Reasoning and Proof Process Standard

This article focuses on four key concepts:

All students, from the pre-kindergarten classroom through senior year of high school should be able to:

1. Understand proof and reasoning methods as a ways of understanding, explaining and comprehending mathmatics.
2. Develop mathematical conjectures based on the problem given
3. Develop,analyze and represent mathematical reasoning
4. Understand when and how to utilize the various reasoning methods


Students need to be able talk about mathematics. From an early age children are able to justify and explain what they see in the world around them. One of the first things a child will recognize is a pattern and they can quickly learn to anticipate the next element in a pattern and explain why it is the correct choice. As students get older and enter school this is generally called reasoning or proof. When most students think of "proofs" they think about geometry; however, proofs and justification are a part of daily life and an essential component of mathematics and true mathematical understanding. As students grow older their level of reasoning will become more refined and sophisticated; however, the skills are quite the same.

Every time a teacher asks a student to justify his or her answer the teacher is asking for the reasoning, asking that the student demonstrate the proof or line of though that led the student to the answer. This is a important aspect of mathematics that can often become overlooked in worksheets and endless problems.
The second point revolves around conjecture. Conjecture, or guessing, is a part of mathematics, it is a part of learning. Students have to feel that they are able to guess, make mistakes, be wrong and learn from the experiences. Mathematics is a stressful subject for many students, due in part because they fear being wrong and this fear can stop them for interacting with the material. As defined in the process standard, students learn best when they are able to work cooperatively with their peers and reason through their mistakes.
As students learn how to go through the process of reasoning they need to also learn how to represent their work. In the elementary grades it may be most appropriate to have students draw and color their work. In the older grades, students should practice showing their work with sketches, numbers and even in paragraph form, not only as a two columned proof.
Finally, students need to be able to recognize when and how to utilize various reasoning methods. For instance, the nuances of an algebraic equation may vary considerably from a geometric problem. Essentially, as they progress through their schooling students should start to recognize how best to go about a problem and various strategies that they can use.