Is Mathematics Invented Or Discovered Pdf

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Is Mathematical Knowledge Discovered or Invented?

Recently a heated debate between realists and relativists in science has erupted. The conflict is between

those who see science as a rational description of the world converging on the truth, and those who

argue that it is a socially constructed account of the world, and just one of many possible accounts.

Typically scientists and philosophers of science are realists, arguing that science is approaching a true

and accurate description of the real world, whereas social and cultural theorists support a relativist view

of science, and argue that all knowledge of the world is socially constructed.

What has gone unnoticed in this debate is that there is a parallel and equally fundamental dispute over

whether mathematics is discovered or invented. The absolutist view of mathematics sees it as universal,

objective and certain, with mathematical truths being discovered through the intuition of the

mathematician and then being established by proof. Many modern writers on mathematics share this

view, including Roger Penrose in The Emperor's New Mind, and John Barrow in Pi in the Sky, as

indeed do most mathematicians. The absolutists support a 'discovery' view and argue that mathematical

'objects' and knowledge are necessary, perfect and eternal, and remark on the 'unreasonable

effectiveness' of mathematics in providing the conceptual framework for science. They claim that

mathematics must be woven into the very fabric of the world, for since it is a pure endeavour removed

from everyday experience how else could it describe so perfectly the patterns found in nature?

The opposing is view often called 'fallibilist' and this sees mathematics as an incomplete and everlasting

'work-in-progress'. It is corrigible, revisable, changing, with new mathematical truths being invented,

or emerging as the by-products of inventions, rather than discovered. So who are the fallibilists? Many

mathematicians and philosophers have contributed to this perspective and I will just mention a few

recent contributions. First of all, the philosopher Wittgenstein in his later works such as Remarks on

the Foundations of Mathematics contributes to fallibilism with his claim that mathematics consists of a

motley of overlapping and interlocking language games. These are not games in the trivial sense, but

the rule-governed traditional practices of mathematicians, providing meanings for mathematical

symbolism and ideas. Wittgenstein argues that we often follow rules in mathematical reasoning because

of well-tried custom, not because of logical necessity. So Wittgenstein's contribution is to point out

that it is what mathematicians do in practice, and not what logical theories tell us, which is the engine

driving the development of mathematical knowledge.

Imre Lakatos is another fallibilist, and he argues that the history of mathematics must always be given

pride of place in any philosophical account. His major work Proofs and Refutations traces the historical

development of a result in topology, the Euler Relation, concerning the number of faces (F), edges (E)

and vertices (V) of mathematical solids. For simple flat-sided solids, the relationship is F+V=E+2.

However, proving this fact took over a hundred years as the definitions of mathematical solids, faces,

edges and vertices were refined and tightened up, and as different proofs were invented, published,

shown to have loopholes, and modified. Lakatos argues that as in this example, no definitions or proofs

in mathematics are ever absolutely final and beyond revision.

Philip Kitcher offers a further refinement of fallibilism in his book The Nature of Mathematical

Knowledge. He argues that much mathematical knowledge is accepted on the authority of the

mathematician, and not based on rational proof. Furthermore, even when mathematical results are

proved much of the argument is tacit and draws on unspoken mathematical knowledge learned through

practice, as opposed to being completely written down explicitly. Since the informal and tacit

knowledge of mathematics of each generation varies, mathematical proof cannot be described as

absolute.

In my book Social Constructivism as a Philosophy of Mathematics, I argue that not only is

mathematics fallible, but it is created by groups of persons who must both formulate and critique new

knowledge in a formal 'conversation' before it counts as accepted mathematics. These conversations

embody the process that Lakatos describes in the evolution of the Euler Relation, as well as what goes

on in Wittgenstein's mathematical language games. Knowledge creation is part of a larger overall cycle

in which mathematical knowledge is presented to learners in teaching and testing 'conversations' in

schools and universities, before they themselves can become mathematicians and participate in the

creation of new knowledge. This perspective offers a middle path between the horns of the traditional

objective/subjective dilemma in knowledge. According to social constructivism, mathematics is more

than a collection of subjective beliefs, but less than a body of absolute objective knowledge, floating

above all human activity. Instead it occupies an intermediate position. Mathematics is cultural

knowledge, like the rest of human knowledge. It transcends any particular individual, but not all of

humankind, like art, music, literature, religion, philosophy and science.

