In the study described in this paper, mathematical thinking styles of 15 and 16 year old pupils shall be reconstructed. In the actual discussion on mathematics there already exist classifications of thinking styles: F. Klein (1892, quoted by Tobies 1987) for example, distinguishes the thinking styles of the "analyst", the "geometer" and the "philosopher", while Burton (1995) describes a visual, an analytic and a conceptual thinking style. Some of these classifications were developed intuitively or through empirical examinations, and the study only concluded practising mathematicians but no pupils learning mathematics. In this paper it will be shown among others, how mathematical thinking styles have been reconstructed in the study until present.

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Thematic Group 3

EUROPEAN R ESEARCH IN MATHEMATICS EDUCATION III

R. Borromeo Ferri 1

MATHEMATICAL THINKING STYLES – AN EMPIRICAL STUDY

Rita Borromeo Ferri

Department of Education, University of Hamburg

Abstract: In the study described in this paper, mathematical thinking styles of 15 and

16 year old pupils shall be reconstructed. In the actual discussion on mathematics

there already exist classifications of thinking styles: F. Klein (1892, quoted by Tobies

1987) for example, distinguishes the thinking styles of the "analyst", the "geometer"

and the "philosopher", while Burton (1995) describes a visual, an analytic and a

conceptual thinking style. Some of these classifications were developed intuitively or

through empirical examinations, and the study only concluded practising

mathematicians but no pupils learning mathematics. In this paper it will be shown

among others, how mathematical thinking styles have been reconstructed in the study

until present.

1. Introduction and overview

From our experiences we learned, that there are many ways to explain mathematical

facts and that there are as many ways to understand and to think them through. Some

people for example easier understand mathematical facts by drawing sketches or

other kinds of graphics, while others are tending more to search for structures,

patterns or formulas and it's application. This means that people may have

preferences for the so-called visual or the so-called analytic or so-called conceptual

way of thinking, or they show preferences for two or three of the thinking styles

simultaneously (mixed types).

1

Already in 1892 F. Klein (quoted by Tobies 1987)

distinguished - on an intuitive base – the styles "analyst", "geometer" and

"philosopher". Empirical examinations (Burton, 1995) pointed out, that one may

classify a visual, analytic and conceptual thinking style. Since these classification are

limited to practising mathematicians and their results, they cannot applied directly to

pupils. This is the reason why a special study was carried out which started from the

following research questions:

(a) Can these thinking styles also be reconstructed with 15 and 16 year old teenagers,

who are still in the phase of learning mathematical concepts and methods, but,

compared with practising mathematicians, have much less experience in working

with mathematics?

(b) If so, how can these thinking styles be described ?

1

Mixed type can have two meanings: 1. Mixed type as own style, which concludes characteristics of the visual and

analytic thinking style; 2. Mixed type if ,depending on the situation, one style is chosen.

Thematic Group 3

EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III

R. Borromeo Ferri 2

(c) If these thinking styles can be reconstructed, are there thinking styles which

exclude each other, meaning "polar types" or do "mixed types" also exist, meaning

thinking styles, which conclude characteristics from various thinking styles?

(d) What does it mean to pursue a visual, analytic or conceptual thinking style ? How

can these age-related thinking styles be described?

Research in the area of mathematical thinking processes and cognitive psychology

have shown that there are various ways to reconstruct individual ways of thinking. In

this paper a possible way of reconstructing mathematical thinking styles (analytic,

visual, conceptual and mixed type) will be described.

2. Theoretical foundations and considerations

The theoretical considerations of this study conclude findings of the cognitive

psychology and mathematics didactics. Up to now the concept of 'style' and the

concept of 'thinking style' has been used only occasionally within the mathematics

didactic discussion. Thus, at present the construct of 'style' is discussed mainly in

the field of cognitive psychology (see Sternberg 1996, 1997, 2001). An more ability-

oriented understanding of styles from earlier discussions has been replaced by

conceptions which are emphasising choice and the independence of performance.

Sternberg & Grigorenko (2001) support the following characterisation of style which

is largely agreed at the moment: "reference to habitual patterns or preferred ways of

doing something

[…] that are consistent over long periods of time and across many

areas of activity". For research on teaching and learning it is of central meaning to

distinguish learning styles, thinking styles and cognitive styles, although the

underlying conceptions are often not clear and an overlapping of styles is therefore

unpreventable. This study is focussing on thinking styles and is aimed to develop an

adequate characterization of the construct 'mathematical thinking style' for

mathematics education. Sternberg (1997: 19) takes the construct thinking style as a

"preferred way of thinking" or " preference in the use of abilities we have". There

consists the possibilities of changing thinking styles but they may change depending

on time, environment and life demands. Sternberg states that thinking styles are

acquired at least partly through socialisation. There is almost no study on the

theoretical construct of mathematical thinking styles, especially no empirical one.

However, one can find quite a lot of research which refers to the concept of

mathematical thinking. Schoenfeld (1994) for example, he worked extensively,

theoretically as well as empirically, on the learning of mathematical thinking and its

necessary pre-conditions. Saxe et al. (1996) give special emphasize to the construct's

reliance on culture and context, Dreyfus & Eisenberg (1996) clarify affective aspects,

such as self-confidence or mathematical creativity.

From the mathematics didactics discussion there are known classifications and

typologies of thinking styles (Klein, 1892 (quoted by Tobies, 1987); Ribot, 1909;

Burton, 1995) and of cognitive structures (Schwank, 1996) as well. These

classifications and typologies sometimes offer quite helpful approaches for describing

Thematic Group 3

EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III

R. Borromeo Ferri 3

mathematical thinking styles, for example the empirically verified classifications for

research doing mathematicians from Burton (1999: 95):

Style A: Visual (or thinking in pictures, often dynamic),

Style B: Analytic (or thinking symbolically, formalistically) and

Style C: Conceptual (thinking in ideas, classifying)

Besides the clarification of the theoretical construct 'mathematical thinking style' an

adequate, age-dependend description of the visual, analytic and conceptual thinking

style thinking style shall be generated from the data of this study.

