ARE INTEGRATED THINKERS BETTER ABLE
TO INTERVENE ADAPTIVELY? – A CASE STUDY IN A
MATHEMATICAL MODELLING ENVIRONMENT
Rita Borromeo Ferri and Werner Blum
University of Hamburg and University of Kassel, Germany
A lot of empirical studies have shown that the learning and teaching of mathematics
is highly complex and is influenced by many factors. The teacher, in particular, can
promote qualitative lessons with a high learning outcome, but also the contrary. in
previous studies, preferred mathematical thinking styles and preferred interventions
of teachers could be reconstructed as important influence factors when teaching
modelling in the mathematics classroom. In our empirical study the main question
was whether there is a connection between certain mathematical thinking styles and
certain intervention types of teachers. First results and hypotheses of this
quantitative-qualitatively oriented study are presented in this paper.
1. THEORETICAL BACKGROUND
First, we will briefly give a theoretical background regarding the two basic elements
of our study, mathematical thinking styles and teacher interventions.
A mathematical thinking style is “the way in which an individual prefers to present,
to understand and to think through mathematical facts and connections by certain
internal imaginations and/or externalized representations. Hence, a mathematical
style is based on two components: 1) internal imaginations and externalized
representations, 2) the wholist respectively the dissecting way of proceeding.”
(Borromeo Ferri 2003)
Within the theory of mathematical thinking styles the term “preference” is important,
because a style, speaking in the sense of Sternberg (1997), refers to how someone
likes to do something, whereas an ability refers to how well someone can do
something. So styles are preferences in using abilities. Sternberg’s general theory of
styles influenced also the construct of mathematical thinking styles. On the basis of
empirical studies (see Borromeo Ferri 2003, 2004 and 2007) three mathematical
thinking styles could be reconstructed so far with students from grades nine and ten
and teachers of secondary schools:
Visual thinking style: Visual thinkers show preferences for distinctive pictorial
imaginations and representations as well as preferences for the understanding
of mathematical facts and connections in a holistic way. The internal
imaginations are mainly effected by strong associations with experienced
situations.
Analytical thinking style: Analytic thinkers show preferences for formal
imaginations and representations. They are able to comprehend mathematical
facts preferably through symbolic or verbal representations and prefer to
proceed rather in a sequence of steps.
Integrated thinking style: These persons combine visual and analytic ways of
thinking and are able to switch flexibly between different representations or
ways of proceeding.
In a further empirical study (see Borromeo Ferri 2010), one of the main questions
was whether there is an influence of mathematical thinking styles particularly on the
modelling behaviour of students and teachers. Summarizing the results shortly only
regarding the teachers’ behaviour, it became clear that teachers who differ in their
mathematical thinking styles have preferences for focusing on different parts of the
modelling cycle while helping or discussing with students. More concretely:
Analytic thinkers preferred the “mathematical part” of the modelling cycle and were
interested in formal solutions of modelling tasks, whereas the validation of the real
results was not so important for them. Visual thinkers tended to think in terms of the
“real world” (the given situation in the task) and to enrich the reality with their own
associations, whereas looking on formal solutions was not in their focus. Integrated
thinkers found a balance between reality and mathematics for themselves and in their
behaviour with their students. So mathematical thinking styles seem to have a
substantial influence on the way teachers deal with modelling problems in the
classroom.
Of particular interest is now whether and in which way certain types of teacher
interventions are linked with certain mathematical thinking styles. In the following,
we will refer to Leiß’ (2007) theory of teacher interventions. Leiß characterises
teacher interventions as all verbal, para-verbal and non-verbal interferences of a
teacher in the solving process of students. Adaptive teacher interventions are
characterised as supporting the individual learning and solving process of students in
a minimum way, so that students can work as much on their own as possible
(realising a balance between students’ independence and teachers’ guidance, in the
spirit of Maria Montessori: “Help me to do it by myself”).
Leiß (2007) did his empirical study for reconstructing teacher interventions within
the DISUM-project (see Blum/Leiß 2007b). He created, on the basis of existing
theories of interventions (e.g. Zech 1996) and his own classification (Leiß/Wiegand
2005), a coding scheme for analysing his observations. He distinguished levels,
activators and aims of interventions. These will be explained in the following tables
into more detail, because we also used this coding scheme for our own analyses.
