This summer, I had the privilege of teaching a 3-week institute for eighth grade teachers. One of our aims was to help teachers grapple with mathematics in the Common Core State Standards that is new to (or long forgotten by) the teachers. One of the major changes is the inclusion of a mathematical modeling standard (Standards for Mathematical Practice 4), and in eighth grade, three standards refer to investigating patterns in bivariate data. This includes thinking about whether a pattern fits a linear model, and informally fitting a line to data. Thus, we spent a number of sessions engaged with bivariate data. For purposes of this post, the main point I want to make is how technology has made fairly sophisticated mathematics more accessible, and to briefly describe how we used technology to do mathematical modeling.
As I have written previously, I think the TI-Nspire is worthwhile in spite of its price, and so that was the focus of much of our work. The problem Gate-to-Gate, which I updated and adapted from a problem I found in a book, is a good example of our work. In brief, the goal of the lesson is to build and assess a model that predicts flight times from Chicago given the flight distance from Chicago.
In that problem, we started by making observations about a map produced by http://www.flighttimesmap.com that shows concentric rings labeled with estimated flight times. I used Chicago as the point of origin. First, we collected observations, such as: the rings appear to be circles, the circles appear to be equally spaced, and the first circle is marked as a 1 hour flight. Next, we discussed the meaning of the observations, and we conjectured that the equal spacing is an indicator of a fairly constant flight speed. We also wondered whether the map was completely accurate with its times.
The next step was to have the teachers explore a data set. I gathered actual flight times that I had looked up and put the data in a TI-Nspire file. The teachers were then saved the trouble of typing in their own data. Teachers then created scatter plots, attempted to fit their own informal lines to the data, and ran a linear regression on the data. With the line, they then chose flight destinations, looked up the flight distances (via web search), and compared the flight times predicted by the model to those they found on the web. They also tried to think of cities on the map that would be 3 hours away from Chicago by air, and again compared the real data to the predictions of the model.
Finally, we had a summary discussion about the quality of the model fit, the meaning of the values in the linear equation, and considerations about what is an appropriate domain for the linear function. (What does it mean to have a flight covering a distance of 0 miles?) Some teachers graphed distance as the independent variable, and others graphed time as independent, giving two different equations. This led to different insights from the different slope and intercept values. It was a good discussion and led to good insights about both modeling and the meaning of slope and intercept in context.
Stepping back from the problem, here is a look at how technology enhanced this exploration.
- Data is more easily shared. This saves a tremendous amount of time. I shared the TI-Nspire file as a Dropbox link on a Lino board that I established for the class. This is a long way from having to either plot data points by hand, or even sharing the data but having each person enter data into their own spreadsheet for analysis. If I were not doing this lesson with iPads, I would either have to pre-load data onto handheld calculators, or if those were not available, perhaps give the data in a table and an already-plotted graph (or two graphs, with the two choices of independent variable).
- Data is more easily analyzed. Fitting a line informally can be easily explored with touch screens. And, since the technology handles finding the equation of the moveable line, the focus of the conversation is on the quality of fit of the model, rather than a focus on the procedure of finding the line equation. Computing technology to perform tasks such as regression has been available for many years. Nonetheless, it is a powerful tool, and the ability to use regression with a button click means that there can be a discussion of how our informal lines compared with the regression line. If this were a calculator lesson, we might still use moveable lines, but less easily. And, barring that, we would have spaghetti on a paper graph, but then we would not be able to compute the line equations quickly.
- Access to the web helps make it easier to test a model with real-world data. With access to maps and the ability to look up flights, teachers had a lot of freedom to test their models. If we did not have the web, I would have had to preselect a set of cities, listed with distances and flight times, and use that as the basis for testing the model.
- Sharing results is easier. We used Baiboard, and I selected individual teachers, who then uploaded screen shots of their models and results. This meant that when teachers were sharing, either they or I could add annotations to the screen shots. Moreover, as others shared, we could swap back and forth between the current person’s work and the work already shared by others. If this were a lesson on calculators, teachers would have had to keep a separate handwritten record of their work, and switch back and forth between sharing their written work and sharing the work on the calculator. We would probably have to keep a (partial) record of what was shared on a whiteboard for later reference.
In looking at the effect of technology on the lesson, it is not the case that without iPad technology, the lesson is impossible. Compared with, say, having classroom calculators, it is that the technology makes the lesson run more smoothly and quickly, adds the authenticity of finding one’s own data, and improves the way results can be shared.