combined duration of 25 h, were ana-
lysed. For space reasons, we cannot discuss all five prob-
lems in detail here. Instead, we present three of the five
problems here; the other two can be found in the appendix.
4.1.1 The problems
Regarding the ‘Shortest Detour’ (Fig. 2, top), as long as A
and B are on different sides of the straight line, a line seg-
ment from A to B is the shortest way. When A and B are on
the same side of g, an easy (not the only) way to solve this
problem is by reflecting one of the points, e.g. A, on g and
then constructing the line segment from the reflection of A
to B, as reflections preserve lengths.
Part a) of the ‘Three Beaches’ problem (Fig. 2, bottom),
finding the incircle of an equilateral triangle, should be a
routine-procedure as this topic had been discussed in the
lecture. Students working on part b) of this problem needed
to realize that in an equilateral triangle, all points have the
same sum of distances to the sides (Viviani’s problem). This
could be justified by showing that the three perpendiculars
of a point to a side in such a triangle add up to the height
of this triangle, for example by geometrical addition or by
calculating areas.
Like Problem (4), Problem (3) (Fig. 2, middle) contained
an a)-part which is a routine task—finding the circumcircle
of a (non-regular) triangle—and a b)-part that constitutes a
problem for the students.
These tasks were chosen because they actually repre-
sented problems for our students, and expected PS processes
appeared neither too long nor too short for a reasonable
workload by students and for our analyses. Further, the prob-
lems covered the content of the accompanying lecture, and
the problems could be solved both with and without DGS.
Differences between working with and without DGS: With
DGS many examples can be generated quickly, so that an
overview of the situation and the solution can be obtained
in a short time. For the justifications, however, with and
without DGS, students had to reflect, think, and reason to
find appropriate arguments.
4.2 Framework for the analysis of the empirical
data
For the analyses of our students’ PS processes, we used
the protocol analysis framework by Schoenfeld (1985,
Fig. 2 Three of the five prob-
lems used in our study

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Chapter 9) with adaptations and operationalizations by
Rott (2014), following two phases of coding.
Process coding: With his framework, Schoenfeld
(1985) intended to “identify major turning points in a
solution. This is done by parsing a [PS process] into mac-
roscopic chunks called episodes” (p. 314). An episode is
“a period of time during which an individual or a problem-
solving group is engaged in one large task […] or a closely
related body of tasks in the service of the same goal […]”
(p. 292). Please note, the term “episode” refers to coded
process data, whereas “phase” refers to parts of PS mod-
els. Schoenfeld (p. 296) continued: “Once a protocol has
been parsed into episodes, each episode is characterized”
using one of six categories (see also Schoenfeld, 1992a,
p. 189):
(1) Reading or rereading the problem.
(2) Analysing the problem (in a coherent and structured
way).
(3) Exploring aspects of the problem (in a much less struc-
tured way than in Analysis).
(4) Planning all or part of a solution.
(5) Implementing a plan.
(6) Verifying a solution.
According to Schoenfeld (1985), Planning-Implementa-
tion can be coded simultaneously.
The idea of episodes as macroscopic chunks implies a
certain length, thus individual statements do not comprise
an episode; for example, quickly checking an interim result
is not coded as a verification episode. Also, PS processes
are coded by watching videos, not by reading transcripts
(Schoenfeld, 1992a).
Schoenfeld’s framework was chosen to answer our first
research question, for two reasons. (i) The episode types he
proposed cover a lot of the variability of phases also identi-
fied by us (see Sect. 2.1.3). (ii) Coding episodes and coding
episode types in independent steps offers the possibility of
adding inductively new types of episodes.
After parsing a PS process into episodes, we coded the
episodes with Schoenfeld’s categories (deductive catego-
ries), but also generated new episode types to characterize
these episodes (inductive categories). While coding, we
observed initial difficulties in coding the deductive epi-
sodes reliably; especially differentiating between Analysis
and Exploration episode types was difficult (as predicted by
Schoenfeld, 1992a, p. 194). We noticed that Schoenfeld’s
(1985, Chapter 9) empirical framework referred to his theo-
retical model of PS processes (ibid., Chapter 4) which was
based on Pólya’s (1945) list of questions and guidelines.
