Analysing Mathematical Abilities of High School Graduates

Abstract

In this paper we are trying to differentiate the level of mathematical abilities of high school graduates in Bahrain schools. The mathematical abilities that we are trying to analyse are conceptual understanding, content knowledge and problemsolving skills. content understanding focusses on performing algorithms, while conceptual understanding focus on comprehending the concepts and relations. Problem solving needs both. We will prepare test items to measure graduates on different mathematics content domains. This research will try to identify students’ weaknesses in mathematical and suggest ways to improve the understanding in mathematics.


 


 


Keywords: Conceptual understanding; content knowledge; problem solving.

References
[1] Barnett-Clarke, C., Fisher, W., Marks, R., & Ross, S. (2010). Rational numbers: The big ideas and essential understandings. Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3-5. R. Charles & R.


[2] Canobi, K. H. & Bethune, N. E. (2008). Number words in young children’s conceptual and procedural knowledge of addition, subtraction and inversion. Cognition, 108, 675–686. doi: 10.1016/j.cognition.2008.05.011.


[3] Charlesworth, R. (2012). Experiences in math for young children. Belmont, CA: Wadsworth Cengage Learning.


[4] Desimone, L., Smith, T., Hayes, S. and Frisvold, D. (2005). Beyond Accountability and Average Mathematics Scores: Relating State Education Policy Attributes to Cognitive Achievement Domains. Educational Measurement: Issues and Practice, 24(4): 5–18.


[5] Durkin, K., Rittle-Johnson, B., & Star, J. R. (2011). Procedural flexibility matters for student achievement: how procedural flexibility relates to other outcomes. Paper presented at the 14th Biennial Conference of the European Association for Research on Learning and Instruction, August, Exeter.


[6] Dowker, A. (1998). Individual differences in normal arithmetical development. In C. Donlan (Ed.), The Development of Mathematical Skills (pp. 275–301). Hove: Psychology Press.


[7] Faulkenberry, E. (2003). Secondary mathematics preservice teachers’ conceptions of rational numbers. Unpublished Doctoral Dissertation, Oklahoma State University, Oklahoma.


[8] Garderen, D.V. (2006). Spatial Visualization, Visual Imaginary and Mathematical Problem Solving of Students with Varying Abilities. Journal of Learning Disabilities 39(6): 496– 506.


[9] Garnett, K. G. (1998). Maths Learning Disabilities. Journal of CEC. Retrieved on 3rd Feb 2009 from http://www.idonline.org/ld_indepth/math_skill/garnet.html


[10] Geary, D.C. (2004). Mathematical and Learning Disabilities. Journal of Learning Disabilities 37(1): 4– 15


[11] Ginsburg, D. (2012). Building Procedural Fluency and Conceptual Understanding in Mathematics. Edutopia.org: University of Pheonix.


[12] Hallett, D., Nunes, T., & Bryant, P. (2010). Individual differences in conceptual and procedural knowledge when learning fractions. Journal of Educational Psychology, 102, 395–406. doi: 10.1037/a0017486.


[13] Hasnida, C., Zakaria, E. (1991). Students’ Procedural and Conceptual Understanding of Mathematics, Australian Journal of Basic and Applied Sciences, 5(7): 684-691, 2011 ISSN 1991-8178.


[14] Hiebert, J. Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics:
An introductory analysis. In J. Hiebert (Ed), Conceptual and procedural knowledge: The case of mathematics (pp 1-27). Hillsdale, NJ: Erlbaum.


[15] Hiebert, J., Stigler, J., Jacobs, J., Givvin, K., Garnier, H., Smith, M., Hollingsworth, H., Manaster, A., Wearne, D. and Gallimore,R. (2005). Mathematics Teaching in the United States Today (and Tomorrow): Results From the TIMSS 1999 Video Study. Educational Evaluation and Policy Analysis, 27(2): 111-132.