3. Methodology and design of the study

This study is quality-oriented, and the analytic method shall lead to results and

theories. The aim of the study is to generate hypotheses, but because of its case-

study-like character generalisation going beyond the sample can be done only to a

limited degree.

In its qualitative research this study is applying the Grounded Theory (Strauss &

Corbin, 1996), in which's framework various research methods will be described in

order to develop an inductively derived grounded theory about a phenomenon (see

Strauss & Corbin, 1996: 8). By systematically collecting and analysing data on an

examined phenomenon, data gathering, analysis and theory are mutually connected.

"The aim of Grounded Theory is to create a theory, which is fair to the examined

object and illuminate it." (Strauss & Corbin, 1996: 9). Nevertheless Grounded Theory

does not claim, that the researcher starts one's approach to a research object from a

"tabula rasa" situation. Therefore Strauss & Corbin are emphasising the importance

of the theoretical sensibility which allows them "to develop a grounded, conceptual

dense and well integrated theory – much faster than if this theoretical sensibility is

missing." (ibid 1996: 25). Concerning the systematic data collection, in this study the

following methods were used: video-taping of problem-solving processes, stimulated

recall (Wagner, Weidle, Uttendorfer-Marek, 1977; Schoenfeld, 1985) and focussing

interviews (Flick, 1999).

Altogether 12 pupils, 6 boys and 6 girls in years 9 and 10, who that time were

15 and 16 years old participated in the study. Six pupils from each year group were

arranged in pairs: a pair of boys, a pair of girls and a mixed one. It was paid attention

for the chosen students were accustomed to work together in their lessons. Of course,

this study is aimed to reconstruct individual thinking styles, but the reason for the

methodical decision for problem-solving in pairs (see Goos, 1994) is, that in this way

there is much more verbalisation during the problem-solving process and more

questions arise. There were two sessions for each pair. In each session 4 non-routine

problems were to be solved. In order to get as much as possible information about

the pupils' reflections on problem-solving processes and on their way of thinking the

following 3-steps design of has been developed in accordance with Busse (2001).

The procedure of each session was the following:

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EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III

R. Borromeo Ferri 4

Step 1: Problem-solving process. Each pair of students solved 4 problems, one after

another. They were free to decide how far to work together. The working process was

video-taped.

Step 2: Individually stimulated recall. Afterwards, each student was shown

individually a video-recording of Step 1. Beforehand, the pupils were asked to stop

the video to give them the chance to express their ideas, explanations or difficulties

they had while working on the problem. Additionally, I stopped the video at positions

I wanted to know what they where thinking. This grade of intervention (see

Schoenfeld, 1985) could be justified by the fact that especially through these

enquiring questions I received quite a lot of important information which otherways I

would not have got. This step of stimulated recall was tape-recorded.

Step 3: Individual interview. Each student was interviewed directly after the

stimulated recall. The individual interview was devided into two parts: In part one

they were asked about the problem-solving process and their judgements concerning

the items to be solved. In part two the questions focussed on their image of

mathematics: For example what the pupils' understanding of mathematics is, what

are their preferences or aversions of mathematical topics in school. Data on the image

of mathematics were collected in order to investigate, how stable the reconstructed

thinking style preferences of an individual are. These interviews were tape-recorded

too.

4. Analysing the data

The exclusively verbal data and the pupils' products from this study are analysed and

encoded according to the Grounded Theory: at first by open coding, then by axial

coding and finally by a selective coding. Encoding the data is the basic strategy for

decomposing – or breaking off – the data and then recomposing them in a new way.

Therefore, it is the central process through which theories are developed out of the

data. Before starting the encoding procedure the problem-solving processes of all

pupils must be reconstructed in a sequential way in order to get a better

understanding of the thinking processes. This first step enabled me to divide the

solving process into the following 5 phases: (1) Reading and understanding a

problem (2) First ideas and impressions (3) Searching for ideas (4) Creating solutions

(5) Results and checking. Through these phases the pupils could be compared, in

general and within each phase of the solving processes. By reconstructing the

problem-solving processes I received 4 dimensions which then were used to develop

codes:

1. Internal imagination of a person ("in the brain", "inner eye") while trying and

solving a problem

2. External representation of mathematical facts through a person

3. Wholist – Analyst way of thinking and procedure

Thematic Group 3

EUROPEAN RESEARCH IN MATHEMATICS EDUCATION III

R. Borromeo Ferri 5

4. Image of mathematics as confirmation of stabilized preferences of thinking

styles

These phases of the problem-solving process together with the above listed 4

dimensions served as codes during the open coding phase. The open coding phase is

that part of the analysis which especially refers to the naming and categorising of

phenomena. (see Strauss & Corbin, 1996: 44). In this study the data were

decomposed and categorised by a line-per-line analysis and by comparing single

events, which in the following were categorised together as one similar phenomenon,

for instance as phenomenon which describes an individual's internal imagination

which was often carried out with the help of stimulated recall. Here an example of

Sylvia, 15 years, grade 9 with her first idea to solve one item of the examination with

her internal pictorial imagination

S:" I had directly an pictorial imagination, the numbers were not important for me, I must have an

pictorial imagination."

During the axial coding procedure the data will be put together in a new way, in

which the connections of one code to a sub-code will be investigated. Actually, sub-

codes are codes too, but they specify the main codes more exactly.

