Project Report

Widespread student failure of meeting algebra standards in high schools in the U.S. has illuminated problematic teaching strategies (i.e., introducing algebra late and as an isolated course) and caused an increased desire to provide opportunities to engage in algebraic reasoning in the elementary grades (Carpenter, Levi, Berman, & Pligge, 2005). Further, depriving students of early opportunities to build a foundation of algebra skills can make learning algebra in middle and high schools even more difficult (U.S. Department of Education, 2008). Considering Singapore statistically significantly outperformed the U.S. in fourth grade mathematics on the 2007 International Mathematics and Science Study assessment (TIMSS; Mullis, Martin, & Foy, 2008), it seems the U.S. would greatly benefit from learning from Singapore’s successes. In an effort to further examine algebraic reasoning skills, the following research questions were addressed in collaboration with the National Institute of Education at Nanyang Technological University in Singapore: What successful teaching strategies might elementary school teachers in the U.S. learn from elementary school teachers in Singapore regarding teaching algebraic reasoning skills? What algebraic reasoning skills do students in Singapore possess? These research questions were pursued through a mixed-methods research design, in which both qualitative and quantitative techniques were employed. Four elementary schools in Singapore participated in this research. The research activities included interviewing 14 teachers, observing 36 different mathematics lessons, and assessing the algebraic reasoning skills of 1,619 students. The numbers of first, second, third, fourth, fifth, and sixth grade students assessed were 248, 224, 436, 324, 297, and 90, respectively. Algebraic reasoning skills assessed included open number sentences (i.e., 8+__=15), using variables (i.e., 6+b=9), equivalence (i.e., 8+4=__+5), linear patterns (i.e., 4, 8, 12, __), and nonlinear patterns (i.e., 2, 3, 5, 8, __). Human Subjects approval from the University of Washington, ethics approval from Nanyang Technological University, and clearance from the Ministry of Education in Singapore was obtained prior to commencing the research procedures. Several important findings were discovered. First, when asked about their teaching of algebraic reasoning skills, the teachers responded nearly unanimously that they do not teach such skills because algebra is taught nearly exclusively at Grades 6 and above. Many teachers seemed to contradict this however, by describing their use of the Model Method (i.e., an illustrative problem-solving tool). Although the teachers held a wide range of strong opinions regarding the Model Method, the majority respected its usefulness as a tool regardless of their attitude. Second, Singaporean students appeared to be fairly proficient in many algebraic reasoning skills, especially at grades 3 and above. As examples, the majority (i.e., 89% or more) of Singaporean students in grades 1-6 appeared to have mastered open number sentences and the majority (i.e., 85% or more) of Singaporean students in grades 3-6 appeared to have mastered both linear and nonlinear patterns. Third, despite the fact that many of the teachers stated that the use of variables is not taught at all and that their students would not be able to solve such problems correctly, the majority (i.e., 86% or more) of Singaporean students in grades 2-6 answered problems with variables correctly. First grade students did appear to struggle more with problems using variables than similar problems involving blanks (i.e., 64% of Grade 1 students used variables correctly while 89% of Grade 1 students solved open number sentences correctly). Similarly, there was some controversy among the teachers about whether equivalence problems (i.e., 8+4=__+5) are explicitly taught or not, however the majority (i.e., 74% or more) of Singaporean students in grades 2-6 appeared to have mastered the skill. Overall, it does appear that students in Singapore experience misconceptions in algebraic reasoning skills similar to their student counterparts in the U.S. The difference appears to be, however, that misconceptions appear to be addressed earlier in Singapore. Misconceptions may exist in Grades 1 and 2 but they appear to have nearly disappeared by Grade 3, which is quite the opposite of the misconceptions experienced by students in the U.S., which sometimes persist across the elementary grades. These results were especially interesting because teachers in Singapore do not appear to believe that they explicitly teach algebraic reasoning skills. But although it appears students in Singapore face fewer misconceptions surrounding algebraic reasoning skills than students in the U.S., misconceptions in general remain worrisome because, like Carpenter and colleagues (2005) stated: "If not addressed, misconceptions about equality and variables persist and provide serious impediments for learning high school algebra and other advanced mathematics" (p. 96).

National Science Foundation (NSF)
Office of International and Integrative Activities (IIA)
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Carter Kimsey
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Ralston Nicole C
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