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## Selection of Appropriate Statistical Methods for Research Results ProcessingRezeda M. Khusainova, Zoia V. Shilova & Oxana V. Curteva
pp.
The purpose of the article is to provide an algorithm that allows choosing a valid method of statistical data processing and development of a model for acquiring knowledge about statistical methods and mastering skills of competent knowledge application in various research activities. Modelling method is a leading approach to the study of this problem. It allows us to consider this issue as a targeted and organized process of application of the author’s methodology for the selection of appropriate statistical method for the efficient processing of the research results. The article showcases an algorithm that allows to choose an appropriate method of statistical data processing: general algorithm of statistical methods application in scientific research, statistical problems systematization based on which there have been outlined conditions for specific research methods application. To make a final decision concerning the statistical method at the stage of data received and statistical tasks of the research defined, it is proposed to use an author’s algorithm that allows to competently select the method of processing the research results.
2014 Progress Report of the Arbitration Court of the Kirov region. (2014). Reference Form № 1. from http://kirov.arbitr.ru Biryukov, B. V. (1974). Bluvshtejn, J. D. (1981). Cochran, W. (1976). Ermolaev, O. J. (2006). Mathematical statistics for psychologists: the textbook. Moscow: Flint. Ganieva, Y. N., Azitova, G. S., Chernova, Y. A., Yakovleva, I. G., Shaidullina, A. R., Sadovaya, V. V. (2014). Model of High School Students Professional Education. Glantz, S. (1998). Glass, J. & Stanly, J. (1976). Gmurman, V. E. (2003). Grabar, M. & Krasnyanskaya K. A. (1977). Granichina, O. (2012). Hollender, M. & Wolfe, D. (1983). Kabanova-Meller, E.N. (1981). Krajewski, V. V. (1977). Krutetskiy, V. A. (1972). Landa, L. N. (1966). Leontiev, A. N. (1959). Lerner, I. J. (1981). Litvak, K. B. (1985). Masalimova, A. R. & Nigmatov, Z. G. (2015). Structural-Functional Model for Corporate Training of Specialists in Carrying Out Mentoring. Mikheev, V. (1987). Nikolaev, A. G. & Degtyarev, M. P. (2013). Identification of text files by statistical methods (conventional cases). Novikov, D. A. & Novochadov, V. V. (2005). Novikov, D. A. (2004). Orlov, A. I. (2001). Platonov, A. E. (2000). Polonsky, V. M. (1987). Professional education. (1999). Rosenberg, N. M. (1979). Shilova, Z. V. (2014) Urbach, V. J (1975). Vygotsky, L. S. (1965/1986). Vygotsky, L. S. (1982/2012). Zaripova, I. M., Shaidullina, A. R., Upshinskaya, A. Y., Sayfutdinova, G. B., Drovnikov, A. S. (2014). Modeling of Petroleum Engineers Design-Technological Competence Forming in Physical-Mathematical Disciplines Studying Process. |
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## The Affective Domain in Mathematics LearningNuria Gil Ignacio, Lorenzo J. Blanco Nieto and Eloísa Guerrero Barona
pp.
The present work set out to analyze the beliefs, attitudes, and emotional reactions that students experience in the process of learning mathematics. The aim was to be able to demonstrate that the existence of positive attributes, beliefs, and attitudes about themselves as learners are a source of motivation and expectations of success in dealing with this subject. We used a sample of 346 students of the second cycle of Obligatory Secondary Education (ESO) of high schools in Badajoz. The participants responded to a questionnaire on beliefs and attitudes about mathematics. It was found that neither the students' gender nor their year of studies influenced their beliefs about their self-concept of mathematics.
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## Teaching and Assessing Higher Order Thinking in the Mathematics Classroom with ClickersJim Rubin & Manikya Rajakaruna
pp.
