Descartes Dreamworld

Most of us probably know the name Rene Descartes. In addition to his contributions to the mathematical world (the Cartesian coordinate system is named after him), he is also a well known philosopher, particularly for the quote "I think, therefore, I am". In the first half of the 17th century, Rene Descartes dreamed of a world completely described and governed by mathematics. He saw the trajectory of mathematics, which was, to him and many others, the pure distillation of understanding. He wanted to apply the same logic and precision that mathematics exhibited. He believed that a world dominated by mathematics would be a triumph of understanding and knowledge.

We now sit in the 21st century. Every day, we use innumerable devices and tools that are based upon digitization and mathematics.  Since the early 1600’s, the world has experienced an explosion of mechanization, automation and computerization.  We now live in a world more similar to the world Descartes dreamed about and yet, this world driven by mathematics has not given us the salvation he predicted.  We have adopted a mathematical approach to a number of sciences and arts that would impress Descartes, but there are numerous ways in which numbers and metrics are used to confuse and mislead people.  Whether it is politically, financially or in a wide variety of other fields, numbers, modeling and mathematics in general are used for questionable ends.  Beyond the intentional misuse of mathematics, the act of digitizing and analyzing information creates the feeling of a loss of humanity. 

In order to aid students as they head into the mathematized world, I created a few activities that could be inserted into a unit that might allow students to better understand the pitfalls of the information they see throughout their lives.  I used the unit below, on Linear Equations and their Graphs as a starting point, and created 4 activities that could be used at specific points in the unit. 

 

Unit Warm-Up Activity

Students will write a journal entry reflecting on the following questions. 

• What kind of information, or data, can we put in a graph?

• Graphs and charts are good ways to organize information.  Do you agree or disagree?  Explain.

• How do people use graphs?  Do they help to understand what happened?  Do they predict what will happen next?

6.3 Applying Linear Functions Activity

In order to begin to understand the limitations and specific problems created by linear graphs, students will complete this activity to better recognize what requirements are necessary for real world examples to be accurately modeled using a linear graph. 

The students will create a real world example that can be modeled using a linear graph.  They can use examples from the book as a starting point, as seen in the graph below. 

Once students do this, select students (some selected for correct work, others for incorrect work to “find the mistake”) will share their results with the class. 

Students should work to answer the following question:  What conditions do you need to exactly meet the conditions of a linear relationship?

Students should discuss different features of the situations they created.  The discussion should be guided towards the idea that in order to be a true linear relationship, the change in the dependent variable is proportional to the change in the independent variable.   They should also consider and discuss what types of questions could they answer with the graph they've created.

The text uses the graph below to describe profits from a car wash, based on the number of cars the students wash.  What variables could we introduce that would make this an imperfect linear relationship?

 

6.7 Scatter Plots and Equations of lines

Students will again grapple with the limits of linear modeling, this time by evaluating trend lines in data. 

As a class, students will work in a computer lab to explore this idea.  First, students will research and select a set of data that includes at least 5 points of data, and no more than about 25.  This can come from anywhere, and only the most extreme cases will be disallowed. 

Using Excel, students will enter their selected data into a spreadsheet and create a scatter plot of the data.  They will then create a line of best fit, or trend line, and print out their resulting graph.

Back in class, we will discuss again their findings.  We will try to identify if any students had a true and accurate linear graph, which is unlikely but not impossible.  We will also discuss the student’s graph that is least accurately represented by their trend line.  It will be important to discuss what the trend line (using vocabulary like Slope and Intercept) really represent in each graph, and whether or not that information is valuable. 

The discussion should cover what we mean when we take an average.  We can also discuss statistical ideas like sample size and deviation if possible.  The text introduces a correlation coefficient, which will be a good measure of the accuracy of a trend line.  Additionally, the students should again think and discuss what types of questions they could answer with their data and the corresponding trend line.

As a class, we will gather the heights of all students in the class, and then average the heights.  The average should be a non-integer around 5-6 or 5-7.  I will make the point that even though everyone was accounted for in the experiment, no one in the class is actually represented by the average height.  The class will discuss what is meant by the “average person” or the “average family” (graph here), and what it means to have 3.13 family members.    

At this point, students will have seen the limitations of a modeling from a few different viewpoints.  Students should have a better understanding of how statistics can be used and misused.

Final Project

Students will discuss the ways that data can be skewed and how bias can show up in seemingly scientific and mathematically sound areas.  Students should understand that data can be “spun” in different ways graphically, which can alter the conclusions an audience draws from that graph.    

As a final project, students will research some piece of information that is represented graphically (preferably linear graphs, or a scatter plot with trend line) for which specific data is available.  Students will then, using the same data and information available, re-create the graph in a way that, in their opinion, may change the utility or possible conclusions drawn by the audience.  Students can alter the scale of the graph, the orientation and even the sample of data used, but cannot change the data itself.  Students will also explain how their choices may affect the perceptions created by their new graph.

As a conclusion to the unit, students will return to the journal entries from the beginning of the unit.  They will have the option of editing (or rewriting) their journal entry, or, responding to their “earlier selves” and correcting any mistakes or misconceptions they feel are part of the entry.  These edits and explanations should demonstrate new knowledge and insight learned from the chapter, as well as these lessons.

Note:  The idea for this post came from reading Descartes Dream: The World According to Mathematics by Philip Davis & Reuben Hersh.