**Transforming Math Through Technological Visualizations**

“I don’t understand math because I can’t see it.”

There is an almost magical moment in mathematics when, after struggling with a concept, something inside just ‘clicks’ and suddenly that concept goes from abstract and frustrating to tangible and fascinating.

As an educator, the look in a student’s eye when they finally ‘get it’ and the concepts and their extensions start to make sense can be intensely satisfying. As a life-long student of mathematics who has spent many long, frustrating hours struggling to understand various mathematical problems, experiencing moments when the fog lifts and the problem and its solution start to make sense has an almost addictive quality that makes me want to dive deeper into the fascinating world of mathematics.

The question is: what triggers that moment?

According to research, nearly 65% of the population are visual learners.[1] This means that over half the population learn through seeing concepts visually presented to them. This makes more abstract subjects, such as mathematics, more difficult to understand. Technology has allowed the average person to peer into a world previously reserved for those rare mathematicians who can close their eyes and see the world of math in their minds. Technology’s visual capabilities allows us to play with math concepts in a tangible way and explore a fascinating world through new eyes.

Many areas of mathematics are considered abstract because they are not easily visualized. They are hard to picture, difficult to draw accurately, and almost impossible to construct in our 3D world.

Modern technology has allowed us to see the previously ‘un-seeable’.

Take the concept of the fourth dimension, for instance. We live in a three-dimensional world, meaning we are physically unable to see into higher dimensions. This makes understanding such a concept a highly abstract one and difficult to grasp for most people. We can talk about 3D shadows as representations and compare them to concepts we can represent as lower dimension, but ‘seeing’ the fourth dimension has traditionally been difficult or even impossible.

This is where advances in modern technology have stepped in to help us turn abstract concepts into tangible, visual representations, thereby broadening the reach of mathematics to a whole new audience.

Modern technology has allowed us to see the previously ‘un-seeable’. Going back to the concept of a fourth dimension, Miegakure has created an interactive game that not only explains this concept, but also allows users to ‘play around’ with shapes in the fourth dimension.

Technology has also allowed us to quickly compute and visualize large numbers, such as one million digits of Pi or using individual’s computers to search for larger and larger prime numbers.

We can also play with curved or hyperbolic space and explore the famous Mandelbrot set in great detail and at exponentially zoomed in values.

There are programs available that allow non-mathematicians to quickly and efficiently program and visualize various concepts in math, such as GeoGebra and Wolfram Alpha.

One of my personal favourite mathematical concepts is fractal mathematics, which are dense, lovely, and very difficult to accurately represent by hand-drawn images. Technology has allowed us to quickly and efficiently represent these gorgeous and complex images with little to no programming knowledge. With a few keystrokes or mouse clicks, we can manipulate these patterns to give us an amazing, visual representation of these concepts.

We can also use technology to show simple concepts in a new way. Many people struggle with the infamous trigonometry graphs: sine, cosine, and tangent. The internet is full of interactive unit circles that allow users to trace a circle and show this trace transposed on the three graphs. This simple visualization brings an abstract concept to life, allowing a user to see, step by step, what is occurring without having to draw out complex, individual graphs.

The Golden Ratio is another rich, visual concept that technology has greatly enhanced by simple applets, such as the one found here. This particular visual allows users to play with different numbers to see the effect they have on a pattern.

The benefits of programs like these are they allow people to easily change variables within math equations and concepts, then quickly see the results. Without technology, these alterations would have to be done by hand, taking time, resources, and a deeper understanding from the start of the end product.

Modern technology has allowed us to see abstract concepts in new, easy to understand ways at the click of a button, making it easier for the 65% of the population who learns by seeing to visualize these previously hidden concepts. As Shakespeare wrote in *The Tempest, *“Oh what a brave new world that has such creatures in it.”

[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=587201

## About the blogger:

Pamela BrittainPamela Brittain is the Founder of VedaVox Inc., an online education company dedicated to the idea that anyone can learn, understand, and appreciate mathematics.She has spent the past thirteen years in Education Management, working for various educational centers and programs including the conception, implementation, development, and management of mathematics outreach programs for the Department of Mathematics at the University of Toronto.

Her time with the Math Department showed her a whole new world of mathematics and inspired her to complete a Masters of Mathematics for Teachers (MMT) from the University of Waterloo in order to more fully understand and teach this fascinating and ever-present subject. She is also a certified high school teacher in Ontario with qualifications in Mathematics and Science/Chemistry and will be starting a PhD in Mathematics Education at the Ontario Institute for Studies in Education (OISE) at the University of Toronto in fall of 2017.