Volume 11 Issue 10

(2016)

Volume 11 Issue 9

(2016)

Volume 11 Issue 8

(2016)

Volume 11 Issue 7

(2016)

Volume 11 Issue 6

(2016)

Volume 11 Issue 5

(2016)

Volume 11 Issue 4

(2016)

Volume 11 Issue 3

(2016)

Volume 11 Issue 2

(2016)

Volume 11 Issue 1

(2016)

pp. **3160-3175** | Article Number: **iejme.2016.258**

Published Online: **September 28, 2016**

Article Views: **462** | Article Download: **397**

**Abstract**

**Determining the kind of teachers’ mathematical knowledge for teaching a topic of mathematics at elementary school is substantial most notaby for developing teacher training. The important purpose of this study was to propose a model by applying Delphi method for identifying the horizon mathematical knowledge for teaching fraction division at elementary schools in Indonesian context. The results were the procedure for identification horizon mathematical knowledge and eight statements of competences in mathematical knowledge for teaching fraction division at elementary schools. The recommendations based on the results were the model developed in this study which could be one of the many alternatives for identifying the horizon mathematical knowledge for teaching, and the results should be followed by an empirical validation process to analyze the ultimate effects on students’ learning**

**Keywords: ** Horizon mathematical knowledge for teaching, delphi method, teaching fraction division, elementary schools

**References**

Ball, D. L., Bass, H., Sleep, L., & Thames, M. (2005, May). A Theory of Mathematical Knowledge for Teaching. Paper presented at the The Fifteenth ICMI Study: The Professional Education and Development of Teachers of Mathematics, State University of Sao Paolo at Rio Claro, Brazil. Retrieved from http://stwww.weizmann.ac.il/G-math/ICMI/log_in.html

Bourgeois, J., Pugmire, L., Stevenson, K., Swanson, N., & Swanson., B. (2006). The Delphi Method: A qualitative means to a better future. Retrieved from: http://www.freequality.org/documents/knowledge/Delphimethod.pdf

Charalambous Y. C. (2016). Investigating the knowledge needed for teaching mathematics: An exploratory validation study focusing on teaching practices. Journal of Teacher Education, 67(3), pp. 220-237.

Coddington, L.R. (2014). An Investigation of Teachers’ Noticing, Cognitive Demand, and Mathematical Knowledge for Teaching: Video Reflections in an Elementary Mathematics Context. *CGU Theses & Dissertations.* Paper 88. Retrieved from: http://scholarship.claremont.edu/cgu_etd/88

Delaney, F.D. (2008). *Adapting and using U.S. measures to study Irish teachers’ mathematical knowledge for teaching* (Doctoral Dissertation). Retrieved from: https://deepblue.lib.umich.edu/bitstream/handle/2027.42/60756/sdelaney_1.pdf%3Bsequence=1.

Demir, I. & Kilic, S. (2010). Using PISA 2003, examining the factors affecting students’ mathematics achievement. *H. U. Journal of Education) 38*:44-54. Retrieved from: http://www.efdergi.hacettepe.edu.tr/english/abstracts/38/pdf/%C4%B0BRAH%C4%B0M%20DEM%C4%B0R.pdf

Flake, M.W. (2014). *An investigation of how preservice teachers’ ability to professionally notice children’s mathematical thinking relates to their own mathematical knowledge for teaching *(Doctoral Dissertation). Retrieved from: http://kuscholarworks.ku.edu/bitstream/handle/1808/15149/Flake_ku_0099D_13273_DATA_1.pdf?sequence=1&isAllowed=y

Galant. J. (2013). “Selecting and sequencing mathematics tasks: Seeking mathematical knowledge for teaching”. *Perspectives in Education, 31*(3).

Gencturk, Y.C. (2012). *Teachers' mathematical knowledge for teaching, instructional practices, and student outcomes* (Doctoral Dissertation). Retrieved from: https://www.ideals.illinois.edu/bitstream/handle/2142/31173/CopurGencturk_Yasemin.pdf?sequence=1

Heiko A. G. (2102. May). Consensus measurement in Delphi Sudies: Review and implication for future quality assurance. *Technological Forecasting & Social Change*, *79*(8), pp.1525-1536.

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), pp. 371-406.

Jakobsen, A., Thames, M. H., Ribeiro, C. M., & Delaney, S. (2012). Using practice to define and distinguish horizon content knowledge. In *Proceeding of the 12th International Congress on Mathematics Education*, 8th-15th July, 2012 (pp. 4635 – 4644), COEX, Seoul, Korea.

Schooneveld, J.G. (2014). *The relationship between beginning elementary school teachers' mathematical knowledge or teaching and their instructional design practices* (Doctoral Dissertation). Retrieved from: http://repository.upenn.edu/dissertations/AAI3622674

Kalaian, S. & Rafa, M.K. (2012). Terminating Sequential Delphi Survey Data Collection. *Practical Assessment, Research & Evaluation*, *17*(5). Retrieved from: http://pareonline.net/getvn.asp?v=17&n=5

Lo, J., J. & Luo, F. (2012). Prospective Elementary Teachers’ Knowledge of Fraction Division. *J. Math Teacher Educ,5*:481-500.

Metzler, J., &Woessmann, L. (2010). *The impact of teacher subject knowledge on student achievement: Evidence from within-teacher within-student variation*. IZA Discussion Paper No. 4999 June 2010. Retrieved from: http://ftp.iza.org/dp4999.pdf

Mosvold, R. & Fauskanger, J. (2014). Teachers' beliefs about mathematical horizon content knowledge. International Journal for Mathematics Teaching and Learning, September 25th, pp. 1-16.

Olanoff, D.E. (2011). *Mathematical Knowledge for Teaching Teachers: The Case of Multiplication and Division of Fractions *(Doctoral Dissertation). Retrieved from: http://surface.syr.edu/cgi/viewcontent.cgi?article=1063&context=mat_etd

Hanafin, S. (2004, March). *Review of literature on the Delphi Technique*. Retrieved from: www.dcya.gov.ie.

Shadiq, F. (2013). Peran penting guru matematika dalam mencerdaskan siswanya. *Artikel Pendidikan*. Retrieved from: http://p4tkmatematika.org/2013/04/peran-penting-guru-matematika-dalam-mencerdaskan-siswanya/.

Sugilar. (2015, Maret). *Analisis isi kandungan pengetahuan matematika untuk mengajar dalam bahan ajar S-1 PGSD*. Paper presented at the Seminar Nasional Matematika dan Pendidikan Matematika, STKIP Sumatera Barat, Padang, Indonesia.

Winklbauer,W. T. (2014). Identifying technical competencies: A Delphi Approach. *Asian Journal of Engineering and Technology*, *2*(2), pp. 83-97.