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Algebra Instruction, Then and Now

January 2007

When the National Council of Teachers of Mathematics (NCTM) issued its Curriculum Focal Points document this past September the nation witnessed a flood of media response. Unfortunately, many articles, editorials, and op-ed pieces missed the point. Some gleefully pronounced that the days of “fuzzy math” were over and that the NCTM was “admitting” that it had made a “big mistake” back in 1989 when it released its Curriculum and Evaluation Standards for School Mathematics. Writers in newspapers large and small cast the NCTM’s new Focal Points document as “backpedaling.”

In response to these misunderstandings, NCTM president Skip Fennell explained that the new Focal Points document does not “undo” anything, but rather sets clearer, more common goals for math learning by focusing on a number of crucial, agreed-upon concepts. The document offers paragraph-long descriptions of concepts the authors regard as essential at each grade level, broken out by topics such as numbers and operations, basic algebra, measurement, and geometry. Those expectations become more demanding with each grade.

Impassioned editorial discussion about mathematics standards and instruction is a welcome change from the norm. But it’s important to note that this recent flurry of opinions generally omitted two crucial and equally important parts of reforming mathematics instruction. It’s not just about curricular expectations: Reform also includes getting inside the teachers’ heads to determine what they think about their own practice, and creating better professional development programs that will train teachers to deliver meaningful content in ways that lead to the highest possible student achievement.

WCER and the Standards

In the 1980s UW-Madison education professor Thomas A. Romberg helped direct educators and researchers in producing the document that was released in 1989 as the NCTM Curriculum Standards. WCER research has consistently taken these ideas as foundations for research in a number of research projects. That is still the case today. Far from “turning back” or admitting to some “mistake,” NCTM and education researchers continue to draw attention to, and discover more about, best practices that lead to improved student learning, effective teaching, and better professional development.

As an example, researchers in WCER’s interdisciplinary STAAR project (Supporting the Transition from Arithmetic to Algebraic Reasoning) study middle grade students’ transition from arithmetic into algebraic reasoning. Given the complexity of this topic, the research progresses along three parallel lines of study. At the student level, the research drills down to core algebraic concepts including students’ understanding of equivalence, variable, and representational fluency. Knowing what the equals sign means seems fairly obvious, but students often misinterpret it, and that affects their ability to learn. At the teaching level, STAAR research investigates teachers’ knowledge and beliefs about their own practices. Teachers are humans, and they enter the classrooms with some preconceptions and biases. It’s important to know how those affect student learning, and this project measures those effects. And third, on the level of professional development, the project builds on cumulative insights from the other two levels to help teachers recognize students’ algebraic thinking patterns and to recognize opportunities to foster better mathematical thinking.

WCER researchers Martha Alibali, Pam Asquith, Sharon Derry, Eric Knuth, and Mitchell Nathan are part of the interdisciplinary STAAR team that includes experts in developmental psychology, educational psychology, technology and learning environment design, teaching and teacher education, research methods, and program evaluation. Here are some of their current areas of focus.

Student Understanding

• Equivalence. STAAR research has found that many students misunderstand what the equals sign means. They often think it is a command to “process something,” rather than showing a relationship between two sets of values. This misconception interferes with students’ ability to solve equations. Improving students’ understanding of the equal sign, and their preparation for algebra, may require changes in teachers’ instructional practices as well as changes in elementary and middle school mathematics curricula. For example, teachers can present students with statements of equality in different contexts besides the traditional presentation of operations on the left side of the equal sign and the “answer” to the right of it to further develop students’ notions of equivalence.
• Variable. STAAR research investigates the meanings middle school students assign to symbols used as variables, and how these perceptions change over time. As students move through the middle grades they come to understand that a symbol can stand for multiple values. For example, in the expression 2n + 3, students’ thinking about the symbol progresses from the misconception that n can stand for a single digit only, to an understanding that n can stand for any number.
• Students’ representational fluency is their ability to solve problems using tables, graphs, words, or symbolic representations, and then to translate among these representations. STAAR research looks at how middle school students use graphs to reason about and generalize patterns, with a focus on their speech and gestures. Verbal representations seem to be an effective way for students to learn to reason about patterns algebraically. Teachers and curriculum developers may want to hold back on combining representations for beginning students, as they appear to be most suitable for the most advanced students.

Teacher Practice and Professional Development

STAAR research goes beyond studying students’ development to include teachers’ knowledge, instructional practices, and continuing professional development. For example, what kinds of teacher feedback best helps students repair their mathematical errors? The study has found “mitigated” negative feedback works best: The teacher avoids overtly indicating students’ erroneous answers (to avoid a loss of face for the student) but rather uses confirmation checks, clarification requests, inviting other learners’ ideas, and repetition of the student’s inaccurate utterances. Combining two or more forms of feedback was the most effective way to help students repair their math errors.

STAAR research has found that in some cases teachers did not accurately predict students’ understanding of the equals sign, but in terms of variables, teachers’ knowledge about students’ algebraic reasoning usually does accurately predict students’ responses to written assessments.

STAAR research is exploring two models of professional development for middle school algebra teachers, both aimed at helping teachers promote algebraic thinking in their classrooms. One model, developed by colleagues at the University of Colorado-Boulder, has been implemented with teachers in Colorado. It includes a summer institute and ongoing monthly professional development workshops. The other model, developed by the WCER research team, has been implemented with teachers located in Wisconsin. The program was the first offering of a permanent 3-credit graduate course that will include a summer workshop and monthly workshops during the school year.

A More Constructive Discussion

This continuing research into teacher professional development and teachers’ thinking about student reasoning all comes down to improving student achievement in algebra, in particular, and mathematics, in general.

But too often popular discussions about mathematics education are reduced to the curriculum itself, and editorials offer a false choice between “the basics” on one hand, or “fuzzy math” on the other. In reality, there is no “math wars,” except in the minds of some commentators. And to portray mathematics reform during the 1990s as “a torrent of misguided methods and confusing content” that left behind “flood debris” is simply irresponsible. It does nothing to promote productive public discourse about education reform. Parents, reporters, editors, and policymakers help students most when they consider how different the world will look when today’s students enter their careers, what they’d like them to be able to do in that world, and how education can best get them there.

Funding: The STAAR Project is a grant funded by the Interagency Education Research Initiative (IERI). WCER is home to a number of mathematics research projects. Here is a list of all current projects. Here a list of findings from completed projects.