Recognizing an analogy between Schoenfeld’s framework
and Pólya’s work (see Fig. 1), we were able to operational-
ize their descriptions in a coding manual (see Rott, 2014).
When the deductive episode types did not fit our
observations, we inductively added a new episode type.
This happened three times. Especially in the DGS envi-
ronment, where students showed behaviour that was not
directly related to solving the task, new types of activities
occurred. For example, students talked about the software
and how to use it. This kind of behaviour was coded by
us as Organization. When it took students more than 30 s
to write down their findings (without developing any new
results or ideas), this episode was coded as Writing. Dis-
cussions about things which were not related to mathemat-
ics, but for example daily life, were coded as Digression.
These codings were used only when activities did not align
with numbers 2–6 of Schoenfeld’s list.
This coding of the videotapes was done independently
by different research assistants and the first author. We
then applied the “percentage of agreement” (PA) approach
to compute the interrater-agreement as described in the
TIMSS 1999 video study (Jacobs et al. 2003, pp. 99–105),
gaining more than PA = 0.7 for parsing PS processes into
episodes and more than PA = 0.85 for characterizing the
episode types. More importantly, every process was coded
by at least two raters. Whenever those codes did not coin-
cide, we attained agreement by recoding together (as in
Schoenfeld’s study, 1992a, p. 194).
Product coding: To be able to compare successful and
unsuccessful PS processes, students’ products produced
in the 45-min sessions were rated. Because the focus
was on processes, product rating finally was reduced to
a dichotomous right/wrong coding without going into
detail regarding students’ argumentations (these will be
analysed and the results reported in forthcoming papers).
Rating was done independently by a research assistant and
the first author with an interrater-agreement of Cohen’s
kappa > 0.9. Differing cases were discussed and recoded
consensually.
5 Results of our empirical study
and implications for our descriptive
model
In this section, we briefly illustrate results of our data
analyses, which underline the need to go beyond existing
models. We summarize key findings of our empirical study
and illustrate how these have contributed to the develop-
ment of our descriptive model of PS processes. After this,
we highlight how answering our research questions based
on our theoretical and empirical analyses contributes to the
development of our descriptive model. Finally, we present
and describe our descriptive model.

744 B. Rott et al.
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5.1 Sample problem‑solving processes and codings
to illustrate the procedure of analysis
To illustrate our analyses and codings of students’ PS pro-
cesses, we present three sample processes, the first two
in detail and the third one only briefly. The first two were
paper-and-pencil processes and stem from the same group of
students, belonging to parts a) and b) of the ‘Three Beaches’
problem. The third process shows a group of students work-
ing on the ‘Shortest Detour’ problem with DGS. Our cod-
ings of the different episodes are highlighted in italics.
5.1.1 Group U1‑C, Three Beaches (part a))
After reading the Three Beaches problem (00:25–01:30), the
three students of group C from tutorial U1 try to understand
it. They remember the Airport problem in which they had
to find a point with the same distance to all three vertices
of a triangle and they try to identify the differences between
both problems. The students wonder whether they should
again use the perpendicular bisectors of the sides of the tri-
angle or the bisectors of the angles of the triangle (Analysis,
01:30–05:05). They agree to use the bisectors of the angles
and construct their solutions with compasses and ruler. One
of the students claims that in the case of an equilateral tri-
angle, perpendicular and angle bisectors would be identical
and convinces the others by constructing a triangle and both
bisectors with compasses and ruler (Planning-Implementa-
tion, 05:05–06:05). Finally, the students verify their solution
by discussing the meaning of the distance from a point to
the sides of a triangle, as they initially were not sure how
to measure this distance (06:05–07:40). Even though the
Analysis episode was quite long (see Fig. 3), this part of
the task was actually not a problem for the students as they
remembered a way to solve it.
5.1.2 Group U1‑C, three beaches (part b))
After reading part b) of the problem (07:45–07:55), the stu-
dents discuss whether the requested point is the same as in a)
(Analysis, 07:55–10:25). They agree to try out and construct
a triangle each, place points in it, draw perpendiculars to the
sides, and measure the distances. One student asks whether
it is allowed to place the point on a vertex and thus have two
distances become zero (Exploration, 10:25–15:30). After
this, the students discuss the meaning of distance, particu-
larly the meaning of a distance related to a side of a triangle.
