Desmos and Mathematical Modeling

I first heard about Desmos on Twitter. What an incredible graphing tool. I don’t think any teacher forgets the first time he/she introduces desmos to the classroom; you can feel the room buzzing with excitement. Just amazing. So every year I look forward to the day I introduce my students to graphs and Desmos.

Unlike a lot of physics modelers, I do take the time to go through “unit zero” (scientific reasoning) with my students in Introductory Physics. Perhaps the reason is because I like the development of the linearization concept in graphs (very useful for my IB students), perhaps it’s because I like to kick off the class with the spaghetti bridge lab, or possibly it’s a combination of both. The funny thing though, is that every year I introduce students to graphing in a different way, and I’m always uncertain of the best way to do it. While I have had students produce graphs by hand, I always questioned the validity of hand-drawn graphs in the digital age. I have also tried digital tools like:

  • Excel (too complicated for most students)
  • LinReg (good but not very flexible–students also have to download the program)
  • LoggerPro (love it but there is a learning curve–school/student must have a license)
  • Desmos (Free and very user-friendly)

This year I decided I was going to let students choose whatever method/tool they prefered for graphing, but I was only going to use Desmos when answering questions or showing solutions on the board. Of course I didn’t have to sell Desmos too hard for students to start adopting it.

In my experience, Desmos really excels in helping students do curve fitting. The way students can interact with graphs is incredible. Before students learn how to make graphs and before best fit lines magically appear on a screen, Desmos helps them understand the basic mechanisms of manual curve fitting. So I gave them a set of problems where the objective was to graph the data, linearize it if necessary, and come up with a mathematical model (equation) for the relationship between variables. The purpose of the linearization step is to help find the mathematical model and aid with the analysis of data (which wasn’t part of the assignment). The thing is, with the help of slides in Desmos, students were able to come up with the correct mathematical equation without much trouble and without taking the extra linearization step–the “test plot.” This is when I started rethinking linearization. I wasn’t fully convinced that students were getting it, and I pushed them towards creating a linear plot. Confusion soon emerged. Students weren’t sure why they were trying to produce a linear plot. It took me a couple of hours to straighten things out. Here is an example of the problem I asked my students to work on and the graph made with Desmos.

Linearization graphing excersise

Right now I’m thinking that maybe next year I’ll skip unit “zero” altogether and introduce the topic as it comes up in other units. And instead of talking about linearization, I’ll just let Desmos do its magic. I don’t know how that’s going to pan out, but I think it’s worth a try. The only thing I’m sure of is that Desmos will once again be the gateway to graphing in my class.

Drive. We Need Motivation.

Note: This is the first post in a series on professional books I’ve committed myself to read over the course of the year. Twelve books–one for each month of the year–that’s my goal. This post is a short account of my impressions of the book, from an educator’s perspective.

Book # 1:  Drive: The Surprising Truth About What Motivates Us by Daniel H. Pink.

DriveI can’t recommend this book enough. Does increased motivation sound good to you? I read this book this past January, but the topic is even more relevant now that we’re at the end of another academic year, a time educators are searching for any motivation that’s still left in them. My first exposure to the book was by means of RSA’s entertaining animation of Pink’s work. This short video addresses the main ideas covered in the book. If you haven’t seen it yet, you should. Right now.

This topic fascinates me, and I knew I had to read the book. It was a quick read divided into three parts: the need for a new model for motivation (section 1), the three elements of motivation (section 2), and toolkits to help you (section 3). At the end, the author even summarizes the information for you so you don’t have to. Great read.

Conventional thinking says that rewards (I could insert grades here) motivate people. Well, the truth is–not always. Here’s a quote from the book:

People use rewards expecting to gain the benefit of increasing another person’s motivation and behavior, but in so doing, they often incur the unintentional and hidden cost of undermining that person’s intrinsic motivation toward the activity.

The book refers to this type of “motivation” as “Motivation 2.0,” and sometime it’s effective. It really depends on the task. For routine, straight forward tasks, it works. For creative, cognitive intense tasks, the type of work that most of us (teachers) do, it has the opposite effect.

That’s right–opposite.

