Pythagorus, we have a problem! An Overview of the PrBL approach to Teaching Mathematics

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By: Josh Giebel, Counselor

CSA New Tech

Columbus, IN

giebelj@bcsc.k12.in.us

@askgiebs

“Hey there, my name is Josh. Nice to meet you”

“Oh, hi. I’m Ellen. What do you do for a living Josh?”

“I’m a math teacher.”

“Wow. I could never do that, I am not a math person at all.”

If you are a math teacher or even know one closely, chances are you’ve been a witness to this conversation on multiple occasions. Early in my career, I found a way to just awkwardly shrug it off and laugh a little and move the conversation on. Now I find myself digging deeper, much to the pain of my conversational counterpart. My goal is to get to the root of what makes a person believe they are or are not a math person (by the way, I firmly believe that the notion of a “math person” is absurd and misplaced). Ultimately, most of my conversations boil down to the idea that most people could not see the use for math while they were learning it and, at some point, they reached a topic that they could no longer understand by just following the examples on the board and repeating the process to get an answer.

Every math teacher in the country, at some point in their career, has been presented with the question, “When are we ever going to use this?”. Many math teachers dread this question; some love it, but all of us know it is coming. Over time, we have become better equipped to answer that question. We can list several applications of various standards or we can talk about how learning math is less about the specific skills and more about critical thinking and problem-solving, but the real problem is that many students will not buy it. They may decide not to further pursue the question, but the truth is that many students just do not see the reason they have to learn how to solve a logarithm or prove trigonometric identities. I may get kicked out of the fictitious, yet elite, mathematics educators club for saying this, but our students are often right. In many cases, students will likely never use the specific concepts and skills they encounter in secondary mathematics, at least not in the same way we ask them to use it in our classrooms.  To be clear, I am not saying that secondary mathematics is not used in the real world. In fact, I believe that ever secondary mathematics standard has authentic, real-life applications. I just know that for many of our students, they will not encounter situations outside of school that require them to use those specific skills.

Unfortunately, this presents a problem for math teachers. We know that engaged students learn better and learn deeper. We know that students tend to engage more in authentic learning opportunities (Walker & Leary, 2009). We also know that many students do not see the authentic learning opportunities in math and therefore can struggle to learn the content. So what do we do?

Project-based learning (PBL) allows educators to present real-life authentic problems to students as a means to deliver content standards. There are a few subjects in mathematics that lend themselves particularly well to PBL. I believe that a course in Probability and Statistics is a perfect class to use PBL as the a primary means of instruction because just about every standard has an authentic application. Moreover, learning these standards fits into the concept of authentic learning. For more on the subject of authentic learning, check out my friend and colleague’s recent post on authenticity. Another course that lends itself well to PBL is Geometry. The applications in construction, engineering, and design seem endless. I have had varying levels of success in facilitating every mathematics course from Algebra 1 all the way through dual-credit Honors Calculus with PBL. The challenge comes when you leave the content that has clear applications and start trying to piece together a string of loosely connected standards into an authentic project. At some point, learning the content becomes inauthentic because it just doesn’t quite fit into the project OR you remain in a really authentic project and sacrifice some of the content. This is far from ideal, in fact, it represents an impossible choice for many math facilitators in a PBL setting. Enter the notion of problem-based learning (PrBL). For a technical definition of problem-based learning, check out this link (please note that in this definition, the acronym for problem-based learning is PBL; for the sake of clarity, we use PrBL to differentiate problem-based learning from project-based learning.

Several years ago, I was asked to be part of a team of educators in the New Tech Network to explore the idea of problem-based learning in mathematics courses. Faced with the impossible choice mentioned above, we got to work on a new approach to teaching mathematics. We wanted something that stayed true to the core beliefs of PBL but also allowed math facilitators more flexibility in how they taught the specific standards in their course. The notion of PrBL has been around for a long time. Experts have been calling for increased problem solving in mathematics classrooms for over 30 years. An Agenda for Action,  by the National Council of Teachers of Mathematics (NCTM) pushed educators to focus on problem solving in their instructional practices (1980). This request was reinforced by NCTM, the Common Core, as well as several other iterations of state mathematical standards. Each set of standards begin with mathematical process standards (“Standards for Mathematical Practice | Common Core State Standards Initiative,” n.d.).

The concept of PrBL stems from the idea that teachers should engage students in rich “problems in which the mathematics to be learned is embedded” (Schoen, 2003, p. xi). This same approach is at the core of project-based learning (PBL). We want to engage students in an authentic real-world project that allows students to learn necessary content standards through solving the problem. The major difference in PrBL is the scope of the problems being solved. A typical PBL project may last anywhere from 3 - 6 weeks and cover 5 - 7 content standards. Throughout the project, the different content standards are uncovered by student need to knows (questions) and learning the answers helps students develop a solution to the problem.

In mathematics, it is difficult to find one application that allows students to uncover 5 - 7 different standards and use them to solve the problem. What is more common is that  1 - 2 standards have an authentic connection to the problem while the other 3 - 4 are on the periphery and do not really help the students complete the project task. This causes conflict and brings up the entirely valid question, “Why do we have to learn this?”.