They agree that any point on a side, even the vertex, would
satisfy the condition of the problem, thus being a suitable
site for the ‘house’ (Analysis, 15:30–16:40). The students
wonder why the distance from one vertex to its opposing
side (the height of the triangle) is as large as the sum of the
distances from the centre of the incircle (from part a)). They
remember that the angle bisectors intersect each other in a
ratio of 1/3 to 2/3. Thereafter, they continue to place points
in their triangles (not on sides) and measure their distances.
They finally agree on the [wrong] hypothesis that any point
on the angle bisectors is a point with a minimal sum of the
distances to the sides; other points in the triangle would
have a slightly larger sum [because of inaccuracies in their
drawings]. They realize, however, that they cannot give any
reasons for their solution (Exploration, 16:40–32:30). The
codings are represented in Fig. 3 (right).
5.1.3 Group Ulap2‑TV, shortest detour
Ulap2-TV working on the shortest detour problem (Fig. 4) is
an example of a process with more transitions. The students
solve the first case of the problem (A and B on different sides
of g) within 5 min (Planning-Implementation, Verification)
and then explore the second case (A and B on the same side
of g) for more than 17 min before solving the problem.
We selected these three PS processes from our study, as
they are examples of our empirical data in several aspects:
They illustrate both learning environments (paper-and-pencil
and DGS), they incorporate all types of episodes (except
for Digression) and, therefore, all types of phases discussed
in the PS research literature, and they include linear and
cyclic progressions (see below). The routine process (Three
Beaches, part a)) is rather atypical as the students take a lot
Fig. 3 Process codings of the group U1-C, working on the ‘Three
Beaches’ problem
Fig. 4 Process coding of the group Ulap2-TV, working on the ‘Short-
est Detour’ problem

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of time analysing the task, before implementing routine tech-
niques (Planning-Implementation). The two PS processes
(Three Beaches, part b) and Shortest Detour) are typical for
our students, spending a lot of time in Exploration episodes.
In the DGS environment, we see that the students take some
time to handle the software (Organization). Compared to
free-hand drawings in paper-and-pencil environments, the
students in the DGS environment need to think about con-
structions (Planning) before exploring the situation.
5.2 From theoretical models and empirical results
to a descriptive model of problem‑solving
processes
In the following, the coded episodes from all 33 PS pro-
cesses of our empirical study are used to answer the first
research question. What parts or phases of the established
models are suited to describe the analysed processes? Which
transitions between phases can be observed? The systematic
comparison of PS models from the literature (Sect. 2.1.3) is
the theoretical underpinning of answering these questions.
This process aims at generating a descriptive process model
suitable for representing students’ actual PS processes.
5.2.1 Different types of episodes that are suited
to describing empirical processes
Within the observed processes, all of Schoenfeld’s episode
types could be identified with high interrater agreement.
Thus, based on our data, we saw no need to merge phases
like Understanding and Planning, even though some models
suggest doing so.
More specifically, structured approaches of Planning
could be differentiated from unstructured approaches which
we call Explorations as suggested by Schoenfeld (1985,
Chapters 4 & 9) (in 6 out of 33 non-routine processes, both
Exploration and Planning were coded).
Furthermore, in some processes, Planning and Imple-
mentation episodes can be differentiated from each other
(as suggested by Pólya, 1945); there are, however, processes
in which those two episode types cannot be distinguished as
the problem solvers often do not announce their plans (as
predicted by Mason et al. 1982). In those PS processes, these
two episode types are merged to Planning-Implementation
(as done by Schoenfeld as well).
Verification episodes are rare, but can be found in our
data. As our students do not show signs of trying to reflect
on their use of PS strategies, we decided not to distinguish
this episode type into ‘checking’ and ‘reflection’.
Incubation and illumination could not be observed in our
sample. This was expected as the students did not have the
time to incubate.
