“Geometry will draw the soul toward truth and create the spirit of philosophy.”—Plato. There is a widening gap between the effective teaching of geometry in elementary schools and the geometry skills students need in high school. Often, geometry tasks at the younger grades are limited to identifying shapes or labeling properties; in high school, students are expected to use abstract reasoning to prove a complex relationship. Dynamic, engaging instruction in geometry has traditionally been overlooked during middle school, causing a critical gap between elementary school experiences and the thought processes required in high school. Why do we need geometry in schools? Geometry is all around us. It is part of our daily lives, whether we are at a cafe (see picture below), a construction site, or at the post office. Geometry gives us the tools to engage analytically with our everyday surroundings. It turns out that these tools provide us with more than just an amusing intellectual exercise. A growing body of research links spatial reasoning with future success in other academic pursuits. Interestingly, these pursuits reach far beyond geometry or even mathematics. Elementary school teachers and researchers at the University of Toronto found that students given lots of spatial reasoning exercises ended up doing better in numeracy, patterning and other areas of mathematics. In an unrelated study, researchers showed that spatial thinking skills are strongly related to students’ future success in STEM disciplines. A common notion held by many people is that math is mostly about numbers. This erroneous idea is part of the reason that many people write themselves off as being “not good in math”. Not only... read more

Last week the first ever Julia Robinson Mathematics Festival was held in San Diego. What a great event celebrating the joy of math in a non-competitive atmosphere. About 70 students, mostly from grades 6-8, were in attendance. My favorite part was watching kids be comfortable enough to admit what they don’t remember, and figure it out right there. It was also great to see kids (and adults!) get really silly with math. Activities included a engaging recreational problems from many different areas of mathematics. You can see them in the San Diego Union Tribune gallery. We were honored to host a table with Geometiles.... read more

If you are going to coach a math club, it is likely that you will come across at least some of the following challenges: A classroom full of students at different levels, despite the fact that most students in the math club self-selected to be there Creating an atmosphere where students feel emotionally safe enough to participate in the class– meaning to risk providing wrong answers Assessing your students’ understanding, other than by testing The three challenges are interrelated, and I will describe some ways of addressing them. Most of the ideas are gathered from other successful teachers and were chosen because the issues they address echoed my experiences in Math Club. Students at different levels This is obviously a loaded subject, and I’d like to focus just on the interpersonal dynamics of a class in which students are performing at different levels. Let’s be honest: math clubs tend to attract high performing students, and, in my experience, such students often have a strong desire to demonstrate to everyone else just how high performing they are. If you are starting a math club, you need to be attuned to this from the very beginning. Otherwise, some serious damage can be done to the learning experience of your students who are either not as high performing, or less confident. My personal suggestion is to nip in the bud any behavior that smacks of arrogance. Do not wait until the damage is done. If there is one phrase that I could eliminate from any math club discussion, it would be “This is easy”. This phrase, and its consequences, are addressed in an article by Tracy Zager, a well-known math teacher... read more

Here are some of our blogs about Platonic and Archimedean solids: Public art installation in Los Angeles featuring huge Platonic and Archimedean Solids Examples of Platonic Solids in antiquity More lesson plans... read more

I found out first hand how important art instruction is for teaching math from some of our math competitions. When there were problems involving cubes, such as counting their surface area or volume, the students’ ability to make a sketch in the high pressured environment of the competition came into play. It didn’t have to be a nice looking sketch; but it needed to be coherent enough to be a helpful tool for the student. I liked the sketch above because of the energy it projects and the “Picasso-like” use of perspective. I liked the spirit behind it; the student was obviously determined to conquer the problem at hand. I think that a better understanding of how to depict a cube on a piece of paper would have helped him organize his thoughts. The amount of time students spend with screens these days is not helping them develop the kind of hand-eye coordination required for a quick sketch. They tend to have little or no practice with making approximate, loose drawings. Obviously, one wouldn’t want to start working on these skills in the midst of a math competition. They need to be developed on a regular basis, so that they can be quickly recalled in time-sensitive situations such a tests. Some students enjoy drawing, but insist on making everything as close to “perfect” as possible. Such students need help in letting go of their perfectionism and putting down their visual thoughts on paper as quickly as possible. The following drawing was made by a student who was initially reluctant to make quick sketches and is making good progress with this skill: It is a nice... read more

Nowadays, thanks to modern day computers, we have instant access to the approximation of π (the ratio of a circle’s circumference to its diameter) to any degree of accuracy. But back in antiquity, getting a good approximation of π was elusive and difficult. Over 2000 ago, the Greek mathematician and scientist Archimedes derived an accurate approximation of π using polygons to approximate a circle. Let’s follow in his footsteps with Geometiles as our tool. We’ll start by approximating the circle with a square, then a regular hexagon, and finally a regular dodecagon (that’s a 12-sided polygon; dodeca means 12 in Greek). You can see that the more sides our regular polygon has, the rounder it is. In fact, the dodecagon is so round that it can almost roll like a wheel, as you can see in this Instagram video: For each of our regular polygons, let’s compute the ratio of the circumference to the diameter of its inscribed circle, and see how it compares to π≈3.14. What’s an inscribed circle, you might ask? That’s the largest round cookie you can make out of your polygon. Archimedes used both an inscribed circle and a circumscribed circle (the smallest circle that completely encloses the polygon) in his work on approximating π, but we’ll just stick to the former in this activity. You can do this activity with your students at two levels. For students who have not yet learned about special right triangles Measure the sides of the polygons and the diameter of the cookie cutter, then compute the ratio of the first quantity to the second. Let’s use the fact that the square is about 2.5 inches on the side. This is... read more