Dark Side of the Wall

Catch!

How does our brain compute the complex trajctory of a ball when taking a catch?
Changing the size of the ball can really mess up the calulations - learned that the hard way this summer!

We don't need no education!

"Hey, Sakura! you have almost used up the whole class, may be just give up and let Mr. Kakashi teach." whispered Naruto from the front row.

Let's learn from the experts
- image generated by AI

Sakura was not yet ready to give up. She knew that her classmate Sasuke was wrong. She also believed that her teacher, Mr. Kakashi, was wrong as well. Mr. Kakashi was wrong to assign a homework problem on a topic that he did not teach, and now his mistake was in accepting the incorrect solution posted by Sasuke on the board.

"I still don't agree, Mr. Kakashi!" Sakura piped up for the fourth time today, "I have read the whole chapter on projectile motions a few times to try to understand how to solve this problem. You definitely cannot just apply the acceleration due to gravity to the initial velocity, you need to first split it into horizontal and vertical..."

Kakashi shook his head. "Seventh graders are as dumb as rocks," he thought, "but this student is probably the densest of them all."

"I may not have a degree in Physics young lady, but I am sure too on how to apply Newton's laws." retorted Kakashi.

May be on any other day Sakura would have backed down, but today was different. She felt resolute in her mission. Even if it took the entire class period, she would keep on arguing her points.

Hey! Politicians! Leave our kids alone!

In 2024, at the time of writing this post, the provincial government of Mantioba, Canada, has removed all subject requirements for certications to teach K-8 (kindergarten to grade eight). Teaching at public schools in Canada requires a certification process designed and accredited by the respective provinces (and territories). While that does create a job barrier, this system is supposed to ensure that the teachers at public schools know what to teach and how to teach the students at this level. Do you believe that the removal of subject requirements for the certification helps maintain this purpose?

A quick summary of the relevant changes that were made in the Manitoba K-8 teaching certificate requirements
- information obtained from the Government of Manitoba website, modified and reproduced under the OpenMB Information and Data Use License

There are a lot of nuances to this topic. For example, eventhough the province does not require a teacher to have any math courses for the certificate to teach K-8, it may not mean that schools will start hiring, say, French teachers — with absolutely no background in higher level Mathematics — to teach math classes. After all, school administrators have a duty of due care and due diligence to the pupils. However, school budgets can be funny sometimes, and economic viability may tend to a future where someone with no math background may need to teach a math class.

A follow up question then asks: "why do kindergarten teachers need to know Calculus?" This is not a fair question, since the curriculum for K-8 covers upto, and including, grade eight. Having been taught Physics by a Physical Education teacher in grade seven, and Biology in grade eight by an Accounting teacher, I can attest to the difference it makes to have a non-specialist teach specialized subjects at these levels. Despite their best effort, these teachers are often missing foundational knowledge to properly respond to student questions about the subject. This may also exacerbate a dire situation where the teacher is not keen about the subject that they are teaching, regardless of their motivation towards teaching itself. Consequently the students may become frustrated with the experience and the subject.

Finally, when it comes to math, an assumption is often made that math specialists do not suffer as badly from math anxiety as non-specialists do. It is not far-fetched to assume that a non-specialist may be more intimidated about teaching math, although I have not yet come across research articles which have definitive tests for this hypothesis. Ramirez et al. tells us that "many elementary teachers have higher math anxiety than individuals in other fields of study, such as business, physical science, and health sciences". Nevertheless, this post will try to shed light on how teacher math anxiety may affect student learning in math classes, with a slight bias towards the ansatz that teachers who have never taken math classes at the University level may experience higher levels of math anxiety than those who have done so.

All-in-all, you're NOT just another brick-in-the-wall!

It is certainly concerning to note that the level of math anxiety is the highest amongst Elementary education majors compared to any other college majors, mentioned by both Yanuarto and Ramirez et. al. What would happen if one of these education majors was suddenly thrust in front of a classroom full of young inquisitive minds and asked to teach math? My son, who just started kindergarten this year, often asks deep questions about the math that his sister — in grade five — is learning. Yes, it is important to remember that even kindergarteners are not brought up in isolation; many-a-times they have an older sibling who they try to emulate. If his teacher were to have general math anxiety, then how would they have responded to my son's questions? Negative experiences with learning math at home or in class can lead to poor understanding and development of math skills (Yanuarto). As Beilock and Willingham puts it quite succinctly, "regardless of a student's gender, his or her teacher's math anxiety seems to carry implications for the student's level of math achievement."

The working memory can be overwhelmed by the presence of math anxiety
-photo generated by AI

Research has shown that brighter minds often suffers the most disruption in math performance due to math anxiety (Beilock et. al.). Just to clarify, the stronger students would not necessarily have the highest level of math anxiety, rather that their math learning would suffer the worst from the presence of math anxiety. While this may be counterintuitive, Beilock and Willingham provides a very natural explanation: students with highest level of working memory often adopt advanced problem solving strategies that are cognitively demanding, and math anxiety simply disrupts these strategies. Once caught in the cycle of being anxious about math, students' performance in math may continue to go down due to a self-fulfilling route, where, in the words of Yanuarto:

"...person who feels as if they are going to fail with math, which builds anxiety, which then leads to failure. Failure fosters the idea that they are not good at math which then creates even more math anxiety, creating more failure..."