Altogether, the following theoretically recorded phases
could be identified in our empirical data and are part of our
model: understanding (analysis), exploration, planning,
implementation (sometimes as planning-implementation),
and verification.
5.2.2 Transitions between phases: linearity
and non‑linearity of the processes
Apart from the phases that occur, the transitions between
these phases are of interest. Transitions have been coded
between nearly all possible ordered pairs of episode types.
If the phases proceed according to Pólya’s or Schoenfeld’s
model (Analysis → Exploration → Planning → Implemen-
tation → Verification), we consider this as a linear process.
If phases are omitted within a process but this order is still
intact we regard this process still as ‘linear’. In contrast,
a process is considered by us as non-linear or cyclic, if
this order is violated (e.g., Planning → Exploration). We
also checked whether non-linear processes are cyclic in
the sense of Wilson et al. (backward steps are possible
after all types of episodes), or whether they are cyclic in
the sense of Schoenfeld and Mason et al. (backward steps
only before Implementation).
The first sample process (Three Beaches, part a) illus-
trates a strictly linear approach as in Pólya’s model, rep-
resented in the descending order of the time bars (Fig. 3,
left). The second example (Three Beaches, part b) shows
a cyclic process as after the first Exploration, an Analy-
sis was coded (Fig. 3, right). The third example (Shortest
Detour) starts in a linear way; then, after a first Verifica-
tion, the students go back to Planning-Implementation and
Exploration episodes. Thus, overall, their process is cyclic
(and not in a way that would fit Schoenfeld’s model as the
linear order is broken after a Verification).
We checked all our process codings for their order of
episodes (see Table 1). In our sample, a third of the pro-
cesses are non-linear; thus, a strictly linear model is not
suited to describing our students’ PS processes.
Table 1 Linearity and non-linearity of the coded processes
Linear Non-linear (Non-linear before
Implementation)
Sum
Paper-and-pencil 12 3 (3) 15
DGS 10 8 (1) 18
Sum 22 11 (4) 33

746 B. Rott et al.
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5.3 Deriving a model for describing
problem‑solving processes
Using the results of our empirical study as described
in Sects. 5.1 and 5.2, our findings result in a descrip-
tive model of PS processes. We consider this model as
an answer to our first research question. We identified
phases from (mostly normative) models in our data, then
empirically refined these phases, and took the relevance
of their sequencing into account as illustrated in Fig. 5.
In our descriptive model (see Fig. 5), we distinguish
between structured (Planning) and unstructured (Explora-
tion) approaches in accordance with the model of Schoen-
feld (1985). It is also possible to differentiate between
explicit planning (Planning and Implementation coded
separately) as well as implicit planning, which means
(further) developing a plan while executing it (Plan-
ning and Implementation coded jointly), as suggested
by Mason et al. (1982). Our descriptive model combines
ideas from different models in the literature. Furthermore,
linear processes can be displayed (using only arrows that
point downwards in the direction of the solution) as can
non-linear processes (using at least one arrow that points
upwards). Therefore, with this model, linear and non-
linear PS processes can explicitly be distinguished from
each other. Please note that we use ‘(verified) solution’
with a restriction in brackets, as not all processes lead to a
verified or even correct solution. Our model is a model of
the outer structure as it describes the observable sequence
of the different phases.
In the following, we illustrate how far our descriptive
model can also respond to our second research question.
We use it to describe, as well as to distinguish different
types of PS processes.
6 Using our descriptive model to analyse
problem‑solving processes
Below, we illustrate how our descriptive model (Fig. 5)
can be used to analyse and compare students’ PS pro-
cesses. We first reconstruct different processes of student
groups and then propose a new way to represent typical
transitions in students’ PS processes.
6.1 Representing students’problem‑solving
processes
In contrast to the process coding by Schoenfeld, which
contains specific information about the duration of epi-
sodes, our analyses are more abstract. We focus on the
empirically found types of episodes and transitions
between these episodes. This is done following Schoenfeld
(1985), who emphasised: “The juncture between episodes
is, in most cases, where managerial decisions (or their
absence) will make or break a solution” (p. 300). Focus-
ing on the transitions between episodes is one important
characteristic that distinguishes different types of PS pro-
cesses. Using our descriptive model allows one to do this.
