I am not doing math - so I am happy!

Being chased by the ghosts of Math

The Tune that Sticks

What is 17 times 3?

I loved playing cards when I was young. Contract Bridge, ACOL system - in case you were interested. Inevitably, once in a while, we would be a partner short. So we would often play "Auction Bridge". If you have never heard of auction bridge before, here is an excerpt from Wikipedia:

[A]uction bridge [is] an invention of three members of the Indian Civil Service stationed at an isolated community, designed as a three-handed form of bridge to compensate the lack of a fourth player.

A deck has 52 cards, but 52 is not divisible by three. One logical solution is to use a "dummy" hand in each round, but a more interesting solution is to take away a single card and be left with 51 cards to deal amongst 3 people. Thus, 17 times 3 equals 51 gets etched in the mind of a young boy without having to memorize times tables.

... One Step Back ...

In a previous blog, I questioned the notion of math anxiety being more prevalent in girls than in boys. Some of the earlier papers, specially OG articles by Dreger and Aiken or by Richardson and Suinn, hint that math anxiety scores do not differ significantly for males and females. Nevertheless, the review article "Mathematics Anxiety: What Have We Learned in 60 Years?" by Dowker et. al paints a more complicated story.

The relationship between gender and mathematics anxiety has been studied widely. I plan to go through this topic in detail in a later post, but for now I will rely on some the snippets from the review article. The citations for the primary sources can be found in the article by Dowker et al, and are not explicitly mentioned in the points below.

1. Females tend to rate themselves higher for Math Anxiety. Males, on the other hand, show more confidence.
2. Girls show a relationship between math anxiety and performance even when general text anxiety has been accounted for, while for boys this relationship disappears after controlling for general test anxiety.
3. There are studies that suggest no gender differences exist in math anxiety and performance, although these studies do not account for general text anxiety.
4. Females are often being reminded of the sterotype that males are better at math than females, and due to these gender stereotypes, girls may expect to feel more math anxiety which results in avoiding math activities and courses.
5. There is a lot of room for further research on how the working memory of girls may be affected differently than that of boys by math anxiety.

Overall, the idea that math anxiety is more prevalent in girls than boys is more nuanced than I expected. The review article discusses this in more detail, and covers numerous other aspects of math anxiety. I am not a big fan of summarizing review articles, as that will just be a redundant summary of a summary. Therefore the reader is strongly recommended to read the review article themselves. What are your thoughts on providing key takeaways from review papers in learning blogs? Please let me know in the comments.

Why we do what we do.

Recently, a student dropped by my office to announce that she received a perfect score in her math test. Just a week earlier she had been suffering from severe math anxiety. I had no doubt in the student's math ability, but a single problem that she had trouble understanding was enough to derail an entire practice exam. It brought her to a near standstill, and led her to make mistakes in proceeding problems, problems that she could otherwise solve in her sleep. Thankfully, studying about math anxiety gave me some perspective on what she was facing. It also allowed me to empathize with her, and provide helpful suggestions on sidestepping her math anxiety. Her grades bore the fruit of her hard work and my understanding of math anxiety. Have you been able to help someone overcome their math anxiety? How did you do it?

Removing Math Anxiety

Throwing Caution to the Wind

"I am not doing math - so I am happy!" is a verbatim quote from an acquaintance of mine. How soul crushing would it be if that sentence were to be spoken in front of a group of strugggling students, who are all facing math anxiety to some degree? We really need to be mindful of projecting negative attitude towards math in front of others.

In fact, in their short paper "Math anxiety: who has it, why it develops, and how to guard against it", Beilock et al. list social influences as a cause for math anxiety. In particular, teachers with negative attitude towards math can pass it on to their students (specially female students). What's worse is that it can have a rolling snow-ball effect from one generation to another as the negative vibes keep getting transmitted from teachers to students.

Sale! Sale! Sale!

The images in this section look like typical sales in your neighborhood grocery chain. However, carefully reading the small text tells us that the Classico pasta sauce, if bought individually, is $1.99 each, while the PC Spinach and Cheese sauce can be purchased for $2.29 per single jar. Do you see what is wrong here? How often do you check whether the sale prices actually make sense?

It is imperative that we promote positive math vibes to those around us. We should be encouraging kids to play around with numbers in their heads, so that when they grow up, they are not intimidated to calculate whether they are getting a good deal at the grocery store. Let us tell the next generation how awesome it has been to do math without a calculator, so that they may find their way out of using AI to do all their math.

Courtesy of Reddit user u/dotdoth

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.

Born with a Math Brain

     Make sure your baby is born with a math brain     

A Story for the Ages.

Sylvanna: Nice to meet you. What is your role at the college?

Me: I am a Math Consultant. I help students with math.