Levels of interventions
with regard to the content
Aims of interventions
diagnose
interventions of the teacher referring to the
content, here: the modelling process and the
corresponding mathematics
teacher informs students about the current
state of their solving process
strategic
evaluation/feedback
interventions concerning the meta-level,
that is general aspects of the problem
solving process
teacher gives feedback to solving process
without further information or correction
affective
indirect advise
interventions trying to influence the mental
state of students
subtle hints of the teacher in order to help
students to find the “best way of solving”
according to the teacher
organisational
direct advise
interventions
concerning
the
conditions of students’ working
basic
teacher gives relevant explanations and
information explicitely
conscious non-intervention
no intervention of the teacher although
students may have problems
Table 1: Level and aims of interventions
Activators of intervention
invasive
own initiative of the teacher to intervene in
the solving process of the students
responsive
students ask for advise explicitely
Table 2: Activators of interventions
Main results of Leiß’ study were, among others, that strategic interventions are
included in the intervention-repertoire of the observed teachers only very marginally
and that organisational aims of teachers are communicated to students using direct
instruction.
More sophisticated results were obtained in the so-called main study of the DISUMproject (see Blum 2011). Altogether 25 grade 9 classes participated in this study with
a pre-, post and follow up test design and with ten lessons devoted to modelling
tasks. The research focus was the comparison between two teaching styles: a more
teacher guided “directive” style and an “operative-strategic” style focussing more on
students’ independent work in groups. The results showed a significantly higher
progress of modelling competency of students taught in the “operative-strategic”
style. The best results concerning progress of modelling competencies could be
reconstructed in those classes were the balance between students’ independence and
teachers’ guidance was realised best, based on experts’ ratings. This corresponds to a
constructivist view of learning where students have to build their knowledge and
competencies as actively and independently as possible, supported by the teacher.
More generally, the background model for our studies is a view on “quality
mathematics teaching” characterised by a demanding orchestration of teaching the
mathematical subject matter (by giving students vast opportunities to acquire
mathematical competencies and making connections within and outside
mathematics), by a permanent cognitive activation of the learners and by an effective
and learner-oriented classroom management (see Blum/Leiß 2007b and Blum 2011).
2. RESEARCH QUESTION AND METHOD
The central question of our study was: Do connections exist between teachers’
mathematical thinking styles and types of teacher interventions in mathematical
modelling environments?
In particular the results of the main study of the DISUM-project, mentioned before,
raised new questions. In fact, the “operative-strategic” style was more successful, but
big differences between several teachers concerning the quality of teaching and the
progress of the students could be observed, although all teachers got a special training
for teaching the ten modelling lessons. So an open question is whether these
differences can be explained more deeply with certain individual attributes of the
teachers. The procedure of our study was as follows. Firstly, regarding data
collection: we chose four of the 25 teachers of the main study of the DISUM-project
(for whom a particular lot of video material was available) for analysing the same
two videotaped lessons where the tasks “Filling up I and II” were treated (see
Blum/Leiß 2007a). Additionally, we also conducted focussed interviews (Flick 1990)
with all four teachers in order to reconstruct their mathematical thinking styles. The
guideline for these teacher interviews was developed and used in the recent study
Borromeo Ferri (2010). Examples of interview questions are: What does mathematics
mean for you? Do you think that your attitude concerning mathematics changed
through your teaching life? What is your preferred way of solving tasks (concretely:
the “Filling up” task)? What is important for you when teaching mathematics?
The interviews were transcribed and analysed in combination with the actions of the
teachers in the lessons for reconstructing their mathematical thinking styles. A basic
instrument for the classification of the thinking styles was the coding scheme
developed by Borromeo Ferri (2003). In order to get more validity, two persons
analysed the data independently, in the sense of so-called concurrent coding (Schmidt
1997, 559). With the same method, using the coding scheme of Leiß (2007), teacher
interventions were reconstructed directly from the videos. For analysing the
interventions, a structure was created so that interventions of the teachers could be
better compared in several phases of the lessons. The following six phases were
considered during the whole analysing process: 1. Introduction of the lesson (task
“Filling up I”); 2. Group work; 3. Plenum (discussion about students’ results); 4.
Improving students’ own results; 5. Plenum (reflections on the solution process); 6.