For each process, the transitions between episodes can
be displayed with our model (Fig. 5). In the following, we
consider only the five content-related episode types, but
not Reading, Organization, Writing, and Digression, as
activities of the latter types of episodes do not contribute
to the solution and they are not ordered as in Pólya’s or
Schoenfeld’s phases.
For example, the routine process of group U1-C (Three
Beaches, part a), see Sect. 5.1), starts with an Analysis,
followed by a merged Planning-Implementation and a
Verification or, in short: [A,P-I,V]; thereafter, this process
ends. This means, there are four different transitions in this
process indicated by arrows: Start → A, A → P-I, P-I → V,
and V → End. Thus, in Fig. 6 (left), these transitions are
illustrated with arrows. In this case, these transitions each
occur only once, which is indicated by a circled number 1.
The second example (U1-C, Three Beaches, part b))
consists of the following episodes: Analysis–Explora-
tion–Analysis–Exploration [A,E,A,E]. This means that
there are five transitions in this process: Start→A, A→E,
E → A, A → E, and E → End (see Fig. 6, middle). Please
notice that the transition A → E is observed twice.
The final example shows group Ulap2-TV (Shortest
Detour), which starts with a Planning-Implementation and
proceeds through [P-I,V,P,E,P-I,V] with a total of seven
transitions, two of which are P-I → V (ignoring Organiza-
tion and Writing, Fig. 6, right).
Fig. 5 Descriptive model of problem-solving processes

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This reduction to transitions, neglecting the exact order
and the duration of episodes, enables one to do a specific
comparison of processes and an accumulation of several
PS processes (e.g., from all DGS processes, see Sect. 6.2).
The focus is now on transitions and how often they hap-
pen, which indicates different types of PS processes as
shown below. This ‘translation’ from the Schoenfeld cod-
ing to the representation in our descriptive model has been
done for all 33 processes. The directions of the arrows
indicate from which phase to which the transitions are
occurring, e.g., from analysis to planning; the numbers on
the arrows show how often these transitions were coded
(they do not indicate an order).
The three selected processes already show clearly dif-
ferent paths, for example, linear vs. cyclic (see Sect. 2.2.4).
6.2 Characterizing types of problem‑solving
processes by accumulation
Students’ PS processes can be successful or non-successful
or conducted in paper-and-pencil or DGS contexts. Looking
at different groups of students simultaneously can be fruitful,
as such accumulations allow one to look at patterns in exist-
ing transitions. Our descriptive model allows one to consider
several processes at once, via accumulation.
Representations of single processes, as presented in Fig. 6
and in the boxes in Fig. 7, can be combined by adding up all
coded transitions (which would be impossible with time bars
used by Schoenfeld). For such an accumulation, we count
all transitions between types of episodes and display them
in numbers next to the arrows representing the number of
those transitions. For example, six of the processes in the
outer boxes start with a transition from the given problem to
Planning, while one process begins with an Analysis. This
is shown in the centre box by the numbers 6 and 1 in the
arrows from the given problem to Planning and Analysis,
respectively (see Fig. 7 for the combination of all processes
regarding task 3a). Arrows were drawn only where transi-
tions actually occurred in this task. Looking at the arrows
that start at the ‘given problem’ or that lead to the ‘(verified)
solution’, one can see how many processes were accumu-
lated. All episode types (small boxes) must have the same
number of transitions towards as well as from this episode
type.
To show the usability of our model, we distinguish
between working on routine tasks and on problems in
Sect. 6.2.1; thereafter, the routine processes are not further
considered.
6.2.1 Routine vs. non‑routine processes
In our study, two sub-tasks (3a) and 4a)) were routine tasks
in which the students were asked to find special points in
triangles. If we look at the accumulations of those processes
Fig. 6 Translation from Schoenfeld codings to a representation using the descriptive model; the circled numbers indicate the number of times a
transition occurs

748 B. Rott et al.
1 3
in our model, clear patterns emerge: There are no Explora-
tion episodes at all, either in the seven processes of task 3a)
(Fig. 8, left) nor in the eight processes of tasks 4a) (Fig. 8,
middle). Instead, there are Planning and/or Implementation
episodes in all 15 processes. In some of those processes,
Planning and Implementation can clearly be coded as two
separate episodes. In other processes, it is not possible to
discriminate between these episode types as two distinct
episodes in the empirical data (see Fig. 8).
