Math teachers need to embrace the Principle of Conjecture, according to math teacher/coach Gerald Aungst, author of 5 Principles of the Modern Mathematics Classroom: Creating a Culture of Innovative Thinking (Foreword by Hack Learning creator and podcast host Mark Barnes).
Aungst suggests that no one really hates math–that perhaps they just haven’t been taught in a way that inspires curiosity and conjecture.
“We want kids to be asking a lot of questions and then seeking those answers,” Aungst says in Episode 53 of the Hack Learning Podcast, embedded above.
Create situations that spark conjecture
Aungst learned how to inspire inquiry and conjecture in kids by creating unique situations where “something doesn’t make sense or is not immediately obvious.”
Once students’ curiosity is heightened, Aungst suggests allowing kids to immerse themselves in the problem. Rather than handing them the correct formula and a worksheet, Aungst says the teacher’s job is to ask questions the lead to collaboration and conjecture. Questions like: “What can you figure out? What can you notice here?”
When this strategy is employed, students often solve the problem with little or no teacher intervention, according to Aungst (learn more at time index 4:38 in the audio embedded above).
What to do tomorrow
The best way to hack math instruction tomorrow, according to Gerald Aungst, is to create a mindset of questioning and problem-solving in students from the beginning of the year and continue it throughout the year. Here are some do-tomorrow strategies to create this mindset:
“How do you know?” This is the simplest thing you can do to raise the level of thinking in a classroom or for your child at home. Ask this every time a kid answers a question.
IWonders: create very open-ended challenges for students that promote lots of inquiry and have multiple paths to a valid solution.
- Example: Imagine you decided you were going to read every book in your school library. How old would you be when you finished? Such a simple question, but there’s so much going on here.
- There’s no way to have one “right” answer, so students are going to have to rely on justifying their solutions and choices with solid arguments and evidence.
- Here’s another example that would work well for older students: when does it make sense to take a longer route in order to have one less traffic light? This might require some research or at least some reasonable assumptions.
- For both of these, you and your students can decide how precise the answers need to be, and what level of supporting evidence is needed to justify a solution.
- The same question is easily adapted to different students, too, just by creating different parameters or expectations for the depth of the solution
Find the Problem: Instead of only giving problems and having students work out the solutions, try these variations. All of these are great activities for partner or group work, too.
- Give the problem and the solution without the intermediate steps. Students have to figure out how to get from A to Z.
- Give students just the solution steps and answer. They must deduce the problem. For a greater challenge (and greater opportunity for creative and critical thinking), be more vague in your steps. This is also a great way to illustrate why students need to be clear, detailed, and precise in their solutions!
- Let students start a solution then switch papers periodically. They should then continue the solution from where the previous person left off. You can’t change anything they did, but you can add steps to correct an earlier error.
- Give students random bits of unconnected information and ask them to generate a problem and solution that somehow connects them.
Aungst says teachers should be asking students questions often about the problem-solving process, in order to create this mindset (listen to his list of key questions at time index 10:30 in the audio embedded above).
Gerald Aungst is the supervisor of gifted education and elementary mathematics for the School District of Cheltenham Township, in Pennsylvania, and author of 5 Principles of the Modern Mathematics Classroom: Creating a Culture of Innovative Thinking. Prior to his service as an administrator, Gerald taught mathematics at the elementary level for eighteen years in both the regular classroom and as a gifted support specialist. Most recently Gerald has been part of his district administrative team leading the transition to the Pennsylvania Core Standards and training teachers in high quality math instructional practices.
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