LEARNING AND TEACHING FUNCTIONS AND THE TRANSITION FROM LOWER SECONDARY TO UPPER SECONDARY SCHOOL
|
|
- Philippa Bradley
- 8 years ago
- Views:
Transcription
1 LEARNING AND TEACHING FUNCTIONS AND THE TRANSITION FROM LOWER SECONDARY TO UPPER SECONDARY SCHOOL Hans Kristian Nilsen Sør-Trøndelag University College, Norway (PhD student) INTRODUCTION In Norway, the transition between different phases of schooling, particularly in relation to the learning and teaching of mathematics, is an area where little research has been done. The major part of the international research in this field concerns the transition from upper secondary school to university/university college (often denoted as the secondary-tertiary transition) (Gueudet, 2008; Guzmán et al., 1998). My own experiences as a student and a teacher, at both lower and upper secondary school levels have led me to believe that the traditions and beliefs in these institutions differ in ways which in turn might affect students learning. Kindergarden (1-5 years) Primary school (6-13) Lower Secondary school (14-16) Upper secondary vocational or general study program (17-19) University/univ ersity college (20- ) Figure 1: Transitions in the Norwegian school system. As a PhD student (in my third year), I have chosen this transition as the focus of my research. It is important to note that in Norway, upper secondary schooling is divided in two main programmes: the vocational programmes, which are orientated towards practical professions and the general study program, which aims to prepare students for higher education. The curriculum is different in these programmes and is considered to be more theoretical at the general study program. This is also the case for mathematics as a subject. Both of these programmes are included in this research. Further, I have chosen to focus on functions as this is an area highly relevant to both levels of schooling, and personally I find the development of students conceptual understanding of functions to be an interesting research area. It is also possible to expand this area of research, for example by taking the universities/university colleges into consideration, as the learning and teaching of functions is an important issue in several of these study programmes. RESEARCH QUESTIONS How do the students conceptions of functions develop from the 10th grade at lower secondary school to the 11th grade at upper secondary school? How do the students argue for their conception of functions at lower secondary and upper secondary school? How do the students argue for their conception of the slope of a function at lower secondary and upper secondary school? How do the students at upper secondary, general study programme, relate the slope of a function to the concept of differentiation? 1
2 How is the topic of functions mediated at lower secondary compared to upper secondary school? How is the concept of functions presented at lower secondary compared to upper secondary school? How is the slope of a function presented at lower secondary compared to upper secondary school, and in which way is this related to the concept of differentiation at upper secondary, general study programme? How do the teachers argue for the way that they are teaching functions? How do the students experience these two areas of learning (lower secondary and upper secondary) when it comes to the teaching and learning of mathematics, and functions in particular? THEORETICAL BACKGROUND I illustrate an overview of my theoretical framework by the use of the sketch below: Socio-cultural perspective Institutional perspective Brousseau, G. (1997); Chevallard, Y. (2005); Gueudet, G. (2008) Textbook and task analysis TEACHING Fucntions as a boundary object Star & Griesemer (1989) Historical development: Boyer (1959); Klein (1897); Kleiner (1989 LEARNING Sociocultural teories of learning Vygotsky (1978; 1981; 1987); Cole (1985); Lerman (2000) ; Pozzi et al (1998) Students conceptions: Vygotsky (1987); Pierce (1998); Presmeg (2005); Tall & Vinner (1981); Sfard (1991) Figure 2: An overview of my theoretical basis As indicated in the model above, I will use two perspectives for analyzing respectively what I call the teaching aspect and the learning aspect of the actual transition from lower to upper secondary school. In the teaching category, I will consider aspects which can be regarded as external to the individual student. Examples of this could be the teaching methods used and how the teachers approach the subject of functions. This information is mainly provided through observations and interviews. Another example could be the applied textbooks, and the different exercises given to the students. It will also be interesting to investigate upon whether the (later described) use- and/or exchange value is dominating in the mathematics teaching. Studying these issues, I find it useful to apply the institutional perspective as this provides me with an appropriate analytical tool for analyzing the actual teaching situations. I will also argue that the institutional perspective is coherent with the overarching socio-cultural perspective. When it comes to students learning I will be working within the frames of socio-cultural theory of learning. The core of this will be the students engagement in mathematical activities (provided by 2