Again, Strauss & Corbin emphazise the connection between open and axial coding:

"Although open and axial coding are separate ways of analysis, the researcher is

changing between these two modi during the analysis." (ibid 1996: 77). Referring to

this study, here an example: the internal imaginations of a person find their

expression in various ways in the data. Pupils told that they had strong pictorial

imaginations or that they used more mathematical symbols or terms in their

imaginations to solve a problem such as Jenny, 15 years, grade 9:

J: " I had numbers in my mind, no pictures."

These sub-codes can be 'dimensionalised' further during the researcher's analysing

process, for example into static and dynamic imaginations of the pupils. This putting-

into-relation of a sub-code to a code is done by using questions which describe a form

of relationship. Therefore axial coding represents a complex process of inductive and

deductive thinking.

Finally, selective coding is the process of choosing the central category as well as the

systematically putting-into-relation of the central category to other categories, the

validation of the received relations and the refilling of the categories which then need

to be refined and developed further (see Strauss & Corbin, 1996: 94).

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R. Borromeo Ferri 6

All data of this study are coded with the help of the softare-tool ATLAS.ti, so that an

overview about fixed codes, sub-codes and the creation of code families will make

the analysis comprehensible easily.

5. Reconstructing mathematical thinking styles

As mentioned above and illuminated under methodological aspects in the chapter

before, there are 4 dimensions serving as base for developing codes. Through the

encoding process these dimensions shall help to clarify more deeply the question

what does it mean to practice an analytic or a mixed style of thinking. However, one

should not decide too quickly in classifying a pupil as being exclusively a visual or

analytic thinker just because he or she uses a pictorial demonstration while solving a

problem. From my analyses I learned, that in this regard it must be distinguished

more exactly, if, for instance, pupils put down a graphical demonstration only

because teachers told them to do so and not because that moment they tried to

visualize. For this reason it is important to look at the internal imagination and at the

externalised presentation as well. Different internal imaginations are already

mentioned above in chapter 4. In this context Skemp (1987) is cited, who

distinguishes verbal-algebraic and visual symbols.

The 3

rd

dimensions is about the way pupils are structuring their thinking and the

information during the problem-solving process. This can either be a done in a

wholistic or an analytical way, or if settled between these two extremes, in a mixed

way. This wholist-analytic dimension is more related to an individual's imagination

and will be reconstructed through the problem-solving process in which internal

imaginations are taken into account. By this it shall be found out, if for example, a

person prefers to adapt a visual thinking style, but nevertheless, simultaneously

follows the analytic way. Riding (2001) also distinguishes the dimensions 'wholist'

and 'analytic' which I refer to in my descriptions:

„Wholists see a situation as a whole and are able to have an overall perspective, and

to appreciate its total context. By contrast, analytics see a situation as a collection of

parts and often focus on one or two aspects of the situation at a time to exclusion of

the others. Intermediates are able to have a view between the extremes, which should

allow some of the advantages of both."(Riding 2001: 55-56)

The 4

th

dimension is about the pupils' image of mathematics. It is aimed to stabilise

the found out preferences for one (ore more) thinking style.

6. Results up to now

The results show that distinct preferences for a visual and analytic thinking style, the

so-called "polar-types" can be reconstructed with 15 and 16 year old pupils, but not

distinct preferences for the conceptual thinking style (following Burton

"Conceptual", following Klein "philosopher"). The conceptual thinking style could

only be reconstructed in connection with the other two thinking style, as a so-called

Thematic Group 3

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R. Borromeo Ferri 7

"mixed type". Besides this, other "mixed-types" with only two thinking styles could

be reconstructed. (see Borromeo Ferri, 2002 and 2003)

From my 12 participants 3 girls are "polar types" of the visual thinking style, one girl

and one boy are "polar types" of the analytic thinking style. There's one boy who is a

"mixed type" of the three thinking styles and the remaining pupils are "mixed-types"

of the analytic and visual thinking style. The "mixed types" show a higher degree of

flexibility in solving problems during the examination, as well as in their descriptions

in school about their approach to an item and on how to work on it. The results show

that a learner's internal imaginations must not correspond with his external

representations. Following two quotations of two girls of my research work who are

"polar types" of the analytic and visual thinking style:

Saskia, 16 years, grade 10, "polar type" of the analytic thinking style":

"Yes, one always must think in formulae, I think, because somehow mathematics always has to do

with formulae, even if a teacher does not say so at the beginning, it always has to do with

mathematical formulae!"

Sylvia, 15 years, grade 10, "polar type" of the visual thinking style:

"I only memorize formulae and mostly I didn't understand them, because it does't help me much,

but I know how it pictorial belongs together, but by formulae, no, I can't cope with that."

With reference to the present results of the study I can give the following description

of the concept of "mathematical thinking style":

A mathematical thinking style is an individual's preferred way of thinking and

understanding mathematical facts and connections by using various internal

imaginations and externalised representations.

Due to one's mathematical socialisation, an individual's mathematical thinking style

finds its expression more or less clearly in certain mathematical topic areas, in which

a dependence on items an context is concluded.

Mathematical thinking styles are not the same as structures, but they can help to build

structures in knowledge. Each individual gives preference to one's own thinking style

by which he or she is able to understand mathematical facts and contexts. These

individual preferences help to establish structures within one's knowledge. The

structures are created gradually by the fact that young individuals think through again

mathematical facts, so that the structure of knowledge extends continuously.