Although fallibilist views vary, they all try to account for mathematics naturalistically, that is in a way

that is true to real world practices. Unfortunately, fallibilism is too often caricatured by opponents as

claiming that mathematics may be part or all wrong; that since mathematics is not absolutely necessary

it is arbitrary or whimsical; that a relativist mathematics, by relinquishing absolutism, amounts to

'anything goes' or 'anybody's opinion in mathematics is as good as anybody else's'; that an invented

mathematics can be based on whim or spur of the moment impulse; and that if social forces are what

moulds mathematics then it must be shaped by the prevailing ideology and prejudices of the day, and

not by its inner logic.

However these claims and conclusions are caricatures, and no fallibilist I know would subscribe to

them. Fallibilism does not mean that some or all of mathematics may be false (although Gödel's

incompleteness results mean that we cannot eliminate the possibility that mathematics may generate a

contradiction). Instead, fallibilists deny that there is such a thing as absolute truth, which explains why

mathematics cannot attain it. For example, 1+1=2 is not absolutely true, although it is true under the

normal interpretation of arithmetic. However in the systems of Boolean algebra or Base 2 modular

arithmetic 1+1=1 and 1+1=0 are true, respectively. As this simple example shows, truths in

mathematics are never absolute, but must always be understood as relative to a background system that

provides the meaning for the sentences. Unlike in physics, in which there is just one world to determine

what is true or false, mathematics allows the existence of many different interpretations. So an

assumption like Euclid's Parallel Postulate and its denial can both be true, but in different mathematical

interpretations (in the systems of Euclidean and non-Euclidean geometries). Mathematicians are all the

time inventing new imagined worlds without needing to discard or reject the old ones.

A second criticism levelled at fallibilism is that if mathematics is not absolutely necessary then it must be

arbitrary or whimsical. Relativist mathematics, the criticism goes, by relinquishing absolutism amounts

to 'anything goes'. Therefore an invented mathematics is based on whims or spur of the moment

impulse. For example, Roger Penrose asks, are the objects and truths of mathematics "mere arbitrary

constructions of the human mind?" His answer is in the negative and he concludes that mathematics is

already there, to be discovered, not invented.

Plausible as this view seems at first, it is often argued on mistaken grounds. Mathematicians like

Penrose often contrast necessity with arbitrariness, and argue that if relativist mathematics has no

absolute necessity and essential characteristics to it, then it must be arbitrary. Consequently, they argue,

anarchy prevails and anything goes in mathematics. However as the philosopher Richard Rorty has

made clear, contingency, not arbitrariness, is the opposite of necessity. Since to be arbitrary is to be

determined by chance or whim rather than judgement or reason, the opposite of this notion is that of

being selected or chosen. I wish to argue that mathematical knowledge is based on contingency, due to

its historical development and the inevitable impact of external forces on the resourcing and direction of

mathematics, but is also based on the deliberate choices and endeavours of mathematicians, elaborated

through extensive reasoning. Both contingencies and choices are at work in mathematics, so it cannot

be claimed that the overall development is either necessary or arbitrary. Much of mathematics follows

by logical necessity from its assumptions and adopted rules of reasoning, just as moves do in the game

of chess. This does not contradict fallibilism for none of the rules of reasoning and logic in mathematics

are themselves absolute. Mathematics consists of language games with deeply entrenched rules and

patterns that are very stable and enduring, but which always remain open to the possibility of change,

and that in the long term, do change.

The criticism that relativism in mathematics means that "anything goes" and that "anybody's opinion is

as good as anybody else's" can be countered by using William Perry's distinction between the positions

of Multiplicity and Contextual Relativism. Multiplicity is the view that anyone's opinion is valid, with

the implication that no judgements or rational choices among opinions can be made. This is the crude

form of relativism in which the opposite of necessity is taken as arbitrariness, and which frequently

figures in 'knockdown' critiques of relativism. It is a weak and insupportable 'straw person' position

and does not represent fallibilism. Contextual Relativism comprises a plurality of points of view and

frames of reference in which the properties of contexts allow various sorts of comparison and

evaluation to be made. So rational choices can be made, but they always depend on the underlying

contexts or systems. Fallibilists adopt a parallel position in which mathematical knowledge is always

understood relative to the context, and is evaluated or justified within principled or rule governed

systems. According to this view there is an underlying basis for knowledge and rational choice, but that

basis is context-relative and not absolute.

