Education, Mathematics

Learning in a Content-Saturated Environment

Mastering existing content is no longer enough; in today’s world we also need to master the process of learning, critiquing, using, and contributing to content. We need to develop the confidence to solve complex problems. We need to learn how to work effectively in teams.

In short, we need to think through how we learn best, exercise that approach, and strive to grow and improve.


Here are some practices used by expert learners. Can we deliberately and actively attend to them in today’s classroom practice? In team problem-solving and research? How might we do so?

Practices of Expert Learners

The Beagle Method

At Beagle Learning we believe the key is to practice in the classroom the learning cycle we naturally follow on our own accord outside of the classroom: an iterative cycle of finding, exploring, and questioning content. (Think of how you naturally research a question on the internet.) That way, we can be practicing and reflecting on every part of the learning process with our peers and instructors there to help us learn.

How We Define the Beagle Method

  1. Identify a goal: Identify a topic, question, or challenge to be explored. Then enter a cycle of learning to progress toward the goal, namely:

  2. Explore: Find and explore a piece of content or idea that relates to the goal.

  3. Respond: Have a response to that content — a question or query, an idea for a next possible step, or identification of an idea that needs clarification — and identify a set of natural next questions that are relevant to the declared goal.

  4. Decide: Decide on a next step to follow, the question or idea identified that seems to move best toward the declared goal.

  5. Repeat: Repeat this Explore/Respond/Decide cycle, the inquiry cycle, finding and exploring new content in response to the chosen questions asked.

An Example

Last year I put this learning cycle into practice. I ran a test with a group of self-volunteer high-school mathematics educators, from different parts of the continent, interested in taking a deep dive into the curriculum topic of logarithms. These teachers wanted to understand their mathematical story so as to figure out better ways to teach them in their classes.

We had only six weeks or so of the semester remaining, so we knew this would only be the start to a full experience, simply a test run of the idea of having a purely on-line, iterative, learning conversation. All was done by weekly emails and the sharing of PDF documents. A hand-drawn flow-chart of the conversation was also shared each week.

Identify a Goal: We had a conversation goal. Improve our teaching of logarithms by reviewing the standard curriculum on the topic with an eye for finding a meaningful, student-relevant story-line.

Explore: And to start off the conversation I, as instructor, shared a first piece of content for review. Essay: “A no-fuss explanation of what a logarithm is.” A quick overview of what logarithms are and the historical motivations for them.

Respond: Over the week that followed … Participants read the essay and emailed me their questions about it.

Decide: I shared the questions with all the participants and it was clear from the input of the group… Two question ideas were particularly pressing: “Who was Napier and what big problem was he trying to solve?” and “Why do we care about logarithms?”

I then wrote short essays responding to these two questions and shared them with the group over email. They read and mulled for a week, and …

Repeat: New questions were submitted, new essays in response were written and shared, and the process was repeated.

I have since drawn the conversation chart in Beagle software. You can see it below, with the pressing questions in gray and the title of my essays in response linked to the questions.

(Those cards can be opened to read the essay and watch videos I also made on the topic.)

Logarithms: What Are They?


One can see that the discussion started expanding into three general topics — the mathematics of logarithms, the history of logarithms, and the teaching of logarithms. By keeping this map in mind as we discussed, we started seeing connections between previously asked questions and new content.

If we had the opportunity to continue this exercise further I am sure conversation would have turned to a focus on teaching, the theme tied to our goal. But, as instructor, I also knew there was fixed mathematical content we had to discuss: the standard and expected textbook mathematics of logarithms. I started preparing essays on those topics even though questions about them had not been asked, and was going to bring them into play when the timing seemed appropriate. (I also knew I could exercise my right as instructor to nudge aspects of the conversation in these directions too.)

Of course, all that was presented in this conversation is readily available, in some form, on the internet. I chose to write essays in response to each chosen question and played the role of a “professor” for sure, giving my personal slant on matters. But a much freer, and more appropriate discussion, would be one of having participants find or create content in response to questions asked, and holding online discussions about the depth and clarity of the content presented. I certainly had that in my mind to try to turn the experience that way if it continued. Why didn’t I from the get-go? I think I was nervous about the experiment and wanted to be assured of some initial reliable structure.

But that makes a lovely point. Even if I am a professor nervous about letting go of the reins of professing, this approach offers a valid, and I think worthwhile, foray into non-linear teaching and learning, one that feels natural and easy to put into practice. I felt confident I could make sure we’d cover the required content, and I enjoyed the surprise forays into areas of logarithms I hadn’t really thought about myself.

And those practices of expert learning? I could see them actually being practiced even in this little experiment. They were just naturally being exercised and there was no need to profess about them.

I am eager now to try running a full, non-experiment course on a next deep-dive into mathematics. What would you like to learn about?

End Note

If you are curious about an in-person university research seminar taught via the Beagle Learning method in 2015 check out this blog post.

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About James Tanton

Bringing joyful, genuine, meaningful, uplifting learning to the world is my thing … especially with mathematics. Global Math Project, Beagle Learning & more!
  • Phoenix, AZ