For a number of reasons (experience as a classroom teacher, principal, and supervisor, standards writer and reviewer, Past President of NCTM, etc.), I regularly receive questions about mathematics standards and related in-school mathematics experiences. A while back a colleague sent me the following statement from a group in opposition to the Common Core State Standards for Mathematics (CCSSM).
“The smarter kids’ brains aren’t accepting lines, dots, circles and squares as an acceptable means of learning division. Yes, our children have been reduced to using lines, dots, dashes, squares and circles to learn division.”
Ok, let me try to figure this out. Do we really think that kids aren’t able to use representations, probably area models, groups of objects circled, or number lines to see that, for instance, 38 ÷ 15 is 2 groups of 15 with 8 as a remainder? I also get questions from friends as to why their children, or grandchildren, have to “show” or present a model to represent a problem’s solution.
With friends I can be pretty candid. So, I often ask: “do you even remember understanding ANYTHING you did in mathematics class?” And guess what, number lines, area models, and other representations for operations and both how and why they work have been used for a LONG time – we’re talking decades! What’s different is that the Common Core State Standards for Mathematics actually includes statements like the following from the 4th grade level:
“Find whole number quotients and remainders…using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain calculation by using equations, rectangular arrays, and/or area models” (CCSSI, 2010, page 30).
So, isn’t it about time we value the how and why of things math-wise? What is division anyway? Can you see from hopping back by 15’s on a number line or circling groups of fifteen objects, that one interpretation of 38÷ 15 is really just subtracting groups of 15 from 38? It’s way past time for all learners to have the opportunity to experience the how and why of doing mathematics as they understand concepts and move toward procedural proficiency.
Francis (Skip) Fennell
L. Stanley Bowlsbey Professor of Education and Graduate and Professional Studies Emeritus
Project Director, Elementary Mathematics Specialists and Teacher Leaders Project