As a teacher in a Project Based Learning (PBL) school, I cherished the projects I created. I worked diligently to ensure that all PBL units I developed were rigorous and engaging. They were oftentimes a beautiful marriage between my Schoolwide Learning Outcomes and my state content standards, a labor of love that took weeks to prepare and weeks to implement. I had students usually more drawn to art or reading or athletics declare that mine was the first math class they ever enjoyed. A substantial pillar of this class they so enjoyed was a PBL approach in which students created amazing complex products that aligned with my project goals. In order to develop math fluency, I incorporated consistent math scaffolding into a meaty project tied to the “real world” that would create cognitive connections in students’ brains that would last well after the unit test was completed.
These were good things.
I am now, however, unconvinced that a solely PBL approach in mathematics is always the most effective conduit through which math and 21st Century skills are transmitted.
These are also good things:
Reading critically is good.
My pet theory as to why our math scores went up in year one of switching to PBL wasn’t that they were necessarily better at math, but that since they were used to reading an Entry Document and analyzing it and problem solving without direct instruction, they didn’t freak out when they saw a complex problem on a state test. Instead of simply, say, multiplying together every number in a long word problem, they actually read the problem and tried to parse out what they knew and what they didn’t know.
Writing for explanation is good.
Along with reading critically, writing mathematically is a universal good. It’s why students must be given the explicit task of parsing out a problem. When you look at this example of a Problem Solving Framework intended for student consumption and guidance, you’ll see it’s hard to avoid critical reading and writing in your math class.
Students steeped in inquiry is good.
You always want there to be enough bread crumbs for students to want to follow a particular path. If you can provide that narrative such that students are just barely at the answer, you’ll have them desiring you to teach them. Furthermore, you always want them to be striving towards a higher plane of learning. Oftentimes, once students reach a solution, if you haven’t provided additional routes to a solution, or the potential for multiple solutions, their thinking will cease.
Continual assessment is good.
Either informally or formally, teachers need to be constantly assessing for student understanding. Questioning, journal entries, and formal assessments such as quizzes are ways for you to track your students’ progress and keep them focused on the math at hand. As you know, it doesn’t take much for a student to disengage in math. Sometimes, if a student goes just one day without being assessed in some form or another, there’s a potential to not obtain the requisite mathematics you intended.
Multiple entry points into a math problem are good.
Even if it’s just drawing a diagram, if every student can make an initial attempt at a Project or Problem, then you’ve already avoided immediate disengagement. And once the whole class is engaged at the start of the problem, you can carry that momentum.
Student conversation about math is good.
Another theory: any student who can eloquently describe a complex mathematical process will never, ever fail a state mandated assessment.
These are some of the tenants of a Problem Based Learning approach, which I am convinced is an extremely effective mode of both mathematics instruction and 21st Century skills development.
At the New Technology Network New Schools Training (#NST2011), I had the opportunity to work with math teachers new to an inquiry based model of instruction. I asked them to list the characteristics they would like their students to leave their class with. Among the characteristics that received the most audible support were Confidence, Critical Thinking, Persistence, Math experts, and great Collaborators. Upon reflection, the teachers and I came to the realization that a Problem Based Learning mode of instruction can develop these characteristics in our math students.
I would define a good problem as one that contains multiple attributes of the list of “things that are good” above. Those are sort of my criteria. Even my best Projects saw students go a couple days here and there without them being practitioners of mathematics: requiring students to narrate a voice-over to a video. Even my best Projects were contrived when I tried to fit math into the Project were it doesn’t belong, affecting the authenticity of the Project at its core: requiring parallel lines to be drawn onto an already-created light rail map for some unknown reason. And once the threads of math and authenticity begin to get pulled away from a Project, it can unravel quickly. And then you’re left with a five-week investigation that students don’t fully believe and you don’t fully believe in.
So what is a Problem, in the sense I am talking about a Problem Based approach in Mathematics? What does it look like?
- Students read/view an Entry Event, which launches them into the Problem Scenario. A prescription for a solution is NOT included.
- Students brainstorm “Knows”, “Need to knows”, and “Next steps”, all the while being guided by the facilitator to generate the intended learning outcomes.
- Students work in pairs or groups to solve the problem, beginning with what was brainstormed in the “next steps” section of the entry document. The facilitator prepares workshops and lessons and has helpful resources at the ready, as needed.
- Students present their solution in some form.
- The teacher/facilitator asks guiding questions, prompts generalizations and promotes connections. *
This process is similar to the Projects of PBL, the primary difference being size and scope. A Problem generally focuses on one specific key concept or skill and lasts only 2-5 days, unlike a large swath of standards and 2-5 weeks in the former. This process allows for all the inquiry-based learning of PBL, but on a much more rapid scale to allow for differentiation, assessment, and possibly revisiting a concept or two. Also, by nature of their shortened scope or size, it forces the student to think about the mathematics involved, rather than the final product. This final point is crucial.
What, then, of all the Projects I created as a PBL teacher? I really liked them, and sometimes my students did too. Moreover, they were scaffolded well enough to avoid one of the great pitfalls that can sometimes plague Math PBL: students focusing solely on the product, not the process, thereby avoiding the Math content. Am I just to toss out those Projects that I worked so hard on and engaged students?
Either due to my better judgment or my educational hoarding tendencies (I think most teachers have this disease) I have not and probably will never take the step of dragging any folders into the recycle bin on my desktop. Why? Because I might want to use them someday. Maybe I’ll see a way I can condense those sprawling, hyper-relevant (or pseudo-relevant) projects into a concise, maybe-not-quite-as-relevant problem. Maybe I’ll use one of the scaffolding activities as the basis for a problem. Shoot, maybe I’ll use it as a Project again someday.
I can’t deny that I had students buy into my class through the drug of the product-focused Project. I’m really glad I did, too, because I’m just not that good of a teacher that I can get by solely on math content. If I have to bribe students every now and then with the carrot of them creating a cool animation, video, or presentation in front of community members, so be it. And if you want to do the same, and have the ability to create engaging, authentic, rigorous, and math content-rich PBL units, go for it. And while you’re at it, post it somewhere public so we can all see it and learn from you. But continually ask yourself the following questions: what are the students engaged in right now, the product or the math? When’s the last time I probed each individual student for their math knowledge? And, is this Project really that good, that I can be sure, absolutely sure, that students are becoming fluent in mathematics?
The challenge here is not to make you question the use of Project Based Learning, or even to disregard it as a potential math educational tool. The challenge is to develop Problems that are equally engaging as your projects. So let’s begin that work.
An array of Problems that are equally engaging as Projects, and to get the student to consistently think about math: those would be things that are great.
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* – Update: This 5th step did not appear in the original version of this post, but a friend pointed out that this is as crucial as any step. She was absolutely correct.
emergent math 