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Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions

Year 2021, Volume 9, Issue 4, 283 - 297, 15.12.2021
https://doi.org/10.17478/jegys.1000495

Abstract

This article report on two case studies in which we explored two Grade 6 teachers’ mathematical knowledge for teaching the concept of fraction. We were interested in the mathematical knowledge teachers need to know, and how to use, to teach the concept of fraction. Of the two teachers sampled, one teacher (Rose) tried to unpack the fraction concept of fraction emphasising the understanding of mathematical concepts through using various modes of representation in teaching the concept of fraction and uses fractional manipulatives to emphasise the understanding of the concept. On the other hand, the teacher (Eddy) emphasized procedural knowledge. Eddy used the traditional way of teaching fractions, encouraging learners to memorise rules without understanding. The learners were just blind followers of the rules because they did not tell them where the rules originated, while Rose tried to use various modes of representation in teaching the concept of fraction

References

  • Adler, J., & Davis, Z. (2006). Opening another black box: Research mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education, 37(4), 270-296. https://doi.org/10.2307/30034851
  • Bassarear, T., & Moss, M. (2016). Mathematics for elementary school teachers (6th ed.). Cencage.
  • Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14-17, 20-22, 43-46.
  • Ball, D. L., Thames, M.H.& Phelps, G.(2008). Content Knowledge for teaching. Journal of teacher education,59(5),389-407.
  • Bansilal, S., Mkhwanazi., & Brijlall, D. (2014). An exploration of the common content knowledge of High School teachers. Perspectives in Education,32(1),34-50.
  • Blaise, M. (2011). Teachers theory-making. In G. Latham, M. Blaise, S. Dole, J. Faulkner & K. Malone (Eds.), Learning to teach: New times, new practices (pp. 105-157). Oxford University Press.
  • Carnoy, M., Chisholm, L., & Chilisa, B. (2012). The low achievement trap: Comparing schooling in Botswana and South Africa. HSRC Press.
  • Cazden, C. B. (2001). The language of teaching and learning. Heinemann.
  • Chisholm, L., & Leyendecker, R. (2008). Curriculum reform in post-1990s sub-Saharan Africa. International Journal of Educational Development, 28(2), 195-205. https://doi.org/10.1016/j.ijedudev.2007.04.003
  • Cramer, K., & Wyberg, T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning, 11(4), 226-257. https://doi.org/10.1080/10986060903246479
  • Cramer, K., & Henry, A. (2002). Using manipulative models to build number sense for the addition of fractions. In B. Litwiller & G. Bright. (Eds.). Making sense of fractions, ratios and proportions: 2002 yearbook, 6(16), 41-48. Whitney.
  • Cramer, K., Monson, D., Whitney, S., Leavitt, S., & Wyberg, T. (2010). Dividing fractions and problem-solving. Mathematics Teaching in the Middle School, 15(6), 338-346. https://doi.org/10.5951/mtms.15.6.0338
  • Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School, 13(8), 490. https://doi.org/10.5951/mtms.13.8.0490
  • Davis, B.(1997)Listening for differences: An evolving conception of mathematics teaching: Journal for research in mathematics Education.28(3),355-376.
  • Department of Basic Education. (2011). Curriculum and Assessment (CAPS): Foundation Phase Mathematics, Grade R-3. https://www.education.gov.za/Portals/0/CD/National%20Curriculum%20Statements%20and%20Vocational/CAPS%20MATHS%20%20ENGLISH%20GR%201-3%20FS.pdf?ver=2015-01-27-160947-800
  • DBE (Department of Basic Education) (2011) Report on the Annual National Assessment of 2011.Pretoria: DBE
  • DBE (Department of Basic Education) (2012) Report on the Annual National Assessment of 2012. Pretoria: DBE
  • Ernest, P. (1996). Varieties of constructivism: A framework for comparison. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 335-350). Routledge.
  • Essien, A. A. (2009). An analysis of the introduction of the equal sign in three Grade 1 textbooks. Pythagoras, 2009(69), 28-35. https://doi.org/10.4102/pythagoras.v0i69.43
  • Fazio, L., & Siegler, R. (2011). Teaching fractions. Educational practices series, International Academy of Education-International Bureau of Education. http://unesdoc.unesco.org/images/0021/002127/212781e.pdf
  • Fleisch, B. (2008). Primary Education in Crisis: Why South African schoolchildren underachieve in reading and mathematics. Juta and Company Ltd.
  • Gould, P. (2005).Year 6 students’ method of comparing the size of fractions. In Proceedings of the Annual Conference of the Mathematics Education Research Group of Australia (pp.393-400). MERGA.
  • Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251-283. https://doi.org/10.1207/s1532690xci1403_1
  • Humphreys, C., & Parker, R. (2015). Making number talks matter: Developing mathematical practices and deepening understanding, Grades 4-10. Stenhouse Publishers.
  • Kazima, M., Pillay, V., & Adler, J. (2008). Mathematics for teaching: observations from two case studies. South African Journal of Education, 28(2), 283-299.
  • Kong, S. C. (2008). The development of a cognitive tool for teaching and learning fractions in the mathematics classroom: A design-based study. Computers & Education, 51(2), 886-899. https://doi.org/10.1016/j.compedu.2007.09.007
  • Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 170-193.
  • Lamon, S. J. (2002). Part-whole comparisons with unitizing. In B. Litwiller (Ed.) Making sense of fractions, ratios, and proportions. (pp. 79-86). National Council of Teachers of Mathematics.
  • Machaba, F. M. (2017). Pedagogical demands in mathematics and mathematics literacy. A case of mathematics and mathematics literacy teachers and facilitators. Eurasia Journal of Science and Technology Education, 14(1),95-108.
  • Machaba, M. M. (2018). Teaching mathematics in Grade 3. International Journal of Olivier, A. (1989). Handling pupils’ misconceptions. Pythagoras, 21, 9–19.
  • Olivier, A. (1989). Handling pupils’ misconceptions. Pythagoras, 21, 9–19. Pantziara, M., & Philippou, G. (2012). Levels of students’ ‘conception’ of fractions. Educational Studies in Mathematics, 79(1), 61–83
  • Petit, M. M., Laird, R., & Marsden, E. (2010). Informing practice: They ―Get fractions as pies; Now what? Mathematics Teaching in the middle school, 16(1), 5-10. https://doi.org/10.5951/MTMS.16.1.0005
  • Piaget, J. (1964). Development and learning. In R. E. Ripple & V. N. Rockcastle (Eds.). Piaget Rediscovered (pp. 7 – 20). Ithaca.
  • Pienaar, E. (2014). Learning about and understanding fractions and their role in High School. Curriculum. http://hdl.handle.net/10019.1/86269
  • Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
  • Sarwadi, H. R. H., & Shahrill, M. (2014). Understanding students’ mathematical errors and misconceptions: The case of year 11 repeating students. Mathematics Education Trends and Research, 2014, 1-10. https://doi.org/10.5899/2014/metr-00051
  • Schunk, D. H. (2000). Learning theories. An educational perspective (3rd ed.). Merrill.
  • Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., & Wray, J (2010). Developing effective fractions instruction: A practice guide. National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, 54 Eichhorn U.S. Department of Education. (NCEE #2010–009). https://files.eric.ed.gov/fulltext/ED512043.pdf
  • Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 5(1), 1-23.
  • Shulman, L. S. (1986). Those who understand: Knowledge in teaching. Educational Research, 15(2), 4-14.
  • Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
  • Skemp, R. R. (1977). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-6. https://doi.org/10.5951/at.26.3.0009
  • Sowder, J. T., & Wearne, D. (2006). What do we know about eighth-grade achievement? Mathematics Teaching in the Middle School, 11(6), 285–293. https://doi.org/10.5951/mtms.11.6.0285
  • Stohlmann, M., Cramer, K., Moore, T., & Maiorca, C. (2013). Changing pre-service elementary. teachers' beliefs about mathematical knowledge. Mathematics Teacher Education and Development, 16(2), 4-24.
  • Tall, D. (1989). Concept images, generic organizers, computers & curriculum change. For the Learning of Mathematics, 9(3), 37-42.
  • Taylor, N. & Vinjevold, P. (1999). Teaching and learning in South African schools. Getting Learning Right. Report of the President’s Education Initiative Research Project, 131-162.
  • Van de Walle, J. A., (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Pearson Education.
  • Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching developmentally. Addison-Wesley Longman, Inc.
  • Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centred mathematics: Grade K-3. Pearson/Allyn and Bacon.
  • Van de Walle, J. A., Karp, K. S., & Bay, W. (2010). Developing fractions concepts. In J. A. Van de Walle & A. John (Eds.). Elementary and middle school mathematics. Teaching developmentally, (pp. 286-308). Addison-Wesley Longman, Inc.
  • Van de Walle, J. A. (2013). Elementary and middle school mathematics: teaching developmentally (8th ed.). Pearson Education
  • Van de Walle, J. A. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Pearson Education

