Araştırma Makalesi
Pre-Service Teachers’ Restructuring of Mathematical Content Knowledge in a Learning Trajectories based Instruction
Öz
The purpose of this study
was to explore how elementary pre-service teachers (PTs) restructure their existing
mathematical content knowledge (MCK) for equipartitioning related mathematical
ideas in a Learning Trajectories Based Instruction (LTBI). A teaching
experiment was conducted with nine elementary education PTs with the use of
equiparitioning learning trajectory. The findings of this study briefly
indicated that PTs’ existing misconceptions and mathematical errors were
remediated and underlying reasons behind these misconceptions and errors were revealed
in LTBI. PTs also examined mathematically correct responses. In addition, a
variety of mathematical strategies and representations was utilized, and then
explained by PTs to the peers in mathematically correct ways as they worked on the
activities. These findings suggested that PTs had improved their existing MCK.
The process of these improvements in PTs’ MCK and the observed pattern in the
improvements was categorized under seven restructuring practices. These
practices suggested an emergent framework for MCK restructuring practices.
These seven practices were observed in relation to PTs’ restructuring practices
of Common Content Knowledge, Specialized Content Knowledge, and Horizon Content
Knowledge.
Anahtar Kelimeler
Learning trajectories pre-service teacher mathematical content knowledge restructuring
Kaynakça
- Baki, M. (2013). Pre-service classroom teachers’ mathematical knowledge and instructional explanations associated with division. Eğitim ve Bilim, 38(167), 300-311. Baki A. & Gökçek, T. (2007). Matematik öğretmeni adaylarının benimsedikleri öğretmen modeline ilişkin bazı ipuçları. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 20, 16-25. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. Ball, D.L., & Forzani, F.M. (2010). Teaching skillful teaching. Educational Leadership, 68(4), 40-45. Baştürk, S. (2007). Öğretmen adaylarının öğretmenlik uygulaması dersiyle i̇lgili deneyimleri. 16. Ulusal Egitim Bilimleri Kongresinde sunulmuştur. Gaziosmanpaşa Eğitim Fakültesi, Tokat. Battista, M. T. (2004). Applying cognition-based assessment to elementary school students' development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185-204. Butterfield, B., Forrester, T., McCallum, F. & Chinnappan, M. (2013). Use of learning trajectories to examine pre-service teachers’ mathematics knowledge for teaching area and perimeter: Emerging issues. Paper presented at 36th Annual Conference of the Mathematics Education Research Group of Australia, Melbourne, Australia. Clements, D., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81-89. Clements, D. H., & Sarama, J. (2013). Rethinking early mathematics: What is research-based curriculum for young children?. İçinde L. D. English and J. T. Mulligan (Eds.), Reconceptualizing Early Mathematics Learning (s. 121-147). The Netherlands: Springer. Clements, D., Sarama, J., & Julie, A. (2009). Learning and teaching early math. İçinde The Learning Trajectories Approach (s. 18). New York: Routledge. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational researcher, 32(1), 9-13. Confrey, J. (2008). A synthesis of the research on rational number reasoning: A learning progressions approach to synthesis. 11th International Congress of Mathematics Instruction, Monterrey, Mexico. Confrey, J. (2012). Better measurement of higher-cognitive processes through learning trajectories and diagnostic assessments in mathematics: The challenge in adolescence. İçinde V. Reyna, M. Dougherty, S. B. Chapman, & J. Confrey (Eds.), The adolescent brain: Learning reasoning and decision making. Washington, DC: American Psychology Association. Confrey, J., & Maloney, A. P. (2011). A next generation of mathematics assessments based on learning trajectories: Diagnostic learning profiles. İçinde J. Confrey, A. P. Maloney & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education: Information Age. Confrey, J., Maloney, A. P., Nguyen, K. H., Mojica, G., & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece. Corcoran, T., Mosher, F.A., & Rogat, A. (2009). Learning progressions in science: An evidence based approach to reform. NY: Center on Continuous Instructional Improvement, Teachers College Columbia University. Creswell, J. W. (2007). Qualitative inquiry & research design: Choosing among five approaches (2nd baskı). Thousand Oaks, CA: Sage. Çıkla A. O. & Duatepe, A. (2002). A qualitative study on the proportional reasoning skills of the preservice elementary mathematics teachers. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 23, 32-40. Duncan, R.G., & Hmelo-Silver, C.E. (2009). Learning progressions: Aligning curriculum, instruction, and assessment. Journal of Research in ScienceTeaching, 46, 606–609. Edgington, C., Sztajn, P., & Wilson, P. H., & Confrey, J. (2011). Teachers’ use of a learning trajectory for formative assessment. Thirty-third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, Nevada. Empson, S. B., & Turner, E. (2006). The emergence of multiplicative thinking in children’s solutions to paper folding tasks. Journal of Mathematical Behavior, 25, 46-56. Engelhardt, P. V., Corpuz, E. G., Ozimek, D. J., & Rebello, N. S. (2004, September). The teaching experiment- What it is and what it isn't. İçinde 2003 Physics Education Research Conference (Cilt 720, s 157-160). Eraslan, A. (2009). lköğretim matematik öğretmeni adaylarının öğretmenlik uygulaması üzerine görüs ve değerlendirmeleri. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 3(1), 208-221 Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. İçinde D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. (s. 147-164). New York: Macmillan. Franklin, A., Yilmaz, Z., & Confrey, J. (2010, Kasım). Reconciling student thinking and theory: The delta learning trajectory and the case of transitivity. Thirty- Second Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH. Hacıömeroğlu G., & Şahin Taşkın Ç. (2010). Sınıf öğretmeni adaylarının matematik öğretimi yeterlik inançları. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 23(2), 539-555. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum. Miles, M. B. & Huberman, M. A. (1994). Qualitative Analysis: An Extended Source Book (2. baski) Thousand Oaks, CA:Sage. Mojica, G. (2010). Preparing pre-service elementary teachers to teach mathematics with learning trajectories (Yayınlanmamış doktora tezi). North Carolina State University, Raleigh, NC. Pellegrino, J. W. (2009). The challenges of conceptualizing what low achievers know and how to assess their competence. İçinde M. Perie (Ed.), Considerations for the alternate assessment based on modified achievement standards (AAMAS): Understanding the eligible population and applying that knowledge to their instruction and assessment. New York, NY: New York Comprehensive Center. Philipp, R. (2008). Motivating prospective elementary school teachers to learn mathematics by focusing upon children’s mathematical thinking. Issues in Teacher Education, 17(2), 7-26. Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of learners' mathematical ideas and reasoning using videotape analysis. Journal of Mathematical Behavior, 22, 405-435. Ryan, J. & McCrae, B. (2006). Assessing pre-service teachers' mathematics subject knowledge. Mathematics Teacher Education and Development, 7, 72-89. Sawchuk, S. (2010). Full cost of professional development hidden. Education Week, 30(11), 14-16. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145. Sztajn, P., Confrey, J., Wilson, P. H. & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156. Steffe, L. & Thompson, P. (2000). Teaching experiment methodology: underlying principles and essential elements. In A. Kelly & R. Lesh (Editörler.), Handbook of Research Design in Mathematics and Science Education (s.267 – 306). Mahwah, NJ: Lawrence Erlbaum Associates. Steffe, L. P., & Ulrich, C. (2014). Constructivist teaching experiment. In Encyclopedia of mathematics education (s. 102-109). Springer Netherlands. Stein, M. K., & Smith, M. S. (2011). Five practices for orchestrating productive mathematics discussions. Reston, VA: Corwin. Toluk, Z. & Middleton, J. A. (2001). The development of children’s understanding of quotient: A teaching experiment. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of 25th Annual Meeting of the International Group for Psychology of Mathematics Education. Utrecht, The Netherlands: Hogrefe. Toluk-Uçar, Z. (2010). Sınıf Öğretmeni adaylarının matematiksel bilgileri ve öğretimsel açıklamaları. e-Journal of New World Science Academy, 5(3), 911-920. Wilson, P. H. (2009). Teachers’ uses of a learning trajectory for equipartitioning (Yayınlanmamış doktora tezi). North Carolina State University. Raleigh, NC. Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning trajectories in teacher education: Supporting teachers’ understanding of students’ mathematical thinking. Journal of Mathematical Behavior, 32, 103-121. Wilson, P. H., Sztajn, P., Edgington, C., & Confrey, J. (2014). Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. Journal of Mathematics Teacher Education, 17(2), 149-175. Yilmaz, Z. (2011). Toward an understanding of students’ strategies on reallocation and covariation items: In relation to an equipartitioning learning trajectory (Yayınlanmamış yüksek lisans tezi). North Carolina State University, Raleigh, NC. Zembat, I, O. (2007). Working on the same problem – Concepts; with the usual subjects – Prospective Elementary Teachers. Elementary Education Online, 6(2), 305-312.
