Essentials Of Mathematical Thinking Pdf

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WHAT IS MATHEMATICAL THINKING

AND WHY IS IT IMPORTANT?

Kaye Stacey

University of Melbourne, Australia

INTRODUCTION

This paper and the accompanying presentation has a simple message, that

mathematical thinking is important in three ways.

• Mathematical thinking is an important goal of schooling.

• Mathematical thinking is important as a way of learning mathematics.

• Mathematical thinking is important for teaching mathematics.

Mathematical thinking is a highly complex activity, and a great deal has been written

and studied about it. Within this paper, I will give several examples of mathematical

thinking, and to demonstrate two pairs of processes through which mathematical

thinking very often proceeds:

• Specialising and Generalising

• Conjecturing and Convincing.

Being able to use mathematical thinking in solving problems is one of the most the

fundamental goals of teaching ma thematics, but it is also one of its most elusive goals.

It is an ultimate goal of teaching that students will be able to conduct mathematical

investigations by themselves, and that they will be able to identify where the

mathematics they have learned is applicable in real world situations. In the phrase of

the mathematician Paul Halmos (1980), problem solving is "the heart of

mathematics". However, whilst teachers around the world have considerable

successes with achieving this goal, especially with more able students, there is always

a great need for improvement, so that more students get a deeper appreciation of what

it means to think mathematically and to use mathematics to help in their daily and

working lives.

MATHEMATICAL THINKING IS AN IMPORTANT GOAL OF

SCHOOLING

The ability to think mathematically and to use mathematical thinking to solve

problems is an important goal of schooling. In this respect, mathematical thinking

will support science, technology, economic life and development in an economy.

Increasingly, governments are recognising that economic well-being in a country is

underpinned by strong levels of what has come to be called 'mathematical literacy'

(PISA, 2006) in the population.

Mathematical literacy is a term popularised especially by the OECD's PISA program

of international assessments of 15 year old students. Mathematical literacy is the

ability to use mathematics for everyday living, and for work, and for further study,

and so the PISA assessments present students with problems set in realistic contexts.

The framework used by PISA shows that mathematical literacy involves many

components of mathematical thinking, including reasoning, modelling and making

connections between ideas. It is clear then, that mathematical thinking is important

in large measure because it equips students with the ability to use mathematics, and

as such is an important outcome of schooling.

At the same time as emphasising mathematic s because it is useful, schooling needs to

give students a taste of the intellectual adventure that mathematics can be. Whilst the

highest levels of mathematical endeavour will always be reserved for just a tiny

minority, it would be wonderful if many students could have just a small taste of the

spirit of discovery of mathematics as described in the quote below from Andrew

Wiles, the mathematician who proved Fermat's Last Theorem in 1994. This problem

had been unsolved for 357 years.

One enters the first room of the mansi on and it's dark. One stumbles around bumping

into furniture, but gradually you learn where each piece of furniture is. Finally, after six

months of so, you find the light switch, you turn it on, and suddenly it's all illuminated.

You can see exactly where you were. Then you move into the next room and spend

another six months in the dark. So each of these breakthroughs, while sometimes they're

momentary, sometimes over a period of a day or two, they are the culmination of, and

couldn't exist without, the many months of stumbling around in the dark that precede

them. (Andrew Wiles, quoted by Singh, 1997, p236, 237)

At the APEC meeting in Tokyo in January 2006, Jan de Lange spoke in detail about

the use of mathematics to equip young people for life, so I will instead focus this

paper on two other ways in which mathematical thinking is important.

WHAT IS MATHEMATICAL THINKING?

Since mathematical thinking is a process, it is probably best discussed through

examples, but before looking at examples, I briefly examine some frameworks

provided to illuminate mathematical thinking, going beyond the ideas of

mathematical literacy. There are many different 'windows' through which the

mathematical thinking can be viewed. The organising committee for this conference

(APEC, 2006) has provided a substantial discussion on this point. Stacey (2005)

gives a review of how mathematical thinking is treated in curriculum documents in

Australia, Britain and USA.

One well researched framework was provided by Schoenfeld (1985), who organised

his work on mathematical problem solving under four headings: the resources of

mathematical knowledge and skills that the student brings to the task, the heuristic

strategies that that the student can use in solving problems, the monitoring and

control that the student exerts on the problem solving process to guide it in

productive directions, and the beliefs that the student holds about mathematics, which

enable or disable problem solving attempts. McLeod (1992) has supplemented this

view by expounding on the important of affect in mathematical problem solving.

