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Creating Thinking Classrooms with Visual Math Puzzles
This student-centered approach engages students with math challenges, frees them to explore solutions collaboratively, and then connects the entire process to underlying concepts.
GUEST COLUMN | by Matt Haber
SIMPLELINE
Throughout my years in education, I’ve often heard that a teacher’s role is to fill empty brains. Educators know that is not accurate. Students come to us with so much knowledge, cultural skill, and understanding of the world, and filling “empty” brains doesn’t work anyway. The only way to learn is to connect new experiences with old ones. We can’t shovel math knowledge into students’ brains like we’re filling a hole, but we can provide them with math-based experiences and collaborate with them to make those experiences relevant.
I currently work in Oxnard School District, where we put this idea into action by creating “thinking classrooms” for math learning, based on the book Building Thinking Classrooms. Thinking classrooms also allow students to practice skills that are in high demand among employers, including collaboration, perseverance, resilience, and problem-solving. As this approach to teaching and learning has made its way into other classes, I’ve seen students become more intuitive in subjects such as social studies and science, too.
Here’s how my district helps our students understand math by beginning with what they already know and then encouraging collaboration and exploration.
Engagement, Exploration, and Consolidation
When I was in school, a teacher would teach us how to do something, for example multiplying fractions. Then we would do 20 problems just like the one we’d been shown. There was no thinking involved. We were just mimicking, following a process we were shown with no understanding of the logic behind it. In an effort to move beyond mimicry and encourage deeper understanding, thinking classrooms have three parts: engagement, exploration, and consolidation.
We begin with an engaging and exciting launch. Recently, for example, I put an orange and a small, very light block on opposite ends of a seesaw. Of course, the seesaw tilted all the way toward the orange. The question I asked was, “How many blocks will balance the orange?” I told the class I wouldn’t give them a scale, and asked them what they would need to answer the question. They said they would need the weights of the orange and the block.
It’s important to get started quickly with a conundrum like this that grabs students’ attention. When students arrive in the classroom, I don’t ask them to sit down. I just have them toss their backpacks aside and join me at the front of the classroom. Together, we go through the engagement piece and I provide them with enough information to be successful (without giving them too much detail) in the first three to five minutes. I split them into groups by having them draw cards and putting the kings with the kings, the queens with the queens, and so on. It’s important that groups are random so that students are working with a variety of peers and encountering a range of approaches to the concepts we are learning.
Once in groups, the class moves into the exploration stage, where students use vertical whiteboards in groups of three to work together on solutions. Students talk to each other, begin to develop their voices, and exercise their mathematical agency. I give my students between 15 and 20 minutes to complete their exploration.
Educators are not always accustomed to seeing as much self-directed learning in math as they see in my teachers’ classrooms. My experience has shown me that allowing students to explore rather than “sit and get” will lead to higher interest, deeper engagement, fewer behavior issues, and a space for deeper learning to occur. During exploration, I like to give classes three “slices,” which are three different challenges. Three challenges are important because they give students who figure out the first one something else to focus on while other groups are still working.
To transition from their exploration to solidifying a solution, we move to the consolidation or closure stage, which provides students the ability to present their solutions to the class. This is where we connect the experiences they’ve just had back to math concepts, creating the opportunity to take meaningful notes about what they have discovered and learned together.
Another example of consolidation is to conduct a “gallery walk,” where all students go from board to board, writing on sticky notes to place on the boards. When we sit back down, students write “notes to my future forgetful self,” in which they sum up what they learned during the lesson. One way to encourage more meaningful notes is to have students create a math problem in addition to solving the one I gave them.
Writing their problems and solutions down helps to cement concepts in their minds, and the notes themselves provide excellent exit tickets to help teachers decide which students, if any, need additional support or instruction.
Building Thinking Classrooms with Visual-Spatial Puzzles
One of the most important tools we use to launch thinking classroom experiences during the engagement stage is ST Math, created by MIND Education, which uses visual puzzles to illustrate math concepts. I’ve actually used it with my own children, and have even completed a few puzzles myself to refresh my mind on some middle school math concepts. Oxnard adopted ST Math to support our shift from procedural math to a more conceptually focused approach.
The first time I used it as the engagement piece, I was working with a 5th-grade teacher on a fraction lesson. We selected three puzzles, each increasingly difficult, to serve as the three slices. We printed out the puzzles and made copies. At the end of the lesson, we used the software to show students how their proposed solutions worked.
I was excited because the ideas behind Building Thinking Classrooms became the structure of the class and the visual and engaging puzzles became the curriculum. The prep was as easy as making a handful of copies of the puzzles, and even that could be eliminated by having students bring their computers to the front of the classroom with the appropriate puzzle on the screen.
Since that first experiment, we’ve developed a small cohort of teachers who are continuing to use and refine this approach to engagement.
Tangible Results
To gauge how well our new model for math education was working, I identified everyone among Oxnard’s 800 teachers who use thinking classrooms at least twice a week. I then tracked the change in their Star Assessments scores from fall to spring. Students in classrooms that used thinking classrooms at least twice a week had an average improvement of 10.46 percentage points, compared to just 3.89 points among other students. I did have a control group, but this was not a rigorous study, so I can’t say for certain that the difference is attributable to thinking classrooms. It could be that the teachers using this approach are more conscientious or exceptionally hardworking. Nevertheless, it is promising.
Over the past five years, I have consistently interviewed students about thinking classrooms. One of the most frequent responses I get is that students feel less anonymous when they are standing up and working in small groups than when they are sitting at their desks. They share that they like using whiteboards so they can easily erase mistakes and start anew, making it easier for them to take risks as they work through problems.
Recently, I worked with one of our special education classes that was trying this approach for the first time. The teacher and the aides were really excited after the lesson was over because they had never seen this kind of engagement during math. There were even students who I was told usually could not work together, but who had collaborated beautifully throughout the lesson. It reminded me of the misconceptions we hold about what our students are capable of and how they want to learn. These students can work together in the right context, and many more children can thrive when we center them and unleash their agency than when we sit them at a desk with a worksheet to complete.
Building a thinking classroom like we have is logistically very easy, but as is the case with anything new, it will require practice and the space to do so. The key is shifting our philosophy about how we teach and learn. It’s not the way teachers are used to teaching, and students have been taught to learn in a certain way. If it doesn’t work the first time, that may simply mean that everyone needs a little more practice.
My goal is to create spaces for students to explore and collaborate. It took thousands of years for mathematicians to come up with the division algorithm. With a little practice, we can allow our students to have some of that same fun of discovering mathematics — instead of trying to fill their heads.
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Matthew Haber is manager of mathematics and physical education at Oxnard School District. He has been developing mathematics teachers for more than 25 years. He taught all grade levels in the Los Angeles area, then began leading teachers on special assignments and developing and facilitating professional development. For 10 years, he led mathematics in LAUSD. In 2013, he was recruited by the San Joaquin County Office of Education to improve math instruction in multiple districts. He has written two books, including New School Math for Old School Parents, a title for teachers and parents centered around supporting learners in the 21st century. Write to: mhaber@oxnardsd.org.
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