Many schools have invested in clicker technology, due to the capacity of the software to track formative assessment and the increased motivation that students show for incorporating technology in the classroom. As with any adoption of new software that demands amending pedagogy and learning applications, the extent to which clickers are living up to expectations has not yet become apparent. The present study sought to explore the potential of using clickers to teach the reasoning processes behind solving higher order thinking word problems in a mathematics class. A pilot study was conducted with a college algebra class to refine questions used in the coursework and field test a survey to measure student attitudes towards the teaching methodology. The main study took place over the fall semester with a college algebra class (N=21). Results showed increased student motivation and acumen for using the technology and higher test scores, but frustration on the part of both the teacher and students when trying to apply the pedagogy for the purpose of learning higher order thinking reasoning processes. The potential for the technology to offer an alternative for formative assessment was a strong positive element.
Bender, T.A. (1980). Processing multiple choice and recall test questions. Paper presented at the Berlak, H. (1985). Testing in a democracy. Biggs, J.B. & Collis, K. F. (1982). Caldwell, J. E. (2007). Clickers in the large classroom: Current research and best practice tips. Collis, K. F. (1982). The solo taxonomy as a basis of assessing levels of reasoning in mathematical problem solving. Proceedings from the Collis, K. G., Romberg, T.A., & Jurdak, M. E. (1986). A technique for assessing mathematical problem-solving ability. Common Core State Standards Initiative (2015). DeBourgh, G. A. (2008). Use of classroom “clickers” to promote acquisition of advanced reasoning skills. Douglas, M., Wilson, J., & Ennis, S. (2012). Multiple-choice question tests: A convenient, flexible and effective learning tool? A case study. Dowd, S. B. (1992). Elias, J. L., & Merriam, S. B. (2005). Ennis, R. (1985). Large scale assessment of critical thinking in the fourth grade. Paper presented at Frederiksen, N. (1984). The real test bias, Hansen, J. D., & Dexter, L. (1997). Quality multiple-choice test questions: Item-writing. Hatch, J., Murray, J., & Moore, R. (2005). Manna from heaven or “clickers” from hell: Experiences with an electronic response system. Kolikant, Y.B.D., Calkins, S., & Drane, D. (2010). “Clickers” as catalysts for transformation of teachers. Lin, S., & Singh, C. (2012). Can multiple-choice questions simulate free-response questions? Lockwood, D.F. (2003). Liu, W.C. & Stengel, D. (2011). Improving student retention and performance in quantitative courses using clickers Miller, R. G., Ashar, B. H., & Getz, K. J. (2003). Evaluation of an audience response system for the continuing education of health professionals. National Education Association (2015). Oermann, M. H., & Gaberson, K. B. (2006). Popelka, S. R. (2010). Now we're really clicking! Ray, W. (1978). Writing multiple-choice questions: The problem and a proposed solution. Resnick, L.B. (1987). Ribbens, E. (2007). Why I like personal response systems. Romberg, T.A, Zarinnia, E.A., Collis, K.F. (1990). A new world view of assessment in mathematics. In G. Kulm (Ed.), Teaching Effectiveness Program. (2014). Standards (2012). Retrieved from: www.corestandards.org/ October 30, 2012. Sternberg and Baron. (1985). A triarchic approach to measuring critical thinking skills: a psychological view. Paper presented at symposium, Stuart, S. A. J., Brown, M. I., & Draper, S. W. (2004). Using an electronic voting system in logic lectures: One practitioner’s application. Stupans, I. (2006). Multiple choice questions: Can they examine application of knowledge? Torres, C., Lopes, A., Babo, L., & Azevedo, J. (2011). Improving multiple-choice questions. Uhari, M., Renko, M., & Soini, H. (2003). Experiences of using an interactive audience response system in lectures. Wayne, W. (1982). Relative effectiveness of single and double multiple-choice questions in educational measurement. |
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## Teachers’ beliefs about mathematical knowledge for teaching definitionsReidar Mosvold & Janne Fauskanger
pp.
