Mixed methods study of middle school mathematics teachers’ content knowledge in usa and russia using sequential nested design
Main Article Content
Аннотация
The sequential nested mixed methods study focused on comparative analysis of middle school mathematics teachers’ content knowledge in two countries. The study consisted of two stages: (1) quantitative study of teacher content knowledge; (2) qualitative study of teacher topic-specific content knowledge. The initial sample for the first stage included lower secondary mathematics teachers from the U.S. (grades 6–9, N=102) and Russia (grades 5–9, N=97). The Teacher Content Knowledge Survey (TCKS) was applied to assess teacher content knowledge based on the cognitive domains of Knowing, Applying, and Reasoning, as well as addressing the lower secondary mathematics topics of Number, Algebra, Geometry, Data and Chance. The second stage – an interpretive cross-case study – aimed at the examination of the U.S. and Russian teachers’ topic-specific knowledge on the division of fractions. For the second stage, N=16 teachers (8 – from the U.S., and 8 – from Russia) were selected for the study using non-probability purposive sampling technique based on teachers’ scores on the TCKS. Teachers were interviewed on the topic of fraction division using questions addressing their content and pedagogical content knowledge. The study revealed that there are explicit similarities and differences in teachers’ content knowledge as well as its cognitive types. The study results may inform the field on priorities placed on lower secondary mathematics teachers’ knowledge in the USA and Russia. It also suggests close comparison and learning about issues related to teacher knowledge in both countries with a potential focus on re-examining practices in teacher preparation and professional development.
Article Details
Библиографические ссылки
Adamson B. International comparative studies in teaching and teacher education. Teaching and Teacher Education, 2012, 28(5), P. 641–648.
An S., Kulm G., and Wu Z. The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 2004, 7(2), P. 145–172.
Andrews P. Comparative studies of mathematics teachers' observable learning objectives: Validating low inference codes. Educational Studies in Mathematics, 2009, 71(2), P. 97–122.
Bair S.L. and Ric B.S. Characterizing the development of specialized mathematical content knowledge for teaching in algebraic reasoning and number theory. Mathematical Thinking and Learning, 2011, 13(4), P. 292–321.
Baumert J., Kunter M., Blum W., Brunner M., Voss T., Jordan A., Klusmann U., Krauss S., Neubrand M., and Tsa Y. Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 2010, 47(1), P. 133–180.
Blomeke S., Suhl U., and Kaiser G. Teacher education effectiveness: Quality and equity of future primary teachers’ mathematics and mathematics pedagogical content knowledge. Journal of Teacher Education, 2011, 62(2), P. 154–171.
Blomeke S. Content, professional preparation, and teaching methods: How diverse is teacher education across countries? Comparative Education Review, 2012, 56(4), 684–714.
Blomeke S., Paine L., Houang R., Hsieh F-J., Schmidt W., Tatto M., Bankov K., Cedilllo T., Cogan L., Han S., Santillan M., and Schwille J. Future teachers’ competence to plan a lesson: first results of a six-country study on the efficiency of teacher education. ZDM, 2008, 40(5), P. 749–762.
Bransford J., Brown A., and Cocking R. How People Learn. Expanded Edition. Washington, DC: National Research Council, 2000.
Cai J. and Wang T. Conceptions of effective mathematics teaching within a cultural context: perspectives of teachers from China and the United States. Journal of Mathematics Teacher Education, 2010, 13(3), P. 265–287.
Cai J., Ding M., and Wang T. How do exemplary Chinese and U.S. mathematics teachers’ view instructional coherence? Educational Studies in Mathematics, 2014, 85(2), 265–280.
Chapman O. Investigating teachers’ knowledge for teaching mathematics. Journal of Mathematics Teacher Education, 2013, 16, P. 237–243.
Charalambous C. and Pitta-Pantazi D. Perspectives on priority mathematics education unpacking and understanding a complex relationship linking teacher knowledge, teaching, and learning. In Handbook of International Research in Mathematics Education, 3rd Edition. Edited by Lyn D. English and David Kirshner. New York, NY: Routledge, 2016, P. 19–59.
Cronbach L.J. Coefficient alpha and the internal structure of the tests. Psychometrica, 1951, 16, P. 297–334.
Davis B. and Simmt E. Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 2006, 61, P. 293–319.
Delaney S., Ball D., Hill H., Schilling S., and Zopf D. Mathematical knowledge for teaching: Adapting U.S. measures for use in Ireland. Journal of Mathematics Teacher Education, 2008, 11(3), 171–197.
