By Courtney Koestler posted 05-30-2018 12:50


I advocate for more equitable and just classrooms, schools, and schooling. This work must be addressed at multiple levels of the educational system (e.g., see Aguirre, Mayfield-Ingram, & Martin, 2013, Wager and Stinson, 2012), but in all of the work that I do I always try to highlight the importance of honoring and building on the brilliance of young children’s thinking.

Despite decades of evidence, some adults (teachers, administrators, parents, etc.) are surprised, and often a little incredulous, at the idea that young children can understand complex mathematical concepts and even invent strategies to add, subtract, multiply, and divide without explicitly being told how to do so. I distinctly remember in my third year of teaching watching the mathematics coach at my school engage children in tasks that encouraged them to develop and use the “partial products” algorithm for multiplication.


I was astonished that I had never before seen or used this algorithm myself as a child or as a mathematics teacher, given how much sense it made to me (and my students) and how clearly it mapped onto concrete manipulatives and area representations.


Around this same time, I also was learning about Cognitively Guided Instruction (CGI) as a way to make my mathematics teaching more child-centered. I remember reading the following quote in the introduction to the book Children’s Mathematics: Cognitively Guided Instruction (Carpenter, Fennema, Franke, Levi, & Empson, 1999, p. xiv):

“…Children may actually understand the concepts we are trying to teach but be unable to make sense of the specific procedures we are asking them to use.”

The CGI body of work challenged me to rethink my teaching approach and the kinds of tasks I was asking my students to do. In fact, it also made me question whether I was actually asking my students to problem solve - to actually engage in mathematics - or was I simply asking them to compute? Surely computation is an important part of K-12 mathematics; however, I noticed that we often think about computation as “the mathematics” rather than highlighting all of the sense-making and understanding learners engage in when truly understanding addition, subtraction, multiplication, and division.

I have come to believe it is important not to assume the “standard” algorithm is the standard way to compute for people, especially for children.

We know that children themselves would rarely, if ever, develop the “standard” algorithm as a way to compute numbers. I also have seen firsthand the ways students naturally operate on numbers, by using manipulatives, using what they know about place value, and building on their number sense to invent strategies that make sense. When we support children to do this, we can create more equitable and child-centered classrooms by allowing students more opportunities to make sense of and have a role in developing school mathematics.





I end with a proposition made by John Van de Walle in a presentation at an NTCM Regional Conference in 2004: “The traditional whole-number algorithms for addition, subtraction, multiplication, and division should no longer be taught in schools.” Instead, he argued that alternative, flexible, and/or invented strategies should be encouraged and supported.

What do you think?


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 diverse experiences as a classroom teacher, a mathematics coach, and a university-based teacher educator working alongside teacher colleagues and children in classrooms.



Aguirre, J., Mayfield-Ingram, K., & Martin, D. (2013). The impact of identity in K-8 mathematics: Rethinking equity-based practices. The National Council of Teachers of Mathematics.

Carpenter, T.P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann.

Van de Walle, J. (2004). Believe in the power of computational fluency.NCTM Regional Conference and Exposition. Baltimore, MD.

Wager, A. and Stinson, D. (Eds.) Teaching mathematics for social justice: Conversations with educators. Reston, VA: National Council of Teachers of Mathematics.


1 comment



06-01-2018 03:24

Courtney - this is such an important conversation! I find the work of Van de Walle to be connected to CGI (Levi, Carpenter, et. al.). These conversations are relevant and timeless - I am eager to see how teachers respond to the prompt...!