Transfer to the second task “Filling up II”.
3. RESULTS
On the basis of the interviews and the videos, three of the teachers, Mr. H., Mr. B.
and Mrs. R., could be reconstructed as analytic thinkers and one, Mr. S., as an
integrated thinker. So a visual thinker was not in this (small) sample.
a) Quantitative analyses of teacher interventions (lessons “Filling up I and II”)
The duration of the six phases was different in the fouri classes. So we will mainly
have a look at phases 2 and 3 (see tables 3 and 4), which means the group work and
the first plenum (discussing the results), where we got interesting results how the
teachers acted. Table 3 demonstrates the interventions of the teachers during group
work. In Mrs. R.’s and Mr. S.’ lessons the pupils worked in the sense of the
”operative-strategic” style independently in their groups. Mr. H. and Mr. B. were
teaching in the “directive” style which is their preferred way: students sitting headon, solving the task in the whole class. So the high number of Mr. H.’ interventions
becomes clear, in particular invasive, context-bound and directive interventions.
Looking at table 4 which shows the interventions during the plenum phase, only Mrs.
R. and Mr. S. intervene. Because Mr. H. worked on the modelling problem with the
whole class they only got “one right result” that was not discussed further. Mr. S.
asked every group of students to explain their results, which explains his high number
of interventions. Mrs. R. discussed the different results of the students.
invasive
invasive
responsive
responsive
context
context
strategic
strategic
affective
affective
organisation
organisation
diagnose
diagnose
evaluation
evaluation
indirect advise
indirect advise
direct advise
direct advise
non-interv.
non-interv.
Table 3: Interventions phase 2 (group work)
Table 4: Interventions phase 3 (plenum)
In the following table 5, all interventions of all teachers during all phases are
presented. Mr. H. (90%) and Mrs. R (77%) intervened strongly invasive, Mr. S.
(72%) less invasive. Concerning the level of interventions, Mr. H. intervened 50%
and Mr. S. 43% context-bound. Strategic interventions can be stated for Mr. S. twice
as much as for Mrs. R. and Mr. H.. Concerning affective and organisational
interventions, Mrs. R. has the highest rate with 32% and 27%. Results of t-tests
(confidence-interval 95%) concerning the invasive, context-bound, direct and indirect
interventions showed that these differences are all significant.
invasive
responsive
context
strategic
affective
organisation
diagnose
evaluation
indirect advise
direct advise
non-interv.
Table 5: Interventions in total (absolute incidence)
Comparing direct and indirect interventions (table 6) shows a clear picture about
preferences of the three teachers: Mr. S. practised fewer direct interventions than the
other two and also had twice as much non-interventions.
direct interventions
indirect interventions
Mr. H.
12,00%
45,00%
Mrs. R.
14,00%
41,00%
Mr. S.
15,00%
28,00%
Table 6: direct and indirect interventions
b) Qualitative analyses of teacher-interventions (lessons “Filling up I and II”)
In the following, interventions and connections to mathematical thinking styles only
of Mrs. R and Mr. S (both within the “operative-strategic” design) are reconstructed
because of the restricted space. We do that firstly with the help of a table
summarizing the main characteristics of each teacher and secondly with a comparison
of some concrete actions of the teachers. In the left column one can find the actions
within the lessons and in the right column statements from the interview. We start
with Mrs. R., reconstructed as an analytic thinker.
Interventions and actions (lesson)
Reflections about own teaching (interview)
- lesson is structured
- she introduces new themes in maths by a
- students can work independently in their concrete example and then mainly
algorithms are important for her
groups
- different interventions in the groups: less - she prefers a clear structure of lessons
interventions in groups which work on the - she does not like some mathematical
path preferred by Mrs. R., other groups are contents such as riddles
more or less guided during the solving - she personally likes to solve tasks formally
process; altogether many affective and
organizational interventions
Table 7: Mrs. R. – some characteristics
Interpretation: Mrs. R. prefers and also shows well structured maths lessons.
Analysing the two lessons as well as the other lessons in that teaching unit it becomes
clear that Mrs. R. intervenes regularly and consciously in the thinking and solving
process of the students:
“I went from one group to the other and I always said think about it.”
Mrs. R. intends to have their students learn from mistakes:
“I always tell my students that it is not bad making mistakes, but learning from mistakes is
good.”