Ewha R., Ham S-H., and Paine L. Knowledge expectations in mathematics teacher preparation programs in South Korea and the United States: Towards international dialogue. Journal of Teacher Education, 2011, 62(1), 48–61.
Felbrich A., Kaiser G., and Schmotz C. The cultural dimension of beliefs: An investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries. ZDM, 2012, 44(3), P. 355–366.
Ferrini-Mundy J., Floden R.E., McCrory R., Burrill G., and Sandow D. Knowledge for teaching school algebra: Challenges in developing an analytic framework. East Lansing, MI: Michigan State University, Knowledge of Algebra for Teaching Project. 2005.
Groth R.E. and Bergner J.A. Preservice elementary teachers’ conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 2006, 8(1), P. 37–63.
Hemmi K. and Ryve A. Effective mathematics teaching in Finnish and Swedish teacher education discourses. Journal of Mathematics Teacher Education, 2015, 18(6), P. 501–521.
Hill H., Schilling S., and Ball D. Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal, 2004, 105, P. 11–30.
Hill H.C., Rowan B., and Ball D.L. Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 2005, 42(2), P. 371–406.
Hill H., Ball D., and Schilling S. Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 2008, 39(4), P. 372–400.
Izsac A., Jacobson E., and de Araujo Z. Measuring mathematical knowledge for teaching fractions with drawn quantities. Journal for Research in Mathematics Education, 2012, 43(4), P. 391–427.
Kleickmann T., Richter D., Kunter M., Elsner J., Besser M., Krauss S., and Baumert J. Teachers’ content knowledge and pedagogical content knowledge: The role of structural differences in teacher education. Journal of Teacher Education, 2013, 64(1), P. 90–106.
Kvale S. and Brinkmann S. InterViews: Learning the Craft of Qualitative Research Interviewing. 2nd ed. Los Angeles: CA: Sage, 2009.
LeTendre G. Cross-national studies and the analysis of comparative qualitative research. In Steiner-Khamsi G., Torney-Purta J., and Schwille J. (ed.) New paradigms and recurring paradoxes in education for citizenship: An international comparison (International Perspectives on Education and Society, Volume 5). Emerald Group Publishing Limited, 2002, P. 239–277.
Ma L. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Hillsdale, NJ: Erlbaum, 1999.
Marshall J. and Sorto A. The effects of teacher mathematics knowledge and pedagogy on student achievement in rural Guatemala. International Review of Education, 2012, 58(2), P. 173–197.
Mccrory R., Floden R, Ferrini-Mundy J., Reckase M., and Senk S. Knowledge of algebra for teaching: A framework of knowledge and practices. Journal for Research in Mathematics Education, 2012, 43(5), P. 584–615.
Monk D. and Rice J. Multilevel teacher resource effects on pupil performance in secondary mathematics and science: The case of teacher subject-matter preparation. In R.G. Ehrenberg (Ed.), Choices and consequences: Contemporary policy issues in education. Ithaca, NY: ILR Press, 1994, P. 29–58.
Mullis I.V.S., Martin M.O., Foy P., and Hooper M. TIMSS 2015 International Results in Mathematics. Retrieved from. 2016. URL: http://timssandpirls.bc.edu/ timss2015/international-results/.
Murphy C. The role of subject knowledge in primary prospective teachers’ approaches to teaching the topic of area. Journal of Mathematics Teacher Education, 2012, 15(3), P. 187–206.
Nason R., Chalmers C., and Yeh A. Facilitating growth in prospective teachers’ knowledge: Teaching geometry in primary schools. Journal of Mathematics Teacher Education, 2012, 15(3), P. 227–249.
Ng S. and Rao N. Chinese number words, culture, and mathematics learning. Review of Educational Research, 2010, 80(2), P. 180–206.
Robitaille D.F. and Travers K.J. International studies of achievement in mathematics. In D.A. Grouws (Ed.), Handbook of mathematics teaching and learning. New York: Macmillan Publishing Company, 1992,P. 687–709.
Rowland T., Huckstep P., and Thwaites A. Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 2005, 8, P. 255–281.
Schoenfeld A. The complexities of assessing teacher knowledge. In Measurement: Interdisciplinary Research and Perspectives, 2007, 5 (2-3), P. 198–204.
Senk S., Tatto M., Reckase M., Rowley G., Peck R., and Bankov K. Knowledge of future primary teachers for teaching mathematics: an international comparative study. ZDM, 2012, 44(3), P. 307–324.
Seaberg R. Mathematics lessons from Finland and Sweden. The Mathematics Teacher, 2015, 108(8), P. 593–598.