In my own work, I have found it helpful for teachers to consider that solving

problems with mathematics requires a wide range of skills and abilities, including:

• Deep mathematical knowledge

• General reasoning abilities

• Knowledge of heuristic strategies

• Helpful beliefs and attitudes (e.g. an expectation that maths will be useful)

• Personal attributes such as confidence, persistence and organisation

• Skills for communicating a solution.

Of these, the first three are most closely part of mathematical thinking.

In my book with John Mason and Leone Burton (Mason, Burton and Stacey, 1982),

we provided a guide to the stages through which solving a mathematical problem is

likely to pass (Entry, Attack, Review) and advice on improving problem solving

performance by giving experience of heuristic strategies and on monitoring and

controlling the problem solving process in a meta-cognitive way.

We also identified four fundamental processes, in two pair s, and showed how

thinking mathematically very often proceeds by alternating between them:

• specialising – trying special cases, looking at examples

• generalising - looking for patterns and relationships

• conjecturing – predicting relationships and results

• convincing – finding and communicating reasons why something is true.

I will illustrate these ideas in the two examples below. The first example examines

the mathematical thinking of the problem solver, whilst the second examines the

mathematical thinking of the teacher. The two problems are rather different – the

second is within the mainstream curriculum, and the mathematical thinking is guided

by the teacher in the classroom episode shown. The first problem is an open problem,

selected because it is similar to open investigations that a teacher might choose to use,

but I hope that its unusual presentation will let the audience feel some of the mystery

and magic of investigation afresh.

MATHEMATICAL THINKING IS IMPORTANT AS A WAY OF LEARNING

MATHEMATICS

In this section, I will illustrate these four processes of mathematical thinking in the

context of a problem that may be used to stimulate mathematical thinking about

numbers or as an introduction to algebra. If students' ability to think mathematically

is an important outcome of schooling, then it is clear that mathematical thinking must

feature prominently in lessons.

Number puzzles and tricks are excellent for these purposes, and in the presentation I

will use a number puzzle in a format of the Flash Mind Reader , created by Andy

Naughton and published on the internet (HREF1). The Flash Mind Reader does not

look like a number puzzle. Indeed its creator writes:

We have been asked many times how the Mind Reader works, but will not publish that

information on this website. All magicians […] do not give away how their effects work.

The reason for this is that it spoils the fun for those who like to remain mystified and

when you do find out how something works it's always a bit of a let-down. If you are

really keen to find out how it works we suggest that you apply your brain and try to work

it out on paper or search further afield. (HREF1)

As with many other number tricks, an audience member secretly chooses a number

(and a symbol), a mathematical process is carried out, and the computer reveals the

audience member's choice. In this case, a number is chosen, the sum of the digits is

subtracted from the number and a symbol corresponding to this number is found from

a table. The computer then magically shows the right symbol. The Flash Mind

Reader is too difficult to use in most elementary school classes, the target of this

conference, but I have selected it so that my audience of mathematics education

experts can experience afresh some of the magic and mystery of numbers. As the

group works towards a solution, we have many opportunities to observe

mathematical thinking in action.

Through this process of shared problem solving as we investigate the Flash Mind

Reader, I hope to make the following points about mathematical thinking. Firstly,

when people first see the Flash Mind Reader, mathematical explanations are far from

their minds. Some people propose that it really does read minds, and they may try to

test their theory by not concentrating hard on the number that they choose. Others

hypothesise that the program exerts some psychological power over the person's

choice of number. Others suggest it is only an optical illusion, resulting from staring

at the screen. This illustrates that a key component of mathematical thinking is

having a disposition to looking at the world in a mathematical way, and an attitude of

seeking a logical explanation.

As we seek to explain how the Flash Mind reader works, the fundamental processes

of thinking mathematically will be evident. The most basic way of trying to

understand a problem situation is to try the Flash Mind Reader several times, with

different numbers and different types of numbers. This helps us understand the

problem (in this case, what is to be explained) and to gather some information. This

is a simple example of specialising, the first of the four processes of thinking

mathematically processes.

As we enter more deeply into the problem, specialising changes its character. First

we may look at one number, noting that if 87 is the number, then the sum of its digits

is 15 and 87 – 15 is 72.

