Proof is central to doing mathematics. Yet proving is challenging for most students. In this article we describe the findings of a yearlong teaching experiment that focused on developing proof. We discuss the role of classroom discourse in developing argumentation practices that lead proof using examples from our work. The teacher's role in managing and promoting this discourse is explicitly discussed.
Despina A. Stylianou and Maria L. Blanton
Maria L. Blanton and James J. Kaput
We present here results of a case study examining the classroom practice of one thirdgrade teacher as she participated in a long-term professional development project led by the authors. Our goal was to explore in what ways and to what extent the teacher was able to build a classroom that supported the development of students' algebraic reasoning skills. We analyzed 1 year of her classroom instruction to determine the robustness with which she integrated algebraic reasoning into the regular course of daily instruction and its subsequent impact on students' ability to reason algebraically. We took the diversity of types of algebraic reasoning, their frequency and form of integration, and techniques of instructional practice that supported students' algebraic reasoning as a measure of the robustness of her capacity to build algebraic reasoning. Results indicate that the teacher was able to integrate algebraic reasoning into instruction in planned and spontaneous ways that led to positive shifts in students' algebraic reasoning skills.
Maria L. Blanton and James J. Kaput
Helping teachers learn to identify and create opportunities for algebraic thinking as part of their normal classroom instruction.
June Soares, Maria L. Blanton and James J. Kaput
“There is not enough time in the day to teach all subjects!” This is the cry heard in elementary schools all across the country. With testing and accountability on everyone's mind, teachers are looking for creative ways to teach all subjects. Literacy is on the top of the list for testing, so it seems to get top priority. But how can we make sure that mathematics, especially a crucial area such as algebraic thinking, is a priority as well?
Isil Isler, Tim Marum, Ana Stephens, Maria Blanton, Eric Knuth and Angela Murphy Gardiner
Engage your students in functional thinking—an important precursor to algebra—with this classroom activity.
Ana Stephens, Maria Blanton, Eric Knuth, Isil Isler and Angela Murphy Gardiner
Researchers find that these classroom activities and instructional strategies support the development of third-grade students' algebraic thinking.
Maria Blanton, Ana Stephens, Eric Knuth, Angela Murphy Gardiner, Isil Isler and Jee-Seon Kim
This article reports results from a study investigating the impact of a sustained, comprehensive early algebra intervention in third grade. Participants included 106 students; 39 received the early algebra intervention, and 67 received their district's regularly planned mathematics instruction. We share and discuss students' responses to a written pre- and post-assessment that addressed their understanding of several big ideas in the area of early algebra, including mathematical equivalence and equations, generalizing arithmetic, and functional thinking. We found that the intervention group significantly outperformed the nonintervention group and was more apt by posttest to use algebraic strategies to solve problems. Given the multitude of studies among adolescents documenting students' difficulties with algebra and the serious consequences of these difficulties, an important contribution of this research is the finding that—provided the appropriate instruction—children are capable of engaging successfully with a broad and diverse set of big algebraic ideas.
Maria Blanton, Bárbara M. Brizuela, Angela Murphy Gardiner, Katie Sawrey and Ashley Newman-Owens
The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. We employed design research and analyzed data qualitatively to characterize the levels of sophistication in children's thinking about functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.