Emphasizing Concepts vs Procedures in Math: Student Achievement and Social Justice
Over the past few decades, teachers in the United States have faced increasing pressure to boost American students’ mathematics achievement. Comparisons with other countries have revealed that American students lag behind in math achievement, especially when compared to student performance in Japan and Singapore. Thus, American education leaders and policymakers have prioritized mathematics achievement in part to secure America as a globally competitive and innovative nation (Yu 82).
In their paper, “Teacher support, instructional practices, student motivation, and mathematics achievement in high school,” authors Rongrong Yu and Kusum Singh explain that for the past 20 years, mathematics educators and administrators have been engaged in “mathematics wars,” heated debates that have pitted those who favor procedural teaching against those who favor conceptual teaching (81).
Previous studies have shown that students in highachieving countries like Japan and Singapore “spend more time on inventing, analyzing, and proving, with less time on routine procedures, but U.S. students spend almost all their time on routine procedures” (83). In response to the information gained about mathematics teaching practices in highachieving nations, America’s Common Core Standards (2010) called for “a heavy emphasis on students’ conceptual understanding of mathematical ideas” for students in the United States (82). This kind of teaching deemphasizes speedy computation, algorithms, and procedures, and instead promotes the kind of mathematical pedagogy seen in highachieving countries—an emphasis on “developing problemsolving skills, reasoning mathematically, connecting mathematics ideas, [and] understanding the logical structure of mathematics” (91).
In their study, Yu and Singh examine how procedural teaching and conceptual teaching affect student achievement in mathematics. In particular, they investigate the relationships between procedural teaching, conceptual teaching, teacher support, and student achievement in mathematics for high school students. They note that it is important to understand how to best support student achievement in high school mathematics given that high school educational experiences strongly influence students’ future educational and career paths.
“High school years are particularly critical for students’ future educational trajectories. Mathematics achievement in Grade 9 influences students’ beliefs about their mathematical capability and later choice of higher level mathematics courses, which will further influence their postsecondary and occupational options. Therefore, Grade 9 is a critical point for developing interest and positive attitudes toward mathematics” (82).
The data for their study came from 9,662 ninth grade math students and their math teachers, who completed the High School Longitudinal Study of 2009. Among other things, this comprehensive survey asked students to report on items including student selfefficacy (their belief in their own mathematical abilities), their attitudes toward mathematics, and their views about teacher personal and academic support. Students also completed a 72 question mathematical assessment. Teachers were also asked to report on their use of conceptual and procedural teaching (85).
The Findings
“Conceptual emphasis in teaching had positive influence on student mathematics achievement whereas procedural focus in teaching had negative influence on student mathematics achievement” (89).

 Students who had more positive beliefs about their own mathematical abilities (higher self efficacy) reported greater interest in mathematics (89).

 Student interest in math did not translate to achievement. Student interest in math had “no significant influence on mathematics achievement” (89).

 Students who had higher selfefficacy beliefs had higher mathematics achievement (90).

 Students who had higher selfefficacy beliefs also reported having teachers who they viewed as supportive (89). Teacher encouragement and academic support strengthens student selfefficacy beliefs, which is linked to higher mathematics achievement.
 “Conceptual teaching had a direct positive influence, whereas procedural teaching had a direct negative influence on students mathematics achievement” (89).
“Students with higher levels of family SES (socioeconomic status) and prior achievement were more likely to have teachers who use conceptual teaching strategies” (89).
Yu and Singh emphasize that the teaching of conceptual ideas in mathematics (mathematical reasoning, key concepts, logical structures, the history and nature of mathematics, etc.) does not mean that procedures should be excluded: “Procedural skills and conceptual knowledge are both critical to mathematics proficiency” (91). That said, the teaching of procedures only is demonstrably disadvantageous for students’ understanding of mathematics. Finally, Yu and Singh remark that their research adds insight into the American achievement gap. Namely, that students with lower SES are less likely to have teachers who emphasize conceptual teaching relates to the broader problem of “unequal distribution of opportunities to learn and unequal number of effective teachers” between areas of varying SES in the United States (91).
Paper Title: Teacher support, instructional practices, student motivation, and mathematics achievement in high school
Authors: Rongrong Yu (National Institute of Education Sciences, Beijing, China) and Kusum Singh (Virginia Tech, USA)
Full Paper: https://www.tandfonline.com/doi/abs/10.1080/00220671.2016.1204260?journalCode=vjer20
Published: The Journal of Educational Research, Volume 111, No. 1, 2018, Pages 8194
Photograph: Thanks to Antoine Dautry on Unsplash