Beginning to work systematically leads to evidence of a pattern:

87 8 + 7 = 15 87 – 15 = 72

86 8 + 6 = 14 86 – 14 = 72

85 8 + 5 = 13 85 – 13 = 72

84 8 + 4 = 12 84 – 12 = 72

and a cycle of experimentation (which numbers lead to 72?, what do other numbers

lead to?) and generalising follows.

Of course, at this stage it is important to note the value of working with the unclosed

expressions such as 8+7 instead of the closed 15, because this reveals the general

patterns and reasons so much better. Working with the unclosed expression to reveal

structure is an admirable feature of Japanese elementary education.

87 87 – 7 = 80 80 – 8 = 72

86 86 – 6 = 80 80 – 8 = 72

85 85 – 5 = 80 80 – 8 = 72

84 84 – 4 = 80 80 – 8 = 72

It is also worthwhile noting at this point, that although we are working with a specific

example, the aim here is to see the general in the specific.

This generalising may lead to a conjecture that the trick works because all starting

numbers produce a multiple of 9 and all multiples of 9 have the same symbol. But

this conjecture is not quite true and further examination of examples (more

specialising) finally identifies the exceptions and leads to a convincing argument. In

school, we aim for students to be able to use algebra to write a proof, but even before

they have this skill, they can be produc e convincing arguments. An orientation to

justify and prove (at an appropriate level of formality) is important throughout school.

If students are to become good mathematical thinkers, then mathematical thinking

needs to be a prominent part of their education. In addition, however, students who

have an understanding of the components of mathematical thinking will be able to

use these abilities independently to make sense of mathematics that they are learning.

For example, if they do not understand what a question is asking, they should decide

themselves to try an example (specialise) to see what happens, and if they are

oriented to constructing convincing arguments, then they can learn from reasons

rather than rules. Experiences like the exploration above, at an appropriate level build

these dispositions.

MATHEMATICAL THINKING IS ESSENTIAL FOR TEACHING

MATHEMATICS.

Mathematical thinking is not only important for solving mathematical problems and

for learning mathematics. In this section, I will draw on an Australian classroom

episode to discuss how mathematical thinking is essential for teaching mathematics.

This episode is taken from data collected by Dr Helen Chick, of the University of

Melbourne, for a research project on teachers' pedagogical content knowledge. For

other examples, see Chick, 2003; Chick & Baker, 2005, Chick, Baker, Pham &

Cheng, 2006a; Chick, Pham & Baker, 2006b). Providing opportunities for students

to learn about mathematical thinking requires considerable mathematical thinking on

the part of teachers.

The first announcement for this conference states that a teacher requires

mathematical thinking for analysing subject matter (p. 4), planning lessons for a

specified aim (p. 4) and anticipating students' responses (p. 5).These are indeed key

places where mathematical thinking is re quired. However, in this section, I

concentrate on the mathematical thinking that is needed on a minute by minute basis

in the process of conducting a good mathematics lesson. Mathem atical thinking is

not just in planning lessons and curricula; it makes a difference to every minute of the

lesson.

The teacher in this classroom extract is in her fifth year of teaching. She stands out in

Chick's data as one of the teachers in the sample exhibiting the deepest pedagogical

content knowledge (Shulman, 1986, 1987). Her pupils are aged about 11 years, and

are in Grade 6. This lesson began by reviewing ideas of both area and perimeter. We

will examine just the first 15 minutes.

The teacher selected an open and reversed task to encourage investigation and

mathematical thinking. Students had 1cm grid paper and were all asked to draw a

rectangle with an area of 20 square cm. This task is open in the sense that there are

multiple correct answers, and it is 'reversed' when it is contrasted to the more

common task of being given a rectangle and finding its area. The teacher reminded

students that area could be measured by the number of grid squares inside a shape.

In terms of the processes of mathematical thinking, the teacher at this stage is

ensuring that each student is specialising. They are each working on a special case,

and coming to know it well, and this will provide an anchor for future discussions

and generalisations. I make no claim that the teacher herself analyses this move in

this way.

As the teacher circulated around the room assisting and monitoring students, she

came to a student who asked if he could draw a square instead of a rectangle. In the

dialogue which follows, the teachers' response highlighted the definition of a

rectangle, and she encouraged the student to work from the definition to see that a

square is indeed a rectangle.

S: Can I do a square?

T: Is a square a rectangle?

T: What's a rectangle?

T: How do you get something to be a rectangle? What's the definition of a rectangle?

S: Two parallel lines

T: Two sets of parallel lines … and …

S: Four right angles.

