An Idea for After the AP Exam: Who Takes Calculus?

My last post was about the documentary Counted Out. I am hoping that the Pennsylvania Council of Teachers of Mathematics will be able to hold a virtual showing of this film in August or September. If that happens, I'll be sure to announce it on my blog.

If you are wondering which students are counted out in today's math classrooms, I encourage you and your students to read this blog by Just Equations: (Not Yet) Hidden Figures: Preserving Data for the Future of Education.

The blog states:"One of the motivations for this blog's focus is the knowledge that this data source and other Just Equations has relied on may not contiue to be available, given recent cuts to the U.S. Department of Education." Data related to the demographics of students who complete various math and science courses can be found in this chapter of a report called High School Mathematics and Science Course Completion. If I was still in the classroom, I would probably show my AP Statistics and AP Calculus students a few of the graphs and ask them to tell me the story shown by those graphs. The graphs I found most fascinating were the following:

Figure 2: Percentage of public and private high school graduates who completed selected mathematics and science courses in high school, by race/ethnicity: 2019

Figure 4. Percentage of public and private high school graduates who completed selected mathematics and science courses in high school, by school type: 2019

Figure 5. Percentage of public and private high school graduates who completed selected mathematics and science courses in high school, by percentage of students at their school who were eligible for free or reduced-price lunch: 2019

After discussing the story of the graphs, we would look at the report itself, how the data was gathered and the conclusions that were drawn by the authors. Finally, I would have the students reflect on what surprised them the most about the data and encourage them to share their new understanding with parents and other adults in their sphere.

Note: If you want to learn more about Just Equations and their work to support equity in math education, I would encourage you to read their report called The Mathematics of Opportunity: Rethinking the Role of Math in Educational Equity. This report would be a good companion to accompany a discussion about the documentary Counted Out.

Counted Out: An Urgent Call to Action

Last Wednesday, I saw a screening of the movie Counted Out (www.countedoutfilm.com/) at Lafayette College in Easton, Pennsylvania with about 70 professors and teachers. After watching the movie, I felt hope. Hope is something that is often in short supply these days. The movie blends interviews with mathematicians and math educators along with classroom observations. It also includes chats with students, both children and adults, who have felt the fear, shame and anxiety that is experienced by many during their K-12 schooling. Each story they told was something that has been almost universally felt by students at some point on their math journey.

My earliest memories of math involve drilled timed tests on addition and subtraction facts in second grade. I remember not being able to get the answers fast enough before the disembodied voice on the record player moved onto the next question. Tears and frustration frequently accompanied those tests. Eventually it got better. Was it because the teacher took me aside and told me it is ok to not be fast at math as you are learning? Was it because my parents drilled me on my math facts at home? Was it because eventually I just learned them because I had no choice?

The answer to all of these questions is “no”. I distinctly remember learning the math facts because I saw patterns. I saw that the sum of 9 plus a single digit was a number that started with a 1 and ended in a number that was one smaller than the single digit. For example, 9 plus 7 would end in a 6. The answer was 16. I did not realize it at the time, but I was essentially regrouping and rewriting 9 + 7 as 9 + 1 + 6 = 10 + 6 = 16. I also figured out that it was ok if I didn’t memorize 8 + 5, since I knew 8 + 4 and could add one more. It felt like I had discovered a secret to solving these that I thought might be considered cheating. I didn’t share my discoveries with anyone.

Focusing on just getting the answer and thinking of doing mathematics as a linear path with one right answer is what is causing many students to be “counted out”. The movie emphasizes that new teaching methods must be employed to allow students to have access to math and more options. In addition, math has typically been used to divide children into groups resulting in many students seeing themselves as incapable of math simply because of how they were tracked in math class. 

I did walk away from the movie with a feeling of hope, but also a feeling of being overwhelmed. Helping students and adults see the urgency of why math needs to be viewed differently, essentially changing the views of the masses, involves working with larger systems. How do we get state departments of education to understand that there is no quick fix to challenging problems? How do we get all stakeholders to see that math eduction in many areas has not advanced into the 21st century? How do we get current educators to understand that the textbook they used to learn Algebra 1 in the 1980’s is no longer sufficient for the math needed today? How do we get to the place where it is recognized that all students are capable of learning math when we give them the tools to explore, play, and discover? How do we get everyone to see that access to quality math education is crucial to promoting equity, democracy and a sound economy?

I don’t have the answer to these questions, but I am hoping to host a virtual screening of the movie with the Pennsylvania Council of Teachers of Mathematics in August or September. Stay tuned for details.

Polar Function Intersection?

Although spring is around the corner, polar functions crossed my mind last weekend as temperatures fell into the teens. And when mathematicians think of cold, we think of polar functions. Consider the graph shown. How many points of intersection do you see?

Most people would say there is one point of intersection on the vertical axis. But what if I told you the two equations were the following? How many points of intersection are there on the domain of 0 to 2pi?

When you try to solve this system of equations, you get -2 = 1 + sin𝚹 or -3 = sin𝚹. Since the sine ratio can only vary between -1 and 1, this equation has no solution. So why does the graph show a point of intersection? Or does it? Watch the video below and decide for yourself.

I encourage you to play around with Desmos at desmos.com/calculator. Here are three things for you to explore. 

1. What if r = 2 instead of r = -2?
2. Notice that I added the term 0 times sin𝚹  to r = -2. What happens if that is removed?
3. Can you create a pair of polar functions that look like they intersect in 2 or 3 places, but don’t actually intersect.

Share your observations in the comments.