Mathematical Reasoning: Developing, promoting and expanding thinking in whole group and small group
- Recognizing 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
What does all this really mean? Well in the next few posts I would like to explore this. I believe reasoning is taught over time and reinforced through our daily practices. It involves questioning, prompting, listening, challenging, proving, giving evidence, thinking out loud, thinking with a partner, thinking by yourself, thinking in a group, and thinking publicly to name a few things. Teachers set up environments where this happens. We set students up to reason or not to reason. We set them up to publicly prove what they know or not. We teach them specific reasoning skills or we don’t.
My point is students don’t just start reasoning. This is a “habit of mind” that is developed, practiced, promoted and expanded over time and throughout the years. Schools have to think about how they develop a culture of reasoning. Grade levels should come up with some benchmark routines that promote reasoning. You should think about how reasoning is interwoven throughout the day, week, the month and the year.
Here are some sample weekly routines:
What’s the problem routine? You give the students some numbers, say 10 x 30 and ask the students to make up a story about this.
Convince me paper: You give the students something that they have to convince you about: Convince me that ….. (1/2 is greater than 1/3)
True or False essays: You give the students a statement and they have to decide and defend whether it is true or false. For example: Anytime you multiply a number by 0, you get 0. True or False? Defend your answer with 3 examples. Show your thinking. Explain your thinking.
Teachers can do these at any grade level with the appropriate content. These are great thinking prompts that work across grade levels. It’s important to think about access for all. I encourage teachers to develop different graphic organizers for the responses so that they are tiered and give different entry points. I recommend that you process them in small guided math groups so that you can hear the thinking. I also think there should be some whole group sharing, so that the entire mathematical community in your class can engage together. There are benefits to facilitating both types of conversations and there should be a balanced approach.