Pre Service Teacher Education

he field of teacher education is in a state of flux… some might say upheaval… as a plethora of traditional university-based and “alternative route” pathways for preparing new teachers compete for the attention and support of both policy makers and prospective teacher candidates.    In this context, the emergence of alternative route programs has produced a good deal of what might be termed “horse-race” research comparing outcomes of various teacher preparation models (e.g., Henry, et al, 2013; Xu, Hanaway, & Taylor, 2011).    While results of these studies have varied, the general conclusion of researchers is that there is as much within-group variation in program features and quality as there is between groups—with both alt-route and traditional programs sharing many of the same structural features and curriculum content (Boyd, et al, 2008).  The “horse-race” approach has not been a productive strategy for understanding the features and affordances of varying models of teacher preparation.

Teacher educators in South Africa face challenges of preparing new teachers for an inclusive education system that has been accepted as policy but is not yet fully realised in school contexts. Pre-service teachers entering initial teacher education programmes are themselves a product of a schooling system in which many inequalities and marginalising practices are still prevalent. In this paper, we present an analysis of the extent to which prospective teachers' personal experiences within the schooling system influences their perceptions about the benefits and drawbacks of two common organisational arrangements made for learners who experience barriers to their learning. The analysis of empirical data from a questionnaire and individual interviews reveals that participants who had personally observed or experienced such arrangements had fixed views about their benefits or drawbacks. We consider the implications of this finding for teacher education programmes that seek to produce teachers who can teach inclusively in the complexities of South African schools.

W artykule zostały przedstawione wyniki dwóch studiów empirycznych, których celem było zbadanie stosunku przyszłych nauczycieli do nowych edukacyjnych rozwiązań TIK. Studia ta ukazują, że podejście nauczycieli do TIK kształtowane jest nie – jak zakładało wiele wcześniejszych badań – w placówce w której podejmują oni pracę, ale w jednostkach w których są kształceni pedagogicznie. Ponadto pokazują, że stosunek przyszłych nauczycieli do TIK wynika także z ich edukacyjnych/szkolnych doświadczeń – z potrzeby reprodukowania znanych im (nisko stechnicyzowanych) modeli edukacji. Koniecznym wydaje się więc wprowadzenie do systemu kształcenia nauczycieli nowego modelu wiedzy i kompetencji, uwzględniającego zapotrzebowanie na wiedzę i kompetencje technologiczne. Takim modelem może stać się tytułowy TPACK (Technological, Pedagogical and Content Knowledge model) – jego podstawowe założenia zostały przedstawione w ostatniej części tekstu, jako swego rodzaju rekomendacja wynikająca z przytoczonych badań.

Background: In the intermediate phase of schooling, learners’ academic success is largely dependent on their ability to read and write academic texts. Teachers need to teach academic language intentionally and explicitly. In order for teachers to do this, they themselves need knowledge of academic language and its features. Teacher education students, therefore need to be explicitly taught about academic language and provided with sufficient opportunities to develop their own proficiency.

Aim: This article aimed to explore the academic language proficiency of a cross-sectional sample of teacher education students at a Johannesburg university.

Setting: This study took place at a South African university that implements a universityaccredited primary school teacher education qualification. The university is located in an urban area, but attracts students from both urban and rural contexts.

Methods: Students’ test scores on a core academic language skills instrument were utilised as data for this study, with descriptive and inferential statistical analyses procedures used to make sense thereof.

Results: Findings from a cross-sectional analysis between first- and second-year students’ scores indicated that students’ academic language proficiency does not appear to improve after their initial year of study.

Conclusion: The article concludes with a discussion of the implication hereof for teacher education and for the profession.

