# Uncertainty about Uncertainty in IB Science

I have a student that is taking both IB Physics with me and IB Chemistry with another science teacher. The first units in both courses have touched on managing uncertainty in data and calculations, so she has had the pleasure (horror) of seeing how we both handle it. For the most part, our references and procedures have been the same.

Today we worked on propagating error through the calculation $\Delta x = \frac{1}{2}at^2$ with uncertainties given for acceleration and time. The procedure I've been following (which follows from my experiences in college and my IB textbooks) is to determine relative error like this:

$\frac{\delta x}{\Delta x} = \frac{\Delta a}{a} + 2 \cdot \frac{\Delta t}{t}$

In chemistry, they are apparently multiplying uncertainty by 0.5 since it is a constant multiplying quantities with uncertainty. On a quick search, I found this site from the Columbia University physics department that seems to agree with this approach.

My student is struggling to know exactly what she should do in each case. I told her that everything I've seen from the IB resources I have in physics supports my approach. The direct application of the formula suggests that an exact number (like 1/2) has zero uncertainty, so it shouldn't be involved in the calculation of relative error. That said, the different books I've used to plan my lessons agree with each other to around 95%. There is uncertainty about uncertainty within the textbooks discussing how to manage uncertainty. Theory of knowledge teachers would love the fact that teachers of a generally objective field (such as science) have to occasionally acknowledge to our students that textbooks don't tell the entire story.

The reality is that there are a number of ways to handle uncertainty out in the world. Professionals do not always agree on the best approach - this conversation on the Physics Stack Exchange has a number of options and the mathematical basis behind them. For students that are used to having one correct answer, this is a major change in philosophy.

Thus far in my teaching career, I haven't delved this deeply into uncertainty. The AP Physics curriculum doesn't require a deep treatment of the concepts and roughly ignores significant figures as well. I talked about some of the issues with uncertainty with students, but I never felt it was necessary to get our hands really dirty with it because it wasn't being assessed. We also learned error analysis in my experimental design courses in college, and it was part of the discussion there, but it was never the class discussion. It's really interesting to think about these issues with students, but it's also really difficult.

It seems that the questions that have resulted both from class and for my own understanding are exactly the style of conflict that the IB organization hopes will result from its programs. The way this student throws her hands up in the air and asks 'so what do I do' and managing the frustration that results is the same difficulty that we as adults face in resolving daily problems that are real, and complex.

The philosophy that I shared with the students was to be aware of these issues, but not to fear them. It should be part of the conversation, but not its entirety, especially at the level of students that are new to physics. I'm confident that some of the discomfort will melt away as we do more experimentation and explore physics models that tend to describe the world with some level of accuracy. The frustration will yield to the fact that managing uncertainty is an important element of describing how our universe works.

# Graduated Assessment & Web Design

I decided to try teaching programming this year as a class, specifically HTML, CSS, and Javascript. My hope is to get students to the point that they can put together a basic Meteor app by the end of the year, along with a good set of skills for building web pages from scratch.

My assessment scheme for the class is through a series of projects. Some are small, some are bigger and more open ended. I noticed during class on Tuesday that students are cutting corners by copying HTML from files we used in earlier classes. I admittedly do this all the time, but I pay the price for doing so in time spent cleaning up fragments of code that I really could have written from scratch in less time. In the general series of trying to teach good habits, I decided to give a graduated assignment that went through the series of HTML concepts we have learned over the past two weeks.

The requirement is that students need to make one HTML file for each of the ten steps in this file. They have to start from a blank HTML template that has nothing more than the basic head and body elements. I also had students write out the code for the first three steps by hand.

The effect today in class was a clear measurement of where each student stands in understanding how to piece together HTML from scratch. I collected the folders of files from students at the end of class and can see precisely where their difficulties are. Some of this comes from just knowing what step they were on at the end of class, but I also had some good conversations with students throughout the period today. My class, thankfully, was pretty honest in showing what they do and don't understand how to do. While there has been some code sharing in the class before, they seemed to appreciate this opportunity to step their way through the progression of what we've learned so far.

I think there's an analogue in math and science here - I've given leveled practice sheets before in various mathematics topics. The willingness to push through misunderstanding and admit difficulties seems to be a lot more substantial in a programming context.

Here's the full exercise:
day5-Step Instructions

By my last week in the states this summer, I had made it to San Francisco. Before getting to the hard work of eating sourdough and tinkering my way through the Exploratorium, I made two stops that were really special.

