Affective systems of Norwegian mathematics student/teachers concerning 'unusual' problem solving.
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Affective systems of Norwegian mathematics student/teachers
concerning 'unusual' problem solving.
Citation for published version (APA):
Pepin, B. (2012). Affective systems of Norwegian mathematics student/teachers concerning 'unusual' problem
solving. In Proceedings of ICME 12 : the 12th International Congress on Mathematics Education, 8-12 July
2012, Seoul, South Korea [TSG27-16]
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Download date: 20. Sep. 202012th International Congress on Mathematical Education
Topic Study Group 27
8 July – 15 July, 2012, COEX, Seoul, Korea
AFFECTIVE SYSTEMS OF NORWEGIAN MATHEMATICS
STUDENT/TEACHERS CONCERNING ‘UNUSUAL’ PROBLEM
SOLVING
Birgit Pepin
Sør-Trøndelag University College, Trondheim, Norway
The research reported in this paper draws on a small intervention study on teachers’ problem
solving beliefs. Three groups of teachers, each at different stages of their educational and
professional development, were asked to solve a carefully designed mathematical ‘unusual’
problem, and write down their reactions at different stages of solving the problem. The results
provide (1) insights into teachers’ affective systems linked to their engagement in the task;
and (2) how their problem solving affective systems are likely to be influenced by ‘emotional
knowledge’ and skills. It is argued that affective systems in relation to mathematical problem
solving activities (in particular ‘unusual problems’) can be regarded as an interplay between
emotions, experiences in actions, and engagement structures.
Key words: affective systems- teachers- unusual problem-solving
BACKGROUND
A number of studies in mathematics education have indicated that teachers’ beliefs about
mathematics and its teaching and learning play a significant role in shaping their patterns of
behaviour in the classroom (Peressini, Borko, Romagnano, Knuth & Willis 2004; Fang 1996;
Thompson 1992; Ernest 1989). These beliefs make up an important part of teachers’ general
knowledge through which teachers perceive, process and act upon information in the
classroom and in other contexts. They can be embodied, amongst others, in the teacher’s
theories about a particular subject area’s learning and teaching (Fang 1996), or about
particular practices, such as problem solving.
The literature is clear that learning is situated, that is, how a person learns a particular set of
knowledge and skills, and the situation in which the person learns, are a fundamental part of
what is learnt (Greeno, Collins & Resnick 1996). A situated perspective on teacher learning
suggests that knowledge and beliefs, the practices they influence, and the influences
themselves, are inseparable from the situations in which they are embedded (Putnam & Borko
2000; Borko & Putnam 1996). For example, Ernest (1988) is in no doubt that there is a
general socialising influence across particular contexts in a school mathematics department,
and this departmental influence often overrides a teacher’s individual beliefs.
Talking about beliefs and ‘engagement structures’ (with which beliefs are intertwined),
Goldin, Epstein, Schorr & Warner (2011) argue that when students work on conceptually
challenging, and different, mathematics, they have to engage in non-routine thinking where
“mathematical meanings matter more than procedures” (p.547), or to problem-solve where an
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apparently ‘impossible situation’ necessitates the construction of new representations and/or
thinking. This is likely to be true for teacher beliefs when they work on non-standard
mathematical problems. Putting teachers in such affective/social interactions, they respond
differently (than to standard questions which they are used to in their classrooms) and it is
likely to bring out a wider and perhaps more complex spectrum of beliefs during the
‘in-the-moment’ engagement.
Thus, the aim of the study is to explore in which ways students and teachers (work with and)
reflect on their work with an ‘unusual mathematical problem’, and in which ways this may
provide insights into their affective systems.
THEORETICAL FRAMEWORK
Beliefs have been defined in different ways. Schoenfeld (1985) perceives mathematical
beliefs as “the set of understandings about mathematics that establish the psychological
context within which individuals do mathematics” (p.5). In a wider context Aguirre & Speer
(2000) contend that beliefs are “conceptions, personal ideologies, world views and values that
shape practice and orient knowledge” (p.328). The important point in all these definitions,
and relevant for this study, is that beliefs are linked to actions and practice in mathematics.