3 conversations, interviews and handwritten material). Important here is the students concept formation and their conceptual understanding of the function concept. It is also of interest, as stated in my research questions in the previous section, how these conceptual understandings actually develop. The socio-cultural perspective serves as a fundamental basis in this research, overarching both teaching and learning. The socio-cultural perspective Acknowledging the fact that learning is a complex issue, which takes place in a certain social context within a given culture, this perspective to a great extent matches with my own assumptions and beliefs. In addition, it is evident that the concept of mediation is an essential part of this research. The following can serve as examples of psychological tools, and their complex systems: language; various systems for counting; mnemonic techniques; algebraic symbol systems; works of art; writing; schemes, diagrams, maps, and mechanical drawings; all sorts of conventional signs; and so on (Vygotsky, 1981c, p. 137) The quotation above contains some examples of what Vygotsky described as meditational means. Students hand-written materials, their work at computers, their answers and arguing during interviews and conversations, all related to the learning of mathematics (and in this case, functions) are all examples of such mediating means. Hence, in addition to the personal convictions mentioned above, this important role of mediation also brings in the pragmatic dimension in the construction of my theoretical platform Concept formation By basing my argumentation on the Vygotskian understanding of signs as mediating tools, I will approach concept formation from a perspective more in line with the socio-cultural way of thinking. Rooted in Pierce (1998), Presmeg (2005) describes signs through a triad, consisting of a representamen, an object and an interprentant. One can regard the representamen as the sign itself, for example the linear expression y = 2x 3. A classification of this expression (sign) in terms of being a function, an algebraic expression or a linear equation will relate to the object. Interpreting this sign, in terms of acting on it through different representations, for example to draw a straight line through a two-dimensional plane intersecting the y-axis at -3, making a value table or performing algebraic manipulations will all be acts of the interpretant. Figure 3: A representation of a nested chaining of three signs. (Presmeg, 2005, p. 107) 3
4 This interpretant involves meaning making: it is the result of trying to make sense of the relationship of the other two components, the object and the representamen. It is important to note that the entire first sign with its three components constitutes the second object, and the entire second sign constitutes the third object, which thus include both the first and the second signs. Each object may thus be thought of as the reification of the processes in the previous sign The role of students own interpretations in forming mathematical concepts is prominent most of Vygotskys work. Vygotsky separates between pseudoconcepts, concepts as we might use them in our everyday language and true concepts as they are defined and used for example within science and scientific research (Vygotsky, 1987). Working with students understanding of the function concept, it was apparent to me that some students maintained different pseudoconcepts, some of them pointing more in the direction of what function and functions mean in everyday life, than what it actually mean in a pure mathematical sense. In the possible transition from pseudoconcepts to true concepts, Vygotsky emphasises the importance of instruction: Conscious instruction of the pupil in new concepts (i.e. new forms of the word) is not only possible but may actually be the source for a higher form of development of the child s own concepts, particularly those that have developed prior to conscious instruction! (Vygotsky, 1987, p. 172) Functions as a boundary object I think it is an advantage that the focus of the mathematical content considered in such a comparative study as this, is regarded as relevant to both the parties involved. In accordance to the elaborations above, it is evident that functions are a major subject within school mathematics (as within mathematics as a whole). It is also evident (from the Norwegian curricula), that functions are prominent at both lower and upper secondary school. Boundary objects are objects that are both plastic enough to adapt to local needs and constraints of several parties employing them, yet robust enough to maintain a common identity across sites (Star & Griesemer, 1989, p. 46) The notion of boundary concept is used by Star & Griesemer (1989) as an analytical concept which both inhabit several intersecting worlds and satisfy the informational requirements of each of them (p. 393). It is my intention that conceiving of functions as a boundary object will justify the mathematical focus of this study as it (hopefully) creates a common ground for teachers and researchers interesting in developing mathematics teaching and the related transition between lower and upper secondary school. The institutional perspective I find the study of transitions, implying students shifting between two different institutions to be valuable not only for the sake of comparison, but also because it enables the researcher to study the issues of teaching and learning from different perspectives. An institutional perspective opens up for this, as it provides the researcher with the possibilities of analysing different cultural aspects of transition. As the cultural context are investigated and clearly defined to play a role in students learning, I find it evident that this perspective is in accordance with the underlying assumptions of the socio-cultural perspective. Questioning this change of cultures can lead researchers to consider precise mathematical content, and develop detailed transposition studies. It can also lead researchers to study more general institutional expectations (Gueudet, 2008, p. 245) 4
