Reasoning can be thought of as the process of drawing conclusions on the basis of evidence or stated assumptions

Martin, G.W and Kasmer, L “Reasoning and Sense” in teaching children mathematics (2010)

What is mathematical reasoning?

Fluency

Gaining mathematical fluency allows children to have quick and accurate recall of factual knowledge. This includes number, but also other strands of maths including geometry and measure. Fluency of recall alone does not allow children to solve problems.

Problem Solving

Strategies for solving problems and applying facts to different problems are vital. Ways to do this include making a problem simpler to start with, or recording information systematically in a table. Students will strengthen their understanding of the best way to solve problems by having a go and justifying their answers to other students. Some of the problems below allow children in mixed attainment classes to participate and be successful. Much has been written about Low Threshold, High Ceiling Tasks (LTHC) and many resources are available to support this approach to problem solving.

Reasoning

No matter how good a problem is, if students solve problems alone, they miss out on the opportunity to strengthen and challenge their own learning if they have no opportunity to compare their strategies with others. Verbally reasoning gives great opportunities for children to gain more confidence in their answers or to reconsider, given the strength of someone else's reasoning. There is nothing better than a mathematical disagreement for encouraging thoughtful probing of the underlying methods withing a supportive classroom environment. Sentence starters, such as the ones below, offer a structure for reasoning which, when combined with mathematical fluency, allow children to solve problems in an increasingly efficient and accurate way.

Some of the materials below are from the OU's ME831, Developing Maths for the Primary Classroom.

A selection of puzzles from Mathsphere is here.