Things that are Good: A Problem Based Learning Approach in Mathematics

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.

Unavoidable literacy.

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?

  1. Students read/view an Entry Event, which launches them into the Problem Scenario. A prescription for a solution is NOT included.
  2. Students brainstorm “Knows”, “Need to knows”, and “Next steps”, all the while being guided by the facilitator to generate the intended learning outcomes.
  3. 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.
  4. Students present their solution in some form.
  5. 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.

————————————-

* – 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.

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19 Responses to Things that are Good: A Problem Based Learning Approach in Mathematics

  1. jybuell says:

    Excellent reflection.

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  3. adamhendry says:

    Thanks for the post. I really enjoyed the reflection as it mirrors almost entirely our experience at my school. In 2008, at our school, in Sydney, Australia, we embarked upon Project based-learning in stage 5 (ages 14-16) in most subject areas except maths (or ‘math’ as you yankees call it!). We utilised the PBL approach undertaken by New Tech as well. Maths was left out deliberately at the beginning due to what we considered the apparent difficulties inherent within the PBL (Project) approach in this subject area within a state-sponsored syllabus. In the last couple of years we have brought mathematics in to the program but we have been using a problem- based approach.

    The problem-based approach is also the method we adopted in stage 6 (ages 16-17) of the senior school preliminary course due to the nature and quantity of content that students had to ‘acquire’ to meet the requirements of a state-wide standardised testing in the following, final year, of their schooling journey. We adopted the ‘one problem in one day approach’ that Republic Polytechnic, Singapore, utilises. As the name suggests, the curriculum is restructured into a series of authentic, ill-structured and engaging problems that run over one full school day (with some modification due to the differences in indicative hours of instruction required between Singapore and the state of New South Wales, Australia). Students present their solution to the problem at the end of the day.
    Within this pedagogical approach we are still continually fine tuning for mathematics in particular; including breaking the ‘problem’ down into a series of mini-problems throughout the day!

    Anyways, thanks for letting me rant and good luck with your maths classes!

    • emergentmath says:

      Hi Adam, thanks for the encouragement! I’m glad to hear You guys are able to combine the best aspects of PBL into a rigorous math curriculum using Problem Based stuff. If you’re interested, I highly recommend the book “What’s Math Got To Do With It” by
      Jo Boaler. She outlines what good math instruction really looks like and is a fan of an inquiry based approach.

      On a side note, did you go to any of the New Tech trainings here in the States? I’ve been employed with New Tech for a few months in part to help address the inherent problems of math and PBL, and I’ve met several of your collegues I think, including some participants in our leadership summit last week.

  4. adamhendry says:

    Hello emergentmath,

    Yes, I trained there in ’08 during the New Tech summer training course. Over the last couple of years, however, I have been primarily involved in problem-based learning in the senior school. I teach Ancient History and other humanities courses but have also taught senior maths this year. I am keen to hear how you progress with addressing the problems of maths and PBL. A number of colleagues in the maths department would also been keen to hear some of your experiences as well!

    Yes, you would have met my school executive and other colleagues recently. They enjoyed the trip immensely and are reassured that we are on the right track!

    • emergentmath says:

      Awesome! I was at the New Schools training this past year as well as the annual conference in Michigan and I had the pleasure of meeting with a couple of your math teachers – and going out to dinner with them as well! From everything I hear, things are going well using a Problem Based approach. Tim has emailed a few times, and the leaders that were here last week said he was doing quite well.

      Feel free to email me if you’d like feedback or suggestions or anything like that (gkrall@newtechnetwork.org). Or maybe even we could skype sometime to talk about mathematics instruction…..

      -Geoff

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  8. blaw0013 says:

    Yes on test improvement conjecture, based on my extensive experiential data in PrBL environments. Great blog–very glad to have found it!

  9. Thanks for sharing this reflection. I agree with a lot of what you said and feel that project based learning is a great way to support the Standards for Mathematical Practice. I suspect the biggest problem with implementing this approach will be with Step 5. Students may not be prepared to think on their own and teachers need to know how to strike a balance between giving them enough information to get started yet not enough that there is little left for the student to think about.

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