As we are living amid an unprecedent environmental crisis, the need for schools to empower students into environmental citizenship is intensifying. Teachers are considered as the main driving force in fostering students’ environmental citizenship. However, a critical question is how teachers conceive environmental citizenship and whether their perceptions of environmental citizenship are well-informed. There is an urgent need to investigate teachers’ perceptions, considering their crucial role in the formation of students’ environmental citizenship. This study examines teachers’ perceptions of environmental citizenship through a systematic review and thematic analysis of relevant empirical studies. The selected studies (n = 16) were published in peer-reviewed journals during the timespan of the last twenty-five (25) years (1995–2020). The thematic findings of this review revealed that teachers’ perceptions: (a) manifest a relatively decreased understanding of environmental citizensh...

The pivot from a hybrid learning environment to a fully online one brings to the fore questions of fairness and equity for marginalized individuals and groups of students at our minority-majority university. Because effective distance education requires adjustments to instruction at all levels, the shift to remote teaching raises questions about effective preparation of in-service and preservice teachers who work with students from diverse cultural, linguistic, and ability backgrounds. In this paper I address some of the difficulties faced by students, teachers, and families in New Mexico during the sudden transition to remote education. I outline educator perspectives related to the graduate and undergraduate technology integration courses I teach, emphasizing the need to support students from diverse linguistic, sociocultural, and ability backgrounds, particularly those living in geographically isolated communities.
Available at:
https://www.digitalcultureandeducation.com/reflections-on-covid19/new-mexico

The terms “teacher professionalism” and “professional dispositions” are often used when assessing preservice and practicing teachers.  But what is “teacher professionalism”?  What dispositions are the “correct” ones for teachers?  And who gets to decide these things?  This chapter excerpt explains why the Danielson Framework for Teaching is still being used.

Previous publications in applied linguistics concerning nonnative English speaker teachers (NNESTs) (e.g., Kramsch, 2003; Llurda, 2015; Pennycook, 1994; Rajagopalan, 2005; Train 2007) have analyzed and documented how ideologies rooted in the myth of the native speaker and associated ideologies engender discrimination and affect students and teachers’ sense of self-worth. This qualitative study examined a telecollaboration experience to determine how Colombian English pre-service teachers’ interactions with U.S. heritage Spanish speakers (HSSs) influence the Colombian future teachers’ self-perceptions as (non) native speakers. This research joins the call scholars have made to expand telecollaboration into critical perspectives to challenge pervasive and harmful language beliefs and attitudes (e.g., Helm, 2017; Train, 2005). Findings suggest that Colombian participants’ cooperative relationships with U.S. peers provided them affective and knowledge-based resources to build more favorable views of themselves, attitudes to confront the detrimental effects of nativespeakerism ideologies, and informed judgments to dismantle them.

Bu çalışmanın amacı Bruu-Carver, Castera, Gericke,Menezes-Evangelista ve El-Hani(2017) tarafından geliştirilen Genetik ve Genetik Uygulamalarına Yönelik Tutum ve Anlayış Ölçeği’nin (GGUYTAÖ) Türkçe uyarlaması geçerlik ve güvenirlik araştırmasını yapmaktır. Ölçek, fen eğitimi ve bilim okuryazarlığı üzerine yapılan araştırmalarda kullanılmak üzere genç yetişkinlerin genetik determinizm düzeylerini, gen-çevre etkileşimi ve modern genetik uygulamalarına yönelik bilgi düzeylerini, aynı zamanda modern genetik uygulamalarına yönelik tutumlarını belirlemek ve bilgi düzeyleri ile tutumları arasındaki ilişkileri incelemek için geliştirilmiştir. Ölçeğin Türkçe uyarlaması iki devlet üniversitesinin eğitim fakültelerinin farklı bölümlerinde öğrenim görmekte olan 682 birinci sınıf öğrencisine uygulanmıştır. Veriler üzerinde doğrulayıcı faktör analizi (DFA) yapılmıştır. Doğrulayıcı faktö r analizi sonuçları GGUYTAÖ’nin ilk 3 bölümü için (genetik determinizm, gen-çevre etkileşimi ve modern uygulamalar hakkında genel bilgi) boyutları için 4 boyutlu bir yapıyı desteklediği ve GGUYTAÖ’nün son bölümü olan modern genetik uygulamalara yönelik tutumlar bölümünde ise 4 boyutlu bir yapıyı desteklediğini gösterdi. Ölçeğin bölümlerinin Cronbach alpha değerlerinin sırasıyla 0.71, 0.71, 0.68 ve 0.76 olduğu hesaplanırken tekrar test güvenilirliğinin 0.95 olduğu belirlenmiştir. Bu bulgulara dayanılarak bu çalışma, ilgili alan yazına Genetik ve Genetik Uygulamalarına Yönelik Tutum ve Anlayış Ölçeğinin geçerli ve güvenilir bir uyarlama örneğini sunmaktadır.

