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. 

The World Fertility Rate: A Relevant Mathematics Topic

I have decided to begin to blog again. As someone who is very task oriented, I was always excited to be able to check of "write blog" from my To Do list, especially in the days when I blogged each day for one full school year. I really have no idea how I found time to do that. It must be that I created a 25-hour day. As of September of 2024, I am no longer in the classroom and I wondered what topics would be in my future blogs. You'll need to read future blogs to find out! 

Besides the near daily question of "Did I miss anything?" from students returning from an absence, the second most asked question high school math teachers hear is "When will I ever use this?" This is often heard for the first time in Algebra classes, where math begins to seem like an exercise in algebraic origami. For those of us that have an innate love of math, (I wore a red t-shirt that boldly declared “I Love Math” when meeting my colleagues for the first time at Champlain Valley Union High School in Vermont.) we don’t see the need for all math to be relevant. Math is beautiful and should be studied to improve reasoning and thinking. But sadly, it seems as if much of the United States feels that math must be relevant to be worthy of being a part of the curriculum.

And this leads me to the topic of today's blog - The World Fertility Rate: A Relevant Mathematics Topic. As I was looking at a daily news email a few days ago, I clicked on a link with the following headline:

Charting global fertility rates, which have declined by half since 1965

I won’t go into the all the details of what I discovered and ruin the fun for you. But I did find the following two graphs to be quite interesting and learned that the “Global Replacement Fertility Rate” of 2.3 children was reached in 2015. This is the fertility rate at which the population says constant. You might be wondering why it isn’t 2, but that is because we need to take into account the deaths of the female population before the end of their childbearing years. 

If I was using these graphs in my classroom, I would give students 5 minutes to look at both and write down what they notice and wonder about the graphs. Then, I would have them share their ideas with a partner for about 5 minutes. Next, we would debrief as a class for about 10-15 minutes. From our list of wondering questions, students would be asked to choose 1 or 2 of the questions and work in groups to try to find the answer to those questions. 

Possible questions could include the following: 

1. Why is China’s fertility rate lower than the fertility rate in the US now than before 1980?
2. If this trend were to continue, when would a global average fertility rate of 2.3 be reached?
3. Do countries with higher (or lower) fertility rates have common characteristics?

  

The data, graphs and analysis can be found at https://ourworldindata.org/global-decline-fertility-rate.

If you use this graph with your students, post how things went in the comments. Math educators sharing and learning from each other makes the world brighter.