However, this statement is in a certain contrast to her real actions in classroom. In all
analysed lessons Mrs. R. was trying to prevent students from making mistakes and to
guide them to her preferred solution. We interpret her well structured lessons in
combination with her analytic thinking style as a kind of aplomb while teaching and a
sign of her intention to avoid ambiguity regarding mathematical contents. She likes to
have the control of as many aspects during lessons as possible, especially concerning
the way of solving tasks:
“We have a certain ritual of solving tasks and I exercise this with my students.”
For Mrs. R. visualising is also relevant, but her preferred way of thinking becomes
evident when she talks about solving a linear equations task:
“Personally I do not see the straight line, I see the actual data. Of course I know that
behind is the system of equations.”
Altogether, according to our interpretation, Mrs. R. tries to compensate a certain onesidedness, caused by her mathematical thinking style, by planning carefully each step
during a lesson. She emphasised in the interview that acting this way is keeping her
from ambiguity.
Mr. S. was reconstructed as an integrated thinker
Interventions and actions (lesson)
Reflections about own actions (interview)
- lesson is structured
- mathematics is for him “everything”,
- students can work independently in their training thinking processes as well as
applications
groups
- indirect interventions for getting partial - he likes diversity in teaching maths and
solution of students without guiding them developed special methods for several
phases within a lesson
much
- new ideas of students are picked up with - during university he had problems with the
formal exactness of mathematics
extra information
- helping the students to do things by - he sees a balance for himself between
themselves; he gives the same attention to formal and visual acting
all students and praises solutions not only - he welcomes different solutions of students
when they are correct
- aim of intervention is predominantly
diagnosis or indirect, also often noninterventions
Table 8: Mr S. – some characteristics
Interpretation: The main intentions and actions of Mr. S. can be summarized in one
sentence: he tries to help the students to do it by themselves. For many students that
is enough for being active during the whole lesson. Students are not afraid asking
something. For him “the main principle is to inspire students for mathematics.” So
that is why he created various methods. At the beginning of lessons he gives students
sometimes a riddle or a logical story.
“Pupils should see that one can solve problems with the help of mathematics and that
mathematics is training thinking processes.”
His own problem with the exactness of mathematics during his university studies has
influenced his way of teaching and assessment at school. This becomes clear while
analysing the two lessons:
“Students who understood the problem will not get a worse mark if they don’t do it
totally formally correct.”
For Mr. S. it is important to support sustainable knowledge so he creates his lessons
usually very open and uses several methods. He also welcomes individual solution
processes of the students, for instance in the case of linear equations the whole
spectrum from formal equations over tables to graphs. So we reconstructed in his
lessons that Mr. S. is listening to students’ ideas and is doing some kind of “juggling”
with these ideas. He is encouraging every group to think in different directions and
also to think about solutions of other groups.
c) Reconstructed connections
Connections between interventions and mathematical thinking styles were
reconstructed on the basis of the cases shown here, both with respect to qualitative
and quantitative aspects. The teachers’ mathematical thinking styles were clearly
correlated with their actions in lessons. The integrated thinker, Mr. S., had a more
flexible repertoire of intervening, including strategic interventions, he let students
work much more independently and develop more individual solutions. The analytic
thinkers tried a lot more to guide students according to the teachers’ preferred ways
of solving the tasks. Now what was the effect of these ways of teaching? Comparing
the learning progress of the students (see the remark in section 1; for more details see
Schukajlow et al., 2009), Mr. S.’s students learned significantly more during this ten
lesson teaching unit than the students of the other three teachers, in particular
concerning modelling.
So our central hypothesis on the basis of our case study is:
Integrated thinkers are better able to intervene flexibly and minimal-adaptively and
thus get better results concerning students’ learning progress.
Of course, this result ought to be replicated with a bigger sample of teachers before
implications for teacher education can be drawn.
An open question is how teachers’ mathematical thinking styles and interventions are
connected with their subject-related professional knowledge and competencies and
with their epistemological convictions and beliefs. Perhaps the guiding forces behind
both thinking styles and interventions are the same. This should also be subject of
further studies on teachers.
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i
From now on, only the analyses for three teachers are shown. The fourth teacher, Mr. B., will be included in the final
version of the paper presented at CERME-7.