After much positive feedback from Learning to Teach Science, I am proud to present the 2nd Edition, with a new collection of pre-service teacher-authored stories. This year, we begin to link the practice in the stories to underlying theory, with the chapter titles shaped by current research at Monash University and expert introductions to each chapter written by esteemed academics.

Purpose – There is limited discussion in the teacher education literature about the experiences of preservice black male teachers generally and the ethnic diversity among black male pre-service teachers specifically. Thus, this paper aims to explore the experiences of Frank, a black male refugee health education major attending an historically black college and university (HBCU).

Design/methodology/approach – This research study is theoretically guided by selected tenets of Bush and Bush’s (2013) African American male theory and Goodman et al.’s (2006) transition framework and uses a
qualitative approach to explore Frank’s transition experiences when coming to America, attending college and engaging in his student teaching experience.

Findings – Frank experienced some difficulty transitioning to America, as a result of not having a strong financial foundation. During his college transition, Frank believed that the HBCU environment was nurturing;
however, he encountered numerous ethnocentrically charged hostile confrontations from US-born black students at his university because of his accent. While he had some disagreements with the US education system in terms of discipline, Frank believed that his accent served as an asset during student teaching.

Originality/value – This study adds to the burgeoning research that explores the intersectional identities among pre-service black male teachers. As we argue in this paper, researchers, policymakers and
practitioners cannot treat black male teachers as a monolithic group and must contemplate the unique supports needed that can attend to the racial and ethnic needs of black male teachers.

Gran parte de las transformaciones curriculares recientes en materia de formación inicial docente tienden a incrementar el peso relativo de su componente práctico. A pesar de ello, nuestra región cuenta con escasos trabajos empíricos sobre las prácticas docentes en la formación inicial. Este artículo sistematiza las investigaciones anglófonas sobre la tríada de las prácticas (compuesta por el docente en formación, el profesor de prácticas y el docente co-formador u orientador) entre 2000 y 2014. El corpus (84 artículos) se organiza en cuatro ejes temáticos: (1) el proceso de convertirse en formador de prácticas, (2) el papel formativo de los profesores de práctica y docentes orientadores, (3) la retroalimentación en las prácticas docentes y (4) los conflictos entre los miembros de la tríada. El trabajo busca contribuir con evidencia empírica a la discusión regional sobre las prácticas docentes.

In this book review of Breaking the Mold of Preservice and Inservice Teacher Education, the author explains the structure and organization of the book, its strengths and limitations, and the appropriate audience for its content. The author highlights the content of several of the chapters. The author also situates this book in current theories within the complex field of teacher education.

As we are living amid an unprecedent environmental crisis, the need for schools to empower students into environmental citizenship is intensifying. Teachers are considered as the main driving force in fostering students’ environmental citizenship. However, a critical question is how teachers conceive environmental citizenship and whether their perceptions of environmental citizenship are well-informed. There is an urgent need to investigate teachers’ perceptions, considering their crucial role in the formation of students’ environmental citizenship. This study examines teachers’ perceptions of environmental citizenship through a systematic review and thematic analysis of relevant empirical studies. The selected studies (n = 16) were published in peer-reviewed journals during the timespan of the last twenty-five (25) years (1995–2020). The thematic findings of this review revealed that teachers’ perceptions: (a) manifest a relatively decreased understanding of environmental citizensh...