Öğrenme Rotaları Temelli Öğretimde Öğretmen Adaylarının Matematiksel Alan Bilgilerini Yeniden Yapılandırmaları
Öz
Çalışmanın amacı sınıf
öğretmeni adaylarının eşpaylaşım konusu ile ilgili matematiksel alan
bilgilerini (MAB) öğrenme rotaları temelli bir öğretim ortamında nasıl yeniden yapılandırdıklarını
incelemektir. Bu bağlamda dokuz öğretmen adayı ile bir öğretim deneyi eşpaylaşım
öğrenme rotası kullanılarak gerçekleştirilmiştir.
Çalışmanın bulguları, adayların mevcut kavram yanılgı ve hatalarını
düzelttiklerini; bu yanılgı ve hataların altında yatan sebepleri açığa çıkardıklarını
ortaya koymaktadır. Aynı zamanda, adayların doğru matematiksel sonuçların neden
doğru olduklarını da irdeledikleri; etkinliklerde farklı matematiksel
stratejileri ve gösterimleri kullandıkları; kendilerinin ve diğer adayların
matematiksel stratejilerini derinlemesine irdelerken doğru matematiksel
terminolojileri kullandıkları gözlenmiştir. Bu bulgular, adayların MABlerinin öğretim
deneyi başına kıyasla geliştiğini göstermektedir. Bu gelişmenin nasıl bir
örüntü içerisinde gerçekleştiği yeniden yapılandırma eylemleri olarak kodlanmıştır.
Çalışmanın sonuçları, adayların MABlerini yeniden yapılandırma eylemleri için gelişmekte olan bir çerçeveyi ortaya
koymaktadır. Bu çerçeve, adayların MABlerini yedi farklı eylem ile yeniden
yapılandırdıklarını göstermektedir. Bu eylemler Genel Alan Bilgisini, Özel Alan
Bilgisini ve Yatay Alan Bilgisini yapılandırma eylemleri olarak alt
kategorilerde ele alınmaktadır.