8. Conclusions

This paper is aimed to show one way how different mathematical thinking styles can

be reconstructed by applying the methodology of the Grounded Theory and what are

the underlying theoretical approaches of this study. Furthermore, these results

obtained up to present indicate a highly didactical relevance of this kind of study: Its

significance for mathematics lessons is obvious. Pupils who are not sharing the

mathematical thinking style with their teacher may have problems of understanding,

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R. Borromeo Ferri 8

but if the teacher is conscious of his own style and arranges mathematical facts in

different ways, problems of understanding could be prevented.

"My previous teacher explained fast and much and did not make any drawings and then one time I

got a six

2

for a maths test and then I only got a four and then I thought I don't know maths. […]

And that was I couldn't cope with. My new teacher always makes a drawing and now I understand

how to come to the result, not like only by formulae and calculation and for the first and third test I

got a one." Vera, 15 years, grade 9

These results correlate with results from other empirical studies, among others that of

Zhang & Sternberg (2001), who pointed out:" Findings from a third study indicated

that teachers inadvertently favoured those students whose thinking styles were similar

to their own" (2001: 204). Therefore, it is necessary that teachers become conscious

about their own thinking style, on the one hand in order to guarantee equality of

chances among pupils, and on the other hand to develop their own mathematical

potentials.

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... Mathematical thinking can be enhanced by solving problems carefully, transferring the outcomes to the experiences, establishing relations between what is thought and what is applied, making exercises on problem solving processes and making sense of the relation between mathematics and real life (Keith, 2000). In other words, mathematical thinking can be improved by addressing the problem and scrutinizing it from multiple perspectives beyond considering what the answer to the problem is (Ferri, 2003). Quite effective solutions can be achieved by observations on or interviews about the problem solving processes of the students to determine and understand how they defined a problem, and to evaluate their current knowledge (Kükey, 2018). ...

... Matematiksel düşünme, problemlerin dikkatli olarak çözülmesi, elde edilenlerin deneyimlere aktarılması, düşünülenlerle uygulamalar arasında bağlantı kurulması, problem çözme süreçleri üzerinde uygulamalar yapılması ve matematikle gerçek hayat arasındaki ilişkinin anlaşılmasıyla geliştirilebilir (Keith, 2000). Yani matematiksel düşünme, problem çözme sürecinde problemin cevabının ne olduğundan öte, problemin çeşitli boyutlarıyla ele alınıp incelenmesiyle geliştirilebilir (Ferri, 2003). Öğrencilerin bir problemi nasıl tanımladıklarını belirlemek, anlamak ve var olan bilgilerini değerlendirmek amacıyla, problem çözümlerine yönelik olarak gözlem veya görüşme yapılmasıyla oldukça etkili çözümlere ulaşılabilmektedir (Kükey, 2018). ...

  • Ebru KÜKEY
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  • Tayfun Tutak Tayfun Tutak

Bu çalışma, ortaokul öğrencilerinin matematiksel düşünmenin varsayımda bulunma bileşeni kapsamında kullanmış oldukları problem çözme stratejilerini incelemek amacıyla yapılmıştır. Araştırma nitel araştırma yöntemlerinden durum çalışması olarak tasarlanmıştır. Çalışma, 96 ortaokul öğrencisi ile yürütülmüştür. Araştırmada veriler, matematiksel düşünmenin varsayımda bulunma bileşenine yönelik olarak hazırlanan rutin olmayan problem ile elde edilmiş ve elde edilen verilerin analizi aşamasında içerik analizi kullanılmıştır. Öğrencilerin problemi çözerken kullanmış oldukları stratejiler incelendiğinde, denklem kurma ile tahmin ve kontrol stratejilerini kullandıkları belirlenmiştir. Bu doğrultuda öğrencilerin problem çözme stratejilerinden iki tanesini kullandıkları görülmüştür. Bu nedenle diğer problem çözme stratejileri de kullanabilmeleri için öğretim sürecinde bu noktaya dikkat edilmesi gerektiği düşünülmektedir.

... Setiap murid mempunyai kecenderungan yang berlainan dalam gaya pemikiran matematik, sama ada secara visual atau pun analitik. Proses penstrukturan ini seterusnya secara berperingkat akan menggalakkan pemikiran murid terhadap fakta atau formula matematik, maka struktur pengetahuan yang terbina sebelumnya akan terus berkembang seiring pembelajaran murid (Ferri, 2015). Seterusnya Stacey (2014) menyimpulkan bagaimana proses matematik murid ini amat penting lantaran ia menyediakan murid dengan kemampuan untuk memanfaatkan pengetahuan matematik yang dipelajari, sekali gus boleh menjadi hasil pembelajaran bermakna yang dibawa selepas habis alam persekolahan. ...

Proses matematik merupakan kemahiran penting yang perlu dikuasai murid sebagai hasil pembelajaran matematik. Murid dengan kemahiran proses matematik yang baik boleh mengorganisasikan struktur pengetahuan mereka dengan menghubung kait, mewakilkan, berkomunikasi secara matematik, menaakul serta menyelesaikan masalah. Pengetahuan berkaitan tahap kemahiran proses matematik murid bukan sahaja memberi maklumat berkaitan pembelajaran murid, malah ia membolehkan pelbagai langkah intervensi dirangka, serta turut menjadi bukti empirikal kepada pembuat dasar. Namun demikian, bagi mengukur kemahiran proses matematik murid ini, tidak adil bagi murid-murid untuk dinilai secara total sebagai betul atau salah sahaja. Murid juga perlu diberi kredit bagi usaha atau proses yang berjaya ditunjukkan, walaupun belum dapat mencapai jawapan yang tepat. Justeru, artikel ini bertujuan untuk menyumbang literatur kajian berkaitan model-model pembangunan instrumen khususnya rubrik dan tugasan pentaksiran. Penelitian terhadap teori, model dan standard berkaitan akan mencadangkan satu kerangka pembangunan instrumen bagi mengukur proses matematik murid yang disasarkan.

