Mathematical Proof and Reasoning: What, Why, and How?
Jinfa Cai
Professor of Mathematical Sciences
“Reasoning” and “proof” are important mathematical elements. Mathematical reasoning is both
a process and a skill to make sense of mathematics. Mathematical proof is the process of
developing arguments consisting of logically rigorous deductions of conclusions from
hypothesis. In the discipline of mathematics, proof holds a privileged position as the heart of
mathematical activities.
In this seminar, we hold a position that even very young children have the ability to successfully
engage in mathematical reasoning and proof activities - appropriate to the age levels, as
mathematical reasoning and proof are habits of mind.
The Common Core State Standards includes eight Standards for Mathematical Practice, which
describe the expertise that mathematics teachers should seek to develop in their students. Most of
these practices are fundamental aspects of mathematical reasoning and proof, including making
sense of problems and persevering in solving them, constructing arguments and critiquing
others’ reasoning, reason abstractly and quantitatively, and modeling with mathematics.
Mathematically proficient students should be able to construct viable arguments and critique the
reasoning of others. Elementary students can construct arguments using concrete referents such
as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even
though they are not generalized or made formal until later grades. In later grades, students learn
to construct and understand mathematical proofs. Students at all grades can listen or read the
arguments of others, decide whether they make sense, and ask useful questions to clarify or
improve the arguments.
This seminar is designed in such a way that the Fellows will be able to:
(1) Understand what constitute mathematical reasoning and proof
(2) Learn techniques to understand and construct mathematical proof and then develop their
mathematical reasoning and proof skills
(3) Use mathematical reasoning and proof skills to learn mathematical contents;
(4) Apply mathematical reasoning and proof skills to solve real-world problems.
The design of this seminar is content-focused, aligned with the CCSS. The techniques and
methods of doing mathematical reasoning and proof will be embedded in various content areas,
such as fractions, ratio and proportion, algebra, geometry, statistics, and mathematical
modeling. Thus, fellows not only develop their skills for reasoning and proof, but also develop
their understanding in these content areas.
Fellows in this seminar will develop units to enable all students to-•Recognize reasoning and proof as fundamental aspects of mathematics;
•Make and investigate mathematical conjectures;
•Develop and evaluate mathematical arguments and proofs;
•Select and use various types of reasoning and methods of proof.