Today, we’re diving deep into a topic that has truly transformed the way I think about teaching math. We’re going to explore Peter Liljedahl’s book, Building Thinking Classrooms in Mathematics. This book is packed with practical ideas and thought-provoking strategies, and I can’t wait to share some of the highlights with you.
Let’s start with a question: have you ever tried using brain teasers or puzzles in class—activities that don’t seem directly related to the textbook? Well, those activities can actually make a huge difference in how students grasp those tougher math concepts. It’s like giving their brains a good stretch before a workout. Instead of jumping straight into complex equations, we’re warming up their thinking muscles first.
Here’s the cool part: these warm-ups don’t even have to be math-related. Any challenge that encourages critical, strategic, or creative thinking can work. For example, picture this: you show students a grid of squares, but some of those squares are combined to make larger ones. You ask, “How many squares can you find in total?” At first, students might count just the individual squares, but as they look closer, they’ll discover the hidden larger squares. This kind of task builds skills like recognizing patterns, thinking spatially, and viewing problems from different perspectives. It’s a perfect example of what Liljedahl calls a “low floor, high ceiling” task—accessible to everyone but with room for advanced exploration.
These tasks do more than just get students excited. They help develop essential mental habits that transfer directly to traditional math concepts. For instance, with that “how many squares” challenge, students practice breaking problems into smaller parts, recognizing patterns, and considering multiple perspectives. These are skills they’ll use in algebra, number theory, geometry—you name it.
Now, let’s talk about middle schoolers. Here’s a great example: give them two hourglasses, one that measures 7 minutes and one that measures 4 minutes. The challenge? Time exactly 3 minutes using only those hourglasses. It’s a real head-scratcher! What’s important here isn’t the hourglasses themselves but the thought process—thinking strategically, using trial and error, and maybe even working backward from the goal.
For high schoolers, logic puzzles can be especially engaging. Imagine this scenario: a guest arrives at a hotel, and their credit card is declined. The receptionist is suspicious, and students are given a few clues to solve the mystery. This type of task fosters deductive reasoning—analyzing information, identifying patterns, and making logical connections—all skills essential for advanced math concepts like proving theorems or solving complex equations.
Now, you might be wondering: how do we bridge the gap between these fun, non-curricular activities and the actual curriculum? That’s where scripted curricular tasks come in. Instead of just presenting formulas or equations, these tasks guide students toward discovering core mathematical concepts themselves. For example, when teaching the Pythagorean theorem, instead of starting with the formula, you could present students with several right triangles and their side lengths. Then ask, “What patterns do you notice?” By investigating, they’ll likely discover the theorem on their own.
These scripted tasks help students experience the joy of figuring things out for themselves, leading to deeper and more lasting understanding. And here’s the connection: when students practice problem-solving strategies during non-curricular tasks, they carry that flexible, creative mindset into curricular challenges.
You might worry about finding time for these activities in an already packed curriculum. But even dedicating just 5-10 minutes at the start of a lesson to a non-curricular thinking task can make a big difference. And you don’t have to create everything from scratch—there are plenty of resources and communities out there full of ideas you can adapt for your own classroom.
So, what’s the takeaway? By incorporating non-curricular warm-ups, scripted discovery tasks, and thoughtfully modified textbook problems, we can build a classroom culture where thinking and problem-solving thrive. It’s not about perfection—it’s about creating a space where curiosity, exploration, and a love of learning are at the center.