This position weakens the criticism from absolutists that an invented mathematics must be based on

whims or spur of the moment impulses, and that the social forces moulding mathematics mean it can

blow hither and thither to be reshaped accorded to the prevailing ideology of the day. The fallibilist

view is more subtle and accepts that social forces do partly mould mathematics. However there is also a

largely autonomous internal momentum at work in mathematics, in terms of the problems to be solved

and the concepts and methods to be applied, and criteria of proof and truth. The argument is that these

are the products of tradition, not of some externally imposed necessity. Some of the external forces

working on mathematics are the applied problems that need to be solved, which have had an impact on

mathematics right from the beginning. Many examples can be given, such as the following. Originally

written arithmetic was first developed to support taxation and commerce in Egypt, Mesopotamia, India

and China. Contrary to popular opinion, the oldest profession in recorded history is that of scribe and

tax collector! Trigonometry and spherical geometry were developed to aid astronomy and navigational

needs. Later mechanics (and calculus) were developed to improve ballistics and military science.

Statistics was initially developed to support insurance needs, to compute actuarial tables, and

subsequently extended for agricultural, biological and medical purposes. Most recently, modern

computational mathematics was developed to support the needs of the military, in cryptography, and

then missile guidance and information systems. These examples illustrate how whole branches of

mathematics have developed out of the impetus given by external needs and resources, and only

afterwards maintained this momentum by systematising methods and pursuing internal problems.

This historical view of fallibilism also partly answers the challenge that John Barrow issues to

'inventionism'. He asks if mathematics is invented how can it account for the amazing utility and

effectiveness of pure mathematics as the language of science? But if mathematics is seen as invented in

response to external forces and problems, as well as to internal ones, its utility is to be expected. Since

mathematics studies pure structures at ever increasing levels of abstraction, but which originate in

practical problems, it is not surprising that its concepts help to organise our understanding of the world

and the patterns within it.

The controversy between those who think mathematics is discovered and those who think it is invented

may run and run, like many perennial problems of philosophy. Controversies such as those between

idealists and realists, and between dogmatists and sceptics, have already lasted more than two and a half

thousand years. I do not expect to be able to convert those committed to the discovery view of

mathematics to the inventionist view. However what I have shown is that a better case can be put for

mathematics being invented than our critics sometimes allow. Just as realists often caricature the

relativist views of social constructivists in science, so too the strengths of the fallibilist views are not

given enough credit. For although fallibilists believe that mathematics has a contingent, fallible and

historically shifting character, they also argue that mathematical knowledge is to a large extent

necessary, stable and autonomous. Once humans have invented something by laying down the rules for

its existence, like chess, the theory of numbers, or the Mandelbrot set, the implications and patterns that

emerge from the underlying constellation of rules may continue to surprise us. But this does not change

the fact that we invented the 'game' in the first place. It just shows what a rich invention it was. As the

great eighteenth century philosopher Giambattista Vico said, the only truths we can know for certain

are those we have invented ourselves. Mathematics is surely the greatest of such inventions.

... Based on my lived and living experiences and contradictions [3] and various literatures e.g. [2], [4], [5], [6], [7], [8], [9], [10], & [11], I realize that both history and philosophy of mathematics are therefore basis for mathematics education. ...

... The absolute nature of mathematics is universal, objective and certain, with mathematical truths being discovered through the intuition of the mathematician and then being established by proof while the fallible nature of mathematics is an incomplete and everlasting work-in-progress, and is corrigible, revisable, changing, with new mathematical truths, being invented, or emerging as the by-products of inventions, rather than discovered [9]. According to Luitel [11], the pure nature of mathematics gives rise to an exclusive emphasis on an ideology of singularity, epistemology of objectivism, language of universality and logic of certainty whilst developing curriculum, conceiving pedagogies and implementing assessment strategies in school mathematics education and mathematics teacher education programmes while the nature of impure mathematics while the impure nature of mathematics to represent a host of local knowledge traditions and typologies e.g. ...