5 This institutional perspective is rooted in Brousseau s (1997) Theory of didactical situations and Chevallards anthropological theory of didactics. A key notion in Brousseau s theory is didactical transposition. Teachers isolate certain notions and properties, taking them away from the network of activities which provide their origin, meaning, motivation and use. They transpose them into a classroom context. (Brousseau, 1997, p. 21). Use- and exchange value Studying the transition between school and college in England, Hernandez-Martinez (2009) suggests that The Maths discourse at school is about exchange value, [as opposite to use value] which is influenced by the performativity system in which schools compete. Further he suggests that the Maths discourse at college is about use value. Students are asked for a certain level of abstraction and understanding of the mathematical concepts to be used, all in a relatively short period of time. The Marxist terms use value and exchange value are used to separate between the purposes of the mathematical discussions at the institutions. It would be of interest to see if similar findings may also apply for this study. METHODOLOGY As I believe in the multiple constructed nature of social phenomena, this makes me positioned in the ontology of constructivism (or constructionism). Constructionism is an ontological position (often also referred to as constructivism) that asserts that social phenomena and their meanings are continually being accomplished by social actors (Bryman, 2004, p. 17) Within this paradigm the methodology is qualitative, based on the hermeneutic tradition, where contextual factors are described (Mertens, 2005; Geertz, 1973). As a basis for my subsequent analysis I draw upon what Mertens (2005, p. 8) labels to the constructivists paradigm, the naturalistic paradigm of Lincoln & Guba (1985). The well-elaborated principles of this paradigm consist of five axioms: 1) Realities are multiple, constructed, and holistic. 2) Knower and known are interactive, inseparable. 3) Only time- and context bound working hypotheses (idiographic statements) are possible. 4) All entities are in a state of mutual simultaneous shaping, so that it is impossible to distinguish causes from effects. 5) Inquiry is value-bound. (Lincoln & Guba, 1985, p. 37) Samples and data collection Five different classes in five different lower secondary schools participated in this research. Two of these schools are private schools while the other three are public. The private schools were included in an attempt to seek some diversity in the sample, while the public schools were somewhat randomly selected, with the only criteria being that they, due to practical reasons, were located within a reasonable distance from my working place. As the Norwegian school system is quite homogenous I believe that these schools are representative to their area. The headmasters were contacted via telephone and their school was invited to participate. The number of students willing to participate from each class varied from three to ten. In total 33 students participated and I am currently conducting follow-up research on 12 of these as they entered upper secondary school. I have chosen the follow-up students on the basis of three criterions: equal gender distribution, students at both vocational and general study programmes, and variations of skills (on the basis of their marks). My purpose is to gain a rich material with some diversity. My data collection at lower secondary school mainly consisted of five phases : Observations of the teacher teaching, recorded conversations with the students engaging in mathematics in the classroom, interviews with the students, collection of students handwritten material and an interview with their teacher. This 5
6 provides me with a diverse material which allows me to study mathematics education from various perspectives. The data collection at upper secondary school is done in a similar way. My use of research instruments did vary somewhat from school to school, primarily due to the fact that some teachers imposed restrictions for example on my use of a video camera. I have mainly applied semistructured interviews (Kvale, 1997). The figure below shows how the distribution of the 12 participating students. Figure 4: Distribution of the 12 participating students. (The number in parenthesis indicates the number of students in each class. LS: Lower secondary, US: Upper secondary, VS: Vocational study programme and GS: General study program). ANALYSIS [These days (May, 2010) I am working with my data-analysis. As this work is conducted in this very moment it is hard to elaborate on my finding in this paper. Hopefully some of this work is ready for discussion at the summer school, and I hope to have the opportunity to discuss also this aspect with you in Palermo, even if no written material is provided at this point.] 6
7 REFERENCES Bos, H. (1980). Mathematics and Rational Mechanics. In Rousseau, G. S. & Porter R. (Eds.) Ferment of Knowledge (pp ). Cambridge University Press. Boyer, C. B. (1949). The history of the calculus and its conceptual development. Mineola, NY: Dover. Brousseau, G. (1997). Theory of Didactical Situations in Mathematics. Dordrecht: Kluwer Academic Publishers. Bryman, A. (2004). Social Research Methods. 2 nd ed. New York: Oxford University Press. Chevallard, Y. (2005). Steps Towards a New Epistemology in Mathematics Education. In Bosch, M. (Ed.) Proceeding of the fourth congress of the European Society for Research in Mathematics Education, CERME 4, Sant Feliu de Guíxols, Spain. Cole, M. (1985). The zone of proximal development: where culture and cognition create each other. In J. V. Wertsch (Ed.), Culture, Communication and Cognition: Vygotskian Perspectives. Cambridge: Cambridge University Press. pp Geertz, C. (1973). The Interpretation of Cultures. New York: Basic Books. Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), Guzmán, M., et al. (1998). Difficulties in the Passage from Secondary to Tertiary Education. Documenta Mathematica, 901(3), Hernandez-Martinez, P. (2009). Transition to post-compulsory education: the case of algebra as a boundary object between school and college. Presentation held at the European Conference on Educational Research: Vienna, Klein, F. (1897). Mathematical Theory of the Top. New York: C. Scribner s Sons. Kleiner, I. (1989). Evolution of the Function Concept: A Brief Survey. The College Mathematics Journal, 20(4), Kvale, S. (2007). Det kvalitative forkningsintervju. (Translation from S. Kvale, InterViews An Introduction to Qualitative Research Interviewing, 1997). Oslo: Gyldendal Norsk Forlag AS. Lerman, S. (2000). A case of interpretations of social: a response to Steffe and Thompson. Journal for Research in Mathematics Education, 31(2), Lincoln, Y.S. & Guba, E.G. (1985). Naturalistic Inquiry. California: Sage Publication. Mertens, D.M. (2005). Research and Evaluation in Education and Psychology. 