T: So is that [square] a rectangle?

S: Yes.

T: [Pause as teacher realises that student understands that the square is a rectangle, but

there is a measurement error] But has that got an area of 20?

S: [Thinks] Er, no.

T: [Nods and winks]

Other responses to this student would have closed down the opportunity to teach him

about how definitions are used in mathematics. To the question "Can I do a square?",

she may have simply replied "No, I asked you to draw a rectangle" or she might

have immediately focussed on the error that led the student to ask the question.

Instead she saw the opportunity to develop his use of definitions. When the teacher

realised that the student had asked about the square because he had made a

measurement error, she judged that this was within the student's own capability to

correct, and so she simply indicated that he should check his work.

In the next segment, a student showed his 4 x 5 rectangle on the overhead projector,

and the teacher traced around it, confirmed its area is 20 square cm and showed that

multiplying the length by the width can be used instead of counting the squares,

which many students did. In this segment, the teacher demonstrated that reasoning is

a key component of doing mathematics. She emphasised the mathematical

connections between finding the number of squares covered by the rectangle by

repeated addition (4 on the first row, 4 on the next, …) and by multiplication. In her

classroom, the formula was not just a rule to be remembered, but it was to be

understood. The development of the formula was a clear example of 'seeing the

general in a special case'. The formula was developed from the 4 x 5 rectangle in

such a way that the generality of the argument was highlighted.

The teacher paid further attention to generalisation and over-generalisation at this

point, when a student commented: 'That's how you work out area – you do the length

times the width'. The teacher seized on this opportunity to address students' tendency

to over-generalise, and teased out, through a short class discussion, that LxW only

works for rectangles.

S1: That's how you work out area -- you do the length times the width.

T: When S said that's how you find the area of a shape, is he completely correct?

S2: That's what you do with a 2D shape.

T: Yes, for this kind of shape. What kind of shape would it not actually work for?

S3: Triangles.

S4: A circle.

T: [With further questioning, teases out that LxW only applies to rectangles]

In the next few minutes, the teacher highlighted the link between multiplication and

area by asking students to make other rectangles with area 20 square centimetres.

Previously all students had made 4 x 5 or 2 x 10, but after a few minutes, the class

had found 20 x 1, 1 x 20, 10 x 2, 2 x 10, 4 x 5 and 5 x 4 and had identified all these

side lengths as the factors of 20. Making links between different parts of the

mathematics curriculum charac terises her teaching.

Then, in another act of generalisation, the teacher begins to move beyond whole

numbers:

T: Are there any other numbers that are going to give an area of 20? [Pauses, as if

uncertain. There is no response from the students at first]

T: No? How do we know that there's not?

S: You could put 40 by 0.5.

T: Ah! You've gone into decimals. If we go into decimals we're going to have heaps,

aren't we?

After these first 15 minutes of the lesson, the students found rectangles with an area

of 16 square centimetres and the teacher stressed the important problem solving

strategy of working systematically. Later, in order to contrast the two concepts of

area and perimeter, students found many shapes of area 12 square cm (not just

rectangles) and determined their perimeters.

Even the first 15 minutes of this lesson show that considerable mathematical thinking

on behalf of the teacher is necessary to provide a lesson that is rich in mathematical

thinking for students. We see how she draws on her mathematical concepts, deeply

understood, and on her knowledge of connections among concepts and the links

between concepts and procedures. She also draws on important ge neral mathematical

principles such as

• working systematically

• specialising – generalising: learning from examples by looking for the

general in the particular

• convincing: the need for justification, explanation and connections

• the role of definitions in mathematics.

Chick's work analyses teaching in terms of the knowledge possessed by the teachers.

She tracks how teachers reveal various categories of pedagogical content knowledge

(Shulman, 1986) in the course of teaching a lesson. In the analysis above, I viewed

the lesson from the point of view of the process of thinking mathematically within the

lesson rather than tracking the knowledge used. To draw an analogy, in researching a

students' solution to a mathematical problem , a researcher can note the mathematical

content used, or the researcher can observe the process of solving the problem.

Similarly, teaching can be analysed from the "knowledge' point of view, or analysed

from the process point of view.

For those us who enjoy mathematical thinking, I believe it is productive to see

teaching mathematics as another instance of solving problems with mathematics.