Indra Mani Shrestha

Abstract Both human civilization and mathematics evolved together, so far. The history of mathematics has been playing a key role in humanizing mathematics education conceiving of it as historical, social and cultural productions to help students understand the meanings of aims, values, concepts, methods, and proofs in different social practices involving mathematics. The philosophy of mathematics is to account for nature of mathematics through its epistemology so as to address the issues on mathematical knowledge claims and justification. Both history and philosophy of mathematics are therefore basis for mathematics education. In this regard, this paper explores the roles of both history and philosophy of mathematics in mathematics education, thereby further highlighting the prevailing philosophy of mathematics education across the world. For this, I will portray my lived experiences and contradictions of teaching and learning of mathematics in relation to history and philosophy of mathematics and philosophy of mathematics education. Keywords: absolute and fallible mathematical knowledge, auto/ethnography, epistemology, humanizing mathematics education

... As a field of study, mathematics often enjoys a distinctive status among the sciences. Such high status comes from the idea accepted by most practicing mathematicians that mathematics is universal, objective, and certain (Ernest, 1999), or, in other words, mathematics comprises a priori knowledge, that deals with truths that are true by virtue of necessity, and with objects that are abstract (Linnebo, 2017). These ideas, however, are not unanimously shared, and disagreement about such assumptions can be traced back to the work of Plato and Aristoteles (Machado, 2013). ...

Sabrina Bobsin Salazar

In this work I investigate the connection between teaching practices and institutional racism. I combine concepts from critical realism and critical race theory to develop a theory to better describe how local social interactions that occur in a mathematics classroom can disrupt common interactions that lead to the reproduction of the racial structure that permeates contemporary U. S. society. Drawing primarily on the concept of norm circles, I discuss how specific mathematical instructional practices supported the creation of a conflictive normative spaces inside of a classroom in which local disruption of racism are more likely to occur. I apply the theory in an empirical experiment to refine and improve it. I analyzed episodes of instruction from an elementary mathematics laboratory classroom. The application of the theoretical framework consisted first of identifying instances in which the teacher enacted a teaching practice that counter an expected action. The expected action was guided by the literature review on teaching Black children and positioning Black girls in a classroom. Then I checked the normativity of the teacher action to confirm it as a regular instructional practice in this classroom. I also checked the norm circle the endorsed such practice by identifying the members of the circle and how the teaching practice was reinforced. I identified four instructional practices that locally disrupted racism: (1) regulating student seating; (2) keeping the focus on mathematics; (3) regulating speaker and audience participation; and (4) responding to student's thinking. All these practices, in connection with the conception of mathematics endorsed in this classroom, supported the creation of intersectional normative spaces in which Black children were more likely to engage in doing mathematics and to expect and be expected to do so. In these spaces, they were also less likely to be disciplined or have their thinking immediately evaluated and corrected. In these spaces, Black children, in particular Black girls, were often actively and deliberately being positioned as academically and mathematically smart. A second set of findings from this work center on the methodological operationalization of the framework. The strength of the framework rests in the normativity of the teaching practice, therefore the necessity of verifying the norm circle that locally endorses each instructional practice. This dissertation contributes to theory and method related to the study of how racism permeates teaching practice. Connecting analyses of institutional racism to classroom micro-interactions, this study tests and articulates a theoretical framework that also offers practical leverage for developing approaches to disrupting racist patterns in instruction.

... The hottest debate now is that mathematics is absolute in nature or fallible in nature. However, my assumption is that mathematics is fallible in nature as it is corrigible through social discourses (Ernest, 1996) because we lived mathematics, are living mathematics and will be living mathematics. In this context, birds and frogs have significant roles of working together to overcome the culturally decontextualised mathematics education in Nepal by linking academic mathematics to their cultural practices (Luitel, 2012). ...

Indra Mani Shrestha

How did/do/will I learn Mathematics? How did/do/will I teach Mathematics? In what ways my ways of learning and teaching Mathematics got transformed? Who/what is/are responsible for my personal transformation? Am I a transformative learner, teacher and practitioner-researcher? Is transformative learning changing my ways of knowing, ways of being/becoming, ways of valuing and ways of sensing? What will be my ways of living? Through an autoethnographic inquiry under transformative education research, these issues are discussed in three phases (retrospective, introspective and prospective) of learning and teaching of mathematics.

... For a relevant discussion on whether mathematics is a discovery or invention, seeErnest (1999),Fine (2012),Rowlands and Davies (2006).10 For some other works as case examples in sociology of science, seeBassett (1999), BellamyFoster and Clark (2008) andCollins and Restivo (1983).Content courtesy of Springer Nature, terms of use apply. ...