Cet article présente les résultats d'une étude sur la manière dont les enseignants de français conceptualisent les contenus didactiques en fonction des interactions en classe. La représentation qu'ont les enseignants de leur gestion des interactions a été rapportée au moyen de protocoles verbaux concomitants à une tâche de planification et au moyen de protocoles rétrospectifs immédiats. Pour répondre aux réactions des élèves, le fonctionnement fréquemment adopté par l'enseignant semble «improvisationnel». La planification de l'enseignement est fondée sur des connexions conceptuelles. Elle relie les contenus les uns aux autres selon une appréciation des besoins potentiels des apprenants.
Sur le plan cognitif, la planification est la résultante d'une accommodation entre
des éléments didactiques organisateurs et les apports potentiellement désorganisateurs
du contexte d'interaction. Ces résultats descriptifs corroborent ceux qui ont été obtenus
dans d'autres disciplines et acquièrent de ce fait une certaine généralisabilité.

With the sudden interest in fake news, there is an outcry for the need for media literacy in schools.  While this is a good starting point, I argue that this does not go far enough -- as educators and researchers, we must advocate for critical media literacy in the classroom. This paper aims to delineate the differences between media literacy and critical media literacy.

ÖZ
Bu çalışma ile okul öncesi öğretmen adaylarının erken okuryazarlık kavramına ilişkin
görüşlerinin ve staj uygulamalarında erken okuryazarlık becerilerini desteklemeye yönelik
yer verdikleri etkinliklerin sayı ve çeşitliliğinin incelenmesi amaçlanmıştır. Çalışmanın
verileri araştırmacılar tarafından hazırlanan sorular yoluyla toplanmıştır. Çalışmaya
Ankara’da bulunan bir devlet üniversitesinde öğrenim gören 81 okul öncesi öğretmen
adayı katılmıştır. Veriler içerik analizi yöntemi kullanılarak incelenmiştir. Çalışma
bulguları öğretmen adaylarının büyük bir kısmının erken okuryazarlık konusunda yeterli
bilgi düzeyine sahip olmadığını göstermiştir. Ayrıca, öğretmen adayları staj
uygulamalarında sınırlı sayıda ve çeşitte erken okuryazarlığı destekleyecek etkinliklere
yer verdiklerini ve bu etkinlikleri planlama ve uygulama aşamalarında kendilerini yeterli
görmediklerini belirtmişlerdir. Elde edilen bulgular ilgili alanyazın ışığında tartışılmış ve
öneriler sunulmuştur.
Anahtar kelimeler: erken okuryazarlık, okul öncesi, öğretmen adayı görüş, uygulama

ABSTRACT
The present study aims to investigate pre-service preschool teachers’ opinions on early
literacy notion and their school practices to enhance children’s early literacy skills. The
participants of the study were 81 pre-service teachers who are enrolled in a state
university in Ankara. Data were collected through a question form. The form was
developed by the reseachers. Data were analyzed by using content analysis. The results
revealed that pre-service teachers have limited knowledge about the early literacy notion.
In addition, they applied limited number and kinds of early literacy activities in their
school practices. They indicated that they face some problems about preparing and
applying early literacy activities. The findings were discussed based on the related
literature and Turkish context. Implications and recommendations were addressed for
early childhood teacher education programs.
Keywords: early literacy, preschool, pre-service teacher opinions, practices