Schmidt W., Cogan L., and Houang R. The role of opportunity to learn in teacher preparation: An international context. Journal of Teacher Education, 2011, 62(2), P. 138–153.
Shulman L. Those who understand: Knowledge growth in teaching. Educational Researcher, 1986, 15(2), P. 4–14.
Sorto M., Marshall J., Luschel T., and Carnoy M. Teacher knowledge and teaching in Panama and Costa Rica: A comparative study in primary and secondary education. Revista Latinoamerica de Investigacion en Matematica Educativa, 2009, 12(2), P. 251–290.
Stein M., Smith M., Henningsen M., and Silver E. Implementing standards-based mathematics instruction: A casebook for professional development. Foreword by Deborah Ball. New York, NY: Teachers College Press, 2000.
Steinberg T., Haymore J., and Marks R. Teacher’s knowledge and structuring content in mathematics. Paper presented at the Annual meeting of American Educational Research Association, Chicago, 1985.
Stigler J.W. and Perry M. Cross-cultural studies of mathematics teaching and learning: Recent finding and new directions. In D. Grouws and T. Cooney (Eds.), Effective mathematics teaching directions. Reston, VA: National Council of Teachers of Mathematics, 1988, P. 194–223.
Tashakkori A. and Teddlie C. Handbook of Mixed Methods in Social and Behavioral Research. Thousand Oaks, CA: Sage, 2003.
Tatto M. and Senk S. The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics (TEDS-M), Journal of Teacher Education, 2011, 62 (2), P. 121–137
Tatto M.T. (Ed.). The Teacher Education and Development Study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics in 17 countries. Technical report. Amsterdam: IEA, 2013.
Tchoshanov M.A. Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 2011, 76(2), P. 141–164.
Tchoshanov M., Cruz Quinones M., Shakirova K., Ibragimova E., and Shakirova L. Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics. Journal of Mathematical Behavior, 2017, 47, P. 54–69.
Tchoshanov M., Cruz Quinones M., Shakirova K., Ibragimova E., and Shakirova L. Examination of lower secondary mathematics teachers’ content knowledge and its connection to students’ performance. International Journal of Science and Mathematics Education, 2017, 15(4), P. 683–702.
Wagner S., Lee H., and Ozgun-Koca S. A comparative study of the United States, Turkey, and Korea: Attitudes and beliefs of preservice mathematics teachers towards mathematics, teaching mathematics, and their teacher preparation programs. Proceedings of the Annual Meeting of Association of Mathematics Teacher Educators. Chicago, IL, 1999.
Wang J. and Lin E. Comparative studies on the U.S. and Chinese mathematics learning and the implications for standards-based mathematics teaching reform. Educational Researcher, 2005, 34(5), P. 3–13.
An S., Kulm G., and Wu Z. The pedagogical content knowledge of middle school, mathematics teachers in China and the U.S. Journal of Mathematics Teacher Education, 2004, 7(2), P. 145–172.
Andrews P. Comparative studies of mathematics teachers' observable learning objectives: Validating low inference codes. Educational Studies in Mathematics, 2009, 71(2), P. 97–122.
Bair S.L. and Ric B.S. Characterizing the development of specialized mathematical content knowledge for teaching in algebraic reasoning and number theory. Mathematical Thinking and Learning, 2011, 13(4), P. 292–321.
Baumert J., Kunter M., Blum W., Brunner M., Voss T., Jordan A., Klusmann U., Krauss S., Neubrand M., and Tsa Y. Teachers’ mathematical knowledge, cognitive activation in the classroom, and student progress. American Educational Research Journal, 2010, 47(1), P. 133–180.
Blomeke S., Suhl U., and Kaiser G. Teacher education effectiveness: Quality and equity of future primary teachers’ mathematics and mathematics pedagogical content knowledge. Journal of Teacher Education, 2011, 62(2), P. 154–171.
Blomeke S. Content, professional preparation, and teaching methods: How diverse is teacher education across countries? Comparative Education Review, 2012, 56(4), 684–714.
Blomeke S., Paine L., Houang R., Hsieh F-J., Schmidt W., Tatto M., Bankov K., Cedilllo T., Cogan L., Han S., Santillan M., and Schwille J. Future teachers’ competence to plan a lesson: first results of a six-country study on the efficiency of teacher education. ZDM, 2008, 40(5), P. 749–762.
Bransford J., Brown A., and Cocking R. How People Learn. Expanded Edition. Washington, DC: National Research Council, 2000.
