This project develops images, extended examples, and principles that illustrate how the articulation, representation and justification of general claims about operations evolve in the elementary grades and how this work supports the transition from arithmetic to algebra in the middle grades. A Sourcebook developed by staff and teacher-writers provides an account of this work across the school year in grades 1-6 and commentaries that help teachers consider the underlying pedagogical and mathematical aspects of the work. An online course uses the Sourcebook as a text to enegage teachers in implementing these ideas in their instruction.
A collaboration of researchers—Susan Jo Russell (TERC), Deborah Schifter (Education Development Center), and Virginia Bastable (Mt. Holyoke College)—and 27 teachers investigated how students in the elementary grades come to understand the meaning and properties of the operations in the course of their study of arithmetic. The project developed professional development materials—a book and online course—to help teachers integrate this focus into their instruction in order to support students’ development of ideas that are both foundational to their study of arithmetic and also connect arithmetic to algebra. The book, Connecting Arithmetic to Algebra is organized around classroom cases of teaching and learning about the properties and behaviors of the operations, primarily in grades 1-6. The course guide provides facilitators in schools or universities the materials for offering the course online or as a face-to-face book study, using the book as the course readings. (www.heinemanncatalogs.com/lg_display.cfm/catalog/Prof_Resources_K-12_Fall_2012/page/122) The book and course emphasize noticing, describing, representing, and explaining consistencies across many problems. In the elementary grades, students often notice, for example, that when they change the order of the numbers in an addition or multiplication expression, the sum or product does not change. The book and course support teachers to learn how such observations, embedded in the arithmetic students are already doing in class, can be investigated explicitly. In these investigations, students deepen their understanding of the meaning of the operations, develop the habit of noticing regularities about the operations, articulate general claims about these regularities, and construct mathematical arguments about these claims. A particular emphasis of the book and course is on the range of learners in the classroom—those who struggle with grade-level computation and those who excel. Teachers learn to see how all students can participate in productive discussions about significant mathematical ideas, and how teachers can support learning opportunities for students with different strengths and needs. The book and course examine the connections of this grades 1-6 focus to middle grades mathematics—how lack of understanding of the operations is often what underlies the difficulties students have in algebra. Researchers studied the learning of teachers participating in the online course and their students. Teachers who took the course improved significantly in the breadth of generalizations they could generate and articulate, in their abilities to represent mathematical ideas using notation and representations, and in their use of mathematical language. Students in participating’ teachers classrooms also improved significantly, especially in their ability to refer to mathematical structures in their explanations. For example, in deciding whether 6 ´ 7 is equivalent to 3 ´ 21, one might simply compute to determine that the two expressions have the same product. However, at the end of the course, grades 2-5 students in classrooms of teachers who participated, were significantly more likely than students in a comparison group to use explanations such as, "in multiplication, if you double one number and cut the other in half, you still get the same answer." Online learning is a format that has the potential to accommodate teachers’ complicated work and life schedules, reach larger numbers of participants at once, and bring together teachers who are geographically dispersed. For school systems that already have a network in place that includes the possibility of both synchronous (e.g., webinars) and non-synchronous (e.g., bulletin boards, discussion forums) interaction, online learning can be a cost-efficient and effective form of professional development and can be combined with face-to-face study groups or grade-level teams. Some of our participating school or system teams reported this combination to be particularly powerful for them. Our foray into online learning was undertaken, in part, in order to investigate whether this possibly more cost-efficient and manageable form of professional development for school systems could maintain high quality and improve the learning of both teachers and students. Our success in doing so may be of use to school systems grappling with how to afford and structure needed professional development.
