“3 and a bit”

How is pizza related to pi? (image source: Veritasium youtube channel)

We all know about pi and its common numerical equivalent for practical purposes (3.14). But where did pi come from, and why? The development of pi over the centuries is an example of mathematical innovation. As you know, today is Pi Day. If you’re interested in links to help students engage in pi for Pi Day, skip to the bottom. If you want to know a bit more about pi … keep reading.

First, pi is the ratio of a circle’s perimeter to its diameter.

The short history is pi was first approximated by the area of polygons with n=6 sides (a hexagon) that is inscribed (inside a unit circle with diameter = 1) and circumscribed (around the same circle). Pi was approximated to be between 3 and 4. Over a long stretch of time the polygon was bisected and bisected and bisected … and you get the point. The last time this method was used to approximate pi is in 1630 using a polygon with n=10⁴⁰. At this time, pi had 38 digits to the right of the decimal, but it took over 25 years to make this approximation.

That’s a long time to simply calculate a very precise numerical representation of pi. Here is where the innovation happens. In 1666 Sir Isaac Newton, while recovering from Bubonic plague (what discoveries may come from the conditions Covid-19 placed on society?) decides there has to be a better way. Through widespread understanding of Pascal’s triangle and the development of the binomial theorem, Newton “breaks” the rules of algebra, and applies calculus to the geometric theories of circles, and discovers the irrational nature of pi allowing anyone with a computer to calculate pi to whatever level of precision they desire. I’m sure everyone can find pi on a calculator.

This isn’t about how many digits of pi we can calculate, but rather to recognize that something as mundane as pi has a rich history of innovation that allows anyone to use it in its most simplistic form (3.14) in the discovery, creation and invention of both simple and complex ideas. Newton provides an example of how the following three aspects of mathematical innovation come together for a discovery that is now used by every student across the globe.

Arte Scienza – development of balance between science and art, logic and imagination. [Pascal’s triangle and the binomial theorem]

Conneccione – everything is connected to everything else. [pi is an amalgamation of algebra, geometry and calculus]

Curiosita – the curiosity to find the connections. [Newton was curious about how to calculate pi without the arduous task of bisecting polygons]

This line of thinking allows mathematical and scientific innovation to propagate into other disciplines. Here is a brief timeline of pi and it’s uses in other innovations.

Archimedes uses the geometry of a circle to introduce the concept of pi …

Keplar’s laws of planetary motion …

Galileo’s pendulum …

Euler’s use of algebra, trigonometry and geometry to develop Euler’s constant and what some consider the most beautiful formula in mathematics using 5 important constants:

(more on this in a future post)

Gauss’ normal distribution …

Einstein’s theory of relativity!

To hear more about the history of pi and it’s discovery check out the Veritasium youtube channel.

For resources to connect innovations in mathematics with your students check out MoMath .

If you’re looking for activities for Pi Day, What We Do All Day offers some great projects, and Jo Boaler just updated her youcubed website with this Finding Pi activity.

The Green (Hat) Apple

In September 2017, I defended my dissertation on the perceived value of procedural fluency vs. conceptual understanding among secondary mathematics teachers. I know – BOOOORING (my oldest son actually fell asleep during my presentation).

Non-mathematicians (and I’m sure some mathematicians) will likely find it difficult to see the connection between mathematics and creativity. At some point, we’ve all sat in a math classroom and witnessed a teacher demonstrate basic algorithms like addition and subtraction. What most of us missed was the opportunity to explore the creativity associated with discovering how and why the algorithms worked … we just trusted that they did, and there was sufficient evidence to back the teacher’s claims.

At the conclusion of my defense, my professor, Dr. Fredricka Reisman, walked me to her office – exclaimed that I was her last doctoral student before retiring – and handed me the green glass apple picture above, something she has done for all of her graduates. Throughout my program, which focused on educational leadership and creativity, Dr. Reisman often referred to Dr. Edward de Bono’s Six Thinking Hats framework. The Green Hat (or apple in this story) is “the creative hat, used to generate new ideas and possibilities and to explore alternate courses of action.” [taken from Reisman and Tanner’s “Creativity as a Bridge Between Education and Industry: Fostering New Innovations” – the second book from top in the photo].

Reflect for a moment …

How often do you exercise creativity in your teaching?

How often do you ask your students to be creative?

How close is creativity associated with successful teaching and learning?

The challenge is:

How can we make learning in the classroom more fun?

One of the best ways to get inspired is to look outside your context. When working on new design challenges, IDEO designers often use analogous inspiration to gain fresh perspective. For example, emergency room doctors can get insights about organizing their medical supplies by spending time with a Nascar pit crew and an airline employee might get ideas about check-in by observing a hotel front desk.

IDEO U Blog

IDEO is a design firm that has their fingerprints on every industry, including education. We have used their work as inspiration in bringing the concept of design thinking (remember “Launch” and “Empower” by AJ Juliani and John Spencer) to some of our classrooms. Using the suggestion above, we stepped outside the context of education to examine this challenge of making learning more fun and engaging while also invoking curiosity and celebrating creativity.

David Kelley (not the creator of Doogie Howser), founder of IDEO and the d.school at Stanford University, writes in his book Creative Confidence, “[a] creative mindset can be a powerful force for looking beyond the status quo. People who use the creative techniques we outline are better able to apply their imagination to painting a picture of the future (p.18)”.

When is the last time you thought about this question from George Couros’ The Innovator’s Mindset (p.39), “would I want to be a learner in my own classroom?”

David Kelley and IDEO have a lot of resources to help us think through the challenge of (re)designing a classroom that brings joy and confidence to both the teachers and the students.

We’re building a team!

Do you own a Green Hat?

If yes (your answers to the questions throughout this post were in the affirmative), then GREAT we’d like you to join us.

If you don’t own a Green Hat, would you like to? We’d like you to join us also!

Kevin (Murphy) and myself will be visiting your school soon to talk about a collaboration with the MCIU and FluxSpace (a makerspace “sandbox”). We’re hoping to include some interested teachers in exploring the creation of lessons embing computer science, makerspace, design thinking and/or project based learning experiences in core content teaching and learning. Our work will begin with grades K-4, but we will be expanding once we get things off the ground.

This is the first in a series of blog posts related to innovative teaching and learning , so stay tuned.