How many of us ask students to solve a problem and then stop when the student produces a correct answer? Give students a good rich task but don’t ask how they arrived at the answer or how they know it is correct? This is where reasoning and proof comes in.
I have heard so many teachers (in fact just about all teachers) say, “Students have no number sense!” Well, my question is why not and where do they think number sense comes from (hmmm...a little self-reflection might be in order)? There is much written about how students come into school with a sense of numbers and problem solving ability and we teach it out of them! We teach process and procedure without context, without asking students to explain their thinking or how they arrived at an answer, or without asking how they know their answer is correct and we, by doing nothing else, are teaching number sense right out of our students.
Let’s look at what some of the research says about reasoning and proof. In Elementary and Middle School Mathematics, John Van de Walle says that, “If problem solving is the focus of mathematics, reasoning is the logical thinking that helps us decide if and why our answers make sense. Students need to develop the habit of providing an argument or rationale as an integral part of every answer. Justifying answers is a process that enhances conceptual understanding. The habit of providing reasons can begin in kindergarten. However, it is never too late for students to learn the value of defending ideas through logical argument.” In Adding it Up, researchers call reasoning, “...the glue that holds everything together, the lodestar that guides learning.” Both publications very clearly and explicitly state that students who disagree about a mathematical answer shouldn’t have to run to the teacher to sort it out, but should be able to use reasoning and justification (proof) to determine the validity of their solution. Adding it Up also states that it is not sufficient to justify a procedure just once. The development of proficiency occurs over an extended period of time. Students need to use new concepts and procedures for some time and to explain and justify them by relating them to concepts and procedures that they already understand. For example, it is not enough for students to do only practice problems on adding fractions after the procedure has been developed. If students are to understand the algorithm, they also need experience in explaining and justifying it themselves with many different problems and in many different contexts.
I am going to use another fraction example to highlight how the lack of reasoning and proof in our classrooms affects our students. I often see students who are having difficulty comparing and ordering fractions and what I am about to tell you is a true anecdote. I was working with 4 fifth grade boys on comparing and ordering fractions. I put four fractions up on my white board, 1/4, 2/5, 5/6, and 1/2 and I asked them what they could tell me about the fractions just by looking at them. They could tell me very little. Of course they wanted to cross-multiply (ugh!) and when I told them that they could only use that process if they could explain the math behind it, that strategy was out. They knew that sixths were the smallest pieces and the halves were the largest pieces, but beyond that they had nothing. So we spent two or three (maybe even four) 30-minute sessions reasoning through fractions and then proving their sizes with models and number lines. In the end they could reason through and justify to each other (and me) the order of the fractions from smallest to greatest without performing a paper-pencil “process”. They could explain, for example, that while sixths were the smallest pieces, 5/6 was the largest fraction because 6/6 would be one whole and 5/6 was only 1/6 away from the whole, a much smaller piece away from its whole than the other three fractions were away from their wholes. Now that’s what I’m talking about!
Reasoning and proof (or justification) is not only an integral part of problem solving, you can see how it is the “glue” that holds our math together and is also the part of our mathematics that develops and solidifies our number sense! Please, for the sake of number sense everywhere (and so that teachers will quit asking why their students don’t have any number sense), teach your students to reason through problems and justify their answers!