Most processes (12 out of 15) show no need for ana-
lysing the task but start directly with Planning and/or
Implementation. Even though there are five Verification
episodes, these verifications are often only short checking
activities with no reflection in the sense of Pólya; how-
ever, the length and quality of an episode cannot be seen
in the model. Additionally, all of these 15 processes are
linear (as can be seen by the arrows, which point only
downwards).
Fig. 7 Centre rectangle: Accumulation of seven different group processes regarding task 3a)
Fig. 8 Accumulation of seven processes for the routine task 3a) (left) and eight processes for task 4a) (middle), 15 processes in total (right)

749
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In contrast to these routine tasks, non-routine processes
are often non-linear and contain at least one Exploration
episode. In Fig. 9, in direct comparison to Fig. 8, the seven
PS processes of problem 3a) (left), the eight PS processes of
problem 3b) (middle), and an accumulation of the 15 PS pro-
cesses (right) are shown. Overall, in these 15 processes, 17
Exploration episodes were coded, which can be seen in Fig. 9
(right): 4 processes start with an Exploration; 12 times there
is an Exploration after an Analysis episode; and once after
Planning-Implementation.
In Fig. 10 (right), an accumulation is given of all 33 PS
processes of all five problems. The differences of the routine
and the PS processes (e.g., the latter containing Exploration
episodes and being cyclic) can be seen by comparing Figs. 8
and 9.
6.2.2 Successful and unsuccessful problem‑solving
processes
One of Schoenfeld’s (1985) major results was the impor-
tance of self-regulatory activities in PS processes. Schoe-
nfeld was not able to characterize successful PS processes;
however, he identified characteristics of processes that did
not end in a verified solution. The unsuccessful problem
solvers were most often those who missed out on self-regu-
latory activities (i.e., controlling interim results or planning
next steps); they engaged in a behaviour that Schoenfeld
called “wild goose chase” and that he described this way:
Approximately 60% of the protocols were of the type
[...], where the students read the problem, picked a
Fig. 9 Accumulation of seven processes for problem 3b) (left) and eight processes for problem 4b) (middle), 15 problem-solving processes in
total (right)
Fig. 10 Accumulation of transitions in problem-solving processes, paper-and-pencil (left) vs. DGS (middle); all problem-solving processes
(right)

750 B. Rott et al.
1 3
solution direction (often with little analysis or rational-
ization), and then pursued that approach until they ran
out of time. In contrast, successful solution attempts
came in a variety of shapes and sizes—but they con-
sistently contained a significant amount of self-reg-
ulatory activity, which could clearly be seen as con-
tributing to the problem solvers’ success. (Schoenfeld,
1992a, p. 195)
We made similar observations looking at the processes of
our students; several of them, who did not show any signs of
structured actions or process evaluations, were not able to
solve the tasks. Thus, to test if this observation was statisti-
cally significant, we had to operationalize the PS type “wild
goose chase”, as Schoenfeld had provided no operational
definition for this phenomenon. A process is considered by
us to be a “wild goose chase”, if it consists of only Explora-
tion or Analysis & Exploration episodes, whereas processes
that are not of this type contain planning and/or verifying
activities (only considering content-related episode types).
In our descriptive model, by definition, wild goose chase
processes look like the process manifested by U1-C (Three
Beaches, part b) (Fig. 6, middle).
To check if the kind of behaviour in these processes is
interrelated with success or failure of the related products
(see Sect. 4.2), a chi-square test was used (because of the
nominal character of the process categories, no Pearson or
Spearman correlation could be calculated). The null hypoth-
esis was ‘there is no correlation between the PS type wild
goose chase and (no) success in the product’.
The entries in Table 2 consist of the observed numbers
of process–product combinations; the expected numbers
assuming statistical independence (calculated by the mar-
ginal totals) are added in parentheses. The entries in the
main diagonal are apparently higher than the expected
values. The test shows a significant correlation (p < 0.01)
between the problem solvers’ behaviour and their success.