... 2021, 11, 289 3 of 15 and illustrated were not reconstructed with pupils at school. Based on an empirically grounded description, the different thinking styles are defined as follows [8,15]: ...

School is a space where learning mathematics should be accompanied by the student's preferences; however, its valuation in the classroom is not necessarily the same. From a quantitative approach, we ask from the mathematical thinking styles (MTS) theory about the correlations between preferences of certain MTS and mathematical performance. For this, a valid test instrument and a sample of 275 16-year-old Chilean students were used to gain insight into their preferences, beliefs and emotions when solving mathematical tasks and when learning mathematics. The results show, among other things, a clear positive correlation between mathematical performance and analytical thinking style, and also evidence the correlation between self-efficacy, analytical thinking and grades. It is concluded that students who prefer the analytical style are more advantageous in school, since the evaluation processes have a higher valuation of analytic mathematical thinking.

... Yansıtıcı düşünme becerisi sayesinde birey; duygularını, hislerini, tutumlarını, düşünce ve değerlendirmelerini yansıtarak problemlere bulduğu çözüm yollarını görmekte ve çözümün uygunluğunu değerlendirerek tekrar yapılandırma yapmaktadır (Ersözlü, 2008). Ferri (2003) de bu görüşe paralel olarak yansıtıcı düşünme becerisinin bir problemle karşılaşıldığında cevabının ne olduğunu bulmaktan öte, problemin çeşitli boyutlarıyla ele alınarak incelenmesi olarak ifade edildiğini belirtmiştir. Mezirow'a (1991) göre ise yansıtıcı düşünme becerisi, problem çözme becerisi ya da ortaya konulan varsayımların değerlendirilmesini içermektedir. ...

... Their previous study discovered that the difference between the successful and less successful students was on the interpretation in mathematical problems solving task which is closely related to thinking style. . Research indicated that teaching and learning styles contribute a big impact onthinking style (Borromeo Ferri, 2004;Cilliers & Sternberg, 2001;Grigorenko & Sternberg, 1997). Sternberg and Grigorenko (1993) found that certain thinking styles correlated positively to student's achievement in a several learning domains, whereas other thinking styles tended to correlated negatively to achievement in the same domain. ...

Algebra is regarded as one of the primary learning domains as it represents the fundamental entry knowledge to higher forms of learning in mathematics, science, technology and engineering. The acquisition of Algebraic knowledge depends greatly on learners' Mathematical Thinking Style (MTS). Therefore, this study aimed to investigate the university students' MTS through the algebraic problem-solving topic. In general, the MTS was composed of analytical, visual, and integrated thinking styles. In this research, mixed method research design was adopted in which questionnaire and assessment test were used for data collection. A total of 248 engineering students in Universiti Tun Hussein Onn Malaysia were involved in the data collection process. Descriptive statistics was used to analyse the quantitative data obtained from the questionnaire, whereas the written answers from the assessment test were analysed using document analysis technique. The findings indicated that majority of the participating students perceived that they commonly practiced the Analytical Thinking Style in learning Algebra. However, analysis on the students' problem solving steps in the assessment test revealed that the students were actually applying Visual Thinking Style in Algebraic problem solving tasks instead of Analytical Thinking Style. This study revealed that there was a difference between students' perception and the analysis outcomes from the assessment test which reflected the actual MTS of the students. This difference reflects the fact that how an individual perceives his behaviour might not be consistent with his actual behaviour. In conclusion, instructors should know the actual MTS of the students instead of finding out students' perception on their thinking style. This will help instructors design more relevant learning activities that beneficial to students.

... Yansıtıcı düşünme becerisi sayesinde birey; duygularını, hislerini, düşüncelerini yansıtarak problemlere bulduğu çözüm yollarını görmekte ve çözümün uygunluğunu değerlendirerek tekrar yapılandırma yapmaktadır (Ersözlü, 2008). Ferri (2003) de bu görüşe paralel olarak yansıtıcı düşünme becerisinin, bir problemle karşılaşıldığında probleme cevap bulmaktan öte, onun çeşitli boyutlarıyla ele alınıp incelenebilmesi olduğunu belirtmiştir. Shermis'e (1992, s.51) göre "Yansıtıcı düşünme becerisinin amacı bir problemle karşılaşıldığında en uygun çözüm yolunu bulmaktır ve yansıtıcı düşünme becerisi en iyi problem çözme sürecinde gözlenebilmektedir". Mason (2009) yansıtma yapmanın öğrenene problemin çözüm süreci boyunca yaptığı hatalarını belirleme imkânı verebildiğini belirtmektedir. ...

