A deer in headlights

"I have said in the past that it is not clear whether intelligence has any long-term survival value - bacteria multiply and flourish without it."

"Oh dear, just as I was about to unveil the universe's mysteries!"

Picture Concept: Mukto Akash, with special thanks to Car Headlights Stock photos by Vecteezy

For the love of Math

Something rare happened at work recently. I was running a Math drop-in session for an event called "Long Night Against Procrastination", and there were about six or eight students sat a table near me. Turns out, this group was just waiting for pizza to be served, and their college program had no math courses. However, they mentioned that they were all fascinated by math, and asked me give them a math problem to solve while they waited for the pizza.

The first song

Intrigued, I inquired about their level of proficiency in math. They mentioned that they studied mathematics at the intermediate level. Astute readers may have noted that this verbiage alludes to a particular education system - please comment below if you are familiar with what "intermediate level" means. Anyways, back to the story. I asked them to solve the following relatively challenging calculus problem:

`\int ( \sin x + \cos x )^{2} dx`.

It took some good old-fashioned team-work, perseverance, and a little guidance from yours truly, but the problem was solved within 30 minutes. Pizza still wasn't here. To my astonishment, this group of students loved working on the problem so much that they asked me for a second problem!

Encore

This time they requested a Statistics problem as that was their forte. In the fashion of a true Bond villain, I gave them a problem involving conditional probability:

Suppose in an epidemic, it is known that about 5% of the population is sick. Scientists have developed a rapid test which gives 3% false positive and 2% false negative results. Given an individual has tested negative, what is the probability that they are sick?

Of course the students were stumped. This was not an easy problem. Nevertheless, they were keen to understand how to work through the problem. Once I explained the idea of tree-diagrams in conditional probability, they were soon able calculate marginal probabilities. After a brief demonstration of Bayes' rule using the tree-diagram (I never wrote down the formula for Bayes' rule for these students), they were able to get the solution. Their excitement was pulpable. Alas, pizza arrived soon after, so math took a backseat. Nevertheless, they stayed back to shake my hands after finishing their meal.

What I saw in those students that day was pure love and joy from discovering mathematics. Remember, these students have not taken, and will not be taking, any college math courses. It was unusual that they would want to solve math problems, and downright improbable that they would enjoy the process. This experience brought me an exuberance that I haven't felt in quite some time. It's always great to find others who enjoy your craft. Wouldn't it be lovely if all of my engagements were so positive?

Where's the beef? - Defining Math Anxiety.

How did the above math problems make you feel?

Did you find yourself in one of the groups below?

1. You felt an urge to solve the problems right-away. May be you are able to solve these problems in your head, or at least immediately know how to arrive at the solution.
2. You recognized the problems. They seem familiar, and with some effort, you know how to solve them. Something that you may do later when you have more time at hand.
3. You briefly glanced at the problems and moved on without sparing a thought. Those math problems are only accessories to the story, so why pay any attention, right?
4. You cringed, or perhaps even cowered, looking at those problems. Math isn't really your thing. How can someone even enjoy math like that?

These are all valid reactions, but the last two groups may be exhibiting signs of math anxiety. The purpose of this blog is to learn about math anxiety, so you have come to the right place. In this post, let us define math anxiety. Our primary sources will be papers from 1972 (Richardson and Suinn) and 2002 (Ashcraft). You will find the references to these papers in the citations page attached to this blog.

The math demon is out to get you!
Photo Credit - Kindel Media

Key notes from the Papers

According to Marc-Antoine Crocq, the word anxiety is derived from a Latin word which means "to constraint", and is concerned with the psychological and cognitive aspects of worrying and unrest. Browsing the definitions of anxiety on Google tells us that anxiety is dread, fear, and uneasiness from real for perceived threats, and consequently leads to avoidance behavior. Math anxiety is considered to be an independent phenomenon from general anxiety.

Here are some of my key takes on the definition of math anxiety from the papers. Some of these points are direct quotes, and others are portmanteaux of quotes combined from the papers. In other words, the points below are not my own words or thoughts, but my perception of what the authors are saying.

• Richardson and Suinn states Mathematics anxiety involves feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations.

  Ashcraft refines the definition to say Math anxiety is commonly defined as a feeling of tension, apprehension, or fear that interferes with math performance.

• Highly math anxious individuals avoid math. They take fewer elective math courses than people with low math anxiety. And when they take math, they receive lower grades. Thus, mathematics anxiety may prevent a student from passing fundamental mathematics courses or prevent his pursuing advanced courses in mathematics or the sciences.

