The Common Core State Standards in mathematics ask students to be able to reason abstractly and quantitatively. But what exactly is mathematical reasoning? In basic terms, reasoning is the ability to come to a solution using critical-thinking skills. The idea is that instead of learning a memorized set of multiplication tables or algebraic formulas, students should understand why certain math functions work the way they do and how to use those functions to come up with the correct answer. So, whether students are in elementary school math or high school calculus, they can use mathematical reasoning. But, as many educators have found, mathematical reasoning isn't an easy skill to teach. Here are three ideas for improving students' mathematical reasoning:

**1. Help students ask 'why?'**

The most important way to teach mathematical reasoning is to instruct students to justify their answers. If they can verbalize how they arrived at their answer, they can more easily pin point the logical thinking that was involved. For example, say you ask students to solve this equation:

12 + X = 73 + 15

Logically, students could reach the answer in a few different ways. First, since 12 is only 3 less than 15, the numbers are relatively easy to compute. So, after that reasoning, students could conclude that the answer must be 73 added to 3, or 76. Or, since logically X must be equal to 12 less than the sum of 73 and 15, students could first add the larger numbers to 88, then subtract 12 from 76. As much as possible, have students explain their thought processes in this way, and make sure they show their work on assignments and tests to practice this line of thinking.

**2. Teach proofs**

Geometric proofs are a practical application of mathematical reasoning. They ask students to write down first what they are given in a geometry problem, then what they suspect. Then, in a second column, students must write out why each statement is true. Geometric proofs force students to look at problems in small increments, rather than quickly solving them in their head without thought. In that way, they help students understand the reasoning behind solving the problem.

**3. Have students work together**

To help students practice reasoning, have them work in pairs or groups. When they work together on a math problem, they'll be able to justify to each other how they got an answer, and they'll also be able to analyze and critique the other students' reasoning.