The Difference Between Doing Math and Using Math

Using math is an essential skill in many fields to bring about change. Scientists, engineers, and other professionals apply mathematics to aid and clarify research. I teach math to grade 4 students as being comprised of two main components.
One is using math, which I define as adopting the thinking and algorithms others have developed and applying these to mostly routine problems. Most people use math on a regular basis in this way—calculating a tip, figuring out the timing between events, calculating a dose of medicine, etc. The other main division I use is doing math, and I liken this to the process of invention and discovery in which professional mathematicians engage.

Doing math in this sense is a process of discovery, and it prepares students to become the kind of people who can grapple with challenging questions and persist in the face of obstacles. Doing math is the process of looking at phenomenon in the world, attempting to notice patterns, and testing noticings for consistency. When students have the opportunity to find and notice interesting patterns in numbers such as prime, composite, multiples, triangular numbers, etc., they are placed at the center of discovering.

By noticing, checking, and discussing with peers, and verifying new ideas repeatedly, sometimes alone and often in teams, students are working to change their views and understandings in ways similar to the great discoveries of the past in mathematics—albeit usually on a much humbler scale. This process of noticing, checking for consistency, skeptically examining your own thinking, and critically examining the noticings of others is essential work. It strengthens the mind of each engaged person individually, and it directs collaborative conversations on to a plane of both concrete examples and logical abstractions that can be checked and rechecked for factual accuracy.

This enables students to learn to articulate their thoughts and to skeptically question their own thoughts and those of their peers. It allows students to learn in a polite, critical, and rigorous community. It also prepares students to dig for understanding when ideas are formalized along the lines of mathematical tradition. Enabling students to have the “eureka!” moment of discovering an idea and demonstrating its truth before formalizing it is much more exciting than simply being informed of a principle described by another.

In many ways, our use of technology in grade 4 has developed into an extension of math class with an emphasis on discovery-based learning. Students learn a variety of applications such as BirdBlox, Cargo-bots, Turtle Art, and Explain Everything. Most learning of the apps starts with a discovery process to find the tools within and is then followed by a phase of sharing discoveries with others. Then, teachers help refine tool usage on a more specific need basis to achieve goals imagined by individual students and groups of students.

They use these apps to create both with math and about math. Students use geometry and some arithmetic, computational thinking, and precise language to create art and add robotic motion to 3D art using Turtle Art and BirdBlox. Some robotic elements they make interact with the environment by responding to light, temperature, sound, and distance. On one project, girls learned about environmental design in architecture from outside experts, Darren Lloyd and Azizan Aziz, and then they applied such ideas in model size with the Hummingbird robotics kit. This year, students made a miniature ceiling fan that turned on at 80˚ F and an example clock that turned office lights on and off during certain times of the day. Applying their learning in this way is fun and exciting for students and truly engages them with their own individual learning processes.