Year 2021, Volume 9, Issue 4, 283 - 297, 15.12.2021
https://doi.org/10.17478/jegys.1000495

Abstract

References

  • Adler, J., & Davis, Z. (2006). Opening another black box: Research mathematics for teaching in mathematics teacher education. Journal for Research in Mathematics Education, 37(4), 270-296. https://doi.org/10.2307/30034851
  • Bassarear, T., & Moss, M. (2016). Mathematics for elementary school teachers (6th ed.). Cencage.
  • Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 29(1), 14-17, 20-22, 43-46.
  • Ball, D. L., Thames, M.H.& Phelps, G.(2008). Content Knowledge for teaching. Journal of teacher education,59(5),389-407.
  • Bansilal, S., Mkhwanazi., & Brijlall, D. (2014). An exploration of the common content knowledge of High School teachers. Perspectives in Education,32(1),34-50.
  • Blaise, M. (2011). Teachers theory-making. In G. Latham, M. Blaise, S. Dole, J. Faulkner & K. Malone (Eds.), Learning to teach: New times, new practices (pp. 105-157). Oxford University Press.
  • Carnoy, M., Chisholm, L., & Chilisa, B. (2012). The low achievement trap: Comparing schooling in Botswana and South Africa. HSRC Press.
  • Cazden, C. B. (2001). The language of teaching and learning. Heinemann.
  • Chisholm, L., & Leyendecker, R. (2008). Curriculum reform in post-1990s sub-Saharan Africa. International Journal of Educational Development, 28(2), 195-205. https://doi.org/10.1016/j.ijedudev.2007.04.003
  • Cramer, K., & Wyberg, T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning, 11(4), 226-257. https://doi.org/10.1080/10986060903246479
  • Cramer, K., & Henry, A. (2002). Using manipulative models to build number sense for the addition of fractions. In B. Litwiller & G. Bright. (Eds.). Making sense of fractions, ratios and proportions: 2002 yearbook, 6(16), 41-48. Whitney.
  • Cramer, K., Monson, D., Whitney, S., Leavitt, S., & Wyberg, T. (2010). Dividing fractions and problem-solving. Mathematics Teaching in the Middle School, 15(6), 338-346. https://doi.org/10.5951/mtms.15.6.0338
  • Cramer, K., Wyberg, T., & Leavitt, S. (2008). The role of representations in fraction addition and subtraction. Mathematics Teaching in the Middle School, 13(8), 490. https://doi.org/10.5951/mtms.13.8.0490
  • Davis, B.(1997)Listening for differences: An evolving conception of mathematics teaching: Journal for research in mathematics Education.28(3),355-376.
  • Department of Basic Education. (2011). Curriculum and Assessment (CAPS): Foundation Phase Mathematics, Grade R-3. https://www.education.gov.za/Portals/0/CD/National%20Curriculum%20Statements%20and%20Vocational/CAPS%20MATHS%20%20ENGLISH%20GR%201-3%20FS.pdf?ver=2015-01-27-160947-800
  • DBE (Department of Basic Education) (2011) Report on the Annual National Assessment of 2011.Pretoria: DBE
  • DBE (Department of Basic Education) (2012) Report on the Annual National Assessment of 2012. Pretoria: DBE
  • Ernest, P. (1996). Varieties of constructivism: A framework for comparison. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 335-350). Routledge.
  • Essien, A. A. (2009). An analysis of the introduction of the equal sign in three Grade 1 textbooks. Pythagoras, 2009(69), 28-35. https://doi.org/10.4102/pythagoras.v0i69.43
  • Fazio, L., & Siegler, R. (2011). Teaching fractions. Educational practices series, International Academy of Education-International Bureau of Education. http://unesdoc.unesco.org/images/0021/002127/212781e.pdf
  • Fleisch, B. (2008). Primary Education in Crisis: Why South African schoolchildren underachieve in reading and mathematics. Juta and Company Ltd.
  • Gould, P. (2005).Year 6 students’ method of comparing the size of fractions. In Proceedings of the Annual Conference of the Mathematics Education Research Group of Australia (pp.393-400). MERGA.
  • Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251-283. https://doi.org/10.1207/s1532690xci1403_1
  • Humphreys, C., & Parker, R. (2015). Making number talks matter: Developing mathematical practices and deepening understanding, Grades 4-10. Stenhouse Publishers.
  • Kazima, M., Pillay, V., & Adler, J. (2008). Mathematics for teaching: observations from two case studies. South African Journal of Education, 28(2), 283-299.