Anahtar Kelimeler
Öğrenme rotaları öğretmen adayı matematiksel alan bilgisi yeniden yapılandırma
Kaynakça
- Baki, M. (2013). Pre-service classroom teachers’ mathematical knowledge and instructional explanations associated with division. Eğitim ve Bilim, 38(167), 300-311. Baki A. & Gökçek, T. (2007). Matematik öğretmeni adaylarının benimsedikleri öğretmen modeline ilişkin bazı ipuçları. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 20, 16-25. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389-407. Ball, D.L., & Forzani, F.M. (2010). Teaching skillful teaching. Educational Leadership, 68(4), 40-45. Baştürk, S. (2007). Öğretmen adaylarının öğretmenlik uygulaması dersiyle i̇lgili deneyimleri. 16. Ulusal Egitim Bilimleri Kongresinde sunulmuştur. Gaziosmanpaşa Eğitim Fakültesi, Tokat. Battista, M. T. (2004). Applying cognition-based assessment to elementary school students' development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185-204. Butterfield, B., Forrester, T., McCallum, F. & Chinnappan, M. (2013). Use of learning trajectories to examine pre-service teachers’ mathematics knowledge for teaching area and perimeter: Emerging issues. Paper presented at 36th Annual Conference of the Mathematics Education Research Group of Australia, Melbourne, Australia. Clements, D., & Sarama, J. (2004). Learning trajectories in mathematics education. Mathematical Thinking and Learning, 6(2), 81-89. Clements, D. H., & Sarama, J. (2013). Rethinking early mathematics: What is research-based curriculum for young children?. İçinde L. D. English and J. T. Mulligan (Eds.), Reconceptualizing Early Mathematics Learning (s. 121-147). The Netherlands: Springer. Clements, D., Sarama, J., & Julie, A. (2009). Learning and teaching early math. İçinde The Learning Trajectories Approach (s. 18). New York: Routledge. Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational researcher, 32(1), 9-13. Confrey, J. (2008). A synthesis of the research on rational number reasoning: A learning progressions approach to synthesis. 11th International Congress of Mathematics Instruction, Monterrey, Mexico. Confrey, J. (2012). Better measurement of higher-cognitive processes through learning trajectories and diagnostic assessments in mathematics: The challenge in adolescence. İçinde V. Reyna, M. Dougherty, S. B. Chapman, & J. Confrey (Eds.), The adolescent brain: Learning reasoning and decision making. Washington, DC: American Psychology Association. Confrey, J., & Maloney, A. P. (2011). A next generation of mathematics assessments based on learning trajectories: Diagnostic learning profiles. İçinde J. Confrey, A. P. Maloney & K. H. Nguyen (Eds.), Learning over time: Learning trajectories in mathematics education: Information Age. Confrey, J., Maloney, A. P., Nguyen, K. H., Mojica, G., & Myers, M. (2009). Equipartitioning/splitting as a foundation of rational number reasoning using learning trajectories. 33rd Conference of the International Group for the Psychology of Mathematics Education, Thessaloniki, Greece. Corcoran, T., Mosher, F.A., & Rogat, A. (2009). Learning progressions in science: An evidence based approach to reform. NY: Center on Continuous Instructional Improvement, Teachers College Columbia University. Creswell, J. W. (2007). Qualitative inquiry & research design: Choosing among five approaches (2nd baskı). Thousand Oaks, CA: Sage. Çıkla A. O. & Duatepe, A. (2002). A qualitative study on the proportional reasoning skills of the preservice elementary mathematics teachers. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 23, 32-40. Duncan, R.G., & Hmelo-Silver, C.E. (2009). Learning progressions: Aligning curriculum, instruction, and assessment. Journal of Research in ScienceTeaching, 46, 606–609. Edgington, C., Sztajn, P., & Wilson, P. H., & Confrey, J. (2011). Teachers’ use of a learning trajectory for formative assessment. Thirty-third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Reno, Nevada. Empson, S. B., & Turner, E. (2006). The emergence of multiplicative thinking in children’s solutions to paper folding tasks. Journal of Mathematical Behavior, 25, 46-56. Engelhardt, P. V., Corpuz, E. G., Ozimek, D. J., & Rebello, N. S. (2004, September). The teaching experiment- What it is and what it isn't. İçinde 2003 Physics Education Research Conference (Cilt 720, s 157-160). Eraslan, A. (2009). lköğretim matematik öğretmeni adaylarının öğretmenlik uygulaması üzerine görüs ve değerlendirmeleri. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 3(1), 208-221 Fennema, E., & Franke, M. L. (1992). Teachers’ knowledge and its impact. İçinde D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics. (s. 147-164). New York: Macmillan. Franklin, A., Yilmaz, Z., & Confrey, J. (2010, Kasım). Reconciling student thinking and theory: The delta learning trajectory and the case of transitivity. Thirty- Second Annual Conference of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH. Hacıömeroğlu G., & Şahin Taşkın Ç. (2010). Sınıf öğretmeni adaylarının matematik öğretimi yeterlik inançları. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 23(2), 539-555. Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah, NJ: Lawrence Erlbaum. Miles, M. B. & Huberman, M. A. (1994). Qualitative Analysis: An Extended Source Book (2. baski) Thousand Oaks, CA:Sage. Mojica, G. (2010). Preparing pre-service elementary teachers to teach mathematics with learning trajectories (Yayınlanmamış doktora tezi). North Carolina State University, Raleigh, NC. Pellegrino, J. W. (2009). The challenges of conceptualizing what low achievers know and how to assess their competence. İçinde M. Perie (Ed.), Considerations for the alternate assessment based on modified achievement standards (AAMAS): Understanding the eligible population and applying that knowledge to their instruction and assessment. New York, NY: New York Comprehensive Center. Philipp, R. (2008). Motivating prospective elementary school teachers to learn mathematics by focusing upon children’s mathematical thinking. Issues in Teacher Education, 17(2), 7-26. Powell, A. B., Francisco, J. M., & Maher, C. A. (2003). An analytical model for studying the development of learners' mathematical ideas and reasoning using videotape analysis. Journal of Mathematical Behavior, 22, 405-435. Ryan, J. & McCrae, B. (2006). Assessing pre-service teachers' mathematics subject knowledge. Mathematics Teacher Education and Development, 7, 72-89. Sawchuk, S. (2010). Full cost of professional development hidden. Education Week, 30(11), 14-16. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145. Sztajn, P., Confrey, J., Wilson, P. H. & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156. Steffe, L. & Thompson, P. (2000). Teaching experiment methodology: underlying principles and essential elements. In A. Kelly & R. Lesh (Editörler.), Handbook of Research Design in Mathematics and Science Education (s.267 – 306). Mahwah, NJ: Lawrence Erlbaum Associates. Steffe, L. P., & Ulrich, C. (2014). Constructivist teaching experiment. In Encyclopedia of mathematics education (s. 102-109). Springer Netherlands. Stein, M. K., & Smith, M. S. (2011). Five practices for orchestrating productive mathematics discussions. Reston, VA: Corwin. Toluk, Z. & Middleton, J. A. (2001). The development of children’s understanding of quotient: A teaching experiment. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of 25th Annual Meeting of the International Group for Psychology of Mathematics Education. Utrecht, The Netherlands: Hogrefe. Toluk-Uçar, Z. (2010). Sınıf Öğretmeni adaylarının matematiksel bilgileri ve öğretimsel açıklamaları. e-Journal of New World Science Academy, 5(3), 911-920. Wilson, P. H. (2009). Teachers’ uses of a learning trajectory for equipartitioning (Yayınlanmamış doktora tezi). North Carolina State University. Raleigh, NC. Wilson, P. H., Mojica, G. F., & Confrey, J. (2013). Learning trajectories in teacher education: Supporting teachers’ understanding of students’ mathematical thinking. Journal of Mathematical Behavior, 32, 103-121. Wilson, P. H., Sztajn, P., Edgington, C., & Confrey, J. (2014). Teachers’ use of their mathematical knowledge for teaching in learning a mathematics learning trajectory. Journal of Mathematics Teacher Education, 17(2), 149-175. Yilmaz, Z. (2011). Toward an understanding of students’ strategies on reallocation and covariation items: In relation to an equipartitioning learning trajectory (Yayınlanmamış yüksek lisans tezi). North Carolina State University, Raleigh, NC. Zembat, I, O. (2007). Working on the same problem – Concepts; with the usual subjects – Prospective Elementary Teachers. Elementary Education Online, 6(2), 305-312.
Toplam 1 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | Türkçe |
|---|---|
| Bölüm | Araştırma Makaleleri |
| Yazarlar | |
| Yayımlanma Tarihi | 15 Mart 2018 |
| Yayımlandığı Sayı | Yıl 2018 Cilt: 17 Sayı: 1 |