7 Ways to Increase Mathematical Reasoning.

It's March, and we've started going over practice questions for the states required standardized test.  I give one problem a day, and ask students to explain their answer.  Unfortunately, they are still at the "I guessed." or "It popped into my head." or "I'm Shaman.".  I've tried to explain none of those really tell me anything because there was no real thought behind it.  No real reasoning.

This is the first way for students to improve their mathematical reasoning.  When they explain or justify their answer, they are able to examine the logic used during their thinking.  It is important to have students show their thinking process from start to finish be it verbally during exercises or in written form for daily work or even tests. Every time they show their work, they are communicating their thinking.

2. Use geometric proofs or some sort of two column proof.  In the first column, they write down what they are given, then what they suspect while in the second column, they explain why each statement is true.  Doing geometric proofs in this format force them to look at the small steps in solving it.  This is another way to help students see their reasoning.

3. Have students work together because it allows them to justify their thinking to each other.  In the process they can analyze and critic each other's thinking.   Set up "Brain Talk" where students use modeling, verbalization, or other ways to show their understanding and justify their position.  The teacher may need to ask questions such as "What is the same?"  "What is different?"  What do you know?"  to help them get their thinking going. The important thing with this discussion is students have to feel safe and they need a chance to come up with hypothesis and solutions.

4. It is also necessary to come up with agreed upon mathematical terms across the grades because it leads to less confusion.  This makes it easier to for students to continue developing their reasoning.  When I was in school, they used the term "Borrowing" when you needed to "Regroup" as they say in today's math language.  It is also important to encourage students to use mathematical language when they explain their thoughts because the more precise they can be, the better they understand the concept.

5.  Take time to have students look at problems done incorrectly and identify the mistake.  This process adds to developing student reasoning because it teaches them to really look at the process and numbers used to solve the problem.

6.  Encourage students to find two or more ways to represent any problem since its important to "see" things in a different aspect.  This helps students move to a different process if the first choice does not work.

7.  Encourage students who struggle.  Let them know that struggle is a normal part of learning math and developing their reasoning.  The struggle is when they develop their reasoning and as they work on solving problems, their reasoning improves and it becomes so much easier.

You don't have to implement all of these at once but use one or two to start with on a regular basis till students are more comfortable with showing their understanding.  Let me know what you think, I'd love to hear.  Have a great day.