I love doing three act problems. This fact should surprise nobody that regularly reads my blog.
In tasks that involve prediction or measurement from a range of sources, I see lots of tables of values made by students that stay in the notebook. I have always wanted to get that data in the hands of the rest of the students to use, or not use, as they see fit. In one of my previous iterations, I pasted the data into a shared Google spreadsheet that students could then paste into a Desmos graph, again, if they felt doing so would be helpful. This was incredibly rich source of material for conversations between students. Still, that extra step of having to paste from one collaborative document (Google) to a non-collaborative one (Desmos Calculator) was one more step than I felt was needed.
Of course, you’re now screaming at the screen. “Calling out Desmos for being non-collaborative is entirely off base, Evan” , you say. I agree to an extent. Their own activities share data collected by individual students, and on the teacher side, the Activity Builder does the same thing for letting teachers see student data all in one place. They do this incredibly well. Students also get to see each others answers when teachers let them. What doesn’t happen right now is students seeing each other’s graphs, tables, and lists of expressions.
This, along with a desire to play with the Desmos API, is why I created DataTogether (Github repository here), a hacky way to make Desmos data collaborative. The page is written in React, and uses Firebase to do the realtime data connection.
Dan Meyer tweeted shortly after that these changes might be somewhere in the pipeline already:
@emwdx On the one hand, I admire your industry. On the other, I'm mad that we make this so difficult. Promise to make this unnecessary soon.
This is probably why I may not be adding a lot of code comments to my code in the near future.
I did my Two Lane Road 3-act with a small group of students this morning on account of tenth graders being out for the PSAT. After the standard Act 1 conversation, and a really great conversation about agreements between groups on collecting data from the video, the students began collecting data on the red and blue cars.
The students were efficiently able to collect data together on separate computers after profuse apologies for the limitations of my code:
I then had each student use the tools within Desmos to construct a linear model from the data. The fact that two computers were looking at the same data, but in different Desmos windows, paid significant dividends when two students on the same team created their models in different ways. One student made a regression. Another created a line that went through one set of points perfectly, but missed another. Math class conversation gold right there.
I exported both of their data through the console (code shown below) and pasted it into Desmos. I then put together a simulation of the red and blue car so that the teams could see what their car looked like in simulation.
This allowed us to make a prediction directly off of their models that looked like the original video.
We ran out of time in the end to do much more than sharing predictions and watching the third act, but I’m pretty pleased with how things went overall. My paper handouts with three printed color frames of the video went unused. I think
A big shout-out of thanks to everyone for helping test the data collection tool I shared earlier.
Here’s a screenshot of our age vs. teaching years data:
The data can be downloaded from the DataTogether page, loading data set 3DR9, and then by going to the console and entering the code below:
I created an interactive lesson called Thinking Machine for use with a talk I gave to the IB theory of knowledge class, which is currently on a unit studying mathematics.
The lesson made good use of the Meteor Blaze library as well as the Desmos Graphing Calculator API. Big thanks to Eli and Jason from Desmos for helping me with putting it together.
I was asked by a colleague if I was interested in speaking to the IB theory of knowledge class during the mathematics unit. I barely let him finish his request before I started talking about what I was interested in sharing with them.
If you read this blog, you know that I’m fascinated by the intersection of computers and mathematical thinking. If you don’t, now you do. More specifically, I spend a great deal of time contemplating the connections between mathematics and programming. I believe that computers can serve as a stepping stone between students understanding of arithmetic and the abstract idea of a variable.
The fact that computers do precisely what their programmers make them do is a good thing. We can forget this easily, however, in our world that has computers doing fairly sophisticated things behind the scenes. The fact that Siri can understand what we say, and then do what we ask, is impressive. The extent to which the computer knows what it is doing is up for debate. It’s pretty hard to argue though that computers aren’t doing similar types of reasoning processes that humans do in going about their day.
Here’s what I did with the class:
I began by talking about myself as a mathematical thinker. Contrary to what many of them might think, I don’t spend my time going around the world looking for equations to solve. I don’t seek out calculations for fun. In fact, I actively dislike making calculations. What I really enjoy is finding interesting problems to solve. I get a great deal of satisfaction and a greater understanding of the world through doing so.