Cai J. and Wang T. Conceptions of effective mathematics teaching within a cultural context: perspectives of teachers from China and the United States. Journal of Mathematics Teacher Education, 2010, 13(3), P. 265–287.
Cai J., Ding M., and Wang T. How do exemplary Chinese and U.S. mathematics teachers’ view instructional coherence? Educational Studies in Mathematics, 2014, 85(2), 265–280.
Chapman O. Investigating teachers’ knowledge for teaching mathematics. Journal of Mathematics Teacher Education, 2013, 16, P. 237–243.
Charalambous C. and Pitta-Pantazi D. Perspectives on priority mathematics education unpacking and understanding a complex relationship linking teacher knowledge, teaching, and learning. In Handbook of International Research in Mathematics Education, 3rd Edition. Edited by Lyn D. English and David Kirshner. New York, NY: Routledge, 2016, P. 19–59.
Cronbach L.J. Coefficient alpha and the internal structure of the tests. Psychometrica, 1951, 16, P. 297–334.
Davis B. and Simmt E. Mathematics-for-teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 2006, 61, P. 293–319.
Delaney S., Ball D., Hill H., Schilling S., and Zopf D. Mathematical knowledge for teaching: Adapting U.S. measures for use in Ireland. Journal of Mathematics Teacher Education, 2008, 11(3), 171–197.
Ewha R., Ham S-H., and Paine L. Knowledge expectations in mathematics teacher preparation programs in South Korea and the United States: Towards international dialogue. Journal of Teacher Education, 2011, 62(1), 48–61.
Felbrich A., Kaiser G., and Schmotz C. The cultural dimension of beliefs: An investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries. ZDM, 2012, 44(3), P. 355–366.
Ferrini-Mundy J., Floden R.E., McCrory R., Burrill G., and Sandow D. Knowledge for teaching school algebra: Challenges in developing an analytic framework. East Lansing, MI: Michigan State University, Knowledge of Algebra for Teaching Project. 2005.
Groth R.E. and Bergner J.A. Preservice elementary teachers’ conceptual and procedural knowledge of mean, median, and mode. Mathematical Thinking and Learning, 2006, 8(1), P. 37–63.
Hemmi K. and Ryve A. Effective mathematics teaching in Finnish and Swedish teacher education discourses. Journal of Mathematics Teacher Education, 2015, 18(6), P. 501–521.
Hill H., Schilling S., and Ball D. Developing measures of teachers’ mathematics knowledge for teaching. Elementary School Journal, 2004, 105, P. 11–30.
Hill H.C., Rowan B., and Ball D.L. Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 2005, 42(2), P. 371–406.
Hill H., Ball D., and Schilling S. Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 2008, 39(4), P. 372–400.
Izsac A., Jacobson E., and de Araujo Z. Measuring mathematical knowledge for teaching fractions with drawn quantities. Journal for Research in Mathematics Education, 2012, 43(4), P. 391–427.
Kleickmann T., Richter D., Kunter M., Elsner J., Besser M., Krauss S., and Baumert J. Teachers’ content knowledge and pedagogical content knowledge: The role of structural differences in teacher education. Journal of Teacher Education, 2013, 64(1), P. 90–106.
Kvale S. and Brinkmann S. InterViews: Learning the Craft of Qualitative Research Interviewing. 2nd ed. Los Angeles: CA: Sage, 2009.
LeTendre G. Cross-national studies and the analysis of comparative qualitative research. In Steiner-Khamsi G., Torney-Purta J., and Schwille J. (ed.) New paradigms and recurring paradoxes in education for citizenship: An international comparison (International Perspectives on Education and Society, Volume 5). Emerald Group Publishing Limited, 2002, P. 239–277.
Ma L. Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Hillsdale, NJ: Erlbaum, 1999.
Marshall J. and Sorto A. The effects of teacher mathematics knowledge and pedagogy on student achievement in rural Guatemala. International Review of Education, 2012, 58(2), P. 173–197.
Mccrory R., Floden R, Ferrini-Mundy J., Reckase M., and Senk S. Knowledge of algebra for teaching: A framework of knowledge and practices. Journal for Research in Mathematics Education, 2012, 43(5), P. 584–615.
Monk D. and Rice J. Multilevel teacher resource effects on pupil performance in secondary mathematics and science: The case of teacher subject-matter preparation. In R.G. Ehrenberg (Ed.), Choices and consequences: Contemporary policy issues in education. Ithaca, NY: ILR Press, 1994, P. 29–58.