This study focused on job satisfaction of Teacher Educators as well as to compare the various level of job satisfaction of teacher educators who are working in different Govt.-aided, Autonomous and Self-Financed B.Ed. Colleges in West Bengal. Teacher is the backbone of any nation. If education is viewed because the backbone and energizing force for the national progress then the exalted and pivotal position of the teacher can't be ignored as she/he has been assigned the responsibility of educating the future generation. The progress and development of a nation largely depends on its teachers' community due to their noble and large contributions in nation building. They have remarkable contributions in making and shaping each and every person of the society. The investigators choose various factors to measure the Level of Job Satisfaction of Teacher Training Colleges on the basis of multi dimensions that are Working Conditions,

Scientific literacy which is basically synonymous with the public's understanding of science is necessary for people in this age of technology to enable them use, cope and adapt to the technological inventions of the time. Education which has remained to be a social process in capacity building, sustaining humanity and indeed the vehicle for acquiring the requisite scientific literacy in any society has not been well positioned in Nigeria. Specifically, the basic education sector, which is the foundation for all forms of education, has been inundated with a myriad of challenges which militate against the realization of the scientific literacy goals. Central among the challenges is the problem of teacher production for the Basic Education sector. There has been a glaring dislocation/mismatch between the type of teachers trained in Colleges of Education (COEs) and the work they do at Basic Education level. The training programs are devoid of synchronization between the content and pedagogical knowledge with pitiable implementation affecting the quality of their service delivery, thus compromising the goals of the Universal basic education (UBE). The National Commission for Colleges of Education (NCCE) within its constitutional mandate responded to the circumstance and radically overhauls the old Nigeria Certificate in Education (NCE) curriculum to a new service focused one. Salient features in the renewed hope include; differently packaged areas in which prospective teachers are expected to specialize, paradigm shift from teaching to learning in which pre-service teachers will be provided with opportunities to acquire professional skills and attitudes and a well articulated implementation framework. The overall implication of the reform program will be re-orientating the pre-service teacher production pipeline for a solid foundation of scientific literacy delivery. This paper therefore mirrors the interplay of the reform effort at the basic education level with the probable improved state of education quality in general as precursor to sound scientific literacy base.

Whereas the motto of the United Nations’ 2030 Agenda for Sustainable Development is “Leave no one behind”, it serves as a guiding principle for all to work towards an inclusive society.As Physical Education (PE) serves as a fertile ground for expressions of supportive milieu, the new generation of PE teachers should seize this opportunity to become advocates of mainstreaming students with additional needs in order to achieve substantive equality for all in schools. Pursuing this endeavor, understanding the meaning of their experiences in mainstreaming may give a glimpse of the current perspectives and viewpoints that are vital in the formulation of appropriate intervention for improved inclusive instruction in the 21st century. This Heideggerian phenomenological study using one-on-one unstructured interviews was used to explore the meaning of the lived experience of the seventeen (17) purposively selected millennial teachers in mainstreamed PE classrooms in Cebu City and Cebu Province, Philippines. Using hermeneutical analysis, it was deduced that while they see mainstreaming as a wonderful concept, their experiences on it had apparently captured A Reflection of a Wider Problem on Inclusion in Academia as an overarching theme that encapsulated how PE teachers today subsisted the multifaceted issues and challenges on mainstreaming. This journey was specified through the subthemes: A Test of Pedagogical Competence, Multifaceted Contextual Inclusion Challenges, Fete of Teacher’s Complex Support Roles and Risk of Curriculum Infidelity. The findings implicate that pedagogies that “fit all” do not guarantee the best possible learning outcomes and, teachers who represent only as “teacher” do not foster inclusive ethos to their students. Achieving these expectations can be daunting, thus, it is recommended that education administrators to conduct capacity building opportunities to the faculty and staff for indepth theoretical knowledge and practical understanding on the concept and application of inclusive education.

Keywords:- Asia, Inclusive Education, Mainstreaming, Millennial, Phenomenology, Physical Education, Philippines.

This report presents the results of the trends mapping study on the future of technical and vocational education and training (TVET) teaching, conducted by Gita Subrahmanyam (with research assistance from Bindu Law) on behalf of UNESCO-UNEVOC. The study engaged the international TVET community to: (i) improve understanding of the implications of global disruptions; (ii) gather knowledge, insights, experiences; and (iii) highlight promising practices in preparing TVET teaching staff to deliver the skills needed in the 21st century and beyond.

