Education, Mathematics

Teaching Trigonometry with Inquiry Learning: A First Step?

Inquiry-based learning, project-based learning, and exploratory learning are terms we often hear today. Even if we don’t know what they mean in practice (I confess, I am not sure that I personally do), we do know that they each refer to a transformation of how teaching and learning is conducted in the classroom. And we do know that they are attending to important attitudes and skills perhaps ignored in traditional lecture halls.

11 Attitudes and Skills That Transform Learning

When I reflected on this for a moment, eleven ideas came streaming from my brain — and I am sure some might come from your brain too. And why were these ideas in my head? Well, because they are precisely the practices I conduct when actually doing mathematics! (What skills do you regularly practice?)

  1. Developing a sense of confidence with not knowing and taking steps to move forward nonetheless
  2. Being curious about what we do know and how we know it
  3. Developing the tenacity to persist with problems and to handle the false leads and dead ends they might present
  4. Recognizing nuance and being patient with ambiguity
  5. Learning to adjust one's perspective on a problem to find deeper insight
  6. Conducting meaningful play with ideas
  7. Learning how to ask questions and adjust these questions to ones that are manageable for progress
  8. Seeing connections to and between previous known questions to ones that are manageable for progress
  9. Seeing connections to and between previous known content and answers, along with seeing the "big picture" of matters
  10. Learning how to assess answers
  11. Learning and practicing content (and perhaps do the fluency drills to master it) once the content has context

So here’s an idea … even if I don’t know what it means to conduct an “exploratory learning” course, what if I simply presented the content outline of a course as a non-linear mind-map? What mindset would it, by default, provoke for my course?

An Example

On the left, is the Table of Contents for my book on high-school trigonometry (published by the Mathematical Association of America). Onthe right, the same content presented as a mind-map through Beagle’s software on the right.


  • Even if I march through the content in a linear way while teaching a professional development class on high-school trigonometry, has not a two-dimensional image alone spoken a powerful one thousand words?

  • Have I not implied that a standard school topic, such as mathematics, has an organic, interconnected nature to it?

  • And have I not subtly invited the participant to consider inter-connections, tangential themes, and the possibility that there might be even more to explore?

  • Might too have I broken the mindset that one must completely master one topic before moving onto the next, knowing that later content is connected to earlier topics and so might offer further clarity and nuance to beginning ideas?

My sense is that a traditional table of contents in a syllabus can leave an authoritarian impression, giving a student no sense of permission to take ownership of his or her thinking, and so might colour our in-class discussions.

Imagine projecting an image of non-linear course map on a screen during each class. Does not the map alone invite questions about interconnections and possible extensions?

(And dare I mention that the Beagle software is interactive? With access to the map course participants can add comments and questions, conduct discussions, upload additional content, draw connections, and rearrange the visuals. Matters are set to provide another easy step towards exploratory teaching — when I am up for it!)

Author image

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