Therefore, the null hypothesis can be rejected, there is a cor-
relation between showing wild goose chase-behaviour in PS
processes and not being successful in solving the problem.
6.2.3 Paper‑and‑pencil vs. DGS environment processes
Looking at the processes of the non-routine tasks indi-
cates that the tasks were ‘problems’ for the students, as
these processes showed no signs of routine behaviour (see
Sect. 6.2.1). Instead, we see many transitions between differ-
ent episodes and the typical cyclic structure of PS processes.
Comparing accumulations of all 15 paper-and-pencil with all
18 DGS PS processes, we see some interesting differences,
which our model helps to reveal (see Fig. 10). The time
the students worked on the problem was set in the tutori-
als and, therefore, identical in both environments and in all
processes. At the end of this paper, we discuss three aspects
that our comparisons revealed; more detailed analyses are
planned for forthcoming papers.
(1) We coded more transitions in DGS than in paper-and-
pencil processes (73 transitions in 18 DGS processes,
in short: 73/18 or on average 4 transitions per DGS
process compared to 52/15 or 3.5 transitions per paper-
and-pencil process). If transitions are a sign of self-reg-
ulation (Schoenfeld, 1985; Wilson et al. 1993), our stu-
dents in the DGS environment seem to better regulate
their processes (please note that Organization episodes
are not counted here; including them would further add
transitions to DGS processes). However, there might be
more transitions (and thus episodes) in DGS processes
because of having more time for exploring situations
and generating examples, which does not take as much
time as in paper-and-pencil processes.
(2) We see more Planning (and Implementation) episodes
in DGS than in paper-and-pencil processes (9/18 or
Planning in 50% of the DGS processes compared to
2/15 or 13% in paper-and-pencil processes). Using
Schoenfeld’s conceptualization of Planning and Explo-
ration episodes, the DGS processes seem to be more
structured—especially since there are less Exploration
episodes in DGS than in paper-and-pencil processes
(17/18 compared to 21/15), even though there are more
episodes in the DGS environment (see above). There
seems to be a need for students in the DGS environ-
ment to plan their actions, especially when it comes
to complex constructions that cannot be sketched
freehandedly as in the paper-and-pencil environment.
Considering the success of the students (6 solutions in
the DGS environment compared to 3 in the paper-and-
pencil environment), this hypothesis is supported. As
already existing research indicates, better regulated PS
processes should be more successful. Please note that
Table 2 Contingency table—
process behaviour and product
success
Process/product categories No/wrong solution Correct solution Sum
Wild goose chase 21 (17.5) 3 (6.5) 24
Miscellaneous 3 (6.5) 6 (2.5) 9
Sum 24 9 33
χ2
=7.14 p=0.0075 Fisher exact Prob. Test p=0.0047

751
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successful solutions cannot be obtained by stating only
correct hypotheses, which would favour the DGS envi-
ronment; solutions coded as ‘correct’ had to be argued
for.
We double checked our codings to make sure that
this result was not an artefact of the coding, that the
students actually planned their actions, not only using
the DGS (which was coded in Organization episodes).
This result could be due to our setting, as our student
peer groups had only one computer and thus needed to
talk about their actions. In future studies, it should be
investigated if this phenomenon can be replicated in
environments in which each student has his or her own
computer.
(3) We also observed more Verification episodes in DGS
compared to paper-and-pencil processes (7/18 or 39%
compared to 2/15 or 13%). There could be different
reasons for this observation, e.g., students not trusting
the technology, or just the simplicity of using the drag-
mode to check results compared to making drawings in
the paper-and-pencil environment.
The results of using our descriptive model for compari-
sons of PS processes appear to be insightful. The purpose
of this section was to illustrate these insights and the use
of our empirical model of PS processes. Accumulating PS
processes of several groups is a key to enabling comparisons
such as the ones presented.
