Teaching from Patterns to Functions in Algebraic Thinking Process
Year 2010,
Volume: 9 Issue: 1, 213 - 228, 26.06.2010
Tangül (uygur) Kabael
,
Dilek Tanışlı
Abstract
Function concept that mathematics educators has worked on importantly for years is still a source of
difficulties for students. Because the functional relationship is a prerequisite notion for the function concept and the functional
relationship should be gained from the outset, teaching of the function concept should be considered not in the high school
mathematics, but in the early grades mathematics. In this study, the relationship between the pattern and function concepts
and teaching strategies of them in the algebraic thinking process are investigated based on literature, and the results obtained
from this investigation are given by supporting with the researchers’ recommendations.
References
- Bakar, M. & Tall, D. (1991). Students’ mental prototypes for functions and graphs, Proceedings of PME 15, Assisi, 1, 104–111.
- Billstein, R., Libeskind S. & Lott J. W. (2004). A problem solving approach to mathematics for elementary school teachers. (8th Ed.) New York: Addison- Wesley.
- Blanton, M. L. & Kaput, J. (2004). “Elementary grades students’ capacity for functional thinking.” In M. J. Hoines ve A. Fuglestad (Ed.), Proceeding of The 28th Conference of the international Group for the Psychology of Mathematics Education, 2, 135-142. Bergen Norway: International Group For The Psychology of Mathematics Education.
- Breidenbach, D., Dubinsky, E.,Hawks, J.,& Nichols, D. (1992). Development of the process conception of function, Educational Studies in Mathematics, 23 (1992), 247-285.
- Cai, J., Lew, H. C., Morris, A. Moyer, J. C. Ng, S. F. & Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross- cultural comparative. ZDM. 37 (1), 5-15.
- Carraher, D. W. & Martinez, M. V. (2007). Early algebra and mathematical generalization. ZDM. 40 (3), 1-22.
- Cathcart, W. G., Pothier, V. M., Vance, T. H. & Bezuk, N. S. (2003). Learning mathematics in elementary and middle schools. (3th Ed.) River, N.J: Merrill/Prentice Hall.
- Clement, L. (2001). What do students really know about functions? The Mathematics Teacher, 94 (9), 745.
- Confrey, F. & Smith, E. (1991). A framework for functions: Prototypes, multiple representations and transformations. In R.G. Underhill (Ed.), Proceedings of the 13th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 57-63, Blacksburg: Virginia Polytechnic Institute and State University.
- Davidenko, S. (1999). Building the concept of function from students’ everyday activities. In B. Moses (Ed.), Algebraic Thinking Grade K-12 (140-145). National Council of Teachers of Mathematics Reston, Virginia.
- Driskol, M. & Moyer, J. (2001). Using students’ work as a lens on algebraic thinking. Mathematics Teaching In The Middle School, 6 (5), 283-287.
- Dubinsky, E. & Harel, G. (1992). The nature of the process conception of function, In G. Harel and E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA notes 25 (pp.85- 106). Mathematical association of America, Washington.
- English & Warren (1999). Introducing the variable through pattern exploration. In B. Moses (Ed.), Algebraic Thinking Grade K-12 (140-145). National Council of Teachers of Mathematics Reston, Virginia.
- Hargreaves, M., Shorrocks-Taylor, D. & Threlfall, J. (1998). Children’s strategies with number patterns. Educational Studies. 24(3), 315-331.
- Hargreaves, M., Shorrocks-Taylor, D. & Threlfall, J. (1999). Children’s strategies with number patterns. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (67-83). London and New York: Cassell.
- Hatfield, M. M., Edwards, N. T., Bitter, G. G., & Morrow, J. (2004). Mathematics methods for elementary and middle school teachers. (5th Ed.) USA: Wiley&Sons.
- Ley, A. F. (2005). A cross-sectional investigation of elementary school student’s ability to work with linear generalizing patterns: The impact of format and age on accuracy and strategy choice. Masters Abstract International, 44 (02), 124. (UMI No: AAT MR07303).
- Montiel,M., Vidakovic,D. & Kabael,T.(2008) Relationship between students’ understanding of functions in cartesian and polar coordinate systems. Focus on Learning Problems in Mathematics, 1(2), 2008.
- Mor, Y., Noss, R., Hoyles, C., Kahn, K. ve Simpson, G. (2006). Designing to see and share structure in number sequences. International Journal for Technology in Mathematics Education. 13(2), 65-78. (NCTM). (2000).
