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*Prism*, a biannual journal, publishes works that study the shaping influence of traditional literature and culture on modern and contemporary China. The journal will be included in the e-Duke Journals: Expanded collection.

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*Tikkun ***ceases publication**

*Tikkun*

The quarterly journal *Tikkun* will cease publication with volume 33, issue 4, at the decision of its owner, the Institute for Labor and Mental Health. Institutions that previously purchased the journal will continue to receive perpetual access through Duke University Press. Archival content for *Tikkun* will also continue to be hosted on the Project MUSE platform.

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**Change in frequency for ***History of Political Economy*

*History of Political Economy*

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indeed this is an important article for mathematical and as say prof dr mircea orasanu and prof horia orasanu and for holistically founds

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in this case we consider some that as say prof dr mircea orasanu and prof horia orasanu as followd and with JOURNAL OF MATHEMATICS AND LAGRANGIAN

ABSTRACT While this view of mathematics as a practical and useful tool might hold for those who reported liking, for others who dislike the subject, mathematics is perceived to be irrelevant and comprises

‘a lot of things which I never used’ (R492)

Mathematics is regarded as important and essential to learn because mathematical knowledge is necessary and useful in both daily life and at work. Some reported images of mathematics strongly identified with particular uses such as ‘banking account’ (R494) or ‘VAT receipts’ (R459).

Symbolic view: mathematics is perceived as a collection of numbers and symbols, or rules and procedures to be followed and memorised.

Some examples are mathematics is viewed as comprising or represented by:

‘numbers and equations’ (R005)

‘figures and sums’ (R340)

‘multiply, minus, add, divide’ (R453)

For many of these participants, mathematics is seen as sets of rules and procedures to be followed and memorised. For some people, this is a pleasure because mathematics is

‘formulae, involved and exciting’ (R038)

and some of them just

‘like playing around with numbers, equations, finding solution to problems’

(R119, text-unit 3, likes maths).

But to others, mathematics is

‘rules, formulae learnt before understanding’ (R109)

This is well described by a middle-aged housewife who reported disliking mathematics.

‘Sometimes if you can’t remember the formula then you don’t know how to get the answer,…’ (text-unit 5) and ‘…you got to stare on the wall. You are mentally blackout – it is all gone!’ (text-unit 13) As a result, she said, ‘ I have

1 INTRODUCTION

‘logical stimulation’ (R100) as well as ‘logical – organises things in order’ (R122).

Subsequently, mathematics is viewed as a means to model the world. A few respondents expressed this as: maths is

‘a way to model the physical world’ (R301) and

‘problem solving – explaining physical processes’ (R113).

Associated with these images is the view that mathematics learning is ‘learning to think correctly and logically’ (R384) and it is possibly hierarchical in the sense that, one needs to ‘take small steps to understand difficult problems’ (R122). Moreover, mathematics learning is all about ‘making order out of chaos’ (R118).

This problem solving view is more often held by those who reported liking than those who reported dislike of mathematics. For the former, this was also given as one of the main reasons for liking mathematics. Perhaps they enjoyed the challenge in searching for solutions to mathematical problems, and felt a sense of satisfaction when they found a solution. In contrast, for those reporting dislike mathematics, learning mathematics is like ‘solving a complicated puzzle: there is an answer but it takes long time to find it’ (R361) Many of them find mathematics ‘difficult’ (R434) and learning mathematics is more like ‘passing hurdles’ (R470) than doing something enjoyable for them.

Enigmatic view: mathematics is seen as mysterious but yet something to be explored and whose beauty is to be appreciated.

For these respondents, mathematics is seen as mysterious, foreign and incomprehensible but yet, it is also

‘like a sunset – unique and beautiful’ (R168).

There are on one hand, those who like mathematics because, as they report:

‘…I like the elegance of mathematics. The proofs and theories are very elegant. There is …like recognising the patterns of mathematics, I found it very interesting’ (R193, text-unit 3, IT trainer, male, 21-30, like maths).

Due to the elegance and aesthetic appeal of mathematics, and for others, the mysterious nature of mathematics, learning mathematics becomes ‘an exploration into another world’ (R113) or ‘ a voyage of discoveries’ (R116) that is ‘fun and challenging’ (R140) for those who reported a liking of mathematics.

