I have had some great discussions with teachers in my courses lately about the power of providing opportunities for students to *see* and *manipulate* mathematics as a way to test out their ideas, play with patterns, and develop their own rules and understandings. Visualization, manipulation, experimenting – all contribute to students developing deeper understanding and their own ‘algorithms’, and because of these contextual experiences, they are much more likely to recall how to do a math process than if they were just given the rules/algorithm to memorize.

In a recent final reflection, one teacher wrote, “As a high school teacher, I have always stayed away from using manipulates for fear they were “too elementary” for my classroom.” This attitude – that older students don’t need those physical objects or need to *see* – that they just need to memorize rules and practice – is sadly still prevalent today. Which is frightening really. I experienced these same attitudes and beliefs over 2 decades ago when I was teaching in middle and high school, and bringing out my two-colored chips, algebra tiles, and Sketchpad. Allowing students to play with math, to use physical objects, and virtual objects, to represent the math and then be able to manipulate change and see what happens was always considered ‘babying’ them. Clearly that attitude is still going strong today, since as you read above, I hear it in the courses I teach with current classroom math teachers. This despite even more tools being available to provide a way for students to experiment, play, discover, create and find the mathematical patterns and rules themselves. The tendency to just give them the rules and the process and the definitions and have them memorize and regurgitate is still very much a part of our mathematical education. What we really want to do is provide multiple ways to look at and explore math concepts, so that when students ‘forget’, they have that experience where they built the understanding to recall where they can rebuild it again. Much easier to recall something they saw or something they physically moved and connected to than an isolated, memorized fact.

In most typical high school classrooms I visit and work with these days, it is rare to find physical manipulatives (more often in Geometry, but much more rare in an Algebra 2 or Pre-calculus class for example). But – there is almost always a technology tool – whether that be the teachers projector attached to the internet, or students on tablets/laptops, or more often the case, graphing calculators of some sort. Which means there is no excuse NOT to be providing students the opportunity to visually see the mathematics, and to manipulate and explore to come up with those algorithms they are often asked to just memorize. Meaning: *use the technology for more than checking answers!* Use it to help students find the patterns and connections and create their own algorithms and definitions, use it to delve deeper into the math, to gain insight, to test out conjectures and really get a sense of what all those numbers and variables mean and how they interact with each other to change the shape of a graph and what that might mean in a application of that math in the real world. Use the tools to manipulate and *see* the math; technology allows for students to test a conjecture quickly, make predictions and check if they are right, and explore very large and very small numbers, etc.

As an example of this, I am going to use the Graphing Calculator App (for mobile devices), since I haven’t previously used this before in any of my videos, to show the power of visualization and technology to make conjectures and immediately test them with modifying features/dynamic math capability. You can do this on our hand-held Prizm series graphing calculators (handhelds and emulators).

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