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