As always. I request the reader to consult the source materials and to take everything in these blog posts with a grain of salt. The topic of Math Anxiety is something I enjoy learning about, and these blog posts are simply companions to my journey.

There is a dedicated page for the bibliography containing citations to the papers mentioned in this blog. This is a living document that gets updated as I publish newer entries for the blog. The hope is to expand this list to a recommended reading set for scholars of Math Anxiety knowledge.

Catch!

Ramirez et. al. carried out a study in 2018 to investigate how teacher's math anxiety may affect student math performance. The subject of their study was ninth grade mathematics. Their paper is a fascinating read and I highly recommend you to read the source material for unique insights into the matter. However, be forewarned, the authors lean a bit too heavily on fixed versus growth mindset to my liking. Some recent developments into the theory of mindsets indicate the need for a closer look at this concept. Furthermore, as Max Lionel mentions so eloquently in episode one of his podcast, there may be other types of mindsets to consider when it comes to teaching and learning methodology.

Let us look at the objective teaching deficiencies that arise from teachers having math anxiety. Combining ideas from Yanuarto and Ramirez et. al., we can summarize that math teachers with math anxiety may

• be less confident about successfully carrying out their teaching responsibilities,
• respond angrily to students request for help with mathematics, or spend less time attending to students' questions
• project their anxiety on students,
• have a tendency to accept responses quickly without checking solutions and correcting mistakes, which may create an environment that devalues sense-making,
• ask for a single solution to a problem, with emphasis on rote learning and algorithmic thinking.

Ramirez et. al. have shown that higher teacher math anxiety leads to lower student math performance, regardless of students' prior math achievements. Furthermore, students can quickly pick up on the teacher's attitude towards math, and this has a significant affect towards their math learning. This resonates with Yanuarto's viewpoint that "math anxiety is taught in the classroom". Well then, what can be done to ensure that teachers feel confident in their preparation to teach math? Beilock provides a simple suggestion that has been proven effective in addressing math anxiety amoing pre-service teachers: get the teachers to take a course focused on how to teach math concepts. In fact, this strategy works better than requiring teachers take a course focused directly on math concepts themselves.

How will the removal of the core-requirements of languages, sciences, and history and geography affect math teaching? In my experience, a well-rounded math teacher needs to have some understanding of these subjects as well. The relation between the sciences and mathematics is fairly obvious,  mathematics being the lingua-franca for modeling scientific phenomenon. History and geography benefit from many mathematical concepts such as graphs, maps, trees, etc. However, the most crucial connection may be between the languages and mathematics. The theory of Mathematics is rich with linguistic features, and the struggles of learning mathematics run parallel to those of learning a second language. In fact, I believe that the phenomena of "Foreign Language Anxiety" and "Math Anxiety" are closely related. This is a broader topic that I wish to investigate in the future, but for now, in my opinion, math teachers not taking language courses at the University level may be a bigger loss than anticipated. What are your thoughts on this?

Where does this leave us in our criticism of the Manitoban government's change in the requirements for K-8 teaching certificates? As with every idea, there are pros and cons. The obvious positive of this change is reflected by the reason this change was introduced: to make it easier to qualify for the Manitoba K-8 teaching certificate. Whether that will increase the quantity of available teachers is a question for time, but the more pertinent question is what it may do to the quality of teachers who meet the new certification requirements? In particular, what will this do to the math teachers in the K-8 level? From a math anxiety point of view, this makes the math teachers more susceptible to math anxiety. Which then creates a snowball effect perhaps leading to generations of low math achievers from Manitoba. That is the fear, but as always, let's hope for the best.

Baggage

It's all fun and games 'til Erdős comes with his --- knocking on your door!

Get Knocked Down

Jolpona clenched her fists. Her face as white as the Canadian Winter, she slowly put her hand up. Mr. Andrews saw the timid hand go up and frowned. With a glaring stare and wrinkled eyebrows, he asked:

"Do you need me to explain the problem again?"

" Umm, ..., no ..." came a soft reply.

"Well then, hey Jahangir, do you have the answer yet?" called out Mr Andrews.

"Wait, what?" Jolpona screamed in her head! "I was ready to answer that question, why did the teacher suddenly decide to skip me and go to Jahangir?"

It took her a lot of effort to raise her hand earlier, bringing it down felt so much simpler.

A brutal projection from a teacher.

Image generated by AI.

Jahangir wasn't faring much better either. He had no idea how to solve this problem. The teacher had only gone over the idea once, and way too fast for Jahangir, who already struggles with Mr. Andrews' Welsh accent. However, Jahangir was not ready to be a martyr by asking Mr. Andrews to reiterate the  problem. The last person to do so was summarily shot out of a canon, never to be seen again. So Jahangir just hung his face low, quietly begging the ground to split up and absorb him - so that he would not need to bear this ignominy in front of his peers.