7 Discussion
The goal of this paper was to present a descriptive model
of PS processes, that is, a model suited to the description
and analyses of empirically observed PS processes. So far,
existing research has mainly discussed and applied norma-
tive models for PS processes, which are generally used to
instruct people, particularly students, about ideal ways of
approaching problems. There exist a few, well accepted,
models of PS processes in mathematics education (Fig. 1);
however, these models only partly allow represention of
and emphasis on the non-linearity of real and empirical PS
processes, and they do not have the potential to compare
processes across groups of students. For the generation of
our descriptive model of PS processes, following our first
research question, (1) the existing models were compared. It
turned out that similarities and fine differences exist between
the current normative models, especially regarding the
phases of PS processes and their sequencing. We identified
which elements of the existing models could be useful for
the generation of a descriptive model, linking theoretical
considerations from research literature with regard to our
empirical data. Analysing PS processes of students working
on geometric problems, we observed that distinctive epi-
sodes (esp. the distinction between Planning and Explora-
tion) and transitions between episodes, were essential. Clas-
sifying the episodes was mostly possible with the existing
models, but characterising their transitions and sequencing
required extension of the existing models, which resulted
in a juxtaposition of components for our descriptive phase
model (e.g., allowing us to code, separately or in combina-
tion, Planning-Implementation or to regard the (non-)linear-
ity of processes).
Our generated descriptive model turned out not only to
provide valuable insights into problems solving processes
of students, but also with respect to our second research
question, (2), to compare, contrast, and characterise the idi-
osyncratic characteristics of students’ PS processes (using
Explorations or not, linear or cyclic processes, including
Verification and Planning or not). Our developed descriptive
model can be used to analyse processes of students ‘at once’,
in accumulation, which allowed us to group and characterise
comparisons of students’ processes, which was not possi-
ble with the existing models. As demonstrated in Sect. 6.2,
our model further allows one to distinguish students’ PS
processes while working on routine versus problem tasks.
Applying our descriptive model to routine tasks, we detected
linear processes, whereas in problem tasks cyclic processes
were characteristic. Furthermore, in routine tasks, no
Exploration episodes could be coded. Most of the students
expressed no need for analysing the task but started directly
with Planning and/or Implementation.
Our descriptive model also allows one to recognize a type
of PS behaviour already described by Schoenfeld (1992a)
as “wild goose chases”. Our data illustrated that wild goose
chase processes are statistically correlated with unsuccessful
attempts at solving the given problems.
In addition, our descriptive model indicated differences
between paper and pencil and DGS processes. In the latter
context, students showed more transitions, more Planning
(and Implementation), and more Verification episodes. This
result revealed significantly different approaches that stu-
dents embarked on when working on problems in paper and
pencil or DGS environments. These findings might indicate
that in the DGS environment in our study, students better
regulated their processes (cf. Schoenfeld, 1985, 1992b; Wil-
son et al. 1993)—a hypothesis yet to be confirmed.
A limitation of our study might be the difficulty of the
problems given to our students; only 9 of 33 processes ended
with a correct solution. In future studies, problems should be
used that better differentiate between successful and unsuc-
cessful problem solvers. Also, our descriptive model has so
far been grounded only in university students’ geometric
PS processes. Even though geometry is particularly suited
for learning mathematical PS in general and heuristics in
specific (see Schoenfeld, 1985), other contexts and fields

752 B. Rott et al.
1 3
of mathematics might highlight other challenges students
face. Further empirical evidence is needed to see how far
our model is also useful and suitable to describe other con-
texts with respect to specifics of their mathematical fields.
Following some of our ideas and insights, Rott (2014) has
already conducted such a study: fifth graders working on
problems from geometry, number theory, combinatorics, and
arithmetic. Similar results as in the study presented here,
were seen and indicate the value of our descriptive model.
More research in this regard is a desideratum.
Regarding teaching, using our model can be helpful to
discuss with students on a meta-level these documented dis-
tinct phases of PS processes, transitions between them, and
the possibility of going back to each phase during a PS pro-
cess. This might help students to be aware of their processes,
of different ways to gain a solution and justification, and to
be more flexible during PS processes. More reflection on this
aspect is also a desideratum for future research.
Supplementary Information The online version contains supplemen-
tary material available at https://doi.org/10.1007/s11858-021-01244-3.
Funding Open Access funding enabled and organized by Projekt
DEAL.
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