Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Defining Reasoning and Proof
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In addition to modeling fluent and effective use of both reasoning and proof, the teacher has two tasks in the high school classroom in relation to this standard: helping students understand the sequential nature of mathematical argumentation and introducing and developing an understanding and of mathematical rigor.
In practice, many types of reasoning are used at the same time on the same mathematical task. Students work back and forth along a chain of argumentation, checking cases, seeing if there are counterexamples, refining definitions and symbolic representations. Mistakes are likely, and they often may be profitable. This is "messy" work, but it is productive educationally. When students encounter a proof or a derivation of a theorem in a textbook however, they see a much more "directional" type of mathematical thought. There premises are precise and begin the argument, which proceeds cleanly and elegantly to a conclusion. There are certainly no mistakes. This presentation is in some sense "backwards" for pedagogical purposes. It does not emphasize the discovery that led to the result or the motivation for that discovery. These are aspects of the reasoning and proof concept that are best supplied by the teacher and as he or she is in a position to understand where in an argument a student is working, and indeed what level of sophistication in reasoning a student brings to a task.
The teacher also has a responsibility to establish and maintain an appropriate level of mathematical rigor in the class and an appreciation for its value. Students naturally have a range of informal and heuristic approaches to mathematics; these approaches should not be discouraged; even the greatest mathematicians have often used informal or idiosyncratic ways to represent and solve problems. But students should also understand that mathematical arguments ultimately must be clear and convincing, particularly so in the context of formal proof. This requires use of accurate vocabulary, for instance, understanding and respecting the difference between "if and only if" and "if." Students should also understand the distinction between an informal approach to an idea, for instance, the "set of all integers," and how and why such a concept could be formalized in abstract mathematical symbolism such as set notation.
Rigor goes beyond helping students to learn the meanings of mathematical symbols and how to use them accurately. Rigor also requires that we foster development of mathematical ability in a way that builds both conceptual skill and the ability to connect conceptual insight to the abstract language of formal mathematics. This approach aims to give students full access to the tools of mathematics -- the ability to capture ideas in often small powerful statements and use those statements accurately. For instance, when students are introduced to functional notation, they must understand the role and purpose this notation plays and why it has value in mathematical expression. This kind of rigorous understanding will serve them in good stead in later. On the other hand, if they are hazy about functional notation and think of it as some kind of vague mathematical shorthand, it may become a recurring stumbling block.
A final point about rigor: when we think about rigor teachers can justifiably ask, "this too?" With the big task that teachers already have, demanding content and classroom challenges, why lay rigor on top of this, too? particularly when the majority of students will not go on to post-secondary work in mathematics? Why even worry about reasoning and proof to the extent that the standards recommend? One response is that rigorous reasoning and proof is part of the meaning of mathematics. To understand its value is to understand mathematics. If it's missed as a concept, our students don't have the full picture. Another answer is that rigor allows us to talk frankly about the limits of mathematics, what a mathematical argument does and does not show, and, possibly, how it can or cannot be applied.
This is an approach that serves all students well, both those who will go into mathematics, and fields which rely heavily on it such as computer science, but also those who are the "consumers" of mathematical ideas and the fruits thereof, a group that includes all of us. It is compatible with the spirit and content of the NCTM standards, which support a rich and rigorous curriculum for all students.
Watch the video segment (duration 0:23) at left to hear reflections from educator Henry Kepner.
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