Modeling Math Using PhET Simulations
I was talking with a former teacher recently, and she commented on how easy it is for today’s teachers to access resources from all over the world. We really are fortunate to live in the age of widespread digital technology. On the other hand, the sheer volume of educational materials out there can be paralyzing! It frequently takes me as long to locate highquality resources as it does to create my own by the time I filter through the pseudoeducational junk. So when I find something I like, I get excited to share it with other teachers. In this post, I want to tell you about an awardwinning resource called PhET sims that you can start using tomorrow to help your students model mathematics.
Let me explain why this resource is going to be effective. (For more on evaluating the effectiveness of tech tools, check out these sites on choosing tech wisely and applying technology in learning math.)
Grounded in Learning Theory
In the book Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching, researchers make sense of how students learn math based on countless hours analyzing videos in which students wrestle with elementary but powerful underlying math concepts during reallife problem solving situations. They observed that learners first understand the math only in the context of the given scenario but then gradually extend their understanding to more general situations as their ideas become “more shareable, more transportable, and more reusable.” In other words, acquiring new math is the process of formalizing and abstracting what we already understand to be true based on our concrete experiences. It’s not that at one moment you don’t understand and then the next moment you do. It’s that our understanding gradually becomes more accurate, sophisticated, and usable as we grapple with the concept in multiple ways. In Beyond Constructivism, they call this “gradual increases in local competence.” I like The Lesh Translation Model as a visual reminder to promote moving between multiple representations, both concrete and abstract, to help students increase competency.
The Lesh Translation Model
Does all this sound familiar? Representations is an NCTM process standard. See pgs. 6771 and 280284 in Principles and Standards for School Mathematics to learn more. Take this quote from pg. 280 for instance, “Students can develop and deepen their understanding of mathematical concepts and relationships as they create, compare, and use various representations.”
What does this have to do with PhET? I’m so glad you asked. The PhET team partnered with math education researchers to develop interactive math simulations targeted at helping students discover the mathematics. (You may have heard of PhET sims if you or a partner teacher also teach science. Did you know there are now 38 simulations specifically related to math concepts? I didn’t until I had a chance to work with their team for a week this summer!) While there, Dr. David Webb explained how their sims fit into the learning process as described by the book, How People Learn, as “progressive formalization.” Sounds a lot like “gradual increases in local competences” doesn’t it? He went on to explain how our understanding starts out as informal, grows through the preformal stage (the bridge stage supported by the simulations), and culminates with formal understanding. I adapted this table from his slides.
Informal Representations

Preformal Representations

Formal Representations

 From concrete realworld or realistic contexts and no generalization to other tasks unless the mapping is very 1to1
 Represent math principles in the students’ own language but lack formal notation or structure
 Students’ emerging explanations for natural phenomenal
 Real life situations in the translation model (See figure above)

 Representations and strategies are starting to be generalizable across tasks
 Connections are made and patterns are established; mathematical structure is emerging
 Representations and strategies become more meaningful and contextual cues become less needed
 Manipulatives and pictures in the translation model (See figure above)

 Strategies are generalizable across many tasks
 Mathematical abstractions, conventions, and abbreviations are understood and meaningful
 Contextual cues aren’t needed (e.g. “efficient algorithms”)
 Written and verbal symbols in the translation model (See figure above)

Take a look at these two screenshots from the Expression Exchange sim and note how PhET sims put theory into practice.
Screen 1: Students explore a familiar realistic scenario, a collection of coins.
Screen 2: After students make sense of the coins, they can flip to the variable view for a more formalized representation of the same scenario.
Engaging for Students
I’m a huge proponent of learning by doing, and that’s another reason I like PhET sims. Every student or group of students with a device can explore and discover at their own pace. It leaves the teacher free to offer just in time feedback to support meaningful learning. (More about that in the next post!) Every sim is tested with student interviews. Amanda McGarry explains in their 2017 Stem For All Showcase Video, “If a simulation is working, I see a student leaning in, asking questions, and making mathematical connections.” Because they’re vetted by students themselves, every simulation is highly interactive and engaging.
Easy for Teachers to Implement
No matter how amazing a tool may be, if I can’t use it easily in my classroom, then it probably won’t become a part of my repertoire. But don’t worry, the PhET team thought about that when they created the “For Teachers” section for every simulation. It contains a pdf and video overview of the sim including sim features and insights into student thinking. (You probably won’t need it for the features. The sims are designed to be intuitiveyet another thing I like about them. Who wants to read instruction manuals?!?!) The best part is that each sim comes with ready made lesson plans that structure times of exploration and thoughtprovoking questions to help students efficiently wrestle with the intended math. Some of the lessons have been created by the PhET staff and others have been submitted by teachers like you and me. You can also search for plans based on the simulation, topic, subject, and/or grade level. For example, here is a list of lesson plans for middle school math. See? I told you that you could start using this tomorrow!
PhET Sims are just one of my favorite resources for modeling mathematics. Do you have other effective, easy to use, researchbased tools? Share them with us in the comments!