  • Kong, S. C. (2008). The development of a cognitive tool for teaching and learning fractions in the mathematics classroom: A design-based study. Computers & Education, 51(2), 886-899. https://doi.org/10.1016/j.compedu.2007.09.007
  • Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 170-193.
  • Lamon, S. J. (2002). Part-whole comparisons with unitizing. In B. Litwiller (Ed.) Making sense of fractions, ratios, and proportions. (pp. 79-86). National Council of Teachers of Mathematics.
  • Machaba, F. M. (2017). Pedagogical demands in mathematics and mathematics literacy. A case of mathematics and mathematics literacy teachers and facilitators. Eurasia Journal of Science and Technology Education, 14(1),95-108.
  • Machaba, M. M. (2018). Teaching mathematics in Grade 3. International Journal of Olivier, A. (1989). Handling pupils’ misconceptions. Pythagoras, 21, 9–19.
  • Olivier, A. (1989). Handling pupils’ misconceptions. Pythagoras, 21, 9–19. Pantziara, M., & Philippou, G. (2012). Levels of students’ ‘conception’ of fractions. Educational Studies in Mathematics, 79(1), 61–83
  • Petit, M. M., Laird, R., & Marsden, E. (2010). Informing practice: They ―Get fractions as pies; Now what? Mathematics Teaching in the middle school, 16(1), 5-10. https://doi.org/10.5951/MTMS.16.1.0005
  • Piaget, J. (1964). Development and learning. In R. E. Ripple & V. N. Rockcastle (Eds.). Piaget Rediscovered (pp. 7 – 20). Ithaca.
  • Pienaar, E. (2014). Learning about and understanding fractions and their role in High School. Curriculum. http://hdl.handle.net/10019.1/86269
  • Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
  • Sarwadi, H. R. H., & Shahrill, M. (2014). Understanding students’ mathematical errors and misconceptions: The case of year 11 repeating students. Mathematics Education Trends and Research, 2014, 1-10. https://doi.org/10.5899/2014/metr-00051
  • Schunk, D. H. (2000). Learning theories. An educational perspective (3rd ed.). Merrill.
  • Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., & Wray, J (2010). Developing effective fractions instruction: A practice guide. National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, 54 Eichhorn U.S. Department of Education. (NCEE #2010–009). https://files.eric.ed.gov/fulltext/ED512043.pdf
  • Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 5(1), 1-23.
  • Shulman, L. S. (1986). Those who understand: Knowledge in teaching. Educational Research, 15(2), 4-14.
  • Siebert, D., & Gaskin, N. (2006). Creating, naming, and justifying fractions. Teaching Children Mathematics, 12(8), 394-400.
  • Skemp, R. R. (1977). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-6. https://doi.org/10.5951/at.26.3.0009
  • Sowder, J. T., & Wearne, D. (2006). What do we know about eighth-grade achievement? Mathematics Teaching in the Middle School, 11(6), 285–293. https://doi.org/10.5951/mtms.11.6.0285
  • Stohlmann, M., Cramer, K., Moore, T., & Maiorca, C. (2013). Changing pre-service elementary. teachers' beliefs about mathematical knowledge. Mathematics Teacher Education and Development, 16(2), 4-24.
  • Tall, D. (1989). Concept images, generic organizers, computers & curriculum change. For the Learning of Mathematics, 9(3), 37-42.
  • Taylor, N. & Vinjevold, P. (1999). Teaching and learning in South African schools. Getting Learning Right. Report of the President’s Education Initiative Research Project, 131-162.
  • Van de Walle, J. A., (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Pearson Education.
  • Van de Walle, J. A. (2007). Elementary and middle school mathematics: Teaching developmentally. Addison-Wesley Longman, Inc.
  • Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centred mathematics: Grade K-3. Pearson/Allyn and Bacon.
  • Van de Walle, J. A., Karp, K. S., & Bay, W. (2010). Developing fractions concepts. In J. A. Van de Walle & A. John (Eds.). Elementary and middle school mathematics. Teaching developmentally, (pp. 286-308). Addison-Wesley Longman, Inc.
  • Van de Walle, J. A. (2013). Elementary and middle school mathematics: teaching developmentally (8th ed.). Pearson Education
  • Van de Walle, J. A. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). Pearson Education