Araştırmanın amacı matematik problemlerini çözmede başarılı 8. sınıf öğrencilerinin problem çözmeye yönelik yansıtıcı düşünme becerilerinin incelenmesidir. Bu bir durum çalışmasıdır. Araştırmanın amacına en uygun çalışma grubunun başarılı problem çözücüler olduğu ve bu öğrencilerden zengin veri elde edilebileceği düşünüldüğünden dolayı amaçlı örneklem yoluna gidilmiştir. Araştırmanın çalışma grubunu matematik problemlerini çözmede başarılı üç 8.sınıf öğrencisi oluşturmuştur. Veri toplama aracı olarak kaynak taraması sonucu oluşturulan problem çözme etkinlikleri (KTPÇE), bağlamları değiştirilen problem çözme etkinlikleri (BADPÇE), yarı yapılandırılmış görüşme formu, video ve ses kaydı kullanılmıştır. Elde edilen veriler betimsel analiz yaklaşımı kullanılarak çözümlenmiştir Öğrenciler genel olarak belirlenen temalar çerçevesinde başarılı bir şekilde yansıtma yapmakla birlikte, öğrencilerin belirlenen göstergeler dâhilinde yansıtma yapmakta zorlandıkları veya eksik yansıtma yaptıkları olmuştur. Öğrencilerin yansıtma yapmakta en çok zorlandığı şey, seçtikleri stratejiye bir isim vermek olmuştur. Öğrencilerin yansıtmaları incelendiğinde, buldukları sonucun doğru olması ile mantıklı olmasını aynı şeymiş gibi düşündükleri görülmüştür. Öğrenciler verilen probleme benzer problem ile önceden karşılaşmış olmanın işlerini kolaylaştırdığını belirtip, deneyim boyutu ile ilgili başarılı yansıtmalar yapmışlardır. Öğrenciler grup çalışmasının farklı çözüm yolları ortaya koyabilme, bulunan sonuçtan daha emin olabilme gibi olumlu yanları olduğunu düşünmekle birlikte, ortak bir çözüm yolunu bulmanın uzun sürmesinden hoşlanmadıklarını belirtmişlerdir. İki öğrenci problemi bireysel çözerken daha mutlu olduklarını belirtirken bir öğrenci de farklı fikirler olduğundan dolayı grupla problem çözmenin daha iyi olduğunu belirtmiştir. Öğrenciler seçtikleri farklı çözüm yollarından kendisine göre daha iyi olanı nedenleriyle birlikte ifade edebilme konusunda genel olarak başarılı şekilde yansıtma yapmışlardır.

The aim of this study is to investigate the effect of HTTM learning environment on the preservice science teachers' perceptions of mathematical thinking and mathematical modelling skills. The sample of the study was comprised of the 27 students studying at science education department of a state university. The study was designed around one-group pretest-posttest quasi-experimental method and an intervention including HTTM learning environment was implemented in the experimental group. The data collection tools consisted of mathematical thinking scale, Alexandria Lighthouse HTTM activity and mathematical modelling rubric. Descriptive and inferential statistics techniques were used in the analysis of quantitative data. According to the results obtained, the HTTM learning process has improved the pre-service science teachers' perceptions of mathematical thinking in terms of both general level and dimensions (high level thinking, reasoning, mathematical thinking skills, problem solving). Likewise, HTTM learning process has contributed to the preservice teachers' mathematical modeling skills in terms of both general level and dimensions (understanding the problem, determining the essential strategic factors in the problem, creating assumptions, using mathematical symbols appropriately, determining the necessary mathematical concepts, creating effective problem solving strategy, creating appropriate mathematical models, reaching the desired solution and different results from mathematical models, interpretation of the results according to real world situations, trying verify the obtained results in different ways). Accordingly, it can be ensured that teachers or prospective teachers are exposed to HTTM learning process to improve their technological pedagogical content knowledge and skills. / Bu çalışmanın amacı, HTTM (History/Theory/Technology/Modeling) öğrenme ortamının fen bilgisi öğretmeni adaylarının matematiksel düşünmeye ilişkin algılarına ve matematiksel modelleme becerilerine etkisinin incelenmesidir. Araştırmanın katılımcılarını bir devlet üniversitesinin fen bilgisi öğretmenliği programının bir şubesinde öğrenim gören 27 öğrenci oluşturmaktadır. Ön test-son test tek gruplu yarı-deneysel yöntemin benimsendiği araştırmada deney grubu ile HTTM öğrenme ortamını içeren bir eğitim gerçekleştirilmiştir. Veri toplama araçları; matematiksel düşünme ölçeği, İskenderiye Deniz Feneri HTTM etkinliği ve matematiksel modelleme rubriğidir. Nicel verilerin analizinde betimsel ve vardamsal (yordamsal) istatistik tekniklerinden yararlanılmıştır. Elde edilen sonuçlara göre, HTTM öğrenme süreci, fen bilgisi öğretmeni adaylarının matematiksel düşünmeye ilişkin algılarını hem genel hem boyutlar (üst düzey düşünme, akıl yürütme, matematiksel düşünme becerisi, problem çözme) düzeyinde geliştirmiştir. Benzer şekilde, HTTM öğrenme süreci öğretmen adaylarının matematiksel modelleme becerilerini hem genel hem boyutlar (problemi anlamlandırma, problemdeki gerekli stratejik etkenleri ortaya koyma, varsayımlar oluşturma, matematiksel sembolleri uygun bir şekilde kullanma, gerekli matematiksel kavramları belirleme, etkili problem çözme stratejisi ortaya koyma, uygun matematiksel modelleri oluşturma, matematiksel modellerden istenen çözüme ve farklı sonuçlara ulaşma, elde ettiklerini gerçek yaşam durumuna göre yorumlama, elde ettiklerini farklı yollarla doğrulamaya çalışma) bazında geliştirmiştir. Bu doğrultuda öğretmen veya öğretmen adaylarının teknolojik pedagojik alan bilgi ve becerilerinin geliştirilmesi için HTTM öğrenme süreci ile baş başa bırakılmalarının önemli olacağı söylenebilir.

