Teaching Math with Reasoning
Pascopella, Angela, District Administration
THE NATIONAL COUNCIL OF Teachers of Mathematics wants high school students to make sense of their math. The organization is excited about the recent unveiling of Focus in High School Mathematics: Reasoning and Sense Making, a book published last October (and part of what will become a full series), in which NCTM builds on three decades of advocacy for standards-based mathematics learning of the highest quality for students.
It is a follow-up to the NCTM's 2006 document, Curriculum Focal Points for Pre-Kindergarten through Grade 8 Mathematics, which offered grade-by-grade math content standards. This new book offers a different perspective, proposing curricular emphases and instructional approaches that make reasoning and sense making the foundation to the content taught in high school.
DA recently spoke to Hank Kepner, NCTM's president, about this book, which he co-authored with a team of mathematicians and math teachers, and what it means for high schools across the nation.
Q: How has NCTM shifted its focus with respect to math?
HK: At each grade level, the 2006 Curriculum Focal Points said here are three key ideas that students in prekindergarten through eighth grade should know. And math was criticized for being a mile wide and an inch deep. So NCTM responded to the question from our high school teachers, "What are you going to do for these students?" by publishing this book.
What the council said was that no matter what math students are taking in high school, they should be engaged in reasoning and sense making every day. No matter what the content topic, this should be a part of it. And students should be asking themselves, "Why is this working? How is this connected? If I'm looking at an equation, I wonder how many things change as time goes on?" The idea is that when you are looking at doing something in mathematics, there is always a reason behind doing it. Students don't do a good job of reasoning, or we don't push them enough. The students make no connection between the way they think about multiplying whole numbers and multiplying algebra binomials. But it's the same properties and the same basic ideas.
How will districts use the ideas of reasoning and sense making that are spelled out in the latest NCTM book? HK: While these books in the Focus in High School Mathematics series, including one on algebra and geometry, will be useful to individual teachers, I see them best used in professional development settings where teachers can collaborate in planning lessons focused on reasoning, followed by reflections on their classroom experiences and students' work. Training teachers to ask deeper, more meaningful questions that require students to provide reasoning in their answers is one of the biggest goals in this professional development.
Focused opportunities for teachers talking to each other to expand their questioning techniques and observe students' reasoning on each topic, along with the connections across mathematical topics and applications, would be best.
We don't see this [kind of reasoning] adding content to topics, but every day this should be embedded in that day's lessons. In this way, we can get students to consciously explain their reasoning and recognize initial errors in that reasoning.
For the teacher, it's a way of looking for and then building instruction on misconceptions that might have occurred in the students' heads.
What has the response been to the book? Any buy-in yet?
HK: I think, in the math community, that it's well received. It's making a critical point that reasoning ought to be included in math and for the most part, it's not. There is such a public belief that mathematics is a set of rules and calculation procedures. The "I could never do math" comment is often based on two challenges: First, I don't know which formula to use; and second, I don't remember how to do that calculation. …