Previous research indicates the importance of teachers’ knowledge of mathematical definitions—as well as their beliefs. Much remains unknown, however, about the specific knowledge required doing the mathematical task of teaching involving definitions and the related teacher beliefs. In this article, we analyze focus-group interviews that were conducted in a Norwegian context to examine the adaptability of the U.S. developed measures of mathematical knowledge for teaching. Qualitative content analysis was applied in order to learn more about the teachers’ beliefs about mathematical knowledge for teaching definitions. The results indicate that teachers believe knowledge of mathematical definitions is an important aspect of mathematical knowledge for teaching, but they do not regard it as important to actually know the mathematical definitions themselves.
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## The Effects of GeoGebra On Third Grade Primary Students’ Academic Achievement in FractionsMehmet Bulut, Hanife Ünlütürk Akçakın, Gürcan Kaya & Veysel Akçakın
pp.
The aim of this study is to examine the effects of GeoGebra on third grade primary students’ academic achievement in fractions concept. This study was conducted with 40 students in two intact classes in Ankara. One of the classes was randomly selected as an experimental group and other for control group. There were 19 students in the experimental group, while 21 students in control group. The matching- only posttest- only control group quasi-experimental design was employed. As a pretest, student’s first term mathematics scores were used. Data were collected with post-test about fractions. The post-test consisted of 22 short ended questions. Thanks to the scores weren’t violated the normality, independent
Acar, N. (2010). Akın, P. (2009). Clements, D. H., Sarama, J., & DiBiase, A. M. (Eds.). (2004). Demirdöğen, N. (2007). Erdağ, S. (2011). Goodwin, K. (2008). The impact of interactive multimedia on kindergarten students’ representations of fractions. Gutiérrez, A., & Boero, P. (Eds.). (2006). Kayhan, H. C. (2010). Lee, H.J. & Boyadzhiev, I. (2013). Challenging Common Misconceptions of Fractions through GeoGebra. In R. McBride & M. Searson (Eds.), Proceedings of Society for Information Technology & Teacher Education International Conference 2013 (pp. 2893-2898). Chesapeake, VA: AACE. Martín-Caraballo, A. M., & Tenorio-Villalón, Á. F. (2015). Teaching Numerical Methods for Non-linear Equations with GeoGebra-Based Activities. McNamara, J., & Shaughnessy, M. M. (2010). Mısral, M. (2009). Moyer-Packenham, P. S., Ulmer, L. A., & Anderson, K. L. (2012). Examining Pictorial Models and Virtual Manipulatives for Third-Grade Fraction Instruction. Newstead, K. and Murray, H. (1998). Young students’ constructions of fractions. In A. Olivier & K. Newstead (Eds.), Pesen, C. (2007). Öğrencilerin kesirlerle ilgili kavram yanılgıları [Students’ Misconceptions About Fractions]. Pilli, O. (2008). Pitta-Pantazi, D., Gray, E., & Christou, C. (2004). Elementary school students’ mental representations of fractions. In Reimer, K., & Moyer, P. S. (2005). Third-graders learn about fractions using virtual manipulatives: A classroom study. Sözer, N. (2006). Suh, J., Moyer, P. S., & Heo, H. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. The National Council of Teachers of Mathematics [NCTM]. (2000). Thambi, N., & Eu, L. K. (2013). Effect of Students’ Achievement in Fractions using GeoGebra. Van de Walle, J.A., Karp, K.S. & Bay-Williams, J.M. (2010). Yazgan, Y. (2007). Yumuşak, E. Y. (2014). Yurtsever, N.T. (2012). |
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## Teachers’ Beliefs about the Discipline of Mathematics and the Use of Technology in the ClassroomMorten Misfeldt, Uffe Thomas Jankvist & Mario Sánchez Aguilar
pp.
In the article, three Danish secondary level mathematics teachers’ beliefs about the use of technological tools in the teaching of mathematics and their beliefs about mathematics as a scientific discipline are identified and classified - and the process also aspects of their beliefs about the teaching and learning of mathematics. The potential relationships between these sets of beliefs are also explored. Results show that the teachers not only manifest different beliefs about the use of technology and mathematics as a discipline, but that one set of beliefs can influence the other set of beliefs. The article concludes with a discussion of the research findings and their validity as well as their implications for both practice and research in mathematics education.
