Preparing Boys for Life.
Putting mathematics into practice
Nate Bridge, Upper School Mathematics

Four Upper School teachers embarked on an intensive training session on Project-Based Learning this summer at the University of Pennsylvania Graduate School of Education. In this blog post, Upper School mathematics teacher Nate Bridge shares how he has incorporated what he learned, especially about disciplinary learning and teaching real-world problem-solving skills, into his classroom. 

Last summer, I was part of a team of four faculty at The Haverford School who attended the University of Pennsylvania’s Project-Based Learning Certificate program to learn more about how to infuse this teaching concept in our classrooms. One can think of Project-Based Learning as a strategy that engages students in a complex, inquiry-based task where learning happens through a thoughtful process. The approach isn’t just about incorporating projects as assessment tools for students to demonstrate their learning at the end of a unit. Rather, the learning of the content is constructed while students work through their projects. Additionally, students should be able to make connections between the skills they employ to complete the task and the disciplinary practices they would expect to see from “real world practitioners” of the subject.

Of the four pillars defined in Penn’s Project-Based Learning program - Authenticity, Disciplinary Learning, Collaboration and Iteration - Disciplinary Learning may be the most paradoxical in practice. As a math teacher, I have noted that the way mathematical practices are used in the real world often look nothing like the math we do in the classroom. On the one hand, subject area content and ways of teaching it are well entrenched in the core subject courses we teach. As students, we lived out our experiences in the classroom practicing the skills and memorizing the processes. As educators and parents, we know what a math class should look like, or what an English paper should read like, or what a science lab should feel like. But on the other hand, fundamental to the charge of Project-Based Learning is the idea that students should have classroom experiences that are authentic; the math taught and learned should have explicit parallels to the experiences and adventures they might have outside the walls of the classroom. So the question as it pertains to Disciplinary Learning quickly becomes “what do mathematicians, scientists, or writers do in the real world?” How are the skills and practices they learn in our classrooms actually applicable to the lives they will lead after they graduate?  

Fundamental to the charge of Project-Based Learning is the idea that students should have classroom experiences that are authentic; the math taught and learned should have explicit parallels to the experiences and adventures they might have outside the walls of the classroom.

The perceptions that students may have of a mathematician don’t often align with the careers in actuarial sciences, computer analysis and programming, economics, or engineering and data sciences. Although the math techniques learned in K-12 classrooms are a small step along the way to these mathematical occupations, hardly are these techniques actually employed. Instead, folks in these fields use a diverse range of critical and creative thinking skills, working collaboratively to solve complex problems often across multiple disciplines and areas of expertise.

So how do we re-envision Disciplinary Learning in our classrooms in light of this apparent paradox: the tension between our traditional notion of subject matter teaching and learning, and the occupational draw toward increasingly diverse and interconnected fields of work in adulthood? It begins with renewing our commitment to explicitly defining and challenging our students to employ higher order thinking practices in their problem solving and then to use these critical, creative, and collaborative thinking practices to serve as the vehicle through which to teach the subject matter.

In my own teaching I’ve taken a hard look at how much I actually need to explain before setting students off on an assigned task. For many of us teachers, our first instinct is to say, “Of course I have to show the students an example and explain it before I can expect the students to do it!” But if we explain too much at the outset, it takes away from the students having to problem solve the process for themselves. It takes away from the students needing to work together creatively to overcome challenges. It takes away from the very real experience of failure and trying again. It takes away from the students’ sense of discovery and engagement. In short, it minimizes the opportunities for higher order thinking.  

If we explain too much at the outset, it takes away from the students having to problem solve the process for themselves. It takes away from the students needing to work together creatively to overcome challenges. It takes away from the very real experience of failure and trying again. It takes away from the students’ sense of discovery and engagement.

Another critical element of Disciplinary Learning is selecting a task that is complex enough that it requires real struggle and collaboration. Recently, my colleagues and I in the math department took the traditional word problem of optimizing the volume of a box found in every calculus textbook and made it more complex. At the outset, no student in the class knew what dimensions of a 3D rectangular box would produce the greatest volume. They didn’t even really know how to get started on it. So we did what all math scientists do. We built a bunch of boxes, that is, we collected data. We organized the data in a table and plotted it on a coordinate plane to determine which of the boxes we built produced the largest volume. And it worked, except we didn’t know if there was a box with a still larger volume that wasn’t represented in our data. So we found an algebraic function that modeled our data, graphed it onto our data, and sure enough, we saw there was a maximum point on the curve of our graph in between two data points we had plotted. Now we needed the calculus technique – algebra doesn’t provide us with a tool for optimizing functions. We had already learned the skills of finding where the tangent lines to curves are horizontal, so the students were able to creatively figure out how to use their past learning in a new context in order to successfully determine the dimensions of a box that optimized the volume. 

The key elements of this learning task were that it was complex enough so that no student knew how to solve it at the beginning, and I didn’t teach it to them first. That created the conditions for higher order thinking and disciplinary practices more aligned with what the students might ultimately be faced with if they pursue math-oriented professions. It’s not necessarily a matter of prioritizing higher order thinking over the subject content, but how the thinking practices enhance the deep and critical learning of the subject matter.  

Disciplinary Learning is not training students to be mathematicians, scientists, or writers. It is teaching them to be profound thinkers. I believe that when we take this approach, both students and teachers will feel their experiences in the classroom to be more transformative and rewarding.

The good news for teachers who want to move in this direction is that there are already several curricular sources with well defined higher order thinking standards including the College Board and the Common Core. In these resources, one might find that teachers are tasked with challenging students to “make sense of problems and persevere in solving them” or “construct viable arguments and critique the reasoning of others” or “attend to precision.” The reality is that most teachers over the past 70 years would say that these thinking skills are important for learning the subject matter in a deep and meaningful way. The difference is that now these thinking practices are explicitly defined and made central to the tasks of teaching and learning.

If we approach our subject areas as teachers with the primary goal of supporting our students to develop higher order thinking practices, then we will be able to authenticate our students’ learning experience while ensuring that they master the individual techniques inherent to our subject areas. Disciplinary Learning is not training students to be mathematicians, scientists, or writers. It is teaching them to be profound thinkers. I believe that when we take this approach, both students and teachers will feel their experiences in the classroom to be more transformative and rewarding.

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