• Among nonstudents, mathematics anxiety may be a contributor to tensions during routine or everyday activities, such as handling money, balancing bank accounts, evaluating sales prices, or dividing work loads.

• It has been demonstrated that mathematics anxiety exists among many individuals who do not ordinarily suffer from any other tensions.

• Math anxiety is only weakly related to overall intelligence.

• U.S. culture abounds with attitudes that foster math anxiety: Math is thought to be inherently difficult (as Barbie dolls used to say "Math class is hard"), aptitude is considered far more important than effort, and being good at math is considered relatively unimportant, or even optional.

You must be smart because you study Math!

First, let's acknowledge that the current definition of math anxiety really tracks back to 1972. Although Ashcraft refines the definition somewhat in 2002, it's still fundamentally the same. I checked a couple of recent papers from 2021 (Barroso et. al.) and 2023 (Gonzalez-DeHass et. al.) to see if there is an updated definition, but found out that they both refer back to one of these two definitions. May be the age-old principle of "if it ain't broke, don't fix it" applies here. However, in my opinion, with the advancement in our understanding of cognitive and physical manifestation of anxiety in the past 50 years, there should be a more recent attempt at defining math anxiety. If you know of any other accepted definition of math anxiety, please comment below.

Next, it is quite clear from the discussions of both sets of authors that math anxiety is an independent concept. Certainly it's quite interesting that math-anxious individuals may otherwise not be affected by other types of anxiety, though there is mention of high degree of association between math- and test-anxiety. As I have always strongly suspected, intelligence is not a factor. So no more "you must be smart because you study math" please. Perhaps this also explains the epigraph for this post. For bonus points, can you name the famous scientist who is being quoted, and write the full quote in the comments?

Escaping the Concrete Jungle - where do we go now?

Math-anxiety seems to have an effect at all levels of study. In particular, this phenomena is quite common amongst college population. Ashcraft notes that at high level mathematics it becomes more difficult to separate math anxiety from math incompetence. To get a true picture of math competence for an individual, it becomes important to relieve the anxiety attached to math. Could this be achieved through meditation, cognitive behavioral therapy, or through medication? In their respective papers, Richardson and Suinn goes further into measuring the level of math-anxiety, while Ashcraft discusses personal, educational, and cognitive consequences of math anxiety. I plan to cover both of these topics in a later post.

Finally, I am personally drawn towards the comment about U.S culture by Ashcraft. I was not educated in North America, at least not until I started my undergraduate degree in Physics at a Canadian university. For a long time I was always surrounded by people who love math, and use very high level math in their work. So it was indeed a culture shock when I first encountered a negative attitude towards math. How could someone not love math? Well, turns out that a lot of people don't like it, and are often afraid of it. Is this a cultural phenomenon? May be so. Remember the story from the start of this post? I deliberately did not mention anything about where the students were from, but left a strong hint by indicating the way they referred to their education level. I want to know your perspectives on math. Were you part of a math-culture while growing up? How did the people around you think about math problems? Did that influence you? Let us know in the comments.

Math doesn't care if you are a boy or a girl.
Why should you?
Photo Credit - Katerina Holmes

Ashcraft: As conventional wisdom suggests, math anxiety is somewhat higher among women than men...

It may be a bold move to end the blog on such a heading. Now, to be fair to Ashcraft, he goes on to further clarify: "...the gender difference is rather small, may be particularly apparent in highly selected groups and may be partly attributable to a greater willingness on the part of women to disclose personal attitudes." Moreover, he adds the qualifier that based on the recruitment for his research, there were higher number of women than men at higher math-anxiety levels, and the reverse at lower levels of math anxiety.

Nevertheless, I find the wording "conventional wisdom" to be controversial. Based on what I experienced in school (remember, different culture from a different era), females in general are more competent at math. It certainly was not usual for boys to excel at math in middle or high school. So I don't agree with this "conventional wisdom". I have been told, by my colleagues who attended high school in North America, that there is a prevalent notion here that boys are better at math than girls. This disparity feels like a cultural failure to me, and something that I really feel passionate to pursue further research to understand. I want to know your experience on this matter. Tell us a bit about your cultural and education background, and whether you ever felt external barriers to math success.

Hope you enjoyed reading the post this month. Apologies if the math at the start of the post triggered your anxiety. To summarize what we have learned, now we know that our math anxiety does not define how good we are at math. However, it can certainly influence our math competence. Regardless, it has absolutely no bearing on our intelligence. Avoiding math is not the answer, finding ways to relieve the anxiety will allow us to achieve our true math potential. Until next time, ciao.