Artigue, M. (2002). Learning mathematics in a CAS environment: the genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. Beswick, K. (2005). The beliefs/practice connection in broadly defined contexts. Beswick, K. (2012). Teachers' beliefs about school mathematics and mathematicians' mathematics and their relationship to practice. Blömeke, S. & Kaiser, G. (2015). Effects of motivation on the belief systems of future mathematics teachers from a comparative perspective. In B. Pepin & B. Roesken-Winter (Eds.), Buchberger, B. (2002). Computer algebra: the end of mathematics? Carter, G. & Norwood, K.S. (1997). The relationship between teacher and student beliefs about mathematics. Cooney, T.J., Shealy, B.E. & Arvold, B. (1998). Conceptualizing belief structures of preservice secondary mathematics teachers. De Guzman, M., Hodgson, B.R., Robert, A. & Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. Dogan, M. (2007). Mathematics trainee teachers’ attitudes to computers. In M. Joubert (Ed.), Dreyfus, T. (1994) The role of cognitive tools in mathematics education. In R. Biehler, R.W. Scholz, R. Sträßer & B. Winkelmann (Eds.), Drijvers, P., Doorman, M., Boon, P., Reed, H. & Gravemeijer, K. (2010). The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom. Erens, R. & Eichler, A. (2015). The use of technology in calculus classrooms – beliefs of high school teachers. In C. Bernack-Schüler, R. Erens, T. Leuders & A. Eichler (Eds.), Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.), Fleener, M.J. (1995). The relationship between experience and philosophical orientation: a comparison of preservice and practicing teachers’ beliefs about calculators. Forgasz, H.J. (2002). Teachers and computers for secondary mathematics. Furinghetti, F. & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G.C. Leder, E. Pehkonen and G. Törner (Eds.), Fullan, M.G. (1991). Gill, M.G., Ashton, P.T. & Algina, J. (2004). Changing preservice teachers’ epistemological beliefs about teaching and learning in mathematics: An intervention study. Georgsen, M., Fougt, S.S., Mikkelsen, S.L.S. & Lorentzen, R.F. (2014). Green, T.F. (1971). Goos, M. (2014). Technology integration in secondary school mathematics: the development of teachers’ professional identities. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), Hanzsek-Brill, M.B. (1997). Jankvist, U.T. (2015). Changing students’ images of “mathematics as a discipline”. Jankvist, U.T., Misfeldt, M. & Iversen, S.M. (preprint). When students are subject to various teachers’ varying policies: A bricolage framework for the case of CAS in teaching. Kuhs, T. M., & Ball, D. L. (1986). Kvale, S. (1996). Lagrange, J. (2005). Using symbolic calculators to study mathematics: the case of tasks and techniques. The case of tasks and techniques. In D. Guin, K. Ruthven & L. Trouche (Eds.), Lavicza, Z. (2010). Integrating technology into mathematics teaching at the university level. Leatham, K.R. (2006). Viewing mathematics teachers’ beliefs as sensible systems. Leatham, K.R. (2007). Pre-service secondary mathematics teachers’ beliefs about the nature of technology in the classroom. Leder, G.C. (2015). Foreword. In B. Pepin & B. Roesken-Winter (Eds.), Liljedahl, P. (2009). Teachers’ insights into the relationship between beliefs and practice. In J. Maaß & W. Schlöglmann (Eds.), McCulloch, A.W. (2011). Affect and graphing calculator use. Nabb, K.A. (2010). CAS as a restructuring tool in mathematics education. Op’t Eynde, P., de Corte, E., & Verschaffel, L. (2002). Framing students’ mathematics-related beliefs. In G. C. Leder, E. Pehkonen, & G. Törner (Eds.), Pajares, M.F. (1992). Teachers’ beliefs and educational research: cleaning up a messy construct. Partnership For 21st Century Skills (2011). Partnership For 21st Century Skills (2004). Philipp, R. A. (2007). Mathematics teachers’ beliefs and affect. In F.K. Lester Jr. (Ed.), Rokeach, M. (1960). Schmidt, M.E. (1999). Middle grade teachers’ beliefs about calculator use: pre-project and two years later. Schoenfeld, A.H. (2007). Method. In F.K. Lester, Jr. (Ed.), Skott, J. (2015). Towards a participatory approach to ‘beliefs’ in mathematics education. In B. Pepin & B. Roesken-Winter (Eds.), Swan, M. (2007). The impact of task-based professional development on teachers’ practices and beliefs: a design research study. Tharp, M.L., Fitzsimmons, J.A. & Ayers, R.L.B. (1997). Negotiating a technological shift: teacher perception of the implementation of graphic calculators. Thomas, M.O.J. & Palmer, J.M. (2014). Teaching with digital technology: obstacles and opportunities. In A. Clark-Wilson, O. Robutti & N. Sinclair (Eds.), Thompson, A.G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D.A. Grouws (Ed.), Van Zoest, L.R., Jones, G.A., & Thornton, C.A. (1994). Beliefs about mathematics teaching held by pre-service teachers involved in a first grade mentorship program. Walen, S.B., Williams, S.R. & Garner, B.E. (2003). Pre-service teachers learning mathematics using calculators: a failure to connect current and future practice. Wilkins, J.L.M & Brand, B.R. (2004). Change in preservice teachers’ beliefs: an evaluation of a mathematics methods course. Winsløw, C. (2003). Semiotic and discursive variables in CAS-based didactical engineering. |
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## Pre-Service Math Teachers’ Opinions about Dynamic Geometry Softwares and Their Expectations from ThemHakan Şandır & Serdar Aztekin
pp.
This study was designed to determine the pre-service teachers’ opinions about three dynamic geometry software (Cabri II Plus, the Geometer's Sketchpad, GeoGebra) and influences of gender and academic achievement to these opinions. The researchers also investigated the most important properties that the pre-service teachers expect from a dynamic geometry software. The study was conducted in the 2011-2012 academic year with 64 prospective teachers who had taken a course about math education software during a year in the university. Results revealed that pre-service teachers found Geometers’ Sketchpad more effective than others in the positive development of the students' attitudes and in teaching high level geometry. However, they think that GeoGebra is easier than Cabri II Plus to use and has wide area of use. According to the pre-service teachers; using a native language, screen clarity, a detailed user manual and the ease of use are the most important properties of a dynamic geometry software.
Allison, L. (1995). The status of computer technology in classrooms using the integrated thematic instructional model. Bielefeld, T.G. (2002). On dynamic geometry software in the regular classroom. Daher, W. (2009). Pre-service Teachers' Perceptions of Applets for Solving Mathematical Problems: Need, Difficulties and Functions. Erbas, A. K. & Yenmez, A. A. (2011).The effect of inquiry-based explorations in a dynamic geometry environment on sixth grade students’ achievements in polygons. Gomoll, M. (1999). Choosing Contingency Planning Software. The Ease-Of-Use Issue in Software Selection. Göktaş, Y, Küçük, S., Aydemir, M., Telli, E., Arpacık, Ö., Yıldırım & G., Reisoğlu, İ. (2012). Educational Technology Research Trends in Turkey: A Content Analysis of the 2000-2009 Decade. Guven, B. (2012).Using dynamic geometry software to improve eight grade students’ understanding of transformation geometry. Hull, A. N., & Brovey, A. J. (2004).The impact of the use of dynamic geometry software on student achievement and attitudes towards mathematics. Hohenwarter, M. & Fuchs, K. (2004). Combination of dynamic geometry, algebra and calculus in the software system GeoGebra. ZDM classification: R 20, U 70, Retrieved on 10-November-2014, at URL: http://archive.geogebra.org/static/publications/pecs_2004.pdf Hohenwarter, M., & Lavicza, Z. (2007). Mathematics teacher development with ICT: towards an International GeoGebra Institute. In D. Küchemann (Ed.), Kortenkamp, U., & Dohrmann, C. (2010). User interface design for dynamic geometry software. Isiksal, M. & Askar, P. (2005): The effect of spreadsheet and dynamic geometry software on the achievement and self-efficacy of 7th-grade students. Mackrell, K. (2011a). Design decisions in interactive geometry software. Mackrell, K. (2011b). Finding the area of a circle: Affordances and design issues with different IGS programs. Oldknow, A. (2001). Special group 2: DGS — Dynamic Geometry Software. Oldknow, A. & Tetlow, L. (2008). Using dynamic geometry software to encourage 3D visualisation and modelling Petrovici, A. & Sava, A.T. (2010).CABRI 3D-the instrument to make the didactic approach more efficient. Roberts, D.L. & Stephens, L.J. (1999).The effect of the frequency of usage of computer software in high school geometry. Sträßer, R. (2002). Research on Dynamic Geometry Software (DGS) - Stols, G. & Kriek, J.(2011). Why don't all maths teachers use dynamic geometry software in their classrooms? Weigand, H.-G. & Weth, T. (2002). |
