A New Mindset

In one of my book studies that I'm participating in, I'm forced to reconsider the way mathematics is approached in the classroom. Dr. Jo Boaler is a professor at Stanford University for Mathematics Education. Her passion involves helping math teachers ACTUALLY TEACH mathematics. She strives to encourage teachers to move away from "sit and get" styles to pattern-investigation methods.

"When textbooks introduce only the simplest version of an idea, students are denied the opportunity to learn what the idea really is." Dr. Jo Boaler from Stanford University

What I'm reading

She is breaking down the wall and suggesting new, innovative ways to improve EVERY student's success in maths. She does not support the idea that you have a "maths" brain. She explores various research studies that show any student can learn maths if taught using some of her strategies which can include:

• Show examples and non-examples of definitions
• Rethink homework assignments to be reflection based instead of problem based
• Have students explore different methods and compare and contrast
• To reinforce concepts, have students use the concepts in different ways
• NO memorization - but utilize BOTH sides of the brain
• REMOVE timed-testing and math facts

I know what you're thinking...how can I do this? How can students learn maths if I don't show them the best methods and those precious shortcuts?! 

One of my toughest lessons in Algebra was Completing the Square. Students didn't understand for one thing, why it was even called that! They couldn't remember "all the steps" and couldn't make a connection what was really happening. Students memorized the steps and continued on their year which eventually lead to a brain dump to make room for the next memorization event.

Jo discusses the idea of "compression" in our brains. She explains, "when you learn a new area of mathematics...it takes up a large space in your brain." Once you play with ideas and dig deep, you can "file" them away and "compress" them. My biggest "Ah-ha!" moment was when she states, "Notably, the brain can only compress concepts; it cannot compress rules and methods." 

This allowed me to re-create my lesson on Completing the Square utilizing Algebra Tiles to help students explore the concept and build their own connections. If you've never used Algebra Tiles, I HIGHLY recommend it.

Student Notes (Google Slides)

Students build the polynomial using the tiles to literally make a square and determine how many 1's would it take to "complete" it. Throughout the process, students are reflecting on their learning and creating their own solving steps. For the extension exercise (i.e. homework), they will discover the Quadratic Formula! WHAAAA?!!?! Additionally, they will respond on a Flipgrid about their learning.

Student Reflection assignment

Materials:

• Notes: Complete the Square (1 copy for each student) - Google Slides
• Khan Academy Video (already in slides)
• Flipgrid: Create a Quadratics Topic for students to post their thinking
• Bitmoji: Students add bitmojis on their exercises to show how they feel

My Take-Away: 

Every student felt successful learning this new method. I had students make connections to other topics (graphing quadratics using the vertex) and even preferred this method OVER factoring! I couldn't believe the positive energy that was occurring and for once, the students were the ones doing the thinking! I became a guide for the day and my students didn't feel the need to have 20 identical problems for homework. 

Questions you could ask them for reflection exercises (consider using Flipgrid):

1. What kind of number for "b" makes completing the square easier? Explain your thinking.
2. What do you think would happen if "a" is not 1?
3. Do you think this method could work every single time? Defend your position.
4. Is there a time that this method works better than another? Explain by creating your own example.
5. Compare this method with factoring.