[16] Huang, H.E, Witz, K.G (2012). Children’s Conceptions of Area Measurement and Their Strategies for Solving Area Measurement Problems, Journal of Curriculum and Teaching: ISSN 1927-2677 E-ISSN 1927-2685


[17] Jawhara Tak (1995). Problem solving and creative thinking in education. New York: Oxford University Press language.


[18] Jonassen. D. (2003). Learning to solve problems: an instructional design guide. San Francisco: Willey and Sons, Inc


[19] Jordan, J.-A., Mulhern, G., & Wylie, J. (2009). Individual differences in trajectories of arithmetical development in typically achieving 5- to 7-year-olds. Journal of Experimental Child Psychology, 103, 455–468. doi: 10.1016/j.jecp.2009.01.011.


[20] Kilpatrick, J., Swafford, J. & Findell, B. (2001). Adding It Up: Helping Children Learn Mathematics. Washington DC: National Academy Press.


[21] Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 179-192). Reston, VA: National Council of Teachers of Mathematics.


[22] Lehrer, R., Jaslow, L., & Curtis, C. L. (2003). Developing an understanding of measurement in the elementary grades.


[23] Lim, K. H., Foo, L. K., Nagarajah Lee, Mohd. Jamil M. Noor, & Murshidi Nooriza Kassim (1999). The ability of new teachers in non-routine the problem solve math. Collections of Research Science Teachers College in Bintulu. 57 to 87.


[24] Mack, N.K. (1999). Learning fractions with understanding: Building on informal knowledge. Journal for Research in Mathematics Education. 21 (1), 16-32. Retrieved from: http://www.jstor.org/stable/749454


[25] Mahir, N. (2009). Conceptual and Procedural Performance of Undergraduate students in Integration. International Journal of Mathematical Education in Science and Technology, 40(2): 201-211.


[26] McGehee, J. (1990). Prospective secondary teachers’ knowledge of the function concept. Unpublished Doctoral Dissertation, University of Texas.


[27] Mayer, R. E. (2008). Learning and instruction (2nd ed.). Upper Saddle River, NJ: Pearson Education


[28] McGregor, D. (2007). Developing thinking, developing learning. A guide to thinking skills in education. Berkshire, England: Open University Press


[29] Mellony, G. & Stott, D. (2012). Conceptualising Procedural Fluency as a Spectum of Proficiency. Rhodes University, South Africa.


[30] Nathan V., Lauren, Sarah. L, Adam & Nathan, S. (2002). Difficulties with Maths: What Can Stand in the Way of a Students’ Mathematical Development. Misunderstood Minds. Retrieved on 4th Sept 2009 from http://www.misunderstoodmind/ math_skill


[31] National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.


[32] www.nctm.org.


[33] National Council of Teachers of Mathematics. (2009). Alternate forms of assessment. www.nctm.org.


[34] NAEP (2003). What Does the NAEP Mathematics Assessment Measure? Retrieved from nces.ed.gov/nationsreportcard/mathematics/abilities.asp


[35] Polya, G. (1981). Mathematical Discovery on Understanding Learning and Teaching Problem Solving. NewYork: John Wiley and sons.


[36] Pugalee, D.K. (2004). A comparison of verbal and written descriptions of students’ problem solving processes. Educational Studies in Mathematics: An International Journal, 55, 27-47. http://dx.doi.org/10.1023/B:EDUC.0000017666.11367.c7


[37] Pugalee, D.K. (2005). Writing to develop mathematical understanding (2005). Massachusetts: Christopher Gordon.


[38] Reimer, K. & Moyer, P. S. (2005). Third-graders learn about fractions using virtual manipulates: a classroom study. Journal of Computers in Mathematics and Science Teaching, 24, 5–25.


[39] Siegler, R. S., & Alibali, M. W. (2005). Children’s thinking. J. Gilliland (Ed.). (4th ed. pp. 268-304). Upper Saddle River, NJ: Pearson.