  • Mehmet Kasım Koyuncu Mehmet Kasım Koyuncu

When the strong relationship between mathematics and philosophy, behavioral objectives from the secondary mathematics education curriculum of Ministry of National Education and the goal of the philosophy of mathematics is considered, it is realized that there is no scientific research scrutinizing the philosophy of mathematics activities. From this perspective, it is important to investigate the effects of these activities on the attitudes and beliefs towards mathematics. This study has two aims. One of them is conceptualize the Philosophy of Mathematics Activity and the other one is specify the effects of these activities on 9th grade students' attitudes and beliefs towards mathematics. As the quantitative research methodology pretest-posttest control- experimental design, as the qualitative research methodology, phenomenological study formed the methodology. At the end of the treatment semi-structured interviews were held to collect further data about attitudes and beliefs. At the end of the analyses, it is demonstrated that Philosophy of Mathematics Activities increases students' attitudes and beliefs towards mathematics.

  • Mehmet Kasım Koyuncu Mehmet Kasım Koyuncu

Considering the close relationship between mathematics and philosophy, the aims of philosophy of mathematics and the attitudes aimed to acquire individuals as stated in the general objectives of the secondary education mathematics curriculum of Ministry of National Education, no research has been made with respect to mathematics instruction informed by the activities of philosophy of mathematics, as understood from the review of the literature. It thus matters to investigate the impact of mathematics instruction through activities of philosophy of mathematics on such variables as belief, attitude, and mathematical thinking. The present study aimed to investigate the impact of mathematics instruction through activities complied with philosophy of mathematics on attitude towards mathematics, mathematical belief, and mathematical thinking of students in the 9th grade math class at high school. Adopting a mixed-type research design, the quantitative data of the study were collected from pretest and posttest conducted with the experimental and the control groups, while the qualitative data were obtained from phenomenology analysis. At the end of the treatment the students in the experimental group were individually interviewed using semi-structured interview forms. Additionally, students' study papers were utilized in order to examine the consistency of research findings. During statistical analysis of the quantitative data, dependent and independent t-test were employed, while phenomenological analysis method was applied for qualitative data the findings of which are as follows. The study was conducted with two 9th grade classes of a Ministry of National Education-affiliated private Science High School in the second semester of 2016-2017 academic year. It was carried out during six weeks with the students of the aforementioned school. In the experimental group where mathematical philosophy activities (treatment) were implemented were 8 female, 7 male students, and in the control group where conventional teaching practice took place were 8 female and 7 male students, as well. The Mathematics Attitude Scale Baykul (1990), was used to determine the participants' attitude towards math. The Mathematics Belief Scale Kandemir (2011), was utilized to identify the participants' beliefs, and lastly the Mathematical Thinking Scale Ersoy (2012), was used to evaluate the students' mathematical thinking. The activities of philosophy of mathematics which were carried out with the experimental group were found to be effective in increasing the attitude towards mathematics and mathematics belief, while it was found ineffective in changing mathematical thinking. The findings indicated that learning environment supported with activities of philosophy of mathematics contributed to the students' attitude towards and belief of math at a significant level, whereas it did not lead to a significant change in the students' mathematical thinking. Since the activities of philosophy of mathematics guided the students to the bright world of math that they had never heard before, and the students started to look from quite a different perspective to the questions that they used to solve through formal ways, and to math itself, the experimented activities were found effective in increasing attitude towards mathematics, and belief of mathematics.

Bu araştırmanın amacı, matematik problemlerini çözmede başarılı öğrencilerin problem çözmeye yönelik yansıtıcı düşünme becerilerinin incelenmesidir. Araştırmada, nitel araştırma modellerinden biri olan özel durum çalışması kullanılmıştır. Araştırmanın çalışma grubunu matematik problemlerini çözmede başarılı üç öğrenci oluşturmuştur. Veri toplama aracı olarak bireysel problem çözme etkinlikleri, grup problem çözme etkinlikleri, yarı yapılandırılmış görüşme formu, video ve ses kaydı kullanılmıştır. Elde edilen veriler betimsel analiz yaklaşımı kullanılarak çözümlen-miştir. Öğrencilerin süreç boyunca nasıl yansıtma yaptıklarını incelemek için Hong ve Choi'nin (2011) çalışmalarında oluşturmuş olduğu üç boyutlu kuramsal çerçevenin yansıtmanın konusu ile ilgili olan bilgi, deneyim, his/duygu, bağlam ve grup arkadaşı şeklindeki beş teması alınmıştır. Araştırma sonucunda öğrencilerin bilgi, deneyim ve bağ-lam temalarında başarılı yansıtmalar ortaya koydukları belirlenmiştir. Buna ek olarak, öğrenciler his/duygu ve grup arkadaşı temalarında da yansıtmalar yapabilmiştir. Tüm bunlara rağmen, öğrencilerin çeşitli göstergeler dâhilinde yansıtma yapmakta zorlandıkları, eksik yansıtma yaptıkları veya herhangi bir yansıtma yapamadıkları durumların olduğu da görülmüştür. Anahtar Kelimeler: problem çözme, yansıtıcı düşünme becerisi, yansıtma

