A New Way to Learn: Thinking on Your Feet

Across Lakota, elementary students aren’t just sitting at their desks during math. Instead, they’re on their feet, working together in small groups to solve problems on whiteboards and discover new ways to think. This approach is called vertical thinking and it’s transforming how students learn and work.

How Vertical Thinking is Changing Learning

Imagine a classroom with whiteboards or other erasable surfaces scattered around the room. Students work in groups of three to solve math problems, with one student as the recorder, writing their classmates’ ideas on the board. However, the recorder isn’t allowed to talk. 

If the group thinks something is incorrect, then it can be erased - but only if everyone agrees. 

“When you’re in a group of three, it’s very hard to hide,” said Woodland Elementary's  teacher leader, Kim Carlson, referring to students who may not speak up in a large-group setting. Limiting groups to three students is a core principle in the book Building Thinking Classrooms in Mathematics by Peter Liljedahl, a book that is guiding elementary math-teacher training in Lakota this year. Carlson is leading an optional book study on Building Thinking Classrooms at Woodland this semester.

For students who tend to be more quiet in traditional classrooms, these small groups help build confidence and encourage everyone to share their ideas.

Building Confidence by Asking the Right Questions

When using the vertical thinking strategy, students do most of the thinking. If groups ask for help, teachers respond with “thinking questions” instead of offering hints. For example, the teacher may ask, “what do you think” and then move on to the next group. 

If a group is truly stuck, the teacher might suggest they take a walk around the room to see what strategies other students are trying. 

Some refer to this as “borrowing brilliance.” Third-grade teacher Amy Docherty calls it a “light-bulb moment.” It helps students realize that they can keep going and figure things out for themselves. “They are understanding that they can keep trying (instead of) stopping and saying I can’t do this,” explained Docherty.

Problem Solving Through Someone Else’s Eyes

Students say that vertical thinking makes learning more interesting. And, they’re excited for what they call “whiteboard day.”

“I think it’s cool to see how other people think and different ways to do math,” said Anna, a third grader in Docherty’s class at Woodland. She looks forward to “whiteboard” days and how the randomly chosen groups allow Anna to collaborate with classmates she might not normally partner with. 

Classmate Isha agrees. “It’s good to see how you and your partners think and see if you have the same answers.” 

One of the goals of vertical thinking is that students feel safe to make a mistake. Students are graded on their strategies and reasoning, not just whether or not they have the right answer. This encourages students to be persistent and helps them think critically. “It gives students a voice,” said third-grade teacher Sharon George, explaining how the random assignment of groups helps student confidence.

When groups finish solving a problem early, teachers don’t make them wait. They add a challenge to the original problem. For example, when Anna and Isha explained that after solving a problem about a caterpillar climbing a 70-foot tree by moving two feet per day, Docherty added a twist: the caterpillar slipped down each day. The students had to solve it again, using a different mathematical strategy.

Learning New Tools to Strengthen Classroom Thinking

The Lakota Board of Education approved a new math curriculum for grades K-6 in March 2024. Along with teaching the new material, elementary math teachers are learning how to help students become stronger critical thinkers. Vertical thinking supports this goal by focusing on the process of solving the problem, not just the final answer.

Teachers learn these strategies during district professional development (PD) days. “We’ve had really positive feedback,” said Emily Hermann, Director of Curriculum for grades K-6.

In addition to the district PD sessions, Carlson’s book study is helping Woodland teachers bring vertical thinking into their classrooms in meaningful ways.

Although the vertical thinking concept is from a book about teaching math, Hermann said that it could be used for other subjects as well. “It’s meant to make all kids active in their learning.” It’s a mindset shift that teaches persistence, resilience, communication and confidence.