This places the emphasis not on the static knowledge used in the lesson asabove but

on a process account of teaching. In order to use mathematics to solve a problem in

any area of application, whether it is about money or physics or sport or engineering,

mathematics must be used in combination with understanding from the area of

application. In the case of teaching mathematics, the solver has to bring together

expertise in both mathematics and in general pedagogy, and combine these two

domains of knowledge together to solve the problem, whether it be to analyse subject

matter, to create a plan for a good lesson, or on a minute-by-minute basis to respond

to students in a mathematically productive way. If teachers are to encourage

mathematical thinking in students, then they need to engage in mathematical thinking

throughout the lesson themselves.

References

APEC –Tsukuba (Organising Committee) (2006) First announcement. International

Conference on Innovative Teaching of Mathematics through Lesson Study. CRICED,

University of Tsukuba.

Chick, H. L. (2003). 'Pre-service teachers' explanations of two mathematical concepts'

Proceedings of the 2003 conference, Australian Association for Research in Education.

From: http://www.aare.edu.au/03pap/chi03413.pdf

Chick, H.L. and Baker, M. (2005) 'Teaching elementary probability: Not leaving it to

chance', in P.C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce

& A. Roche (eds.) Building Connections: Theory, Research and Practice. (Proceedings

of the 28

annual conference of the Mathematics Education Research Group of

Australasia), MERGA, Sydney, pp. 233-240.

Chick, H.L., Baker, M., Pham, T., and Cheng, H. (2006a) 'Aspects of teachers' pedagogical

content knowledge for decimals', in J. Novotná, H. Moraová, M. Krátká, & N.

Stehlíková (eds.), Proc. 30

conference e International Group for the Psychology of

Mathematics Education, PME, Prague, Vol. 2, pp. 297-304.

Chick, H.L., Pham, T., and Baker, M. (2006b) 'Probing teachers' pedagogical content

knowledge: Lessons from the case of the subtraction algorithm', in P. Grootenboer, R.

Zevenbergen, & M. Chinnappan (eds.), Identities, Cultures and Learning Spaces (Proc.

annual conference of Mathematics Education Research Group of Australasia),

MERGA, Sydney, pp. 139-146.

Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7), 519

– 524.

HREF1 CyberGlass Design - The Flash Mind Reader. http://www.cyberglass.biz Accessed

28 November 2006.

Mason, J. Burton, L. and Stacey, K. (1982) Thinking Mathematically. London: Pearson.

(Also available in translation in French, German, Spanish, Chin ese, Thai (2007))

McLeod, D.B. (1992) Research on affect in mathematics education: a reconceptualisation.

In D.A. Grouws, Ed., Handbook of research on mathematics teaching and learning, (pp.

575–596).New York: MacMillan, New York.

PISA (Programme for International Student Assessment) (2006) Assessing Scientific,

Reading and Mathematical Literacy. A Framework for PISA 2006. Paris: OECD.

Schoenfeld, A. (1985) Mathematical Problem Solving. Orlando: Academic Press.

Shulman, L.S. (1986) Those who underst and: Knowledge growth in teaching, Educational

Researcher 15 (2), 4-14.

Shulman, L.S. (1987) Knowledge and teaching: Foundations of the new reform, Harvard

Educational Review 57(1), 1-22.

Singh, S. (1997) Fermat's Enigma , New York: Walter

Stacey, K. & Groves, S. (1985) Strategies for Problem Solving. Lesson Plans for

Developing Mathematical Thinking. Melbourne: Objective Learning Materials.

Stacey, K. & Groves, S. (2001) Resolver Problemas: Estrategias. Madrid: Lisbon.

Stacey, Kaye (2005) The place of problem solving in contemporary mathematics curriculum

documents. Journal of Mathematical Behavior 24, pp 341 – 350.

... Mason et al. (2010) argued that there are four important processes in mathematical thinking, namely specializing, generalizing, conjecturing, and convincing. Stacey (2006) states that the framework used by PISA to measure mathematical literacy includes several components contained in mathematical thinking abilities, such as components of reasoning, modeling, and making connections between ideas. Based on the PISA results 2018, the math literacy score of Indonesian students is 379 which is under the average score of OECD member (OECD, 2019). ...

... Mathematical thinking can be constructed through the formation of an appropriate learning atmosphere, such as asking questions, giving challenges and reflecting (Stacey, 2006). Delima et al. (2021) stated that the learning model that can build mathematical thinking abilities is a CMI model. ...