- Curriculum and evaluation standards for school mathematics.
- http://www.nctm.org/standards.htm adresinden 14.09.2005 tarihinde alHnmHJtHr.
- Orton, A. & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (104-120). London and New York: Cassell.
- Samsan, M. C., Linchevski, L. & Olivier, A. (1999). “The influence of different representations on children’s generalisation thinking processes.” Proceedings of the Seventh Annual Conference of the Southern African Association for research in Mathematics and Science Education. Harare, Zimbabwe. 406-415.
- Sheffield, L. J. & Cruikshank, D. E. (2005). Teaching and learning mathematics. Pre-kindergarten though middle school. (5th Ed.) New York : J. Wiley.
- Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity, Educational Studies in Mathematics, 12, 151– 169.
- Usiskin, Z. (1997). Doing algebra in grade K-4. Teaching Children Mathematics, 3, 346-356.
- Van De Walle, J. A. (2004). Elementary and middle school mathematics. (5th Ed.) Boston: Allyn and Bacon.
- Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.
- Vogel, R. (2005). Pattern- a fundamental idea of mathematical thinking and learning. ZDM. 37 (5), 445-449.
- Vollrath, H. J. (1986). “Search strategies as indicators of functional thinking.” Educational Studies in Mathematics. 17, 387-400.
- Zaskis, R. & Liljedahil, P. (2006). “On the path to number theory: Repeating patterns as a gateway.” In R. Zaskis & S. R. Campbell (Ed.), Number theory in mathematics education (99-114). London: Lawrence Erlbaum AssocHates, Publishers.
- Warren, E. & Cooper, T. (2005). Introducing functional thinking in year 2: a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6 (2), 150-162.
- Willoughby, S. S. (1999). Function from kindergarten through sixth grade. In B. Moses (Ed.), Algebraic Thinking Grade K-12 (140-145). National Council of Teachers of Mathematics Reston, Virginia.
Cebirsel Düşünme Sürecinde Örüntüden Fonksiyona Öğretim
Year 2010,
Volume: 9 Issue: 1, 213 - 228, 26.06.2010
Tangül (uygur) Kabael
,
Dilek Tanışlı
Abstract
Matematik eğitimcilerinin uzun yıllardır bir güçlük kaynağı olmaktan çıkamamıştır. Fonksiyonel ilişki, fonksiyon kavramının sahip önkoşul bilgisi dönem ve bu tanımında okulöncesi dönemden kalanı yapılacaktırılması, standart kavramının öğretiminin başlaması ortaöğretimde değil erken öğretim dönemlerinde düşünülmelidir. Fonksiyonel ilişki karakterlerin kullanımı ile başlamak değişkenler arası ilişki ile devam eder. Bu işçi, cebirsel ettirilmesi ile ilgili örüntü ve fonksiyon kavramları konusu ve bu kavramların öğretim stratejileri literatürdeki incelenmiş ve bu inceleme sonunda elde edilen sonuçlar ile ilgili öneriler önerileri ile de desteklenerek verilmiştir.
References
- Bakar, M. & Tall, D. (1991). Students’ mental prototypes for functions and graphs, Proceedings of PME 15, Assisi, 1, 104–111.
- Billstein, R., Libeskind S. & Lott J. W. (2004). A problem solving approach to mathematics for elementary school teachers. (8th Ed.) New York: Addison- Wesley.
- Blanton, M. L. & Kaput, J. (2004). “Elementary grades students’ capacity for functional thinking.” In M. J. Hoines ve A. Fuglestad (Ed.), Proceeding of The 28th Conference of the international Group for the Psychology of Mathematics Education, 2, 135-142. Bergen Norway: International Group For The Psychology of Mathematics Education.
- Breidenbach, D., Dubinsky, E.,Hawks, J.,& Nichols, D. (1992). Development of the process conception of function, Educational Studies in Mathematics, 23 (1992), 247-285.
- Cai, J., Lew, H. C., Morris, A. Moyer, J. C. Ng, S. F. & Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross- cultural comparative. ZDM. 37 (1), 5-15.
- Carraher, D. W. & Martinez, M. V. (2007). Early algebra and mathematical generalization. ZDM. 40 (3), 1-22.
- Cathcart, W. G., Pothier, V. M., Vance, T. H. & Bezuk, N. S. (2003). Learning mathematics in elementary and middle schools. (3th Ed.) River, N.J: Merrill/Prentice Hall.
