In my last post, I ended with a proposition made by John Van de Walle: “The traditional whole-number algorithms for addition, subtraction, multiplication, and division should no longer be taught in schools.” Instead, he argued that alternative and/or invented strategies should be encouraged and supported because they are:
- number-oriented, rather than digit-oriented,
- flexible, rather than rigid,
- “left-handed” rather than “right-handed” (starting with the biggest place value first),
- student-centered, rather than teacher-directed, and
- focused on number sense and mental math (Van De Walle, 2004).
In order to support alternative and invented strategies a lot has to be done: in classrooms, in professional development, and in discussions with families about expectations for mathematics teaching and learning. For teachers, here are a few ways to get started. One approach, as part of a math block, is to simply pose “naked number” problems and ask students to develop their own strategies for solving these problems,. Then, as students develop strategies, teachers can facilitate discussions by asking students questions about why a strategy works and about connections between and among different strategies. Sherry Parrish’s (2014) book Number Talks: Whole Number Computation, Grades K-5 is an invaluable resource to do this. The book provides an overview about why doing this work is so important for developing mathematically competent children. It also provides clear guidance about making purposeful decisions, and about what number sets to pose (and in what order) to support students in developing conceptual understanding of numbers, strategies, operations, and specific concepts.
Another important approach is to develop real story problems about topics that are interesting and relevant to students’ lives. In the United States, there are pervasive myths about story problems, such as the myth that story problems are too difficult for young children to solve and the myth that children must know all of their facts before they can engage in story problems. However, there is considerable evidence that these myths simply are not true. If done well, story problems actually support children in developing understanding of numbers, operations, and fact fluency (see work on Cognitively Guided Instruction - Carpenter., Fennema, Franke, Levi., & Empson, 1999).
Finally, a third approach may be to take textbook lessons and consider modifications and adaptations to make students’ strategies and understanding the focus. For example, instead of showing students the text or explanations of how to solve a problem, teachers could begin by asking for their suggestions of how to begin and allowing their ideas to guide the problem solving process.
To create a more child-centered and equitable classroom, it not only takes approaches like the ones outlined above, but also a shift in the beliefs and practices of teachers. To be able to do this work, teachers must:
- believe that all students can invent their own strategies and algorithms, especially those students most marginalized in classrooms;
- be open to students’ different ways of solving, and be open to the idea that they (teachers) may not understand them (at first);
- give students the opportunity to think and to do the intellectual work of doing mathematics; and
- talk to colleagues and families about the goals of mathematics teaching and expectations for the kinds of learning that will take place in their classrooms.
Now It’s Your Turn:
What approaches have you taken (or do you want to take) to create a more child-centered classroom that reveals, and builds upon, student thinking? What are your beliefs about children’s mathematical thinking?
Courtney Koestler is the director of the OHIO Center for Equity in Mathematics and Science and a faculty member in the Department of Teacher Education in the Patton College of Education at Ohio University. Courtney has considerable and diverse experiences as a classroom teacher, a mathematics coach, and a university-based teacher educator working alongside teacher colleagues and children in classrooms.
Carpenter, T.P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies, grades K-5. Math Solutions.
Van de Walle, J. (2004). Believe in the power of computational fluency.NCTM Regional Conference and Exposition. Baltimore, MD.