... The process of mathematical reasoning requires a guide like a teacher (Brodie, 2010). In addition to this, creating opportunities for students to teach mathematical thinking also requires teachers to think mathematically (Stacey, 2006). Nevertheless, teachers who can use mathematical reasoning effectively can also create learning environments that will allow for the development of this skill (Çiftci, 2015). ...

... Matematiksel akıl yürütme süreci öğretmen gibi bir rehbere ihtiyaç duymaktadır (Brodie, 2010). Bununla birlikte matematiksel düşünceyi öğretmek amacıyla öğrenciler için fırsatlar yaratmak, öğretmenlerin matematiksel düşünmesini de gerektirir (Stacey, 2006). Ancak matematiksel akıl yürütmeyi etkili bir şekilde kullanabilen öğretmen, bu yeteneğin gelişmesini sağlayacak öğrenme ortamlarını da oluşturabilir (Çiftci, 2015). ...

This study was conducted with a total of 174 students studying in the primary school mathematics education and secondary school mathematics education undergraduate programs of the faculty of education of a university in the spring term of the 2015-2016 academic year in order to determine the skill levels of mathematical reasoning levels of primary school and secondary school mathematics education students. The quantitative research approach was adopted in the study. The survey method was used in the study, and the two-stage "Mathematical Reasoning Evaluation Scale" consisting of 20 multiple-choice and six open-ended questions was used as a data collection tool. During the data analysis, the mathematical reasoning score was determined by examining the answers given by each student to the test. According to the results of the study, it is observed that the mathematical reasoning skills of primary school mathematics education and secondary school mathematics education students are at the intermediate level. When primary and secondary school mathematics education students were compared, it was concluded that secondary school students were more successful than primary school students on average. Furthermore, when the dimensions of the scale are considered, it is observed that students' correctly answering percentages decreased in the dimensions of being able to decide the correctness of the solution and the result, developing reasonable discussions for the solution and solving non-routine problems.

... This situation offers an opportunity for thinking. Furthermore, if students refer in their voice-over to special cases or examples, and generalize about the situation shown in the video, trying to convince the viewer that what they are saying is true, then they are using processes which are identified by Mason et al. (2010) and Stacey (2007) as being fundamental processes in thinking mathematically. Moreover, in the group discussion based on students' responses to the task, teachers position students as mathematical thinkers worth listening to because their responses are results of students' thinking. ...

Bjarnheiður Kristinsdóttir

This thesis introduces results from a design-based, task design research study in mathematics education, within which silent video tasks were defined, developed, and implemented in upper secondary school mathematics classrooms. It discusses a research problem concerning the identification of opportunities and challenges that arise from the use of silent video tasks. To tackle that problem, the researcher worked with seven teachers in six Icelandic upper secondary schools who implemented silent video tasks in their classrooms. In short, silent video tasks involve the presentation of a short silent mathematics video clip that students are asked to discuss in pairs as they prepare and record their voice-over to the video. On the basis of students' recorded responses to the task, that are listened to by the whole group, the teacher leads a discussion with the aim to deepen and widen students' understanding of the mathematical topic presented in the video. The idea of silent video tasks is grounded in social constructivist theories. It is considered important that interaction happens between teacher and learners and among learners themselves, who work together (support each other) toward richer understanding of mathematical content. The learner is seen as an active participant in the teaching and learning process and in the case of silent video tasks, learners get an opportunity to become aware of their own and their peers' current ways of describing or explaining mathematical phenomena. Two implementation phases were conducted in 2017 and 2019, during which interview data on teachers' expectations and experiences of using silent video tasks was collected and analysed. In the first phase, four mathematics teachers in randomly selected upper secondary schools in Iceland assigned a silent video task to their 17-year-old students. Results from the first phase indicated that silent video tasks might be a helpful tool for formative assessment. Thus, teachers in the second phase were purposefully selected to work at schools that aim for active use of formative assessment. One teacher assigned three silent video tasks to his 16-year-old students and two teachers assigned one silent video task to their 16-year-old students. Besides interview data, classroom observation protocols were collected during the second phase. Influenced by theory and empirical results, the process of assigning a silent video task developed. To conclude the project, some characteristics that make a video suitable for use in silent video tasks were defined and the instructional sequence of silent video tasks was described. Together with the underlying theoretical and empirical arguments, they form design principles for silent video tasks.

... 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.

... Mathematical thinking is an organized process that the student's mind performs if he faces a mathematical problem that challenges his ability so that he cannot find a present solution to it, which leads the student to think about the problem and review it and arrange his previous mathematical experiences after that, and then he searches for a final solution to that problem. It is a thinking that obliges a person to face mathematical problems and tasks in an attempt to solve them and through which this person depends on many factors related to the mental processes through which the solution process is performed and the logical operations that lead to solving the various types of mathematical operations necessary to solve a problem or answer a mathematical question [28,29]. Mathematical thinking skills: The skill is a specific performance in a situation with mastery and mastery, and this requires training and training as often accompanied by behavioral changes in performance. ...

The aim of the research is to find out the effect of applying classroom assessment techniques (CATs) on both mathematical and logical thinking among fourth-grade scientific students. In pursuit of the research objectives, the experimental method was used, and the quasi-experimental design was used for two equivalent groups, one control group taught in the traditional way and the other experimental taught according to the techniques of classroom structural evaluation. The research sample consisted of (44) students from the fourth scientific grade who were intentionally chosen after ensuring their equivalence in several factors, most notably chronologi-cal age and the level of mathematics, and they were distributed equally among the two groups. To implement the research, three tools were built, represented in the teacher's handbook for applying the class formative assessment, the mathematical reasoning test, and the logical thinking test. The two researchers applied the exper-iment in the first semester of the academic year (2019/2020) AD. The two re-searchers applied the techniques of class formative assessment to the experimental group, while the control group studied according to the usual method, and then the mathematical thinking test and the dimensional logical thinking test were applied. On the experimental and control groups. The results showed that there were statis-tically significant differences between the mean scores of the two groups on the mathematical thinking test, and there were also statistically significant differences between the mean scores of the two groups on the logical reasoning test. The size of the effect of applying the class formative assessment on both thinking was calcu-lated, and it appeared that it had a clear effect on both mathematical thinking and logical thinking. In light of the results, the two researchers recommended a number of recommendations.

... Çünkü matematiksel akıl yürütme süreci öğretmen gibi bir rehbere ihtiyaç duymaktadır (Brodie, 2010). Matematiksel düşünceyi öğretmek amacıyla öğrenciler için fırsatlar yaratmak, öğretmenlerin matematiksel düşünmesini de gerektirir (Stacey, 2006). Ancak matematiksel akıl yürütmeyi etkili bir şekilde kullanabilen öğretmen, bu yeteneğin gelişmesini sağlayacak öğrenme ortamlarını da oluşturabilir (Çiftci, 2015). ...

Anahtar Kelimeler: Matematiksel akıl yürütme becerisi, matematiksel muhakeme becerisi, öğretmen adayları, öğretmenlik uygulaması Makale Türü: Araştırma Öz Bu çalışma, ilköğretim matematik eğitiminde öğrenim gören öğretmen adaylarının öğretmenlik uygulaması deneyimleri kapsamında öğrencilerine sundukları matematiksel muhakeme (akıl yürütme) becerisi fırsatlarını belirlemek amacıyla 4 öğretmen adayı ile yapılmıştır. Araştırmada nitel araştırma yaklaşımlarından durum çalışması yöntemi kullanılmıştır. Araştırmada veri toplama aracı olarak video kayıtları ve gözlem kullanılmıştır. Ders anlatım videolarından elde edilen veriler betimsel analiz yöntemi ile analiz edilmiştir. Araştırma sürecinde kaydedilen videolar yazı haline getirildikten sonra Bergqvist ve Lithner (2012) tarafından tanımlanan analiz sürecine bağlı kalınarak analiz edilmiştir. Araştırmadan elde edilen verilerin analizlerinden öğretmen adaylarının öğrencilere matematiksel akıl yürütme becerisi ile ilgili nadiren akıl yürütme fırsatları sundukları genel olarak düşünüldüğünde sınırlı fırsatlar sundukları sonucuna ulaşılmıştır.

COVID-crisis has made significant changes in the educational process of many coun-tries, including the need for new management decisions that would solve the complex problem of accelerating the development of online resources for distance learning. Management, particularly in education, is valuable when it is able to combine both general and specific goals. Especially when it comes to a specific educational process for training future TV and radio journalists, advertisers and PR-managers, screenwrit-ers and directors, sound directors, TV presenters, film and cameramen. The peculiarity of these professions is the combination of both creative and technological components of production and placement of professional audio-video content, i.e. content pro-duced by TV and radio companies, film or TV studios, advertising agencies, and aimed at a mass audience. One of the basic priorities in training such specialists is, first of all, the practice, which is based on the planned implementation of educational audiovisual projects and the ability to put them into effect in certain circumstances, including COVID-crisis caused by COVID-19 virus. Therefore, the aim of the article is to hypothesize how to build a productive distance educational strategy in the condi-tions of COVID-crisis, which specifically affected the quality of practical training of specialists in the field of audiovisual media and arts in Ukraine.