Sylvanna: Wow! I am so glad you are here. If students ever asked me a math question, I wouldn't know what to do; may be I would cry! I am so terrible at math...

Me: ... No at all. I am sure you use math everyday in some capacity! When  driving to Toronto from Waterloo, you are probably doing all sorts of complicated math in your head to see if you will arrive in time. However, you said you get worried when someone asks you a "math" question. There is a concept that fits well with what you described. It's called math anxiety, and I have just started learning about it ...

Math is not the Boogeyman

Sylvanna is not a real person, but the conversation above is very real. I have had very similar conversations with people from all facets of life. Cab drivers, librarians, even college biology professors have expressed to me that sometimes they just don't know what to do if someone asks them a math question. They feel worried, anxious - as if someone shone a spotlight on their unprepared self. They are just happy that the "math" people are available to help tackle these problems. Perhaps this is fine and fair; not everyone needs to an expert in math, just like how not everyone needs to be Picasso. However, each of us often handle fairly complicated arithmetic in our everyday life - the cab driver finds the shortest route to their next fare, the librarian knows how many copies of a book to order based on the popularity of the book, the biology professor has a fair idea on how the H-index of their recent publication is growing - so why do we get stuck, like a deer frozen in approaching headlights, if we have to suddenly do a "math problem"? Why do many say, with some pride, that they simply aren't a "math person"? Perhaps the root of problem lies with the society around us willing to accept this inability to do math as the norm, and individuals start wearing it as a badge of honour.

Beating the odds

I have studied mathematics for quite some time, and have been good at it most of my life. With a graduate degree in mathematics,  I am fairly confident in my math abilities. I have chosen mathematics teaching and consulting as a profession. However, this does not mean that I have a math brain. In fact, I don't have any natural talent when it comes to numbers or counting. Untalented people like us learn mathematics through dedication, grit, and striving for excellence. I learned math because I liked math, and enjoyed learning math. While preparing for my O-level math exam - where I stamped my first real success at being good at math - my first "mock" test received a score of 58%, the lowest amongst my peers. That was a good time as any to simply give up, to walk away saying that I am not a math person, to claim that I was not born with a math brain. Nevertheless, with a desire to learn from my mistakes, I accepted that I was unprepared for the test. I decided to make changes. Those changes led me to where I am today. The same changes that I now try to instill into struggling students.

Definitely happy to be a Master of Mathematics.

The Devil in the cogs of the Wheel of Anxiety

Last week, a student started his consultation by saying "I am really terrible at Math". However, by the end of the session, our (both the student's and mine) realization was that he was not bad at math at all. He was able to walk through all the difficult algebra and complicated rules of logarithms required to solve his problem. His struggle was with making connections between the steps. Simply trying to think about which rule to use to start solving the problem froze him in fear. A fear that is very real. An anxiety that led him to conclude that he is "terrible at math".

The science of learning is fascinating. Having studied memory curves, learning techniques, and good study habits in high school, I follow the techniques from those lessons to this day. As a math instructor, I learned about effective math teaching and learning techniques - enhancing the learning experience with fun math puzzles, revisiting concepts in a new context and challenging the students to rediscover them - which is incorporated in planning my lessons. With the right guidance, I believe everyone is capable of viewing math as a friend - as something to cherish and love - and not as a fearful entity.

Math consultants have to often convince students that they are not "terrible at math", and that there is no such thing as a math brain. This has led me to investigate why students feel anxious at math. Currently, I know very little about math anxiety. That is exactly why I decided to share my journey into studying math anxiety with everyone. Let us venture together, in no particular order, towards finding the answers to some burning questions about math anxiety.

• What is Math Anxiety?
• Who is affected by math anxiety?
• Do we know the causes of math anxiety (there may not be a definite answer to this)?
• What are some common triggers for math anxiety? What are some contributing factors?
• How do we overcome math anxiety? Can we get rid of our weakness(es) in math?
• Why is being bad at math often accepted socially? Are social expectations shaping how we view our math expertise?
• When and where is math anxiety more/less common? Is there a specific education level that sees a spike in math anxiety? Are there some cultures/countries that are nearly immune to it?
• Is there really such a thing as a "Math Brain"?

Welcome to My Jungle

I don't have the answers to many of these questions yet. My plan is to read academic books and articles on the topic and share my understanding with you through this blog. There will be some personal opinions and commentaries, but with an objective approach. Answering the above questions is just the beginning, let's see where that leads us to.

All readers of this blog are welcome to contribute. Please engage the discussion with comments, constructive criticisms, and suggestions. Feel free to provide expert opinions or contribute to the blog with your own post. There may be some subject matter experts in the audience - please let us know about good resources to learn about math anxiety or correct us if we misunderstand something. Just like a polymath project, let us all contribute, and make this blog a great resource for everyone to learn about math anxiety.