Details

Primary Language English
Subjects Education and Educational Research
Journal Section Differentiated Instruction
Authors

France MACHABA (Primary Author)
University of South Africa
0000-0003-1318-3777
South Africa


Margaret MOLOTO
University of South Africa
0000-0002-8920-045X
South Africa

Publication Date December 15, 2021
Published in Issue Year 2021, Volume 9, Issue 4

Cite

Bibtex @research article { jegys1000495, journal = {Journal for the Education of Gifted Young Scientists}, issn = {}, eissn = {2149-360X}, address = {editorjegys@gmail.com}, publisher = {Genç Bilge Yayıncılık}, year = {2021}, volume = {9}, pages = {283 - 297}, doi = {10.17478/jegys.1000495}, title = {Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions}, key = {cite}, author = {Machaba, France and Moloto, Margaret} }
APA Machaba, F. & Moloto, M. (2021). Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions . Journal for the Education of Gifted Young Scientists , 9 (4) , 283-297 . DOI: 10.17478/jegys.1000495
MLA Machaba, F. , Moloto, M. "Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions" . Journal for the Education of Gifted Young Scientists 9 (2021 ): 283-297 <https://dergipark.org.tr/en/pub/jegys/issue/65384/1000495>
Chicago Machaba, F. , Moloto, M. "Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions". Journal for the Education of Gifted Young Scientists 9 (2021 ): 283-297
RIS TY - JOUR T1 - Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions AU - France Machaba , Margaret Moloto Y1 - 2021 PY - 2021 N1 - doi: 10.17478/jegys.1000495 DO - 10.17478/jegys.1000495 T2 - Journal for the Education of Gifted Young Scientists JF - Journal JO - JOR SP - 283 EP - 297 VL - 9 IS - 4 SN - -2149-360X M3 - doi: 10.17478/jegys.1000495 UR - https://doi.org/10.17478/jegys.1000495 Y2 - 2021 ER -
EndNote %0 Journal for the Education of Gifted Young Scientists Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions %A France Machaba , Margaret Moloto %T Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions %D 2021 %J Journal for the Education of Gifted Young Scientists %P -2149-360X %V 9 %N 4 %R doi: 10.17478/jegys.1000495 %U 10.17478/jegys.1000495
ISNAD Machaba, France , Moloto, Margaret . "Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions". Journal for the Education of Gifted Young Scientists 9 / 4 (December 2021): 283-297 . https://doi.org/10.17478/jegys.1000495
AMA Machaba F. , Moloto M. Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions. JEGYS. 2021; 9(4): 283-297.
Vancouver Machaba F. , Moloto M. Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions. Journal for the Education of Gifted Young Scientists. 2021; 9(4): 283-297.
IEEE F. Machaba and M. Moloto , "Grade 6 teachers’ s mathematical Knowledge for teaching: The concept of fractions", Journal for the Education of Gifted Young Scientists, vol. 9, no. 4, pp. 283-297, Dec. 2021, doi:10.17478/jegys.1000495

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