INTRODUCTION My previous teacher explained fast and much and did not make any drawings and then one time I got a six 1 for a maths test and then I only got a four and then I thought I don't know maths. And if I don't understand just a bit of it I understand nothing at all and I would like to understand everything from the beginning, each small step the teacher does. And that was I couldn't cope with. My new teacher always makes a drawing and now I understand how to come to the result, not like only by formulae and calculation and for the first and third test I got a one. Sarah, 15 years (Original in German translated by authors) From this student's statement, it is clear that Sarah has certain preferred kinds of explanations for mathematical methods. One can infer, from this, that there are kinds of explanations which cause young people to understand mathematical methods well and others through which they only understand a little. Sarah shows that explanations of mathematical facts through drawings, meaning pictorial representations, are apparently much more helpful to her than an exclusively analytic-algorithmically way of proceeding. Of course, this does not mean that the teacher's explanation was faulty or bad. On the contrary this statement suggests another question: Has Sarah been disadvantaged by the way of teaching and thinking of her previous teacher because her own way of thinking did not correspond with that of the teacher ? How many girls and boys of her class experienced the same problem, or are there girls or boys who do not cope with the teacher's thinking style? Traditionally many psychologists and pedagogues share the opinion that success and failure of learning are exclusively caused by individually different learning abilities, although the question how it comes that pupils of one teacher are considered as being intelligent and of another one as being not intelligent remains still unanswered. Similarly, it remains still unanswered why the same pupil produces mediocre results in a multiple-choice task while within a project he or she produces extraordinary results. During the last decades, empirical cognitive-psychological studies made it clear that thinking styles, learning styles and cognitive styles are strongly influencing performance in many fields, and therefore are fundamentally determining pupils' great differences in performance. Sternberg & Zhang (2001) point out that these new approaches which clearly distinguish thinking and learning styles and cognitive styles from the construct of abilities, make prediction or explanation of academic success possible in ways that go far beyond the concept of 'ability'. Furthermore, they point out the pedagogical relevance of these approaches: "They (the styles, the authors) are also of interest, because when teachers take styles into account, they help improve both instruction and assessment. Moreover, teachers 1 Six is the worst mark in a one-to-six assessment scale usual in German teaching.

  • Alan H Schoenfeld Alan H Schoenfeld

Discusses 5 classes of variables that affect the generation of verbal reports—number of persons being taped, degree of intervention, the nature and degrees of freedom in instructions and intervention, the nature of the environment and how comfortable the S feels in it, and task variables. A case study in which 2 students work together on a problem illustrates the trade-offs in analyzing verbal data, focusing on the advantages and disadvantages of cognitive processes in this methodology. (23 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)

  • Merrilyn Goos Merrilyn Goos

The study described in this paper investigated the metacognitive strategies used by a pair of senior secondary school students while working together on mechanics problems. Verbal protocols from think-aloud paired problem-solving sessions were analysed in order to examine the monitoring contributions of each individual student, and the significance of student-student interactions. Although the students were generally successful in coordinating their different, yet complementary, problem-solving roles, their metacognitive decision making was sometimes adversely affected by the social interaction between them. The findings suggest some potential benefits and pitfalls of using small group work for problem solving.

Examines the findings of a series of nine studies conducted on university students or secondary school students (two studies) from mainland China, Hong Kong, and the US to assess thinking styles and their relationships to student learning. One measure of thinking styles (Thinking Styles Inventory), one of learning approaches (Study Process Questionnaire), and one of self-esteem (Coopersmith Self-Esteem Inventory), and a self-rating of abilities were administered to a total of 3,043 Ss. The major contribution of the studies presented is that they show that thinking styles as defined by the theory of mental self-government do matter in that they contribute to academic achievement above self-rated abilities; thinking styles are closely related to students' learning approaches; thinking styles significantly relate to students' characteristics; and thinking styles statistically relate to self-esteem. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

  • Richard Riding

Begins with a consideration of the labels used by investigators to describe cognitive style and proposes a categorization of these into two fundamental style dimensions: the wholist-analytic and verbal-imagery. Methods of assessing style are outlined and a simple direct method is described. Style is then examined within the context of other individual difference variables such as intelligence, gender, and personality to establish its independence of these dimensions. The bipolar nature of cognitive style, which distinguishes it from unipolar dimensions of individual differences such as intelligence, is detailed. The evidence for the physiological aspects of style is described. Next, the relation between the style dimensions and a range of behaviors relevant to education is reported and practical implications are explored. Finally, a model of style within the context of other individual difference variables is considered. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

provide the reader with a background in the literature on [thinking] style research and . . . discuss more recent developments in this area / focuses primarily on a hypothesis of thinking styles called "the theory of mental self-government" the 1st part consists of a review of a variety of theories of styles, followed in the 2nd part by a more detailed discussion of the theory of mental self-government / explore the different aspects of styles emphasized in different areas of psychology / focus on the relation between styles and cognition [cognition-centered approach] / refer back to the pioneer approaches to styles in cognitive psychology as well as to more recent developments / personality psychologists study styles in relation to other individual personality characteristics; we refer to this approach as the "personality-centered approach" / discuss [the] "activity-centered" approach, which focuses on styles in relation to various activities, settings, and environments / this approach is primarily found in educational settings and includes theories of learning and teaching skills present the theory of mental self-government (R. Sternberg, 1988) / Sternberg's theory acts as a wide-angle lens as it combines the thinking of the cognition-centered, personality-centered, and activity-centered traditions / analyze 3 major conceptualizations of cognitive styles / field dependence-independence / cognitive styles in categorization behavior / reflection-impulsivity (PsycINFO Database Record (c) 2012 APA, all rights reserved)

  • Leone Burton

This paper reports on the results of a study of the epistemologies of seventy research mathematicians utilising a model containing five categories, socio-cultural relatedness, aesthetics, intuition, thinking style and connectivities. The perspectives of the mathematicians demonstrate extreme variability from one to another but certain persistent themes carry important messages for mathematics education. In particular, although mathematicians research very differently, their pervasive absolutist view of mathematical knowledge is not matched by their stories of how they come to know, nor of how they think about mathematics.

  • Leone Burton

There is, now, an extensive critical literature on gender and the nature of science three aspects of which, philosophy, pedagogy and epistemology, seem to be pertinent to a discussion of gender and mathematics. Although untangling the inter-relationships between these three is no simple matter, they make effective starting points in order to ask similar questions of mathematics to those asked by our colleagues in science. In the process of asking such questions, a major difference between the empirical approach of the sciences, and the analytic nature of mathematics, is exposed and leads towards the definition of a new epistemological position in mathematics.