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## Linguistic Foundation of Foreign Language Listening ComprehensionAlfiya R. Masalimova, Galina V. Porchesku & Tatiana L. Liakhnovitch
pp.
One of the urgent contemporary educational problems, solving of which is important for foreign language teaching and learning is improving listening comprehension skills as it helps to develop communicative competence of foreign language learners. The aim of the article is to discuss the importance of using linguistic findings in the process of teaching foreign language listening comprehension. Thus, the leading approach to research the problem of the article is the linguistic one. It helps to show the peculiarities of the speech perception process in connection with the type of the language; these peculiarities should be taken into consideration when developing listening comprehension teaching techniques and programs. The article illustrates this approach with the findings on the perception peculiarities of the English words and sentences. The findings are discussed in terms of their implication in foreign language teaching. The materials of this article may be of use to those who are interested in the research on problems of speech perception and improving the existing listening comprehension teaching methods.
Abramov, V. Y. (2004). Asaphova, E. V. Golovanova, I. I. (2015). Competence formation of faculty and teaching staff for the design and implementation of educational program in networking. Baiburova, O. V. (2008). Chiknaverova, K. G. (2015). Organizational didactic conditions for development of foreign language competence of undergraduates when activating their independency. Chugaeva, T. N. (2007). Chugaeva, T. N. (2009). Grigoryeva, E. V., Leyfa, I. I., Yatsevich, L. P., Demyanenko, M.A., Makovey, N.V., Pavlushkina, T. V. & Masalimova, A.R. (2015). Designing technology of English language teaching content based on international component. Gutman, E. V., Masalimova, A. R., Shaidullina, A. R., Nizamieva, A. M. & Mukhamadieva, A. H. (2014). Foreign language discipline integrative potential in the students’ research competence development. Dzhaparidze, Z. N. (1985). Kasevich, V. B. (2010 Krause, M. (2002). Leontyev, A. A. (1965). Lopatina, O. V., Borisov, A. M., Leyfa, I. I., Galimzyanova, I. I., Yatsevich, L. P., Demyanenko, M. A. & Masalimova, A.R. (2015). Role of foreign language teacher shaping students’ research skills. Martynova, A. V. (2012). Novik, N. N. & Podgórecki, J. A. (2015). Model of Developing Communication Skills among Adolescents with Behavioral Problems. Porchesku, G. V. (2013). Rumyantseva, I. M. (2000). Psycholinguistic mechanisms and methods of speech formation (Doctoral dissertation). Moscow, 265. Shcherba, L. V. (1974). Shtern, A. S. (1992) Stahr, L. S. (2009). Vocabulary knowledge and advanced listening comprehension in English as a foreign language. Yachina, N. P. (2015). On the problems of formation of professional competence of future teacher. Yusupova, G. F., Podgorecki, J. & Markova, N. G. (2015). Educating Young People in Multicultural Educational Environment of Higher Education Institution. Vandergrift, L. (2004). Listening to learn or learning to listen. Vandergrift, L. (2007). Recent developments in second and foreign language listening comprehension research. Ventsov, A. V. & Kasevich, V. B. (1994). Zalevskaya A. A. (1988). Zalevskaya A. A. (1996). Zinder, L. R., Shtern, A. S. (1972). Factor which influence the perception of a word. |
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## Patterns of Metacognitive Behavior During Mathematics Problem-Solving in a Dynamic Geometry EnvironmentAna Kuzle
pp.