The first was a chance to meet the Desmos team at their office, arranged by Dan Meyer, who was planning to be at the office. I walked into the office while the team, not two days from the successful release of their newest activity Central Park, was on a conference call discussing the next project. As my time with them went on, I periodically felt a wave of giddiness at the fact that I was sitting with some of the people responsible for making Desmos what it is.

The four people that I met there, two thirds of the entire team, had their hands in making a difference in hundreds (if not thousands) of classrooms around the world. Jason shared that his changes to some code had resulted in a substantial increase in the code speed. Jenny showed her prototypes for a beautiful new user interface. Eric repeatedly referred to the guiding principles of Desmos as they made decisions about moving forward. The careful, deliberate work done by this group of passionate people is the reason Desmos is able to create the collaborative learning experiences for which they are known.

At one point, David Reiman, one of the team members and a former teacher himself, asked me what they could do for me. Honestly, all I could muster was that it was an honor to learn from them and see their workflow. They put a lot of energy into making sure their tools are useful for reaching objectives in the classroom, not for the sake of merely being used in the general category of technology. I really appreciate Dan, Eric, Eli, and the rest of the team for arranging to spend time with me.

The next day involved a visit to Meteor headquarters for their monthly Devshop. This is a meeting that gets Meteor coders and entrepreneurs in one room with the goal of everyone helping each other. It was impressive to meet in person some people that I had really seen only as Twitter handles. They were all incredibly genuine, humble people that worked really hard on work that mattered to them. I gave a lightning talk on coding for the classroom (posted here on YouTube) and using code to make my life as a teacher easier. Mine was one of a series of such talks. They were streamed live on the internet, but it seemed much more intimate in the actual room. Each person had three minutes to talk about an idea that mattered to them. It was also a treat that people that came up to me to chat afterwards - some of them teachers themselves - to talk about teaching, coding, and the challenges of teaching effectively with technology.

The theme that struck me after both days, a theme that I think resonates strongly with the beauty of the existence of the Math-Twitter-Blog-o-sphere, is not just that individuals (and teams) are doing interesting, thought-provoking work. That has been true for a long time.

The people at Desmos and Meteor are designing tools that enable others to not just explore their ideas, but develop, build, and share them. Just as these tools are created iteratively (Meteor released version 0.9 today) they encourage others to make the most of what is out there to put ideas in front of an audience and make them better over time. That audience might be a classroom of students. It might be an audience craving a useful online tool that targets their unique niche. Everyone at these companies (and in classrooms) is hoping that the next idea they try is one that gets more people excited than the last. Teachers work in a similar vein hoping that their next idea for tweaking a lesson gets more students engaged and making connections than the last.

I've spent the past three academic years interacting with people through this blog, Twitter, and other online channels. I've shared ideas here and have gotten feedback on them from a number of different perspectives. All of us are working hard. We have ideas and share them because ideas sprout new ideas. This process is addictive. We all have our pet projects and obsessions, and need to be brought back to reality from time to time about what will really work most effectively. We listen to each other and value the conversations that happen.

As this year is getting underway, I'm going to work to keep something in mind this year. We all have governing principles that help us decide what work to tackle at a given point in time. We often wait to share ideas until they are fully formed, but that's not really when we need the most feedback. I hope to share more ideas when they are raw and still forming. Bad days, especially when they are still smarting from an unsuccessful lesson, are revealing. It's in these situations that we stand to grow the most. What makes innovative companies like Meteor and Desmos successful isn't that they have the best ideas from the beginning. It's that they know how to cultivate ideas from beginning to end, and aren't afraid to make mistakes along the way. They acknowledge that there are lots of starts and stops and hiccups before ending up on the idea that will make a difference to people.

Have a great school year, everybody!

# Picasso's Bull - Not Just for Design Thinking

I came across the New York Times article on the Apple's training program and its use in describing their design process. I hadn't seen it before, but saw it also as a pretty good approximation for mathematical abstraction.

I used the lithographs 1 - 11 from http://artyfactory.com/art_appreciation/animals_in_art/pablo_picasso.htm and put them together like this:

We have shortened classes tomorrow (20 minutes) and I think it might be good material for a way to introduce the philosophy of the IB Mathematics and Math 10 courses. Some potential questions floating in my head now:

• How does this series of images relate to thinking mathematically?
• What does the last representation have that the first representation does not? How is this similar to using math to model the world around us?
• Can you do a similar series of drawings that show a similar progression of abstraction from your previous math classes?

This seems to be a really interesting line of thinking that connects well to the theory of knowledge component of the IB curriculum. I see this as a pretty compelling story line that relates to written representation of numbers, approximations, and the idea of creating mathematical models. Do you have other ideas for how this might be used with students?