What does this process involve? I make observations of the world. I look for situations, ideas, and images that interest me. I ask questions about what I see, and then use my understanding of the world, including knowledge in the realm of mathematics, to construct possible answers. As a mathematical and scientific thinker, this process of gathering evidence, making predictions using a model, testing them, and then adjusting those models is in my blood.
I then set the students loose to do an activity I created called Thinking Machine. I styled it after the amazing lessons that the Desmos team puts together, and used their tools to create it. More on that later. Check it out, and come back when you’re done.
The activity begins with a first step asks students make a prediction of a mathematical rule created by the computer. The rule is never complicated – always a linear function. When the student enters the correct rule, the computer says to move on.
The next step is to turn the tables on the student – the computer will guess a rule (limited to linear, quadratic, cubic, or exponential functions) based on three sets of inputs and outputs that the student provides. Beyond those three inputs, the student should only answer ‘yes’ or ‘no’ to the guesses that the computer provides.
The computer learns by adjusting its model based on the responses. Once the certainty is above a certain level, the computer gives its guess of the rule, and shows the process it went through of using the student’s feedback to make its decision. When I did this with the class, more than half of the class had their guesses correctly determined. I’ve since tweaked this to make it more reliable.
After this, we had a discussion about whether or not the computer was thinking. We talked about what it means for a computer to have knowledge of a problem at hand. Where did that knowledge come from? How does it know what is true, and what is not? How does this relate to learning mathematics? What elements of thinking are distinctly human? Creativity came up a couple times as being one of these elements.
This was a perfect segue to this video about the IBM computer Watson learning to be a chef:
Few were able to really explain this away as being uncreative, but they weren’t willing to claim that Watson was thinking here.
Another example was this video from the Google Deep Thinking lab:
I finished by leading a conversation about data collection and what it signifies. We talked about some basic concepts of machine learning, learning sets, and some basic ideas about how this compared to humans learning and thinking. One of my closing points was that one’s experience is a data set that the brain uses to make decisions. If computers are able to use data in a similar way, it’s hard to argue that they aren’t thinking in some way.
Students had some great comments questions along the way. One asked if I thought we were approaching the singularity. It was a lot of fun to get the students thinking this way, especially in a different context than in my IB Math and Physics classes. Building this also has me thinking about other projects for the future. There is no need to invent a graphing library on your own, especially for use in an activity used with students – Desmos definitely has it all covered.
I built Thinking Machine using Bootstrap, the Meteor Blaze template engine, jQuery, and the Desmos API. I’m especially thankful to Eli Luberoff and Jason Merrill from Desmos who helped me with using the features. I used the APIto do two things:
Parse the user’s rule and check it against the computer’s rule using some test values
Graph the user’s input and output data, perform regressions, and give the regression parameters
The whole process of using Desmos here was pretty smooth, and is just one more reason why they rock.
The learning algorithm is fairly simple. As described (though much more briefly) in the activity, the algorithm first assumes that the four regressions of the data are equally likely in an array called isThisRight. When the user clicks ‘yes’ for a given input and output, the weighting factor in the associated element of the array is doubled, and then the array is normalized so that the probabilities add to 1.
The selected input/output is replaced by a prediction from a model that is selected according to the weights of the four models – higher weights mean a model is more likely to be selected. For example, if the quadratic model is higher than the other three, a prediction from the quadratic model is more likely to be added to the list of four. This is why the guesses for a given model appear more frequently when it has been given a ‘yes’ response.
Initially I felt that asking the user for three inputs was a bit cheap. It only takes two points to define a line or an exponential regression, and three for a quadratic regression. I could have written a big switch statement to check if data was linear or exponential, and then quadratic, and then say it had to then be cubic. I wanted to actually give a learning algorithm a try and see if it could figure out the regression without my programming in that logic directly. In the end, the algorithm works reasonable well, including in cases where you make a mistake, or you give two repeated inputs. With only two distinct points, the program is able to eventually figure out the exponential and quadratic, though cubic rules give it trouble. In the end, the prediction of the rule is probability based, which is what I was looking for.
The progress bar is obviously fake, but I wanted something in there to make it look like the computer was thinking. I can’t find the article now, but I recall reading somewhere that if a computer is able to respond too quickly to a person’s query, there’s a perception that the results aren’t legitimate. Someone help me with this citation, please.