Ulas Basar Gezgin

How a social science of big data would look like? In this article, we exemplify such a social science through a number of cases. We start our discussion with the epistemic qualities of big data. We point out to the fact that contrary to the big data champions, big data is neither new nor a miracle without any error nor reliable and rigorous as assumed by its cheer leaders. Secondly, we identify three types of big data: natural big data, artificial big data and human big data. We present and discuss in what ways they are similar and in what other ways they differ. The assumption of a homogenous big data in fact misleads the relevant discussions. Thirdly, we extended 3 Vs of the big data and add veracity with reference to other researchers and violability which is the current author's proposal. We explain why the trinity of Vs is insufficient to characterize big data. Instead, a quintinity is proposed. Fourthly, we develop an economic analogy to discuss the notions of data production, data consumption, data colonialism, data activism, data revolution, etc. In this context, undertaking a Marxist approach, we explain what we mean by data fetishism. Fifthly, we reflect on the implications of growing up with big data, offering a new research area which is called as developmental psychology of big data. Finally, we sketch data resistance and the newly proposed notion of omniresistance, i.e. resisting anywhere at any occasion against the big brother watching us anywhere and everywhere.

... Further discussion on various paradigms, however, would fall outside the remit of this paper. It is important to note from the above that each paradigm has a different position on what mathematics is and where it comes from, and that even in the circles of professional mathematicians, various epistemological opinions are held (Ernest, 1999). As a result, this diverse mixture of conceptions about the nature of mathematics has influenced the ways researchers in mathematics education, school teachers, and the general public see the teaching and learning of mathematics (Dossey, 1992). ...

Constantinos Xenofontos

Drawing on data from the Republic of Cyprus, this paper uncovers elementary teachers' epistemological beliefs about mathematics. Twenty-two experienced teachers were invited to individual semi-structured interviews. Thematic data-driven analyses identified three themes and eight sub-themes, which I discuss, taking their socio-cultural context into consideration. This study suggests that applying predetermined frameworks directly taken from the literature when examining teachers' epistemological beliefs in mathematics can be problematic, as they might hinder other culturally specific beliefs from emerging. In closing, this paper presents some implications for the results on teacher education and professional development, as well as ideas for future research.

Niroj Dahal

This dissertation portrays my lived experience and is an exploration of my pedagogical practices as a learner, as a teacher and as a facilitator focusing on relationship between teacher and students shifting from traditional to transformative approach (e.g. meaning-centered and life-affirming) in teaching and learning. Based on my lived experiences as a learner from school to master level and as a teacher in different places in different times, I generated my research problems about the paradigm of teaching and learning in different stages with various kinds of relationship in mathematics learning and teaching. The aim of my research project was to examine and explore deep settled behavioral practices and seek to change myself and others towards transformative/constructivist approach of learning in terms of teacher-students relationship to maintain quality of education for future generation. In this study, I used interpretivism, criticalism, and post modernism research paradigms to embrace multi- paradigmatic research design. Interpretive research paradigm helped me to be subjective to address the emergent issues that emerged ii during the research process; critical paradigm enabled me to observe educational phenomenon critically thereby helping me to develop research problems from finger pointing to self and other, un/helpful dualism, envisioning and shifting process. Post modernism helped me to present through multiple genres of writing like poem, narrative, fiction, poster, letter, e-mail, text SMS, etc. to make my text wealthy, and pedagogically thoughtful about my experience regarding paradigm of learning and teaching. Staying myself within multi-paradigmatic research design space, I used autoethnography as a fusion research methodology in my inquiry. Auto-ethnography helped me to place myself within my culture thereby enabling me to explore multilayered pictures of my educative practice of self and others. Auto-ethnographic inquiry also helped me to examine the pedagogical culture and context from different perspectives as students, teachers, and researcher thereby offering space for interpretation, transformation and envisioning. As a mathematics pedagogue, I investigated that traditional teacher-centered, transmission pedagogy, culture and content free mathematics curriculum and practices are some unhelpful dualisms to make our mathematics more meaningful which could only be achieved through better relationship with students, and student-centered pedagogy. I predicted that student's active participation in learning, social and cultural enactment, and transformative pedagogy promote our practice to be more meaningful, and learner-centered which in turn develops cordial relationship. My vision to develop cordial relationship between teacher-students is focused on curriculum a bit differently in this study.

