Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?
The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.
Real-World, Meaningful Problems
This kind of scenario — linking the drama of mountaineering with work on percentages in middle school math — can effectively be used in a problem-based learning (PBL) approach in the classroom. You may have heard a lot about PBL in recent years. It’s an instructional approach where students learn by actively engaging in real-world, meaningful problems. Instead of traditional direct instruction, PBL in math encourages students to explore, discuss and understand mathematical concepts by solving problems collaboratively. It fosters critical thinking, problem-solving skills and a deeper understanding of mathematical principles by placing students as active learners rather than passive recipients of information.
Knowledge Through Experience
The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions, to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information.
You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.
The Traditional Approach
Problem-based learning has a rich history in American education, with John Dewey laying the theoretical groundwork in 1916 and McMaster University pioneering the PBL program for medical education in 1969. More recently, the National Council of Teachers of Mathematics published Principles and Standards for School Mathematics in 2000, setting forth a vision that emphasized problem-solving, reasoning and communication, aligning closely with the principles of PBL. It encouraged teachers to design learning experiences that engaged students in meaningful mathematical thinking and problem-solving.
Yet, PBL as a pedagogy has seen varied levels of adoption in math classrooms across the United States, with many math teachers falling back on traditional teachings of formulas and procedures. In 1998, critical research that examined data from the Third International Mathematics and Science Study (TIMSS) revealed teaching methods varied significantly across cultures, resulting in a “teaching gap.” According to video documentation in the study, U.S. math teaching predominantly focused on procedural skills, where students spent most of their time acquiring isolated skills through repetition. In contrast, Japanese teaching was measurably more focused on deeper understanding, where students engaged more with solving challenging problems collaboratively.
Not “Math People”
The procedural approach can be effective for students who thrive in structured environments and excel at memorizing formulas and algorithms. However, for many students, this approach leads to a sense of disengagement and a belief that they are not "math people," a negative self-perception connected with math anxiety. Math anxiety can increase cognitive load, which is the mental effort required to process information. When students are anxious about math, their working memory resources may be diverted to managing the anxiety rather than focusing on solving mathematical problems, resulting in decreased performance in procedural tasks.
So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.
I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.
Supporting Teachers With Implementation
Why aren’t more math teachers adopting problem-based learning in their class lessons? One main stumbling block is a lack of teacher training in PBL methods. Implementing PBL effectively requires a shift in the role of the teacher from a knowledge provider to a facilitator of learning.
In my experience introducing the problem-based curriculum Imagine IM to teachers, professional learning is key to helping teachers make that shift. Experiencing PBL firsthand allows teachers to develop a deeper conceptual understanding of math while building confidence in their own teaching practices. Teachers are not only mastering content knowledge but also shifting mindsets to a more exploratory and inquiry-based approach. I see the same reactions in teachers as I do from students, the same joy in being creative with math: “Hey, I did it! I figured it out!” And then after that — and this is what we do at Imagine Learning — ongoing support and coaching are critically helpful.
Skills and Understanding
Despite the challenges, the trend toward PBL in math education has been growing, driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.