Hafize Gamze Kirmizigül

İlkokul Sınıf Öğretmenlerinin Üstün Yetenekliler ve Üstün Yeteneklilerin Eğitimi İle İlgili Görüşlerinin Belirlenmesi

In this paper we present a framework for investigating teachers' mathematical pedagogical content knowledge (PCK). The components of the framework permit identification of strengths and weaknesses in the PCK held by various teachers on different topics. It was applied to teachers' written and interview responses addressing a student's difficulties with the standard subtraction algorithm. This enabled comparison of the PCK held by individual teachers, together with analysis of the teachers' PCK as a group. Most understood the issues and knew suitable representations, but occasionally lacked key understanding of students' misconceptions and how to help students recognise them. Investigating teachers' pedagogical content knowledge (PCK) is challenging for many reasons. PCK reveals itself in many places—in teachers' planning, classroom interactions, explanations, mathematical competency, and so on—and a study of only one environment will give a limited perspective. In addition, PCK's multifaceted nature makes considering all aspects time-consuming and complex. This report uses a framework that provides a set of lenses for studying PCK to address this second problem. At the same time we show that one carefully chosen environment can still reveal detail about teachers' PCK.

This paper considers the role of content knowledge and pedagogical content knowledge in the teaching of probability to Grade 5 students. Lessons of two teachers were studied to determine the activities and teaching strategies used to bring out ideas associated with chance, and examine how probability understanding is developed in class. The lessons are shown to be rich in deep probabilistic ideas. The complex interplay between these concepts was sometimes handled well by the teachers, whereas on other occasions gaps in content and pedagogical content knowledge had the potential to cause misconceptions for students. Since the 1990s much research attention has focused on the pedagogical content knowledge of teachers. At the same time, chance and data have become more significant components of the curriculum, particularly at the primary school level. This report takes a closer look at the way two teachers use both content and pedagogical content knowledge to help Year 5 students develop key concepts in probability.

Lee S. Shulman

Lee S. Shulman builds his foundation for teaching reform on an idea of teaching that emphasizes comprehension and reasoning, transformation and reflection. "This emphasis is justified," he writes, "by the resoluteness with which research and policy have so blatantly ignored those aspects of teaching in the past." To articulate and justify this conception, Shulman responds to four questions: What are the sources of the knowledge base for teaching? In what terms can these sources be conceptualized? What are the processes of pedagogical reasoning and action? and What are the implications for teaching policy and educational reform? The answers — informed by philosophy, psychology, and a growing body of casework based on young and experienced practitioners — go far beyond current reform assumptions and initiatives. The outcome for educational practitioners, scholars, and policymakers is a major redirection in how teaching is to be understood and teachers are to be trained and evaluated. This article was selected for the November 1986 special issue on "Teachers, Teaching, and Teacher Education," but appears here because of the exigencies of publishing.

Douglas B. McLeod

begins by considering alternative theoretical foundations for research on affect / Mandler's (1984) theory, an approach to research on affect that is based on cognitive psychology, is selected for further discussion, particularly because it illustrates how affect can be incorporated into cognitive studies of mathematics learning and teaching / presents a framework for research on affect that reorganizes the literature into three major areas: beliefs, attitudes, and emotions / research on a number of topics from the affective domain is summarized, and linked to the proposed framework / explores how qualitative as well as quantitative research methods can be used in research on affect (PsycINFO Database Record (c) 2012 APA, all rights reserved)

Kaye Stacey

This paper reviews the presentation of problem solving and process aspects of mathematics in curriculum documents from Australia, UK, USA and Singapore. The place of problem solving in the documents is reviewed and contrasted, and illustrative problems from teachers' support materials are used to demonstrate how problem solving is now more often treated as a teaching method, rather than a goal in itself. The paper also analyses how the curriculum documents describe the growth of students' abilities in the process areas of mathematics, and assesses the guidance that this provides for teachers. At each stage, the paper suggests directions for research that would be useful in assisting curriculum documents to promote the fundamental but elusive goal of making students better problem solvers.