Indra Mani Shrestha

This thesis portrays my comprehensive and evolving inquiry into the prolonged problems of culturally decontextualised mathematics education encountered by the students of Nepal in schools and colleges. It also depicts how the pupils from the land of diverse cultures are deprived of learning contextualised (culturally embedded) mathematics. I have presented comprehensively my learning practice of mathematics in school, colleges and University, teaching practice of mathematics in secondary school and trainings I have given to primary school teachers based upon the Habermas' three fundamental human interests-technical (controlling), practical (understanding) and emancipatory (independence). To portray my research study, I have chosen an autoethnography, small p philosophical inquiry and Living Educational Theory as my methodological referents comprising narratives such as poems, drama, dialogues and stories visualizing them with interconnected photographs so far. Autoethnography helped me produce the research text of my cultural and professional contexts of learning and teaching mathematics. Small p philosophical inquiry enabled me to generate new knowledge via a host of innovative epistemologies that have the goal of deepening understanding of normal educational practices by examining them critically, identifying underpinning assumptions, and reconstructing them through scholarly interpretations and envisioning. And living educational theory enabled me to inquire lived and living contradictions of our lifeworlds. In order to carry out my ontological (what is reality?) and epistemological (to know reality) activities I have used the paradigms of interpretivism (Habermasian practical interest) and criticalism (Habermasian emancipatory interest). The paradigm of interpretivism helped me in interpreting/explaining the teaching-learning practice of culturally decontextualised mathematics embedded in events or situations of my life. The critical paradigm helped me to identify my research problem, to critically reflect upon my teaching-learning experiences and to transform my teaching/learning from culturally decontextualised to contextualised mathematics education. I also depicted how inclusion of ethnomathematics in academic mathematics helps pupils in understanding (practical) the culturally decontextualised (pure?) mathematics in sharing of knowledge through co-operative learning in an independent environment (emancipatory) rather than in controlled environment (technical). I also portrayed how an inclusion of contextual mathematics in Nepali curriculum prevents pupils from diversion/rejection of academic mathematics as a body of pure knowledge. Indra Mani Shrestha, Degree Candidate December 30, 2011

This chapter aims to show the possibilities of the use of plug-avatars "hhh" technology education as a Service-Oriented Virtual Learning Environment (SOVLE) in Sliding Mode (SM). This allows teachers to create an integrated learning environment using tools that have been selected to best meet their academic requirements and individual abilities of each student's full training in the system of Distance Education (DE). The work reported in this chapter engages with all aspects of Virtual Learning Environment (VLE) design and architecture. Thus, created software of plug-avatars "hhh" technology education for SOVLE are applicable for use in DE processes and in virtual research collaboration works at the Astrakhan State University, Tomsk State University of Control System and Radio Electronics (Russian Federation), at HHH University (Australian Federation and the Republic of Armenia), at Rohilkhand University (India), and at National Central University (Taiwan).

Leah Nichole Shilling

The important role of beliefs in the learning and teaching of mathematics has been largely acknowledged in the literature. Pre-service teachers, in particular, have been shown to possess mathematical beliefs that are often traditional in nature (i.e. viewing teachers as the transmitters of knowledge and students as the passive recipients of that knowledge). These beliefs, which are formed long before the pre-service teachers enter their teacher education programs, often provide the foundation for their future teaching practices. An important role of teacher education programs, then, is to encourage the development (or modification) of beliefs that will support the kind of (reform) mathematics instruction promoted in these programs. In this dissertation I explored the impact of different experiences within teacher education programs, particularly those related to mathematics courses, on the mathematical beliefs of pre-service elementary teachers. This exploration was structured around 3 interrelated strands of work. The first strand drew from the existing literature to illuminate the concept of beliefs and identify ways in which teacher education programs may influence and promote change in the beliefs of pre-service teachers. This review also highlighted the need to further investigate the role and impact of mathematics courses for pre-service teachers.The second strand introduced an analytic framework to examine the different views about mathematics promoted in textbooks used in mathematics courses. The findings demonstrated that the linguistic choices made by textbook authors may promote different views about mathematics and, as a result, create different learning opportunities for pre-service teachers. These findings may have several implications for textbook authors and those in teacher education programs who make decisions about textbook adoption. Finally, the third strand investigated the impact of the curriculum materials and instruction in a research-based mathematics course on the beliefs of 25 pre-service elementary teachers. The findings showed that while beliefs are often highly resistant to change, it is possible to motivate change during a single mathematics course. Specifically, the nature of the curriculum materials and the role of the teacher educator in the course were found to have an important impact on the mathematical beliefs of the pre-service teachers.

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