[40] Star, J. R. & Rittle-Johnson, B. (2008). Flexibility in problem solving: the case of equation solving. Learning and Instruction, 18, 565–579. doi: 10.1016/j.learninstruc.2007.09.018


[41] Star, J. R. & Rittle-Johnson, B. (2009). It pays to compare: an experimental study on computational estimation. Journal of Experimental Child Psychology, 101, 408–426. doi: 10.1016/j.jecp.2008.11.004.


[42] Stendall, R. (2009). Use Your Six Intellectual Factors to Achieve Anything in Your Life. Article-idea. Retrieved on 4th Sept2009 from http://www.article-idea.com/ profile/raystendall


[43] Suh, J.M. & Moyer-Packenham, P.S. (2007). Developing Students’ Representative Fluency using Virtual and Physical Algebra Balances. Journal of Computers in Mathematics and Science Teaching, 26(2), 155-173.


[44] Tay Lay Heong (2005). Problem Solving Abilities and Strategies in Solving Multistep Mathematical Problems among Form 2 Students. Kertas Projek Sarjana. Universiti Malaya.


[45] Van de Walle, John. (2007). Elementary and Middle School Mathematics: Teaching Developmentally. 5th ed. Boston: Pearson


[46] Verschaffel, L., Luwel, K., Torbeyns, J., & Van Dooren, W. (2009). Conceptualizing, investigating, and enhancing adaptive expertise in elementary mathematics education. European Journal of Psychology of Education, 24, 335–359. doi: 10.1007/bf03174765.


[47] Watanabe, T. (2006). The teaching and learning of fractions: A Japanese perspective. Teaching Children Mathematics. 12(7), 368-374


[48] Zbiek (Eds.) (pp. 10-56). Reston, VA: The National Council of Teachers of Mathematics, Inc.


[49] National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. https://doi.org/10.17226/9822.


[50] National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. www.nctm.org.


[51] Smith J. P., (1993). Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition,The Journal of the Learning Sciences, Vol. 3, No. 2. (1993 - 1994), pp. 115-163.


[52] Spooner, (2002). Errors and Misconceptions in Maths at Key Stage 2: Working Towards Successful SATS
•Cummings, Kelsey. (2015). How Does Tutoring to Develop Conceptual Understanding Impact Student Understanding?. In BSU Honors Program Theses and Projects. Item 96. Available at: http://vc.bridgew.edu/honors_proj/96
•Hasnida, C.,& Zakaria, E.,(1991). Students’ Procedural and Conceptual Understanding of Mathematics, Australian Journal of Basic and Applied Sciences, 5(7): 684-691, 2011 ISSN 1991-8178.
•Hull, B & Miles, H. ”What is Conceptual Understanding?”. Available at: http://www.mathleadership.com/sitebuildercontent/sitebuilderfiles/ conceptualUnderstanding.pdf (accessed 12 September 2017). •Jazuli, A., Setyosari, P., Sulthon, Kuswandi, D. (2017) Improving conceptual understanding and problem-solving in mathematics through a contextual learning strategy, Global Journal of Engineering Education, 19 (1), pp. 49-53.
•Johnson, R. B., & Alibali, M. W. (1999). Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91, 175-189.
•National Council of Teachers of Mathematics.,(2000). Principles and standards for school mathematics. Reston, VA: Author. www.nctm.org.
•National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. https://doi.org/10.17226/9822.
•Smith J. P., (1993). Misconceptions Reconceived: A Constructivist Analysis of Knowledge in Transition, The Journal of the Learning Sciences, Vol. 3, No. 2. (1993 - 1994), pp. 115-163.
•Spooner, (2002). Errors and Misconceptions in Maths at Key Stage 2: Working Towards Successful SATS., David Fulton Publishers.
•Wood, David J., (1998). How Children Think and Learn: 2nd Edition, Blackwell Publishing.