This paper describes the problem solving behavior of two preservice teachers as they worked individually on three nonroutine geometry problems. A dynamic tool software, namely the Geometer’s Sketchpad, was used as a tool to facilitate inquiry in order to uncover and investigate the patterns of metacognitive processes. Schoenfeld’s (1981) model of episodes and executive decisions in mathematics problem solving was used to identify patterns of metacognitive processes in a dynamic geometry environment. During the reading, understanding, and analysis episodes, the participants engaged in monitoring behaviors such as sense making, drawing a diagram, and allocating potential resources and approaches that helped make productive decisions. During the exploring, planning, implementation, and verification episodes, the participants made decisions to access and consider knowledge and strategies, make and test conjectures, monitor the progress, and assess the productivity of activities and strategies and the correctness of an answer. Cognitive problem-solving actions not accompanied by appropriate metacognitive monitoring actions appeared to lead to unproductive efforts. Redirection and reorganizing of thinking in productive directions occurred when metacognitive actions guided the thinking and when affective behaviors were controlled.
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## The Perceptions of Teachers and Students on a 21st Century Mathematics Instructional ModelSteve Warner & Abtar Kaur
pp.
Facilitating learning at all levels of the education stratum to create effective 21st Century knowledge creators, inventors and innovative workers is increasingly recognized today as a primary objective of education. Presently, the rapid expansion and availability of knowledge indicates the importance of curriculum and instructions that will empower learners to process knowledge using learner centered strategies rather than merely memorizing facts infused by facilitators. The study applied a qualitative research design. Interviews were completed on teachers and students to determine their perceptions on the effectiveness of the 2T2C model. Teachers’ perceptions indicated that they gained a better perspective from the teaching and learning classroom environment; high-order questioning and thinking were accomplished; the relationship between mathematics through real-world questions was realized by students; communication improved through planned cooperative and collaboration sessions; the use of technology as a resource both in and out of class provided a framework for communication and thinking; and students’ confidence and self-efficacy improved as they took responsibility for their learning. This paper presents how the 2T2C Model was conceptualized and reports on teachers’ and students’ perceptions on the model.
Anderson, L., & Krathwohl, D. (Eds.). (2001). Taxonomy for learning, teaching and assessing: A Revision of bloom's taxonomy of educational objectives. New York: Longman Bandura, A. (1977). Self-efficacy: Toward a unifying theory of behavioral change. Psychological Review, 84, 191-215. Bandura, A. (1986). The explanatory and predictive scope of self-efficacy theory. Journal of Clinical and Social Psychology, 4, 259-373. Bandura, A. (1997). Self-efficacy: The exercise of control. New York: W.H. Freeman. Banikowski, K. (1999). Strategies to enhance memory based on brain-research. Accessed December 2013 from http://sc-boces.org/english/IMC/Focus/Memory_strategies2.pdf Barahal, S. (2008). Thinking about thinking: Pre-service teachers strengthen their thinking artfully. Phi Delta Kappan, 90(4), 298-302. Barriball, K., & While, A. (1994). Collecting Data using a semi‐structured interview: A discussion paper. Beers, S. (2011). Bellanca, J. & Brandt, R. 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