2 nd ed. California: Sage Publications. Peirce, C. S. (1998). The essential Peirce. Selected philosophical writings. Vol. 2 ( ). Bloomington, IN: Indiana University Press. Pozzi, S., Noss, R. and Hoyles, C. (1998). Tools in practice, mathematics in use. Educational Studies in Mathematics 36(2), Presmeg, N. (2005). Metaphor and metonomy in processes of semiosis in mathematics education. I M. H. G. Hoffman, J. Lenhard, & F. Seeger (Red.), Activity and sign Grounding mathematics education (ss ). New York: Springer. Sfard, A. (1991). On the dual Nature of Mathematical Conceptions: Reflections on Processes And Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22(1), Star. S. L. & Griesemer, J. R. (1989). Institutional ecology, translations and boundary objects: Amateurs and professionals in Berkeley'sMuseum of Vertebrate Zoology, Social Stud. Sci. 19. Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 22(2), Vygotsky, L.S. (1978) Mind in Society: The Development of Higher Psychological Processes. Cambridge: Harvard University Press. Vygotsky, L.S. (1981) The instrumental Method in Psychology. In J.V. Wretsch (Ed. & 7
8 Trans.) The concept of activity in soviet psychology ( pp )Armonk, NY: M. E. Sharpe. (Russian original published in 1960) Vygotsky, L. S. (1987). Thinking and speech. In L. S. Vygotsky, The collected works of L. S. Vygotsky, Vol. 1, Problems of general psychology (pp ) (R. W. Rieber & A. S. Carton, Eds.; N. Minick, Trans.). New York: Plenum Press. (Original work published 1934). 8
Getting used to mathematics: alternative ways of speaking about becoming mathematical
Getting used to mathematics: alternative ways of speaking about becoming mathematical Abstract Mathematics learning has a long association with cognitive psychology, with its focus on deep and hidden processes
More informationMATHEMATICS KNOWLEDGE FOR TEACHING WITHIN A FUNCTIONAL PERSPECTIVE OF PRESERVICE TEACHER TRAINING
MATHEMATICS KNOWLEDGE FOR TEACHING WITHIN A FUNCTIONAL PERSPECTIVE OF PRESERVICE TEACHER TRAINING Pedro Gómez Universidad de Granada C/. Alisios 17, Albolote 18820, Spain Phone/Fax: (34)958537304 pgomez@valnet.es
More informationWho we become depends on the company we keep and on what we do and say together
International Journal of Educational Research ] (]]]]) ]]] ]]] Who we become depends on the company we keep and on what we do and say together Gordon Wells Department of Education, Crown College, University
More informationProject description of my Master Thesis in Mathematical Education
Project description of my Master Thesis in Mathematical Education Title: Teachers role in developing mathematical skills with technology - Introducing GeoGebra to mathematics teachers in lower secondary
More informationFor example, estimate the population of the United States as 3 times 10⁸ and the
CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number
More informationResearch into competency models in arts education
Research into competency models in arts education Paper presented at the BMBF Workshop International Perspectives of Research in Arts Education, Nov. 4 th and 5 th, 2013. Folkert Haanstra, Amsterdam School
More informationVygotsky's theory in the classroom: Introduction
European Journal of Psychology of Education 2004. Vol. XIX. n'i. 3-7 2004.1.S.P.A. Vygotsky's theory in the classroom: Introduction Alex Kozulin International Center for the Enhancement of Learning Potential,
More informationMathematics Curriculum Guide Precalculus 2015-16. Page 1 of 12
Mathematics Curriculum Guide Precalculus 2015-16 Page 1 of 12 Paramount Unified School District High School Math Curriculum Guides 2015 16 In 2015 16, PUSD will continue to implement the Standards by providing
More informationSocial Presence Online: Networking Learners at a Distance
Education and Information Technologies 7:4, 287 294, 2002. 2002 Kluwer Academic Publishers. Manufactured in The Netherlands. Social Presence Online: Networking Learners at a Distance ELIZABETH STACEY Faculty
More informationBusiness Process Models as Design Artefacts in ERP Development
Business Process Models as Design Artefacts in ERP Development Signe Ellegaard Borch IT University of Copenhagen, Rued Langgaards Vej 7, 2300 København S, Denmark elleborch@itu.dk Abstract. Adequate design
More informationTEACHER IDENTITY AND DIALOGUE: A COMMENT ON VAN RIJSWIJK, AKKERMAN & KOSTER. Willem Wardekker VU University Amsterdam, The Netherlands
International Journal for Dialogical Science Spring 2013. Vol. 7, No. 1, 61-65 Copyright 2013 by Willem Wardekker TEACHER IDENTITY AND DIALOGUE: A COMMENT ON VAN RIJSWIJK, AKKERMAN & KOSTER Willem Wardekker
More informationChapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School
Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education
More informationTooluse supporting the learning and teaching of the Function concept
Tooluse supporting the learning and teaching of the Function concept Paper for ISDDE2008 Michiel Doorman +, Peter Boon +, Paul Drijvers +, Sjef van Gisbergen +, Koeno Gravemeijer* & Helen Reed + + Freudenthal
More informationChapter 2 Conceptualizing Scientific Inquiry
Chapter 2 Conceptualizing Scientific Inquiry 2.1 Introduction In order to develop a strategy for the assessment of scientific inquiry in a laboratory setting, a theoretical construct of the components
More informationComputer Science Education Based on Fundamental
Computer Science Education Based on Fundamental Ideas A. Schwill Fachbereich Informatik - Universität Oldenburg D-26111 Oldenburg, Germany Phone: +49-441-798-2412, Fax: +49-441-798-2155 e-mail: Andreas.Schwill@informatik.uni-oldenburg.de
More informationCritical Inquiry in Educational Research and Professional Practice
DOCTOR IN EDUCATION COURSE DESCRIPTIONS A. CORE COURSES NEDD 800 Professionalism, Ethics, and the Self This introductory core course will explore and interrogate ideas surrounding professionalism and professionalization.