We’ve all seen the short ‘educational and entertaining films shown on study days of teachers, the ones that show practical, real-world classroom examples, such as teachers demonstrating how to use a new ICT application. This article discusses such examples of educational practice (in short: practical examples) because, if they meet certain conditions, they can inspire teachers to integrate ICT into their daily classroom teaching.

I reviewed literature on teaching abroad to increase global competencies among pre-service teachers, seeking specifically papers which evaluated programmatic models embedded in universities or consortiums. I found a relative lack of publications and, of them, a lack of critical voices speaking to aspects of programming which could be approved, or to the externalities of such programs. I close by posing several questions for scholars working in and around this topic.

Children make up half of the homeless popu- lation in the US, and of those, almost 50 percent are under age six. Homeless children face many different challenges in school. These children and their families have been invisible in school due to the indifference and stereotypes about them. This article focuses on early childhood pre- service teachers’ beliefs over the course of one semester at a private university in the United States. The pre-service teachers explored their own perceptions about homeless children through urban community-based field experiences. The pre-service teachers regularly worked with children in homeless shelter learning centers throughout the semester. The data revealed that the pre-service teachers had fears about interacting with the unfamiliar setting, as well as deficit perspectives about homeless children. However, while they were engaged in the community-based experi- ences, they started to re-examine their deficit views about young homeless children and their families, acquire more accurate information on their complex situations, and positively developed their professional perspectives on these children. This study suggests that raising awareness about homeless children and their families should be a part of both professional development in schools and early childhood teacher education programs

Today audiobooks are recognized as effective didactic tools by foreign researchers and methodologists. However, they are still an unexplored phenomenon in Ukraine. We aim to develop an effective methodological procedure with appropriate listening activities, which can be incorporated into foreign language teaching practice at Ukrainian pedagogical universities. Therefore, the issue of audiobook selection for listening skills development of pre-service English teachers is investigated here. On the basis of recent research into text selection for enhancing students’ listening skills and discourse analysis of contemporary short stories in audiobook format, we have determined criteria for fiction which should be used for intensive and extensive listening practice. They include the following: professional performance and recording, highly artistic works, representativeness, relevance of content, the degree of accessibility and simplicity, contemporary topics, diversity (for intensive listening); professional performance and recording, correspondence of the genre and the content of a short story to students’ interests, the degree of accessibility and simplicity (for extensive listening). Since audiobooks are recognized as the art of today, we insist that the basic criterion for the selection of authentic short stories is their professional narration and recording. Such audiobooks give the best opportunity to enjoy listening to fiction, promoting the
development of aesthetic needs. Based on the discourse analysis of contemporary short stories, we highlight the main features that have a signifi cant impact on the result of listening. They are the following: poetic language, socio-cultural lexis, dialect, slang, terms and professionalisms, social and regional accents, English varieties, psychological
and mystic plots, intertextuality, highly informative plots, exogenous context,
«open» endings, humor, sarcasm, irony, atypical narrative form, atypical composition and architectonics. The analysis of the selected audiobooks suggests that the indicators of their objective complexity vary, therefore, it is necessary to assess the degree of their accessibility and simplicity for a particular group of students in advance, which will help to select an appropriate short story, predict its potential difficulties and organize effective listening practice.
Key words: audiobooks, extensive listening, fiction, intensive listening, listening competence, pre-service teachers, selection criteria, short stories.