On the other hand, the complexity and abstract nature of mathematics also drives away some people’s interest in mathematics because they found mathematics incomprehensible and confusing. They found themselves like ‘groping through fog’ (R412) or ‘wandering in a desert – with the odd oasis of understanding’ (R363).

The view of mathematics as an enigma was expressed by a small minority of the sample, particularly those who reported liking of mathematics and those who has direct involvement in mathematics such as mathematics students and mathematics teachers. Perhaps this might be one of the main reasons that have attracted these people to undertake mathematics-related studies and careers.

However, in general, most people were inclined to hold a composite image made up of elements from several of these five common shared views rather than subscribing to a single view.

3. Image as metaphor

Lakoff and Johnson (1980) point out that “metaphor is pervasive in our everyday life, not just in language but in thought and action” (p.3). Perhaps it is then not unusual for people to express their images in the form of metaphors. In this study, 27% of the respondents expressed their image of mathematics in the forms of metaphors, while 66% of them gave their image

2 FORMULATION

For these people, mathematics is a journey to discover new things, new knowledge and new insights. A middle-aged mathematics teacher described his images of learning mathematics as ‘the best sort of travelling in a new land’ and he explained that,

‘Well, when you are studying a new area and you are having to grasp it sometimes, you know, because you haven’t done that sort of mathematics before and you begin to realise why some statements, some theorems in mathematics are true or you begin to see the use of that theorem can have, you know, the statement is making connection without the thing, and that I find interesting’

(R293, text-unit 9, male, mathematics teacher, 41-50, likes maths)

These results suggest that it was the joy of discovering new understanding in mathematics that attracted them to get interested in mathematics. Even though many of them also found learning mathematics a difficult journey like,

‘a journey through a dark tunnel with a light at the end’ (R139) or

‘walking through sand – hard work but put in effort, you’ll get there’ (R136)

Therefore, there is this sense of achievement and satisfaction that encourage these people to work hard and to strike for the solution. Implicitly these metaphors indicate that there is a definite solution for each mathematics problem. Learning mathematics is ‘a journey through a dark tunnel with a light at the end’ (R139) and there is a destination for you ‘to get there’ (R133, text-unit 13).

There was also a young mathematics student who uses journey metaphor to illustrate her change of view from absolutist to fallibilitist (Ernest, 1991):

‘I mean we always brought up with that of the right and wrong answer and suddenly we were told that was not the most important part of maths, the most important part is how to get there, what kind of strategy to use. You know, that is the important part, how to get it done. …Therefore, mathematics does not have a definite solution, it is really your journey to get there.’

(R133, text-unit 13)

According to her view, she was brought up with an absolutist view, but now she was exposed to an alternative view that there are many strategies and possible answers to a mathematical problem. To her, learning mathematics should be focused on ‘process’ rather than ‘product’. This fallibilist view, however, was only shared by very few of the respondents.

It is interesting to read that some undergraduate students and tutors in Allen and Shiu’s (1997) study also gave the metaphor of mathematics as a journey. They categorised these responses under one of their four categories: ‘struggle leading to success’. Two very similar responses from the tutors are: learning mathematics is like

‘climbing a hill: – hard work where you follow the path you’re on – and then the joy and satisfaction of being at the top ‘ (T3)

‘climbing a hill. The higher you get the clearer the view of surrounding countryside – as you can see more the links and layout and connections become more obvious. (T18). (p.10)

In short, mathematics as a journey metaphor indicates that mathematics learning is a difficult processWissenschaften selbst entspricht. Die einzige vernünftige Antwort auf die Frage (a) ist also eine historische Antwort: das sind die Theorien, die wir heute haben, das sind unsere Forschungsmethoden, das sind die Gründe, warum wir sowohl die Theorien als auch die Methoden für gut halten – aber neue Theorien und neue Methoden können uns jeden Augenblick überraschen.

Unnötig zu sagen, daß Feyerabends Anmerkungen zur Frage (b) für den naiven und insbesondere den naturwissenschaftlich geprägten Denker noch viel verwirrender und beunruhigender ausfallen. Es wäre wenig sinnvoll, sie hier zu paraphrasieren. Eine Didaktik, die ihr Fach auch als Ganzes sieht, kommt aber gar nicht umhin, sich solchen Fragen zu stellen.

Subversive

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