Using rewards to promote effectiveness on creative tasks doesn’t work. This makes me think about the recently revived discussion about teacher merit pay. “Teachers should be paid according to results,” some claim. Professor Roland Fryer from Harvard has conducted a study with over 200 public schools in New York and has found no evidence of any improvement when tying performance to pay. Here’s a quote from Dr. Fryer:

I find no evidence that teacher incentives increase student performance, attendance, or graduation, nor do I find any evidence that the incentives change student or teacher behavior. If anything, teacher incentives may decrease student achievement, especially in larger schools.

To find out more about his study, read his 2011 paper on the subject. Other studies have found similar results. Merit pay just doesn’t work.

And since rewards don’t work for the creative, cognitive work of teachers, we’d be foolish to believe they work for the intellectual work we require of students. Unfortunately, too many educators still use grades as a tool for motivating students. Pink warns us against this when he explain that

Good grades become a reward for compliance–but don’t have much to do with learning. Meanwhile, students whose grades don’t measure up often see themselves as failures and give up trying to learn.

So, what increases intrinsic motivation? What improves performance? According to the book and latest research, three things:

1. Autonomy

2. Mastery

3. Purpose

I could go on and on discussing each one of them. The book does an excellent job explaining how and why autonomy, mastery, and purpose increase motivation. For now, I just want to briefly address autonomy and purpose.

Autonomy is a huge deal in my playbook. I’m fortunate to be a teacher and to have some control over what happens in the classroom, but I know not all teachers are as lucky as I am. I can’t imagine working at a school where you’re given little to no autonomy. I completely agree with the author when he says,

Type I behavior [intrinsic motivation] emerges when people have autonomy over the four T’s:  task, their time, their technique, and their team.


Human beings have an innate inner drive to be autonomous, self-determined, and connected to one another. And when that drive is liberated, people achieve more and live richer lives.

We need to protect teacher’s autonomy in the classroom. Better yet, we need to encourage it. Autonomy can prevent us from going crazy, and it will help us be more effective.

As far as purpose goes, we as educators should be inspired by our ability to positively impact someone’s life. What we need is constant reminders of our purpose so we can keep improving at what we do. It seems like everything surrounding us is conspiring against our focus on purpose. Purpose is extremely important if we are to stay motivated. Purpose is also a constant characteristic of excellence and innovation, and we need more of that in education. On the author’s website, I came across this excellent short video that aids in sharpening our focus on purpose.

While I really like the first question, “what is my sentence?”, I would change the second one. What about asking yourself this question every night: “What did I learn today that will make me better tomorrow?” this way the focus is on learning and improving today and tomorrow.

So, what’s your sentence?

Blog. Why?

Have you heard yourself telling your students, “Don’t just sit there . . . do something!” I know I’m constantly asking my students to create/contribute, maybe because I believe in the learning inherent in the creation process. Whether creation means a blog post, video, spreadsheet, science fair project or something else, I want them to contribute to society. It’s not that I believe society will be the great beneficiary of what they produce (although this certainly can be true), but because they will be better students and more responsible citizens for having produced something.

It makes sense, right? Some of the best learning I have experienced was when I had to think hard about something and had to share new ideas with others. Actually, the sharing is a big piece of the puzzle; you can’t just half-ass your learning when you open yourself up to criticism and engage with others in open discourse. Communicating new learned knowledge requires deep understanding and clarity; perhaps we do our best learning when we contribute and participate in the conversation. Some of you have probably heard of the 1% rule. It is believed that only 1% of internet users create content – 99% merely consume it. There is nothing inherently wrong with being a passive consumer of information, but I believe one can experience rapid professional growth by joining the 1%.

A couple of teachers whom I deeply admire (whom I have linked to below) have attested to what I have suspected to be true for quite some time – you learn more and you learn faster by engaging in the conversation. Starting a blog is one way. Dan Meyer and John Burk recommend that teachers looking for good professional development should start a blog. It makes you think and reflect hard on your practices. Additionally, you open yourself up to feedback. Meyer, a famous and influential math blogger, describes his blogging experience like this:

I’m pretty sure that without my collection of blogs and readers and critical comments, I’d still be back there [in the classroom] totally psyched about that amazing worksheet I came up with.

John Burk also gives a wonderful account of what lead him to blogging. His post on why you should start a blog is totally worth reading.

So here I am, not a writer, not an amazing teacher by any means, just someone who wants to become better at what he does. Blogging is just the start.