However, and connecting to more recent work on attitude (e.g. Di Martino & Zan, 2011), the
construct of attitude “is shaped within the context of social psychology as orientation to
behave in a certain way” (p.473). Recent work in the field refers to a tripartite model where
attitude has a cognitive, an affective, and a behavioural component (ibid). The literature is
clear about the “deep interplay between beliefs and emotions” (e.g. Di Martino & Zan, 2011,
p.473). They quote McLeod (1992) who contends that each construct (beliefs, emotions,
attitude) has a cognitive and an emotional component (though with a varying degree of
importance):
“Beliefs, attitudes, and emotions ... vary in the level of intensity of the affects that they describe,
increasing in intensity from ‘cold’ beliefs about mathematics to ‘cool’ attitudes related to liking or
disliking mathematics to ‘hot’ emotional reactions to the frustration of solving non-routine
problems. Beliefs, attitudes, and emotions also differ in the degree to which cognition plays a role
in the response, and in the time they take to develop.” (p.578)
Encompassing these, Philippou & Christou (2002) coined the term ‘affective systems’ which
includes emotions and feelings, as well as attitudes, beliefs and conceptions. Beliefs in
particular are conceived as “the personal appraisal, judgements, and views that constitute
one’s subjective knowledge about self and the environment”(p.212).
There have been several studies (e.g. Philippou & Christou 1998, Swars, Smith & Smith 2009)
where researchers have developed interventions designed to impact positively on students’
problem-solving beliefs, and these studies identified particular features that appeared to have
succeeded in doing that. These studies often base their thinking on the idea that belief systems
are composed of ‘episodically’-stored material and that such episodic memory is organised
“in terms of personal experiences, episodes or events” (Nespor 1987, p.320, in Stylianides &
Stylianides 2011); thus, it is believed, providing students with ‘influential episodes’ may
shape their interactions with subsequent events.
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Previous studies which have used interventions such as the BHP (e.g. Stylianides &
Stylianides 2011) have based their thinking on Schoenfeld’s ideas on beliefs with respect to
problem solving. According to Schoenfeld (1985) a person’s beliefs about mathematics “can
determine how [s/he] chooses to approach a problem ... and how long and hard [s/he] will
work on it” (p.45). Schoenfeld’s work is clearly relevant here, as a person’s persistence with
as well as the ways of approaching a problem seem to be at stake when working with an
unusual problem. For example, many students believe that mathematical problems have to
contain numbers and formulas, and they are hesitant to work on problems that do not fit their
expectations of a mathematical problem, and thus they may give up earlier if they cannot see
how to progress (e.g. Callejo & Vila 2009).
THE STUDY
This study is based on a design experiment (e.g. Cobb, Cofrey, di Sessa, Lehrer & Schauble,
2003) the author conducted in a master and teacher professional development course, with
three different groups of teachers and student teachers:
(1) a group of experienced teachers (between 3 and 20 years of experience) participating in
a professional development course (18 teachers);
(2) a group of student teachers (0-5 years of teaching experience) at the beginning of their
master study (year 4, at beginning of first semester) (11 teachers); and
(3) a group of student teachers (0-5 years of teaching experience) during their first year of
master study (year 4, 2nd semester) (13 teachers).
The groups were chosen because of their difference, in particular in terms of their educational
and professional background, that is number of years working with mathematics (in education)
and as teachers in the classroom.
The intervention, the Blonde Hair Problem (BHP), was chosen from the literature on the basis
of its ‘usability’ and effectiveness in terms of providing insights into students’ beliefs
regarding problem solving (e.g. Philippou & Christou, 1995; Stylianides & Stylianides, 2011).
It was assumed that this would also be the case for student teachers and experienced teachers
alike. (Please see Philippou & Christou (1995), or Stylianides & Stylianides (2011) for the
description of the problem, the solution outline and its design features).