Mullis I.V.S., Martin M.O., Foy P., and Hooper M. TIMSS 2015 International Results in Mathematics. Retrieved from. 2016. URL: http://timssandpirls.bc.edu/ timss2015/international-results/.
Murphy C. The role of subject knowledge in primary prospective teachers’ approaches to teaching the topic of area. Journal of Mathematics Teacher Education, 2012, 15(3), P. 187–206.
Nason R., Chalmers C., and Yeh A. Facilitating growth in prospective teachers’ knowledge: Teaching geometry in primary schools. Journal of Mathematics Teacher Education, 2012, 15(3), P. 227–249.
Ng S. and Rao N. Chinese number words, culture, and mathematics learning. Review of Educational Research, 2010, 80(2), P. 180–206.
Robitaille D.F. and Travers K.J. International studies of achievement in mathematics. In D.A. Grouws (Ed.), Handbook of mathematics teaching and learning. New York: Macmillan Publishing Company, 1992,P. 687–709.
Rowland T., Huckstep P., and Thwaites A. Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 2005, 8, P. 255–281.
Schoenfeld A. The complexities of assessing teacher knowledge. In Measurement: Interdisciplinary Research and Perspectives, 2007, 5 (2-3), P. 198–204.
Senk S., Tatto M., Reckase M., Rowley G., Peck R., and Bankov K. Knowledge of future primary teachers for teaching mathematics: an international comparative study. ZDM, 2012, 44(3), P. 307–324.
Seaberg R. Mathematics lessons from Finland and Sweden. The Mathematics Teacher, 2015, 108(8), P. 593–598.
Schmidt W., Cogan L., and Houang R. The role of opportunity to learn in teacher preparation: An international context. Journal of Teacher Education, 2011, 62(2), P. 138–153.
Shulman L. Those who understand: Knowledge growth in teaching. Educational Researcher, 1986, 15(2), P. 4–14.
Sorto M., Marshall J., Luschel T., and Carnoy M. Teacher knowledge and teaching in Panama and Costa Rica: A comparative study in primary and secondary education. Revista Latinoamerica de Investigacion en Matematica Educativa, 2009, 12(2), P. 251–290.
Stein M., Smith M., Henningsen M., and Silver E. Implementing standards-based mathematics instruction: A casebook for professional development. Foreword by Deborah Ball. New York, NY: Teachers College Press, 2000.
Steinberg T., Haymore J., and Marks R. Teacher’s knowledge and structuring content in mathematics. Paper presented at the Annual meeting of American Educational Research Association, Chicago, 1985.
Stigler J.W. and Perry M. Cross-cultural studies of mathematics teaching and learning: Recent finding and new directions. In D. Grouws and T. Cooney (Eds.), Effective mathematics teaching directions. Reston, VA: National Council of Teachers of Mathematics, 1988, P. 194–223.
Tashakkori A. and Teddlie C. Handbook of Mixed Methods in Social and Behavioral Research. Thousand Oaks, CA: Sage, 2003.
Tatto M. and Senk S. The mathematics education of future primary and secondary teachers: Methods and findings from the Teacher Education and Development Study in Mathematics (TEDS-M), Journal of Teacher Education, 2011, 62 (2), P. 121–137
Tatto M.T. (Ed.). The Teacher Education and Development Study in Mathematics (TEDS-M): Policy, practice, and readiness to teach primary and secondary mathematics in 17 countries. Technical report. Amsterdam: IEA, 2013.
Tchoshanov M.A. Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 2011, 76(2), P. 141–164.
Tchoshanov M., Cruz Quinones M., Shakirova K., Ibragimova E., and Shakirova L. Analyzing connections between teacher and student topic-specific knowledge of lower secondary mathematics. Journal of Mathematical Behavior, 2017, 47, P. 54–69.
Tchoshanov M., Cruz Quinones M., Shakirova K., Ibragimova E., and Shakirova L. Examination of lower secondary mathematics teachers’ content knowledge and its connection to students’ performance. International Journal of Science and Mathematics Education, 2017, 15(4), P. 683–702.
Wagner S., Lee H., and Ozgun-Koca S. A comparative study of the United States, Turkey, and Korea: Attitudes and beliefs of preservice mathematics teachers towards mathematics, teaching mathematics, and their teacher preparation programs. Proceedings of the Annual Meeting of Association of Mathematics Teacher Educators. Chicago, IL, 1999.
Wang J. and Lin E. Comparative studies on the U.S. and Chinese mathematics learning and the implications for standards-based mathematics teaching reform. Educational Researcher, 2005, 34(5), P. 3–13.
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