Mathematical Thinking.

 

The third part of teaching for mastery is to help students develop mathematical thinking.  This is the process where students look for connections, reason, generalize, and so much more.  Mathematical thinking is also referred to as mathematical reasoning. Although it is better for students to develop this skill in elementary schools, we can help middle and high school students learn it. 

So today, we'll look at ways to help students either improve or work on their mathematical thinking/ reasoning.  It is important for students to have strong reasoning alongside the skills to learn and do well in mathematics.

In fact, mathematical reasoning is often thought of as applying both logic and critical thinking to a math problem to make connections so as to solve it.  It is what bonds the students skills together and connects fluency to problem solving.

There are two types of reasoning, one is inductive and one is deductive.  Inductive reasoning is where one comes to a conclusion based on what they observe so the conclusion may or may not be factual, Deductive reasoning is based on reaching a conclusion based on facts.  Over all, mathematical reasoning requires the use of mathematical vocabulary and active listening.

It is suggested that one start class with a question or activity that requires students to struggle.  In addition, you as the teacher are not the answer key so don't provide answers.  Let students share their original ideas. The question you pose at the beginning of the class could be the objective just rewritten into a question.  Instead of telling students that they will be learning about using a ratio to describe the relationship between two quantities, ask them what it means if you use a ratio of fat to flour of 1 to 2.  

When you change learning objectives into questions, you automatically engage students because they have something to think about.  In addition, they have to show their thinking or reasoning when they find an answer.  When they have to share their thinking, they begin to develop deeper understanding of the material.

Rather than providing hints or answers, help students learn to unpack the question at the beginning of the lesson so they bring their ideas together during the lesson.  This leads to a mathematical discussion where they may have to provide visualizations to show their thinking.  Think about using a "never, sometimes, always" activity to help foster reasoning and problem-solving skills.

So think about starting the lesson with a question, a provocative mathematical statement or mind bender.  Present answers as puzzles rather than just giving the answers. Group students together in threes so one talks, one records, and one listens and watches.  Sprinkle reasoning prompt posters around the room.

Have fun trying some of these ideas.  Let me know what you think, I'd love to hear.  Have a great day.