- Clement, L. (2001). What do students really know about functions? The Mathematics Teacher, 94 (9), 745.
- Confrey, F. & Smith, E. (1991). A framework for functions: Prototypes, multiple representations and transformations. In R.G. Underhill (Ed.), Proceedings of the 13th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, 57-63, Blacksburg: Virginia Polytechnic Institute and State University.
- Davidenko, S. (1999). Building the concept of function from students’ everyday activities. In B. Moses (Ed.), Algebraic Thinking Grade K-12 (140-145). National Council of Teachers of Mathematics Reston, Virginia.
- Driskol, M. & Moyer, J. (2001). Using students’ work as a lens on algebraic thinking. Mathematics Teaching In The Middle School, 6 (5), 283-287.
- Dubinsky, E. & Harel, G. (1992). The nature of the process conception of function, In G. Harel and E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA notes 25 (pp.85- 106). Mathematical association of America, Washington.
- English & Warren (1999). Introducing the variable through pattern exploration. In B. Moses (Ed.), Algebraic Thinking Grade K-12 (140-145). National Council of Teachers of Mathematics Reston, Virginia.
- Hargreaves, M., Shorrocks-Taylor, D. & Threlfall, J. (1998). Children’s strategies with number patterns. Educational Studies. 24(3), 315-331.
- Hargreaves, M., Shorrocks-Taylor, D. & Threlfall, J. (1999). Children’s strategies with number patterns. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (67-83). London and New York: Cassell.
- Hatfield, M. M., Edwards, N. T., Bitter, G. G., & Morrow, J. (2004). Mathematics methods for elementary and middle school teachers. (5th Ed.) USA: Wiley&Sons.
- Ley, A. F. (2005). A cross-sectional investigation of elementary school student’s ability to work with linear generalizing patterns: The impact of format and age on accuracy and strategy choice. Masters Abstract International, 44 (02), 124. (UMI No: AAT MR07303).
- Montiel,M., Vidakovic,D. & Kabael,T.(2008) Relationship between students’ understanding of functions in cartesian and polar coordinate systems. Focus on Learning Problems in Mathematics, 1(2), 2008.
- Mor, Y., Noss, R., Hoyles, C., Kahn, K. ve Simpson, G. (2006). Designing to see and share structure in number sequences. International Journal for Technology in Mathematics Education. 13(2), 65-78. (NCTM). (2000).
- Curriculum and evaluation standards for school mathematics.
- http://www.nctm.org/standards.htm adresinden 14.09.2005 tarihinde alHnmHJtHr.
- Orton, A. & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the teaching and learning of mathematics (104-120). London and New York: Cassell.
- Samsan, M. C., Linchevski, L. & Olivier, A. (1999). “The influence of different representations on children’s generalisation thinking processes.” Proceedings of the Seventh Annual Conference of the Southern African Association for research in Mathematics and Science Education. Harare, Zimbabwe. 406-415.
- Sheffield, L. J. & Cruikshank, D. E. (2005). Teaching and learning mathematics. Pre-kindergarten though middle school. (5th Ed.) New York : J. Wiley.
- Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity, Educational Studies in Mathematics, 12, 151– 169.
- Usiskin, Z. (1997). Doing algebra in grade K-4. Teaching Children Mathematics, 3, 346-356.
- Van De Walle, J. A. (2004). Elementary and middle school mathematics. (5th Ed.) Boston: Allyn and Bacon.
- Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366.
- Vogel, R. (2005). Pattern- a fundamental idea of mathematical thinking and learning. ZDM. 37 (5), 445-449.
- Vollrath, H. J. (1986). “Search strategies as indicators of functional thinking.” Educational Studies in Mathematics. 17, 387-400.
- Zaskis, R. & Liljedahil, P. (2006). “On the path to number theory: Repeating patterns as a gateway.” In R. Zaskis & S. R. Campbell (Ed.), Number theory in mathematics education (99-114). London: Lawrence Erlbaum AssocHates, Publishers.
- Warren, E. & Cooper, T. (2005). Introducing functional thinking in year 2: a case study of early algebra teaching. Contemporary Issues in Early Childhood, 6 (2), 150-162.
- Willoughby, S. S. (1999). Function from kindergarten through sixth grade. In B. Moses (Ed.), Algebraic Thinking Grade K-12 (140-145). National Council of Teachers of Mathematics Reston, Virginia.