Teaching Math with Reasoning
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.
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. 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. Too often, students don't see how the mathematical concepts and a set of rules fit together.
There are so many classroom tests and, in many states, high stakes tests, and students are memorizing calculations. It's the expectation, and students learn, "This is what I'm supposed to do."
Some memorization is necessary. But then days later, the teacher gives a new rule, and that appears to be different from what it was prior. But really, it's the same thing, just generalized. So kids get their head stuffed with formulas, but they don't know how to build strategies or choose the correct formulas.
When we first wrote the 1989 Curriculum and Evaluation Standards for School Mathematics, we did not know if anyone paid attention to it. That turned out to be the start of the standards movement in this country. And it forced all the other disciplines to do the same thing.
I wouldn't say every district used it [the 1989 book], but a lot of districts did, and states started creating math standards. I'd argue that the 2006 Curriculum Focal Points was popular. We knew we had leverage and that this was responding to what NCLB wanted. Almost every state is following the focal points.
How do reasoning and sense-making lessons relate to the math wars, such as the battle over soft math and hard math?
HK: I would argue that I think the book has been received well, from both the soft math and hard math proponents. But some mathematicians are concerned that reasoning and sense making is not formal proof. Sometimes they are concerned that we are pushing students to write mathematical truths. However, there is an acceptance that we are on the right track.
If we prepare students for STEM, this will make them more prepared to do that. Children come to school as willing reasoners. In grades K-1, they can prove to you what they did and why they think it was right. They can talk to you comfortably about their thinking in addressing mathematical challenges.
But we often lose it by high school. We shut them down, and we stop asking, "Why?" And we start looking only for answers. Some parents think the only important part of math is answers. To me, it's more frustrating to watch middle grade students who have several word problems and they have two numbers in front of them, and it's sometimes random in what they do with the numbers because they use no reasoning and no sense making. It's so frustrating that they don't even expect to have to think. We've almost trained them that there is one skill to practice—the rule of the day. We've taken the thinking out.
Could sense making play a role in common core standards?
HK: I just talked with the National Governors Association and the Council of Chief State School Officers, who are working on the common core standards. NCTM is providing feedback on the draft documents, and we have been pushing reasoning and sense making in high school math, which is now a core piece of their draft document. They appear to have accepted the importance of mathematical reasoning, which means students can calculate after thinking.
We have a lot of steam going into 2010. Our job is to make sure there is more opportunity for teachers and examples they could use in their classroom and to get practice on a regular basis.
So will this reasoning and sense making help students become financially smarter than their parents, given some recent press about the importance of teaching financial literacy to K12 students?
HK: I will say they will be smarter if we prepare them to ask the right questions about financial issues. They will have better tools to pursue those answers. In terms of income and family expenses, I would argue that if financial literacy learning helps them ask those questions, the reasoning they learn in math should help them better answer that question. What we need is to change people's perspective of what math is. In a broad sense, if we make people think math is more than what they use in the classroom, then that will be a general success.
In the end, will this hoped-for new approach to teaching help students make better advances in math?
HK: Yes. If we can break that belief that to be good in math is to know what formula or rule to use, then I think we'll have better results. But it's a tough belief to break down. If students learn to reason with a problem, they should have more success at a later time. They will be better decision makers at the problem level.
One of our struggles in the high school community is that it's so skill oriented on calculation, but one can't forget about this reasoning part. This belongs to every kid in a math setting, whether they're in the highest or lowest track in math, and it's important for life in general.
A lot of people go through life and do not think math has anything to do with anything—there is math in credit cards, when you go to a grocery store, get a car loan and get a mortgage. Students need to realize there is a mathematical side to it, and reasoning could help them.
Angela Pascopella is senior editor.