More informationPrentice Hall Connected Mathematics 2, 7th Grade Units 2009
Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems
More informationDoing Online Relearning through Information Skills (DORIS):
(DORIS): A Mutual Shaping Perspective for Information Literacy Research and Practice Juan D. Machin-Mastromatteo and Tallinn University, Institute of Information Studies Introduction Snapshots This PhD
More informationDeveloping Teacher Leadership and its Impact in Schools M. Snoek
Developing Teacher Leadership and its Impact in Schools M. Snoek SUMMARY DEVELOPING TEACHER LEADERSHIP AND ITS IMPACT IN SCHOOLS Introduction Successful school improvement is dependent on schools capacities
More informationAssumptions of Instructional Systems Design
Assumptions of Instructional Systems Design 1 The ISD Model Design Analysis Development Evaluation Implementation 2 ISD is Empirical Science 4 In its classical sense, ISD is a systematic method for designing
More informationEUROPEAN RESEARCH IN MATHEMATICS EDUCATION III. Thematic Group 1
METAPHOR IN YOUNG CHILDREN S MENTAL CALCULATION Chris Bills Oxfordshire Mathematics Centre, Oxford OX4 3DW, UK. Abstract: In this study 7-9 year old children performed mental calculations and were then
More informationPrinciples to Actions
Principles to Actions Executive Summary In 1989 the National Council of Teachers of Mathematics (NCTM) launched the standards-based education movement in North America with the release of Curriculum and
More informationHow To Build Connection With New Arrival Students
Building connection in working with new arrival immigrant and refugee students Jenny Barnett, University of South Australia, South Australia Rosie Antenucci, Department of Education and Children s Services,
More informationSchool of Advanced Studies Doctor Of Education In Educational Leadership With A Specialization In Educational Technology. EDD/ET 003 Requirements
School of Advanced Studies Doctor Of Education In Educational Leadership With A Specialization In Educational Technology The mission of the Doctor of Education in Educational Leadership degree program
More informationINTEGRATING TECHNOLOGY INTO TEACHING: NEW CHALLENGES FOR THE CLASSROOM MATHEMATICAL MEANING CONSTRUCTION
INTEGRATING TECHNOLOGY INTO TEACHING: NEW CHALLENGES FOR THE CLASSROOM MATHEMATICAL MEANING CONSTRUCTION Angeliki Mali*, Irene Biza*, Michalis Kaskadamis**, Despina Potari** and Charalambos Sakonidis***
More informationUnifying Epistemologies by Combining World, Description and Observer
Unifying Epistemologies by Combining World, Description and Observer Stuart Umpleby Research Program in Social and Organizational Learning The George Washington University Washington, DC Umpleby@gwu.edu
More informationModernization of Library and Information Science Education through the Enhancement of Intercultural Communication
Qualitative and Quantitative Methods in Libraries (QQML) 4: 359 364, 2013 Modernization of Library and Information Science Education through the Enhancement of Intercultural Communication Sirje Virkus
More informationMeasurement with Ratios
Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical
More informationMaster of Arts in Business Education (MA) 29 January 2016. Module 1 Introduction to Business Education (6 ECTS) Content. Learning Outcomes F01 BE01
Module 1 Introduction to Business Education (6 ECTS) F01 BE01 I) Basic ideas and fundamental theories of Educational Sciences Introduction to the Study of Learning Behaviourism Information Processing Theory
More informationGRADE 8 MATH: TALK AND TEXT PLANS
GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculum-embedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems
More informationResearch Basis for Catchup Math
Research Basis for Catchup Math Robert S. Ryan, Ph. D. Associate Professor of Cognitive Psychology Kutztown University Preface Kutztown University is a 4 year undergraduate university that is one of 14
More informationTECHNOLOGY AND SEMIOTIC REPRESENTATIONS IN LEARNING MATHEMATICS. Jose Luis Lupiáñez Gómez University of Cantabria, Spain. Abstract
1 TECHNOLOGY AND SEMIOTIC REPRESENTATIONS IN LEARNING MATHEMATICS Jose Luis Lupiáñez Gómez University of Cantabria, Spain Abstract New technologies modify the socioculturals environments. The educational
More informationAssessment Plan Department of Psychology Park University. Preparing learners to think critically. Preparing learners to think
Assessment Plan Department of Psychology Park University The approach adopted by the Department of Psychology stems from the mission of Park University to prepare learners to think, communicate effectively
More informationDoctor of Education - Higher Education
1 Doctor of Education - Higher Education The University of Liverpool s Doctor of Education - Higher Education (EdD) is a professional doctoral programme focused on the latest practice, research, and leadership
More informationResearch Paradigms, the Philosophical Trinity, and Methodology
Research Paradigms, the Philosophical Trinity, and Methodology by Graham Durant-Law BSc, MHA, MKM, Grad Dip Def, Grad Dip Mngt, Grad Cert Hlth Fin, psc. Copyright Graham Durant-Law Presentation Objectives
More informationChapter 2 A Pedagogical Overview of Relevant Literature
Chapter 2 A Pedagogical Overview of Relevant Literature 2.1 Introduction There are many approaches that used various research methods to describe and explain teachers knowledge for teaching mathematics
More informationThe Relevance of Glaserian and Straussian Grounded Theory Approaches in Researching Human Resource Development
2011 International Conference on Financial Management and Economics IPEDR vol.11 (2011) (2011) IACSIT Press, Singapore The Relevance of Glaserian and Straussian Grounded Theory Approaches in Researching
More informationMathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades
Appendix A Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades To respond correctly to TIMSS test items, students need to be familiar with the mathematics
More informationStandards for Certification in Early Childhood Education [26.110-26.270]
I.B. SPECIFIC TEACHING FIELDS Standards for Certification in Early Childhood Education [26.110-26.270] STANDARD 1 Curriculum The competent early childhood teacher understands and demonstrates the central
More informationStudent Conceptions of Integration and Accumulation. Jason Samuels City University of New York BMCC
Student Conceptions of Integration and Accumulation Brian Fisher Lubbock Christian University Jason Samuels City University of New York BMCC Aaron Wangberg Winona State University Prior research has shown
More informationAlgebra I. In this technological age, mathematics is more important than ever. When students
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
More informationMethodology in Social Psychology. Logics of inquiry
Methodology in Social Psychology Logics of inquiry How to carry out scientific research given our understanding of the nature of knowledge. Philosophy of Science clarifies why experimental, scientific
More informationIn mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data.