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A problem-based approach breaks the unit down even further. Rather than looking for an application that connects 5 - 7 standards and takes 3 - 6 weeks, PrBL units are comprised of several smaller authentic problems and tasks that cover 1 - 2 standards and last maybe 2 - 4 days. This allows facilitators to concentrate on finding authentic learning opportunities for specific skills and concepts. An experienced and skillful PrBL facilitator will make connections between a series of PrBL tasks within a unit that allows students to see the bigger picture applications of a particular set of standards. This Venn diagram developed by New Tech Network instructional coach Geoff Krall illustrates that many of the elements we have grown accustomed to in PBL still exist and are used in PrBL(Geoff, 2012).

Facilitating in a PrBL setting requires many of the same skills that facilitating in a PBL setting requires. A PrBL facilitator needs to be able to pose authentic and challenging questions. He/She must maintain the cognitive demand of the task by asking probing questions and sustaining inquiry. He/She collects need to knows and uses them to guide his/her lesson planning and workshops. He/She facilitates productive struggle and demonstrates how to use collaboration and critical thinking to remove barriers from the solution pathway. A PrBL facilitator needs to be able to locate and develop rich, authentic, and meaningful tasks that allow students to engage in the content standards.

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Getting started with PrBL really begins by embracing the spirit of teaching mathematics through problem-solving. This can be done by starting each lesson with a challenging problem that students work to solve and in solving the problem uncover some new mathematical concept or approach. The graphic to the right demonstrates what a typical day in my PrBL classroom might look like (“NAIS - Teaching mathematics through problem solving,” n.d.). Students are asked to learn mathematics as they are struggling to solve a problem. This visual represents the basic approach to PrBL. The added component of PrBL is ensuring that the problem or task is authentic.

In order to generate meaningful learning through a problem-based approach, the task is essential. First and foremost, the task should be connected to the content standards. Beginning with the standards is crucial and will ensure that your students are meeting the expectations of the state. Furthermore, the task should allow students to uncover the mathematical content rather than applying the content after it has already been taught. Put another way, the new content you wish to teach should be necessary to solve the problem and something that your students could ask a need to know about. Lastly, in an ideal setting, the task represents an authentic application of the content standard.

A few of my favorite PrBL tasks are ones that were born out of curiosities. The first one began with my principal and I discussing the fact that camp stoves do not really have a good setting to simmer while cooking. I asked my geometry students to tackle this dilemma by looking at the inner-workings of the gas-stove valve. My colleague and I wrote an article about the task that appeared in the New York State Mathematics Teacher’s Journal. The article is called Using a Student-centered Activity to Develop Mathematical Understanding: The Gas Pipe Valve Task, it appears in Volume 66 Issue 2 of the New York State Mathematics Teacher’s Journal. The journal is for members of the Association of Mathematics Teachers in New York State. If you have access to a database, you might be able to find the article here. Here is the task:

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Another problem that I really enjoy for geometry is analyzing the effectiveness of a rear-windshield wiper. All of us have probably been in a parking lot and walked past a car with a rear window that is covered in mud except for a small streak that has recently been cleaned off by the wiper. Asking students to evaluate the effectiveness of this product is a great launching point for discussions about arcs and sectors of a circle. Better yet, ask the students to design the most effective system for cleaning a rear-windshield!

In Calculus, I ask students about effective ways to share a loaf of sourdough bread for a dinner party. The problem isn’t quite as authentic and has the sound of one of those typical word problems, but man it gets kids engaged. I even have had some of my students tweet me pictures of their bread while they are out to dinner at prom asking me if I could help the restaurant with this issue! Here is a link to the problem.

Developing these tasks may seem daunting, but there are a lot of resources that may help you in this endeavor. Believe it or not, textbooks might be a decent launching point for your journey into PrBL. Many textbooks have some pretty good application problems (and no I absolutely do not mean the ones about buying 52 pineapples and 43 avocados for a party-ha!). The issue with a lot of these problems is that they ultimately take away all of the problem solving by leading students through the solution by including parts a, b, c, and d… all of which are specific steps that are needed to solve the problem. Take away the different parts and pose the original question - then you have a genuine problem that requires students to think critically! There is a great TED talk by Dan Meyer on this very subject: Math class needs a makeover. Dan also has developed a series of 3-act math problems that are available on his website (http://blog.mrmeyer.com/about-2/). These problems are a great example of engaging students in mathematics by using real-world applications and relying on student curiosity to prompt problem-solving, research, and classroom discussion. Another great resource is Geoff Krall’s Problem-Based Curriculum Maps found on his blog (https://emergentmath.com/my-problem-based-curriculum-maps/). Both sources should be used to provide you with an idea of what these tasks might look like.

As with any consumable teacher content, you know your students and know what sparks their curiosity so you should be working to create content that meets the needs of your students. Remember, as you begin this journey, there will be bumps and bruises, but if you keep the spirit of PrBL in mind, you will be developing meaningful learning opportunities for your students. While I did spend this entire post talking about math, I do believe there is a place for PrBL in other content areas as well. With a better understanding of PrBL and some of the resources listed above, you should be ready to get started. Happy planning!


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Josh Giebel is incredibly excited to begin his 9th year as a member of the CSA family and even more excited to transition into his new role as a counselor and administrator after spending the previous 8 years as a math facilitator! He graduated from the University of Wisconsin (Go Badgers!) with a degree in Secondary Mathematics Education and went for a Masters Degree in the same field from Ball State University. He’s also the co-host and co-creator of The PBL Playbook podcast. When he’s not working, you will probably find him outside, reading, or cheering on his favorite Wisconsin sports team!