The study describes the kinds of problems posed by pre-service teachers on the basis of complex solid geometry tasks using the “what if not?” strategy and the educational value of such an activity. Twenty-eight pre-service teachers participated in two workshops in which they had to pose problems on the basis of given problems. Analysis of the posed problems revealed a wide range of problems including those containing a change of one of the numerical data to another specific one, to a proof problem. Different kinds of posed problems enlightened some phenomena such as a bigger frequency of posed problems with another numerical value and a lack of posed problems including formal generalization. We also discuss the educational strengths of problem posing in solid geometry using the “what if not?” strategy, which could make the learner rethink the geometrical concepts he uses while creating new problems, make connections between the given and the new concepts and as a result deepen his u...

Lesson planning for student teachers is often regarded in technical terms, merely as the means to ensure effective classroom performance. This approach limits the possibilities that the process of lesson planning offers to the development of professional competence among student teachers. In particular, student teachers need to begin to develop their pedagogical content knowledge (PCK), the capacity to make pedagogical choices that are logically derived from content and contextual knowledge. This article reports on a Lesson Planning Guideline which is used to scaffold the construction of student teachers’ PCK individually by requiring them to consider the constituent parts of PCK individually and in relation to one another during the planning process. This guideline was developed in response to perceived limitations of existing guidelines used in our institution and found in texts for student teachers. Called a “Rationale for lesson design” the Guideline does not attempt to simplify the planning process, but rather enables students systematically to access the complexities inherent in effective lesson preparation. By requiring students to articulate their content knowledge and narrate their pedagogical reasoning in some detail, the Guideline enables students not only to teach with confidence but also to construct PCK.

This article is grounded in both Ellsworth and Milner’s arguments, while
demonstrating the connections and intersections of their claims in the current
terrain of teacher education. I demonstrate the tension between the prevalent
forms of academic critique and the increasing diversity in pre-service teacher
education programs in Canada and argue that critical discourses may limit the
creation of inclusive educational spaces in diverse teacher education programs.
Such analysis is timely in light of the recent academic debates around
“safe spaces,” “cultural appropriation,” and “trigger warnings” (on which I
will elaborate in the next sections; Boysen 2012; Etzioni 2014; Lukianoff
and Haidt 2015).