In terms of implementation the problem was presented and ‘administered’ exactly the same
way for the three groups. It was important to reassure the students/teachers that they were not
assessed for solving the task, but that this activity was done in order to reflect on ‘beliefs’
(part of their course module):
1. The problem was presented on a handout and read out (but not discussed), without
commenting on whether it was solvable or not. Student/teachers were asked to work
individually and to write down (a) their initial reaction to the BHP; (b) in which ways it
may differ from other problems they have met. Answers were subsequently collected.
2. Students were told that the problem was solvable and they were asked to solve the task,
and they could discuss in small groups.
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3. Once all groups had solved the task, individuals were asked to present their solution/s at
the board. This initiated a discussion about the mathematics which was led by the
instructor.
4. Students were asked to describe/write down their experiences of working on the BHP.
5. Students were asked to read and compare their responses to what they had written at the
beginning.
The responses of all students to all five tasks were collected for analysis, and a discussion was
held about teacher beliefs (as part of the course).
In terms of analysis student/teachers responses to all five tasks/questions offered data for
examining the different facets of student/teacher beliefs when confronted with a ‘different’
problem. All responses were coded independently, and codes compared between
student/teachers and questions.
THE FINDINGS
The findings (for this paper) can be categorised under two headings:
(1) Initial engagement/interest and self- efficacy; and
(2) The role of communication and group work/support.
(1) In terms of initial engagement and interest, and self-efficacy to be able to solve the BHP,
the two student groups differed quite considerably from the teacher group. Whereas most of
the teachers in the teacher group wanted to solve the problem and they thought that they could
do it (with some help or clues), the students reported interest, but were unsure whether the
problem was solvable, and if so, whether they could solve it. Teachers used expressions like
“I want to solve it”, or “interesting task, gives you the wish to solve it”, although they were
not certain how to solve it.
Another interesting feature of teachers’ reactions to the BHP was that they directly connected
it to their classroom teaching and pupil learning:
“Funny ... funnier than the usual context we have in textbooks.” (T2)
“I think that pupils who think that they cannot do mathematics will give up after the 3rd sentence.”
(T7)
“Nice starting point to link maths and everyday.” (T1)
“I had enough information after the first sentence. The next bit of information was just confusing.
I don’t get it ... I guess that this ... [compares] to feelings my students have when solving these
kinds of problems.” (T9)
“I have tried these kinds of problems with my pupils. They really like them. ... it is important that
students know that the problem is solvable.” (T5)
On the other hand, at least half of the students were not sure whether the problem was
solvable (‘not enough cues’), and all were uncertain whether they could solve it. None of
them thought that they could solve it without help (and would give up if they were left to
themselves).
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“They do not tell us whether it is solvable or not, and therefore ... must give up.” (S3)
“It seems difficult because there isn’t enough hint of the problem. ... a fine balance in the task, and
it ‘wakes’ the interest and a kind of urge to solve it.” (S8)
“I found it very different from other problems I have met in school. The problem was presented in
such a way that I felt it was possible for me to find the solution, that I had the same basis as the
others in the room. Thinking back I found it fun and intriguing. It took a while working with the
problem before I realized that I actually was using knowledge and skills I possess in my effort in
solving the problem. At first it felt like I just was trying out different possible solutions, but that it
soon was put in a system... Even though the solution was presented after I left, I certainly felt I had
been brushing up my math skills." (S3)
Students found the task ‘funny’, described it as ‘unmathematical’ and ‘tricky’ and that it did
not seem ‘a ‘proper’ mathematics task’, in the sense that they did not find ‘well known
procedures’ to solve it. This last notion (that no ‘formula’ was provided for solving the task)
was quite prominent amongst students- they clearly wanted a ‘recipe’; that would have made
them feel safe. It appeared that their self efficacy and confidence in problem solving was not
as developed as that of the teachers. It was not clear whether this was due to teachers working
with ‘known’ (school) mathematics every day, whereas students feel that they still have to
learn/revise some of the mathematics for teaching, or whether it was the structure/type of the
task that did not ‘fit into’ their perceptions of mathematical tasks.