MATHEMATICS: THE LEVEL DESCRIPTIONS In mathematics, there are four attainment targets: using and applying mathematics; number and algebra; shape, space and measures, and handling data. Attainment target
More informationSyllabus Master s Programme in Child Studies (60/120 credits)
Syllabus Master s Programme in Child Studies (60/120 credits) Aim The objective of the United Nation s Convention on the Rights of the Child is to protect children, a group considered particularly vulnerable,
More informationDesigning Socio-Technical Systems to Support Guided Discovery-Based Learning in Students: The Case of the Globaloria Game Design Initiative
Designing Socio-Technical Systems to Support Guided Discovery-Based Learning in Students: The Case of the Globaloria Game Design Initiative Rebecca Reynolds 1, Sean P. Goggins 2 1 Rutgers University School
More informationMPHIL PROGRAMME IN CHILDHOOD STUDIES
CHILDHOOD STUDIES SIDE 71 MPHIL PROGRAMME IN CHILDHOOD STUDIES Approved by the Board at NTNU 30.08.2005, with changes made by the Faculty of Social Sciences and Technology Management 9.01.2007. Norwegian
More informationProgramme description for PhD Programme in Educational Sciences for Teacher Education (180 ECTS credits) at Oslo and Akershus University College of
Programme description for PhD Programme in Educational Sciences for Teacher Education (180 ECTS credits) at Oslo and Akershus University College of Applied Sciences Approved by the Oslo and Akershus University
More informationCORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
More informationScience teachers pedagogical studies in Finland
1 Science teachers pedagogical studies in Finland Jari Lavonen Summary An overview of planning, organising and evaluating of science teachers pedagogical studies in Finland is given. Examples are from
More informationEDUCATING MATHEMATICS TEACHER EDUCATORS: A COMPETENCY-BASED COURSE DESIGN
EDUCATING MATHEMATICS TEACHER EDUCATORS: A COMPETENCY-BASED COURSE DESIGN Tomas Højgaard & Uffe Thomas Jankvist Department of Education, Aarhus University, Denmark The paper argues for a three-dimensional
More informationIntroduction to quantitative research
8725 AR.qxd 25/08/2010 16:36 Page 1 1 Introduction to quantitative research 1.1. What is quantitative research? Research methods in education (and the other social sciences) are often divided into two
More informationMaster of Arts in Mathematics
Master of Arts in Mathematics Administrative Unit The program is administered by the Office of Graduate Studies and Research through the Faculty of Mathematics and Mathematics Education, Department of
More informationTeacher Leaders in a Culture of Accountability - Emergent Roles for Transformative Teacher Learning or the New Dispensable Middle Managers?
Teacher Leaders in a Culture of Accountability - Emergent Roles for Transformative Teacher Learning or the New Dispensable Middle Managers? Chair: Alex Alexandrou (Freelance Academic) Discussant: Jason
More informationSTUDENTS DIFFICULTIES WITH APPLYING A FAMILIAR FORMULA IN AN UNFAMILIAR CONTEXT
STUDENTS DIFFICULTIES WITH APPLYING A FAMILIAR FORMULA IN AN UNFAMILIAR CONTEXT Maureen Hoch and Tommy Dreyfus Tel Aviv University, Israel This paper discusses problems many tenth grade students have when
More informationFABRICATION OF KNOWLEDGE: A FRAMEWORK FOR MATHEMATICAL EDUCATION FOR SOCIAL JUSTICE
FABRICATION OF KNOWLEDGE: A FRAMEWORK FOR MATHEMATICAL EDUCATION FOR SOCIAL JUSTICE Brian R. Lawler California State University San Marcos This essay is meant to spark discussion that seeks to pragmaticize
More informationUniversity of Southampton. Conceptual Understanding, Scientific Explanations & Arguments. In First Year University Physics:
University of Southampton Conceptual Understanding, Scientific Explanations & Arguments In First Year University Physics: Development and evaluation of an intervention Foteini Chaimala A thesis submitted
More informationA GENERAL CURRICULUM IN MATHEMATICS FOR COLLEGES W. L. DUREN, JR., Chairmnan, CUPM 1. A report to the Association. The Committee on the Undergraduate
A GENERAL CURRICULUM IN MATHEMATICS FOR COLLEGES W. L. DUREN, JR., Chairmnan, CUPM 1. A report to the Association. The Committee on the Undergraduate Program in Mathematics (CUPM) hereby presents to the
More informationJOINT MASTER OF ARTS IN LEADERSHIP AND EDUCATION CHANGE COURSE DESCRIPTIONS
JOINT MASTER OF ARTS IN LEADERSHIP AND EDUCATION CHANGE COURSE DESCRIPTIONS A. CORE COURSES MALC 801 Perspectives in Educational Leadership Educational leadership is a complex concept, both in theory and
