Open Spaces

When I observe effective teachers of mathematics, there is one characteristic that they all have in common. It is the space they provide for their students to voice their ideas, reflect on their thinking and explore mathematical concepts. This may seem simplistic. But I argue that it is not. Space is time. It is the […]


I am sitting at the back of the class, unnoticed. You instruct us at the front of the class, unnoticed. Patterns and the rich meaning of mathematics pass us by, unnoticed. I feel invisible in this sea of numbers and equations. I am drowning, unnoticed. With just one question you would unlock my curiosity. With […]

Why Use Mathematical Think Alouds?

Students often hold misconceptions about effective thinkers. They may believe that these people never struggle; that they never second-guess their ideas; that “they get it right” immediately. Think alouds can help dispel these misconceptions by opening up discussions about self-talk and enabling all students to access mathematical thinking. Self-talk is simply what we say to […]

When Will I Ever Use This? Beyond Pizza, Tiles, and Fences.

As teachers we continually strive to explain to our students how they will use the mathematics they are learning in their daily lives. Many resources provide examples of laying tile for area, building fences for perimeter, and eating pizzas for the division of fractions as real-life applications of mathematical concepts. I believe that by using […]

Deepening Mathematical Learning with Concept Circles

Recently I was in Nashville, presenting at the NCTM conference, where I shared examples of concept circles and their use. Similar to other presentations, the group asked, What are these concept circles and where can I find more information? I get this request a lot whether I am presenting in North America, Asia, or Europe. […]