(2) Concerning the role of communication and group work/support, more students than
teachers commented on the positive influence of communication/dialogue and group work as
support for their working with the BHP and development of problem solving skills.
“Through common discussions did I understand what the hints meant and thus could get to the
right solution.” (S2)
“I did not think that it was possible to solve the task, but when I started to discuss it together with
the others, did I see quickly that it was possible.” (S7)
“The collaborative work with peer students [was] positive, as [we could] exchange ideas and clear
potential misunderstandings.” (S4)
“We solved the problem together, with my colleagues. We ...” (T1)
This can be seen in the light of the confidence with problem solving or self efficacy – they felt
comfortable and ‘safe’ when working with their peers (see above), but perhaps also in view of
their ‘normal’ working conditions: most teachers are ‘alone’ in teaching their classes, there is
no one else in the classroom. It could be argued that this produces a kind of ‘self-dependency’.
Professional development activities and discussions on particular mathematical tasks are rare.
On the other hand students are used to group work and peer discussions in their
university/college sessions.
DISCUSSION OF RESULTS
Self- belief systems, and these include self efficacy, are part of a person’s general beliefs
system (Philippou & Christou, 2002), and numerous studies claim that confidence is one’s
ability to undertake a certain action. In this study the students’ and teachers’ feelings of
confidence depended on their experiences in connection to related and previous actions, i.e.
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whether they had solved or seen or experienced similar tasks before. It can be argued that
whereas students’ experiences were likely to have been shaped by ‘formula and calculating’
approaches, the more experienced teachers’ experiences were more varied, which helped
them to feel more confident about being able to ‘have a good go’ and perhaps succeed in
solving the task. This was also expressed by, and connected to, their perceptions of whether
the ‘unusual task’ was solvable: students wondered whether it was solvable, and many
decided that it was unsolvable, because they appeared not to have the correct kinds of
information. Most teachers, however, simply assumed that they should have a go before
deciding otherwise, and they applied their skills in trying to solve it- a more practical
approach, which appeared to provide resilience. Bandura (1997) highlights particular sources
of self-efficacy, amongst them ‘mastery experiences’, and it appeared that teachers may have
had more of those experiences.
One of the advantages of construing affect as a representation system is that it “entails the
recognition that affect is not simply an emotional state or an attitude, it is configured” (Goldin,
1998, p.214). This dual role could be traced in this ‘unusual’ problem solving situation, where
for some students the confrontation with an ‘unusual’ problem might have signified ‘giving
up’ or ‘I cannot do it’ (recalling an unsuccessful or uncomfortable situation), whilst for others,
and in particular teachers, the ‘unfamiliarity’ led to new strategies (and perhaps the arousal of
a challenging situation).
However, when being able to work together on the problem in groups, all groups developed
more positive engagement structures around the mathematical problem, and it was interesting
that these were triggered by discussion. It can be argued that Bandura’s (1997) ‘social
persuasion’ was a key element, linking to social behaviour when working with an ‘unusual
problem’. Equally, it can be contended that communication and reflection with others is
known to help to develop deeper mathematical understandings when working on problems.
However, it is argued in this case whether students/teachers had to work alone or in a group
on an ‘unusual problem’ influenced how they construed the situation. It was the combination
of ‘alone’ and ‘unusual’ that were likely to evoke negative ‘desires’ and consequently
negative engagement structures for activation. Thus, ‘alone’ (no communication with others,
and thus likely failure) and ‘unusual’ mathematical problem links to ‘negative engagement’.
In other words, in this case emotions and particular mathematical experiences lead to
engagement (or not):
Emotions/affective system + in-the-moment mathematical experiences/in action - >
engagement structures
The question remains whether the arrow also works in the other direction, i.e. whether
engagement structures can lead to/provoke mathematical experiences/unusual problems and
emotions. Leaning on previous research (e.g. Philippou & Christou, 2002; Di Martino & Zan,
2011) it can be argued that affective systems in relation to mathematical problem solving
activities (in particular ‘unusual problems’) can be regarded as an interplay between emotions,
experiences in actions, and engagement structures.
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