ABOUT WCER NEWS Events Cover Stories Research News International Research Press WHAT'S THE RESEARCH ON...? PROJECTS All Active Projects All Completed Projects PUBLICATIONS LECTURE SERIES PEOPLE Staff Directory Project Leaders ERG - EVALUATION RESOURCES GROUP RESOURCES Conference Rooms Equipment GRANT SERVICES GRADUATE TRAINING SERVICE UNITS Director's Office Business Office Technical Services Printing & Mail EMPLOYMENT CONTACT INFO MyWCER WORKSPACE LOGIN

How can mathematics educators better understand the early stages of algebraic reasoning in children?

How can these stages be developed in the context of elementary school mathematics in ways that reach many schools and are sustainable?

Although elementary teachers are a critical part of algebra reform they typically have little experience with the rich and connected kinds of algebra that need to become the norm in their instruction. James Kaput (University of Massachusetts-Dartmouth) and colleagues at WCER's National Center for Improving Student Learning and Achievement (NCISLA) in mathematics and science are developing ways to enhance teachers' understanding of the kinds of algebraic thinking that need to occur in elementary school mathematics. Their research is funded by the Office of Educational Research and Improvement, U.S. Department of Education.

Kaput pursues an "algebrafication" strategy that embodies three dimensions of teacher-based classroom change:

• using available instructional materials to build opportunities for algebraic reasoning, especially generalization and progressive formalizing of those generalizations.
• building teachers' "algebra eyes and ears" so they can recognize opportunities for generalization-and systematic expression of that generality-and then act on these as they occur.
• creating classroom practice and culture to support students’ purposeful conjecture and argument, so that algebra opportunities occur frequently and are viable when they occur.

Algebrafication is a deliberate capacity-building strategy that enables teachers to expand their curricular and pedagogical capacities and, in so doing, also deepen their content understanding of elementary mathematics. Teachers learn how to use their own instructional resource base and their own classrooms as venues for structuring algebra opportunities in elementary school mathematics.

While working with 15 teachers from six elementary schools, Kaput and colleague Maria Blanton collected written and oral reports and sample student work. They then focused on the classroom practice of one third-grade teacher, "Jan," who was creating the kind of algebrafied instructional context that reflected the researchers’ work with teachers. Compelling evidence was gathered during the past few months to support Kaput’s "algebrafication" strategy and the specific forms that it can take "on the ground."

"In Jan's classroom we see evidence of student achievement that validates our professional development activities with teachers," says Kaput. "We have also used our case study of Jan's practice to identify what it means for a teacher's classroom practice to be 'algebrafied' in a way that is generative and self-sustaining."

The theme of "learning with understanding" guides this project. "The most important feature of learning with understanding is that it is generative," says Kaput. "When students or teachers acquire knowledge with understanding, they can apply the knowledge to learn new topics and solve new and unfamiliar problems. When students or teachers do not understand, they perceive each topic as an isolated skill. They cannot apply their skills to solve problems not explicitly taught to them, nor extend their learning to new topics."

This past spring the team administered a set of 15 fourth-grade MCAS exams (Massachusetts Comprehensive Assessment System) to Jan's class of 14 third-grade students, who completed the exam individually, then again with partners in seven dyads. These third-grade students performed significantly above the "proficient" level on 33 percent of the items and were at or above the "needs improvement" level for 67 percent of the items. These third graders from a low SES neighborhood outperformed district and state means for 4th graders.

Students' individual performances, although varied, also support the promising results indicated by the item analysis. In particular, out of 14 students, 1 student performed at the "advanced" level, 3 students performed at the "proficient" level, and 6 students performed at the "needs improvement" level. Only 4 students ranked lower.

Paired-student results on the assessment were quite exceptional, and reveal perhaps more adequately than individual scores what these students really understand because they were able to externalize their thinking and bring their linguistic resources into play. Out of the seven partner groups, four groups performed at the "advanced" level, two groups performed at the "proficient" level, and one group performed at the level "needs improvement." "The fact that this was a fourth-grade assessment makes these results particularly compelling," Kaput says.

This study of Jan's teaching revealed characteristics of teacher change in the context of an algebrafied classroom practice. Specifically

(a) an ability to generalize an arithmetic activity to an algebraic activity

(b) the use of algebraic conversations in the classroom on a regular basis

(c) the use of algebraic themes over significant periods of time, and

(d) the use of multiple algebraic processes in a single mathematical task.

For more information see the article by Blanton and Kaput, “Characterizing generative and self-sustaining teacher change in a classroom practice that promotes students’ algebraic thinking,” available through the NCISLA web site at www.wcer.wisc.edu/NCISLA.