More informationUtility of a conceptual framework within doctoral study: A researcher s reflections
Issues in Educational Research, 23(1), 2013 1 Utility of a conceptual framework within doctoral study: A researcher s reflections Jeanette Berman University of New England The author of this paper provides
More informationUNIVERSITY OF BELGRADE FACULTY OF PHILOSOPHY. Part two: INFORMATION ON DEGREE PROGRAMS
Part two: INFORMATION ON DEGREE PROGRAMS Part two: Information on Degree Programs Philosophy Bachelor s Degree Philosophy Master s Degree Philosophy Doctoral Degree Sociology Bachelor s Degree Sociology
More informationDEGREE PROGRAMME IN EARLY CHILDHOOD EDUCATION CURRICULUM 2014-2017
DEGREE PROGRAMME IN EARLY CHILDHOOD EDUCATION CURRICULUM 2014-2017 (approved by the faculty council 27.3.2014, updated VAAM044, VAAM045 and VAAM051, VARS030, KTK0006, VARS034 faculty council 26.3.2015)
More informationDoctor of Education (EdD) in TESOL AVAILABLE IN EXETER AND DUBAI
Doctor of Education (EdD) in TESOL AVAILABLE IN EXETER AND DUBAI The Programme The degree of Doctor of Education (EdD) in TESOL aims to provide experienced professionals within the field of language teaching
More informationHow To Be A Mathematically Proficient Person
REPRODUCIBLE Figure 4.4: Evaluation Tool for Assessment Instrument Quality Assessment indicators Description of Level 1 of the Indicator Are Not Present Limited of This Indicator Are Present Substantially
More informationThe Role of Modelling in Teaching Formal Methods for Software Engineering
The Role of Modelling in Teaching Formal Methods for Software Engineering A. J. Cowling Department of Computer Science University of Sheffield Sheffield, England A.Cowling@dcs.shef.ac.uk Abstract. This
More informationStandards for Mathematical Practice: Commentary and Elaborations for 6 8
Standards for Mathematical Practice: Commentary and Elaborations for 6 8 c Illustrative Mathematics 6 May 2014 Suggested citation: Illustrative Mathematics. (2014, May 6). Standards for Mathematical Practice:
More informationBAŞKENT UNIVERSITY INSTITUTE OF EDUCATIONAL SCIENCES MASTER S PROGRAM IN ELEMENTARY SCHOOL MATHEMATICS EDUCATION REQUIRING THESIS
BAŞKENT UNIVERSITY INSTITUTE OF EDUCATIONAL SCIENCES MASTER S PROGRAM IN ELEMENTARY SCHOOL MATHEMATICS EDUCATION REQUIRING THESIS INFORMATION ABOUT THE PROGRAM General Description: The program prepares
More informationUNDERSTANDING THE QUALITATIVE AND QUANTATITIVE METHODS IN THE CONTEXT OF CONTENT ANALYSIS
QQML2009: Qualitative and Quantitative Methods in Libraries, International Conference, Chania Crete Greece, 26-29 May 2009 1 UNDERSTANDING THE QUALITATIVE AND QUANTATITIVE METHODS IN THE CONTEXT OF CONTENT
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationTheme Analysis and Case Studies of Teacher Interview Data
ANALYSING INTERVIEW DATA FOR CLUSTERS AND THEMES LYNDA BALL The University of Melbourne lball@unimelb.edu.au This paper outlines a qualitative approach for analysing interview transcripts. The approach
More informationUnit 7 Quadratic Relations of the Form y = ax 2 + bx + c
Unit 7 Quadratic Relations of the Form y = ax 2 + bx + c Lesson Outline BIG PICTURE Students will: manipulate algebraic expressions, as needed to understand quadratic relations; identify characteristics
More informationA Microgenetic Study of One Student s Sense Making About the Temporal Order of Delta and Epsilon Aditya P. Adiredja University of California, Berkeley
A Microgenetic Study of One Student s Sense Making About the Temporal Order of Delta and Epsilon Aditya P. Adiredja University of California, Berkeley The formal definition of a limit, or the epsilon delta
More informationThe Mathematics School Teachers Should Know
The Mathematics School Teachers Should Know Lisbon, Portugal January 29, 2010 H. Wu *I am grateful to Alexandra Alves-Rodrigues for her many contributions that helped shape this document. Do school math
More informationBiology Program Health Sciences Program Assessment Report 2009-2010
Biology Program Health Sciences Program Assessment Report 2009-2010 I. Introduction The Biology program serves all OIT students wishing to major or minor in the biological sciences, including those entering
More informationDOCTOR OF PHILOSOPHY DEGREE. Educational Leadership Doctor of Philosophy Degree Major Course Requirements. EDU721 (3.
DOCTOR OF PHILOSOPHY DEGREE Educational Leadership Doctor of Philosophy Degree Major Course Requirements EDU710 (3.0 credit hours) Ethical and Legal Issues in Education/Leadership This course is an intensive
More informationHIDDEN DIFFERENCES IN TEACHERS APPROACH TO ALGEBRA a comparative case study of two lessons.
