Musing Mathematically: A Tour

Musing Mathematically is a blog written by Nat Banting, a mathematics teacher, and mathematics education lecturer at the Department of Curriculum Studies at the University of Saskatchewan, Canada.  His blog, which began back in 2011, is centered around the ideas behind teaching and learning.

As he describes on his website, “his academic interests include the ecological and biological roots of cognition (particularly pertaining to the instigation and observation of student problem-posing), the decision-making of teachers and students with(in) high-density mathematics classrooms, and student (and teacher) impressions of probability.” In this tour, I will give a glimpse of some of my favorite recent posts on the blog.

Thoughts on Thinking Classrooms

In this post, Banting discusses his thoughts on the ideas presented in the book  Building Thinking Classrooms by Peter Liljedahl. In this context, a thinking classroom, as defined by Liljedahl himself, means,

A classroom “that’s not only conducive to thinking but also occasions thinking, a space inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion.” – Peter Liljedahl, Building a Thinking Classroom in Math

It has 14 elements, that if your curious, you can see in this wonderful sketchnote by Laura Wheeler. Banting states in his post, ‘I don’t have a thinking classroom’. I was taken aback a bit by this, since I for one, believe many of the mathematics teachers I know strive to create thinking classrooms. He does, however, mention that the statement comes with three qualifications which I encourage you to read. The one that caught my attention the most was the second one in which he points out how the idea of “Thinking Classroom” get’s simplified to  “any class with students standing at whiteboards in random groups.”

The structure above serves as an example of two of the elements of a “Thinking Classroom”:  vertical non-permanent surfaces (VNPS) and visibly random groups (VRG). I found this very insightful since with a lot of active learning scenarios. I know, I’ve counted on these structures creating a conducive environment for learning that is lost in traditional lectures. As Banting  puts it elegantly,

“However, those structures do not teach; they amplify potential teaching moments. We, as teachers, still need to harness them, and Peter develops many ways to do so in the sections of the book. And so when it is proposed that I run a “Thinking Classroom,” I am always careful to interrogate what the proposer has in mind, because I think we have the responsibility to ensure that the term “Thinking Classroom” doesn’t strictly refer to the structure(s) of VNPS and VRG and leave the teaching behind.”

Probability Quizzes with No Questions

In this post, Banting begins sharing his encounter with the following question:

If you chose an answer to this question at random, what will be the chance it would be correct?

A) 25%

B) 50%

C) 60%

D ) 25%

What I appreciated about the post is the idea that, while questions like this are can lead to a lot of mathematical arguments,  it is not the answer itself the end goal. As he mentions,

“The point of the exercise is not to complete the exercise, it’s to dwell a while in the complexities it offers. By constructing the argument, you interact with notions of odds, randomness, probability, and the like.

To engage help students to step out of the usual problem-solving approach (i.e. recall a similar situation and apply it to a new context),  Banting proposes shifting the focus of answering a question to puzzling with it by not offering any questions at all (see Figure 1).

Figure 1. Sample questions from quiz from ‘Probability Quizzes with No Questions’ by Nat Banting.

For example, in a ten-question quiz, seven out of those 10 questions will have the correct answer be C, you may ask students to circle the correct response and ask them to reflect about what are the chances that, in fact, that would happen? What I love most about this idea, is that it throws the student into grappling with uncertainty in a very familiar scenario. You can use mathematics as a way to demystify this feeling of uncertainty. As Banting mentions,

“Mathematics is suggested as a way to dissect the feeling of uncertainty, and this act of justification becomes the focal point. I mean, of course it does. What is left to argue about the solutions of questions 1-7? They were simply the vehicle to encounter an experience, and, in this way, those seven “questions” were never the questions to begin with.”

Since I am a big fan of board games and art, I also enjoyed looking at Project: QuaranTiles and hope to try them out soon. In addition, to Musing Mathematically, he is also the curator of, a really neat website that fosters creative ways to visualize fractions. I loved exploring this website, especially, since oftentimes one of the most memorable early mathematics experiences for students is learning fractions! The ideas behind this website also lead to great implementations of these for the classroom such as Marie Brigham’s Fraction Talk War for fourth-graders and the creation of the Fraction Wars Cards by Carla Dawson

Have an idea for a topic or a blog you would like for me and Rachel to cover in upcoming posts? Reach out in the comments below or on Twitter (@VRiveraQPhD).

About Vanessa Rivera-Quinones

Mathematics Ph.D. with a passion for telling stories through numbers using mathematical models, data science, science communication, and education. Follow her on Twitter: @VRiveraQPhD.
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