"Inconceivable!" exploded Mr. Andrews, "None of you know how to solve this simple problem? You should have already known how to solve quadratic equations since grade school! I knew I should not have agreed to teach such a low-level math class. None of you are smart enough to do math, may be you should try something else."

"You know what?" he continued roaring, "Just to make you all start taking my class seriously, we are going to have a pop-quiz right now on this question. Hmpf!"

 

Get up Again

I hope you felt as uncomfortable reading the above piece as I did writing it. While it is a work of fiction, the scenario is actually built on research findings from the Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College by Carol D. Jackson and R. Jon Leffingwell. 

Undoubtedly teachers play a key role in a person's aspirations and life journey. Great teachers teach us, guide us, aspire us, and inspire us to follow in their footsteps. One such high-school teacher, Mr. Yamin Chowdhury, was the causal link between my siblings and I studying Physics. Let us know in your comments who was that amazing teacher who shone light on your career trajectory?

The late and great, incredibly talented, Yamin Sir

The next few blogposts will be focused on teachers. We started with things that can go wrong. Next we discuss what a teacher can do to reduce math anxiety for their students. Finally, we will end with personal accounts from accomplished mathematicians and instructors on how they dealt with their personal math anxiety, and how they help their own students overcome the fear of numbers.

You are never gonna keep me Down!

This issue on Learning about Math Anxiety will end with a summary on Some Thoughts for Teachers on Mathematics Anxiety by Lorelei R. Brush. Brush analyzed high school and college student responses in 1981 to the original ninety-four item Mathematics Anxiety Rating Scale (see footnote below) to conclude that while students may dislike the details in numeric computations, they are rarely anxious about them. On the other hand, many get anxious when they are forced to do something mathematical that other people with watch or evaluate.

A key observation is that not only do math topics get more difficult from grade six through to grade twelve, but math anxiety also increases for students in general. In contrast, even with difficulty levels increasing, students feel less anxious about English (as a subject). Brush reasons that students perceive that they are limited to learning mathematics that has already been discovered, and the opportunities are few-to-none when it comes to contributing their own ideas to push the envelope of Mathematical knowledge. Once again, comparison is drawn to English studies, which in upper-levels often take the form of English literature, where the students have more opportunities to express their own ideas.

There's more where this came from.

Humble-brag time: At the end of a first year algebra course, which introduced number theory, a student 'claimed' on the discussion board to have discovered the following theorem:

For any prime p, if you write down the prime twice, then the resulting number is composite.

It is quite easy to see that this result is obviously true for any integer n ≥ 2 (not just primes). However, instead of providing a direct response, I let the students play with ideas for coming up with a proof, and prodded them to expand on their ideas in interesting directions. Very soon, the discussion left the realm of the original claim and started focusing on more abstract notions. By the end of the thread, the students were trying to come up with an algorithm to find all prime factors of numbers of the form `10^{2k} + 1`.

The students' excitement in that thread was pulpable. They were keenly challenging each other, and commending each other's efforts. There were lots of 'Great Job!', 'I agree', and '... really interesting.' amongst the replies. It was polymath buzzing with positivity, and I genuinely believe that the experience elevated the students' enjoyment of Math.

quis docebit ipsos doctores

Pardon my Latin, but I could not resist a poor man's rendition of Juvenal's question "quis custodiet ipsos custodes" - who guards the guardians? Brush, in 1981. came up with the following set of suggestions for math instructors. Teachers should:

• choose explanations that maximize understanding.
• ease transition into new topics.
• invite lots of questions about the tough points in a lesson.
• give students [a] perception that their ideas are new.
• include tasks that require creativity.
• design math examples to address issues that adults might deal with in every day life.
• create story plots around problems that capture human interest.
• let students work together towards gathering and analyzing information.
• change classroom practices (e.g., stop forcing students to solve problems on the blackboard in front of their peers) to reduce math anxiety.

These suggestions may seem trivial if you are a good teacher. A lot of these practices have been incorporated in modern teacher training. There are many great instructors who go above and beyond these suggestions and do an excellent job. Nevertheless, there are some teachers who may benefit from being reminded about these points. After all, the 1999 paper by Jackson and Leffingwell mentioned earlier show that only seven percent of their respondents had positive experiences with mathematics from kindergarten through college. It is our job as teachers to improve these numbers to include the other ninety-three percent. To do so, we must teach ourselves to do better.

footnote: The MARS proposed by Richardson and Suinn in 1972 had ninety-eight situations investigated on a likert scale. I have not been able to find the original set of questionnaire anywhere, nor have I found the ninety-four items mentioned by Brush. If any reader is able to direct me to a source where I can access either of these sets, I would be eternally greatful.

There is a 30-item version from Suinn and Winston that is available if the reader wants to take a look. Modern math anxiety measurements are often done using other scales such as the 9-item AMAS. Methods for measuring Math Anxiety will be discussed in a following blog post.

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

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.