HIDDEN DIFFERENCES IN TEACHERS APPROACH TO ALGEBRA a comparative case study of two lessons. Cecilia Kilhamn University of Gothenburg, Sweden Algebra is a multi-dimensional content of school mathematics
More informationMODIFIED TRAINING PROGRAMME FOR TRAINERS AND TEACHERS OF OCCUPATIONAL THERAPY IN TAJIKISTAN
MODIFIED TRAINING PROGRAMME FOR TRAINERS AND TEACHERS OF OCCUPATIONAL THERAPY IN TAJIKISTAN These materials have been prepared within the framework of the project Technical Assistance to the Sector Policy
More informationPresentation. Identity and education: tendencies and challenges
Presentation. Identity and education: tendencies and challenges César Coll Leili Falsafi Universidad de Barcelona. Facultad de Psicología. Departamento de Psicología Evolutiva y de la Educación. Barcelona,
More informationFairfield Public Schools
Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity
More informationMastery approaches to mathematics and the new national curriculum
October 2014 Mastery approaches to mathematics and the new national curriculum Mastery in high performing countries The content and principles underpinning the 2014 mathematics curriculum reflect those
More informationVolumes of Revolution
Mathematics Volumes of Revolution About this Lesson This lesson provides students with a physical method to visualize -dimensional solids and a specific procedure to sketch a solid of revolution. Students
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationB.Ed. Two Year Programme. F.2: Human Development, Diversity and Learning
B.Ed. Two Year Programme F.2: Human Development, Diversity and Learning Maximum Marks: 100 Vision This course will facilitate an understanding of the processes of development and learning and some of the
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationGrade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills
Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate
More informationIndiana Statewide Transfer General Education Core
Indiana Statewide Transfer General Education Core Preamble In 2012 the Indiana legislature enacted Senate Enrolled Act 182, thereby establishing the requirements for a Statewide Transfer General Education
More informationDRAFT. New York State Testing Program Grade 8 Common Core Mathematics Test. Released Questions with Annotations
DRAFT New York State Testing Program Grade 8 Common Core Mathematics Test Released Questions with Annotations August 2014 Developed and published under contract with the New York State Education Department
More informationPre-Algebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationTeacher Education for the Future
Teacher Education for the Future A Policy Document from the Union of Education Norway www.utdanningsforbundet.no Teacher education that emphasises knowledge and quality The education of teachers should
More informationINTRODUCING THE NORMAL DISTRIBUTION IN A DATA ANALYSIS COURSE: SPECIFIC MEANING CONTRIBUTED BY THE USE OF COMPUTERS
INTRODUCING THE NORMAL DISTRIBUTION IN A DATA ANALYSIS COURSE: SPECIFIC MEANING CONTRIBUTED BY THE USE OF COMPUTERS Liliana Tauber Universidad Nacional del Litoral Argentina Victoria Sánchez Universidad
More informationMathematical Knowledge as a Social Construct of Teaching- / Learning Processes The Epistemology Oriented Mathematical Interaction Research
Mathematical Knowledge as a Social Construct of Teaching- / Learning Processes The Epistemology Oriented Mathematical Interaction Research Heinz Steinbring, Universität Duisburg-Essen, Campus Essen 1)
More informationPROGRAMME AND COURSE OUTLINE MASTER S PROGRAMME IN MULTICULTURAL AND INTERNATIONAL EDUCATION. 12O ECTS credits. The academic year 2013/2014
PROGRAMME AND COURSE OUTLINE MASTER S PROGRAMME IN MULTICULTURAL AND INTERNATIONAL EDUCATION 12O ECTS credits The academic year 2013/2014 Oslo and Akershus University College of Applied Sciences Faculty
More informationSouth Carolina College- and Career-Ready (SCCCR) Algebra 1
South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process
More informationSouth Carolina College- and Career-Ready (SCCCR) Probability and Statistics
South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)
More informationAbstract Title: Evaluating Mathematical Knowledge for Teaching Using TKAS
Abstract Title: Evaluating Mathematical Knowledge for Teaching Using TKAS MSP Project Name: Learning Mathematics for Teaching- Teacher Knowledge Assessment System (LMT- TKAS) Author(s): Geoffrey Phelps
More informationFIELD 002: EARLY CHILDHOOD TEST OBJECTIVES
FIELD 002: EARLY CHILDHOOD TEST OBJECTIVES Subarea Multiple-Choice Range of Objectives Approximate Test Weighting I. Knowledge of Child Development 01 02 25% II. Knowledge of Children's Literature and
More informationIndiana State Core Curriculum Standards updated 2009 Algebra I
Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and
More informationWhite paper for the CS Education Research Summit
White paper for the CS Education Research Summit Josh Tenenberg I have two goals for this paper. The first is to go meta, to question the very research questions that we are to generate, by unpacking their
More informationAbstract Title Page. Authors and Affiliations: Maria Mendiburo The Carnegie Foundation
Abstract Title Page Title: Designing Technology to Impact Classroom Practice: How Technology Design for Learning Can Support Both Students and Teachers Authors and Affiliations: Maria Mendiburo The Carnegie
More informationSchool of Advanced Studies Doctor Of Management In Organizational Leadership. DM 004 Requirements
School of Advanced Studies Doctor Of Management In Organizational Leadership The mission of the Doctor of Management in Organizational Leadership degree program is to develop the critical and creative
More informationProblem of the Month: Perfect Pair
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More information