This text is intended as a reference for the first part of the course Teaching Mathematics II for prospective elementary school teachers.  It focuses on fractions of quantities.  The second part focuses on the concepts of rational number, real number, ratio and percent.  It is published separately.
Knowledge presented in the text is meant as a foundation for the “Specialized Content Knowledge” (Ball, Thames, & Phelps, 2008) for the teaching of fractions in elementary schools.  It is intended as teachers’ knowledge, not knowledge that could be imparted, as is, to schoolchildren. It is neither the mathematics of school textbooks for children, nor the mathematics of academic textbooks for future mathematicians. 
Our intention is to look at the content to be taught in elementary school about fractions from the point of view of mathematics itself.  We will strive to clarify the meanings of the fundamental concepts (fraction of a quantity, ratio, proportion, etc.) by formulating definitions and using precise language; we will justify techniques and rules of operations on fractions by reasoning; we will analyze the structure of problems about fractions and classify them into types.  In that sense, it is a mathematical knowledge base for making informed decisions in planning lessons, responding to students’ questions and preparing exercises for them.  We will leave aside, for the time being, the issues of the psychology of teaching and learning fractions: how children learn fractions, what difficulties they experience and what teachers can do to help them overcome these difficulties.  These issues will be dealt with in the course Teaching Mathematics III.
In Part I of the course notes we focus on fractions as a way to represent the multiplicative relationship between two quantities (lengths, areas, volumes, weights, capacities, market values, etc.).  We view fractions as an answer to the question of how many times one quantity is bigger or smaller than another.  We explicitly define what it means for one quantity to be such and such fraction of another quantity.  In concrete problems we identify and name the quantities that are being compared and the units with which they are measured.  The comparison may require change of unit or conversion from one unit to another.  We become aware of these operations, analyze them and learn how to write them.  In brief, in this course, we make explicit what is often left implicit in teaching fractions to children. 
      In Part I of the course notes, we will talk mainly about quantities.  So, for example, instead of talking about abstract numbers, e.g.,  3/4  and  1/8  , and calculating their sum,  3/4+  1/8 = 7/8  , we will be talking about, say, Tom using up  3/4  of a 120-gallon water tank, meaning the quantity of 90 gallons, and Jerry using up  1/8  of the same water tank and asking how much of the tank they used together.  We will be adding quantities, not abstract numbers, expressing this operation in terms of numbers of units (gallons in this case), as
3/4  of 120 Gal + 1/8  of 120 Gal =  7/8  of 120 Gal
or as 90 Gal + 15 Gal = 105 Gal. 
In this text, the concept of fraction of a quantity will be derived from the practices of measurement, particularly the operations of change of unit and unit conversion.  If we want to measure some quantity, the existing units of measurement may not represent it precisely enough, and we have to change those units into some finer ones.  For example, if we measure the length of some tiny piece of metal and find that it measures 1 mm and a little bit and find that 3 such “bits” cover 2 mm, then we will say that 1 bit = 2/3  mm. This is a “conversion equation.” 
The approach to fractions through measurement – we call it the Measurement Approach – proposed in this text, is different from the more common Division Approach to fractions through division of numbers, modeled by division of things (e.g., apples or pizzas), or graphical representations of things.  We explain the differences between the Measurement and the Division approaches to fractions in more detail in Chapter I.
Chapter II focuses on the practical and theoretical foundations of the Measurement Approach.  It starts by explaining how the notion of a fraction of a quantity could be derived from the historical practices of measurement.  Then the meaning of the phrase “one quantity is such and such fraction of another quantity” is defined in two ways.  One is inspired by barter exchanges where integer multiples of two quantities are equated with each other. The other is based on measuring each of the quantities independently with a common unit and then comparing those measurements multiplicatively. 
Because a fraction represents a multiplicative relation between two quantities, the operation of multiplication requires special attention.  When dealing with whole numbers of quantity only, understanding multiplication as repeated addition was enough.  But this meaning cannot be applied to taking, say, three-fifths times some quantity, and we need to redefine the operation of multiplication so that it covers multiplication of both a quantity by a whole number and by a fraction.
The definitions introduced in the second chapter constitute the conceptual basis for the subsequent chapters in which a whole arithmetic of fractions of quantities has been developed.  The various operations on fractions of quantities are studied, such as calculating the fraction of a fraction of a quantity; addition of two fractions of a quantity; their subtraction; division of a quantity by a fraction and division of a quantity by a quantity.  The names of these operations include terms known from the arithmetic of whole abstract numbers (e.g., 2 × 3) or whole numbers of quantities (e.g., 2 × 3 kg), such as “addition”, “subtraction”, “multiplication” and “division”, but the meaning of these terms is extended or modified to apply to the domain of fractions of quantities. 
The purpose of this elaborate development is to provide future elementary teachers with a coherent theory that fulfills a twofold purpose.  On one hand, it may help them develop a mature understanding of fractions, beyond the understanding they have acquired in school (Ma, 1999).  On the other hand, it may equip them with the practical tools and flexibility to operate in a variety of contexts and within, likely, very different developments of the topic of fractions, which they may encounter in their future practice. 
Throughout the text, understanding of fractions of quantities and operations with them is fostered by means of examples, exercises and end-of-chapter “Problem sets.”  Chapters VII (Proportional quantities and proportional reasoning) and VIII (Comparison of fractions of quantities) offer an opportunity to consolidate this understanding in solving somewhat more complex problems. 

NOTE: This is a slightly revised and corrected version of the course notes published in 2014 by Concordia University and used as a text in the course EDUC 387 Teaching Mathematics II taught in the Winter 2015 term. An earlier and rather different version of these course notes – using only one definition of fraction of a quantity, and authored by Anna Sierpinska only – has been published in 2013, and used as a text in EDUC 387 Teaching Mathematics II in the Winter of 2014.