Algebraic Thinking: What It Is and Why It Matters
The average score for eighth-graders on the latest National Assessment of Educational Progress (NAE P) was the highest ever, but only 39 percent scored at or above the proficient level (Lee, Grigg, & Dion, 2007). Even fewer high school seniors (23 percent) were proficient (Grigg, Donahue, & Dion, 2007). “The sharp falloff in mathematics achievement in the U.S. begins as students reach late middle school, where, for more and more students, algebra course work begins,” the National Mathematics Advisory Panel said in 2008 in its final report.
According to the Math Panel, preparing more students for success in algebra and beyond will require the K8 math curriculum to emphasize three critical foundations (whole numbers, fractions and some aspects of geometry). Also, math instruction should address all three elements of proficiency (conceptual understanding, computation and procedural fluency, and automatic recall of facts). One approach to making the math curriculum more cohesive (another of the Math Panel’s recommendations) is to develop students’ algebraic thinking at all grade levels.
A Paradigm Shift
Readers won’t find the phrase “algebraic thinking” in the Math Panel’s report, but researchers and others have used the term to describe “particular ways of thinking, including analyzing relationships between quantities, noticing structure, studying change, generalizing, problem solving, modeling, justifying, proving, and predicting” (Cai & Knuth, 2005).
Asking teachers to develop students’ algebraic thinking represents a paradigm shift in the teaching of mathematics. In the past, math curricula have emphasized arithmetic (calculation) in elementary school and algebra (e.g., exploration of patterns and use of symbolism) in middle school. This sequence makes the transition from arithmetic to algebra difficult for many students because it requires them to make various adjustments. For example, when students study algebra, they no longer focus merely on calculating numerical answers but on understanding and representing relationships through the use of both letters and numbers (Kieran, 2004).
For students, the change in emphasis between elementary and secondary school can create a conceptual barrier to mathematics achievement. To remove this barrier, a new paradigm is evolving in math education—one that calls for teachers at all grade levels to help students develop “habits of mind that attend to the deeper underlying structure of mathematics” (Katz, 2007).
Helping Teachers Make the Shift
The National Research Council reports that the ability of K12 teachers to effectively select tasks and guide student thinking “is highly dependent on teachers’ knowledge of mathematics, pedagogical content knowledge, and knowledge of students in general” (Bransford, Brown, & Cocking, 1999). Unfortunately, not all math teachers are well prepared to teach the content. A recent analysis of the 2003-2004 Schools and Staffing Survey found that 22 percent of secondary school math classes are taught by teachers who didn’t major in math and are not certified to teach it. In high-poverty schools, the percentage (41 percent) nearly doubles (Ingersoll, 2008). Shifts in the way mathematics is taught will require shifts in teacher preparation, support and professional development.
According to the Math Panel’s random national survey of algebra I teachers, student preparation for algebra seems especially weak in three areas: rational numbers, word problems, and study habits. The Math Panel called for research on the use of full-time math teachers in elementary schools to improve students’ preparation for algebra. Also recommended is professional development that helps teachers understand how the content they teach is connected to what students have already learned and what they will learn next.
Carla Thomas McClure is a staff writer at Edvantia (www.edvantia.org), a nonprofit education research and development organization. Joy Runyan (firstname.lastname@example.org) is a math and science specialist for the Appalachia Regional Comprehensive Center at Edvantia.
Bransford, J.D., Brown, A.L., & Cocking, R.R. (Eds.). (1999). How people learn: Brain, mind, experience and school. Washington, D.C.: National Research Council. Retrieved from http://www.nap.edu/openbook.php?record_id=6160.
Cai, J., & Knuth, E. (Eds.). (2005). Developing algebraic thinking: Multiple perspectives. International Review on Mathematics Education, 37(1).
Grigg, W., Donahue, P., & Dion, G. (2007). The Nation’s Report Card: 12th-Grade Reading and Mathematics 2005 (NCES 2007-468). Washington, D.C.: National Center for Education Statistics. Retrieved from http://nces.ed.gov/nationsreportcard/pdf/main2005/2007468.pdf.
Ingersoll, R. (2008). Core problems: Out-of-field teaching persists in key academic courses and high-poverty schools. Washington, D.C.: Education Trust. Retrieved from http://www2.edtrust.org/NR/rdonlyres/0D6EB5F1-2A49-4A4D-A01B-881CD2134357/0/SASSreportCoreProblems.pdf.
Katz, V.J. (Ed.). (2007). Algebra: Gateway to a technological future. Washington, D.C.: The Mathematical Association of America. Retrieved from http://www.maa.org/algebra-report/algebra-gateway-tech-future.pdf.
Kieran, C. (2004). Algebraic thinking in the early grades: What is it? The Mathematics Educator, 8(1), 139-151.
Lee, J., Grigg, W., & Dion, G. (2007). The Nation’s Report Card: Mathematics 2007 (NCES 2007?494). Washington, D.C.: National Center for Education Statistics. Retrieved from http://nces.ed.gov/nationsreportcard/pdf/main2007/2007494.pdf.
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, D.C.: U.S. Department of Education. Retrieved from http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf.