Month: November 2019

I am often asked the question, “How do you do it?!” Now while I don’t think my note-taking strategy is particularly special, I am happy to share! I’ll preface the information by stating what you probably already know: I LOVE to write.* I am a very visual learner and often need to go through the physical act of writing things down in order for information to “stick.” So while some people think aloud (or quietly),

I think on paper.

My study habits, then, are built on this fact. Of course not everyone learns in this way, so this post is not intended to be a how-to guide. It’s just a here’s-what-I-do guide.

With that said, below is a step-by-step process I tried to follow during my final years of undergrad and first two years of grad school.**

‍Step 1

Read the appropriate chapter/section in the book before class

I am an “active reader,” so my books have tons of scribbles, underlines, questions, and “aha” moments written on the pages. I like to write while I read because it gives me time to pause and think about the material. For me, reading a mathematical text is not like reading a novel. It often takes me a long time just to understand a single paragraph! Or a single sentence. I also like to mark things that I don’t understand so I’ll know what to look for in the upcoming lecture.

STEP 2

Attend lecture and take notes

This step is pretty self-explanatory, but I will mention this: I write down much more than what is written on the chalkboard (or whiteboard). In fact, a good portion of my in-class notes consists of what the professor has said but hasn’t written.

‍My arsenal

‍STEP 3

Rewrite lecture notes at home

My in-class notes are often an incomprehensible mess of frantically-scribbled hieroglyphs. So when I go home, I like to rewrite everything in a more organized fashion. This gives the information time to simmer and marinate in my brain. I’m able to ponder each statement at my own pace, fill in any gaps, and/or work through any exercises the professor might have suggested. I’ll also refer back to the textbook as needed.

Sometimes while rewriting these notes, I’ll copy things word-for-word (either from the lecture, the textbook, or both), especially if the material is very new or very dense. Although this can be redundant, it helps me slow down and lets me think about what the ideas really mean. Other times I’ll just rewrite things in my own words in a way that makes sense to me.

A semester’s worth of notes!

As for the content itself, my notes usually follow a “definition then theorem then proof” outline, simply because that’s how material is often presented in the lecture. But sometimes it’s hard to see the forest for the trees (i.e. it’s easy to get lost in the details), so I’ll occasionally write “PAUSE!” or “KEY IDEA!” in the middle of the page. I’ll then take the time to write a mini exposition that summarizes the main idea of the previous pages. I’ve found this to be especially helpful when looking back at my notes after several months (or years) have gone by. I may not have time to read all the details/calculations, so it’s nice to glance at a summary for a quick refresher.

And every now and then, I’ll rewrite my rewritten notes in the form of a SaiBlog post! Many of my earlier posts here at Math3ma were “aha” moments that are now engrained in my brain because I took the time to SaiBlog about them.

STEP 4

Do homework problems

Once upon a time, I used to think the following:

How can I do problems if I haven’t spent a bajillion hours learning the theory first?

But now I believe there’s something to be said for the converse: 

How can I understand the theory if I haven’t done a bajillion examples first?

In other words, taking good notes and understanding theory is one thing, but putting that theory into practice is a completely different beast. As a wise person once said, “The only way to learn math is to DO math.” So although I’ve listed “do homework problems” as the last step, I think it’s really first in terms of priority.

Typically, then, I’ll make a short to-do list (which includes homework assignments along with other study-related duties) each morning. And I’ll give myself a time limit for each task. For example, something like “geometry HW, 3 hours” might appear on my list, whereas “do geometry today” will not. Setting a time gives me a goal to reach for which helps me stay focused. And I may be tricking my brain here, but a specific, three-hour assignment sounds much less daunting than an unspecified, all-day task. (Of course, my lists always contain multiple items that take several hours each, but as the old adage goes, “How do you eat an elephant? One bite at a time.”)

By the way, in my first two years of grad school I often worked with my classmates on homework problems. I didn’t do this in college, but in grad school I’ve found it tricky to digest all the material alone – there’s just too much of it! So typically I’d first attempt exercises on my own, then meet up with a classmate or two to discuss our ideas and solutions and perhaps attend office hours with any questions.

As far as storage goes, I have a huge binder that contains all of my rewritten notes*** from my first and second year classes. (I use sheet protectors to keep them organized according to subject.) On the other hard, I use a paper tray like this one to store my lecture notes while the semester is in progress. Once classes are over, I’ll scan and save them to an external hard drive. I’ve also scanned and saved all my homework assignments.

Well, I think that’s about it! As I mentioned earlier, these steps were only my ideal plan. I often couldn’t apply them to every class — there’s just not enough time! — so I’d only do it for my more difficult courses. And even then, there might not be enough time for steps 1 and 3, and I’d have to start working on homework right after a lecture.

But as my advisor recently told me,”It’s okay to not know everything.” Indeed, I think the main thing is to just do something. Anything. As much as you can. And as time goes on, you realize you really are learning something, even if it doesn’t feel like it at the time.

Alright, friends, I think that’s all I have to share. I hope it was somewhat informative. If you have any questions, don’t hesitate to leave it in a comment below!

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*

Mathematicians are stunned by the discovery that prime numbers are pickier than previously thought. The find suggests number theorists need to be a little more careful when exploring the vast infinity of primes.

Primes, the numbers divisible only by themselves and 1, are the building blocks from which the rest of the number line is constructed, as all other numbers are created by multiplying primes together. That makes deciphering their mysteries key to understanding the fundamentals of arithmetic.

Although whether a number is prime or not is pre-determined, mathematicians don’t have a way to predict which numbers are prime, and so tend to treat them as if they occur randomly. Now Kannan Soundararajan and Robert Lemke Oliver of Stanford University in California have discovered that isn’t quite right.

“It was very weird,” says Soundararajan. “It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”

Surprising order

So just what has got mathematicians spooked? Apart from 2 and 5, all prime numbers end in 1, 3, 7 or 9 – they have to, else they would be divisible by 2 or 5 – and each of the four endings is equally likely. But while searching through the primes, the pair noticed that primes ending in 1 were less likely to be followed by another prime ending in 1. That shouldn’t happen if the primes were truly random – consecutive primes shouldn’t care about their neighbour’s digits.

“In ignorance, we thought things would be roughly equal,” says Andrew Granville of the University of Montreal, Canada. “One certainly believed that in a question like this we had a very strong understanding of what was going on.”

The pair found that in the first hundred million primes, a prime ending in 1 is followed by another ending in 1 just 18.5 per cent of the time. If the primes were distributed randomly, you’d expect to see two 1s next to each other 25 per cent of the time. Primes ending in 3 and 7 take up the slack, each following a 1 in 30 per cent of primes, while a 9 follows a 1 in around 22 per cent of occurrences.

Similar patterns showed up for the other combinations of endings, all deviating from the expected random values. The pair also found them in other bases, where numbers are counted in units other than 10s. That means the patterns aren’t a result of our base-10 numbering system, but something inherent to the primes themselves. The patterns become more in line with randomness as you count higher – the pair have checked up to a few trillion – but still persists.

“I was very surprised,” says James Maynard of the University of Oxford, UK, who on hearing of the work immediately performed his own calculations to check the pattern was there. “I somehow needed to see it for myself to really believe it.”

Stretching to infinity

Thankfully, Soundararajan and Lemke Oliver think they have an explanation. Much of the modern research into primes is underpinned G H Hardy and John Littlewood, two mathematicians who worked together at the University of Cambridge in the early 20^th century. They came up with a way to estimate how often pairs, triples and larger grouping of primes will appear, known as the k-tuple conjecture.

Just as Einstein’s theory of relativity is an advance on Newton’s theory of gravity, the Hardy-Littlewood conjecture is essentially a more complicated version of the assumption that primes are random – and this latest find demonstrates how the two assumptions differ. “Mathematicians go around assuming primes are random, and 99 per cent of the time this is correct, but you need to remember the 1 per cent of the time it isn’t,” says Maynard.

The pair used Hardy and Littlewood’s work to show that the groupings given by the conjecture are responsible for introducing this last-digit pattern, as they place restrictions on where the last digit of each prime can fall. What’s more, as the primes stretch to infinity, they do eventually shake off the pattern and give the random distribution mathematicians are used to expecting.

“Our initial thought was if there was an explanation to be found, we have to find it using the k-tuple conjecture,” says Soundararajan. “We felt that we would be able to understand it, but it was a real puzzle to figure out.”

The k-tuple conjecture is yet to be proven, but mathematicians strongly suspect it is correct because it is so useful in predicting the behaviour of the primes. “It is the most accurate conjecture we have, it passes every single test with flying colours,” says Maynard. “If anything I view this result as even more confirmation of the k-tuple conjecture.”

Although the new result won’t have any immediate applications to long-standing problems about primes like the twin-prime conjecture or the Riemann hypothesis, it has given the field a bit of a shake-up. “It gives us more of an understanding, every little bit helps,” says Granville. “If what you take for granted is wrong, that makes you rethink some other things you know.”

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Jacob Aron*

Yesterday I received a disheartening 44/50 on a homework assignment. Okay okay, I know. 88% isn’t bad, but I had turned in my solutions with so much confidence that admittedly, my heart dropped a little (okay, a lot!) when I received the grade. But I quickly had to remind myself, Hey! Grades don’t matter.

The six points were deducted from two problems. (Okay, fine. It was three. But in the third I simply made an air-brained mistake.) In the first, apparently my answer wasn’t explicit enough. How stingy! I thought. Doesn’t our professor know that this is a standard example from the book? I could solve it in my sleep! But after the prof went over his solution in class, I realized that in all my smugness I never actually understood the nuances of the problem. Oops. You bet I’ll be reviewing his solution again. Lesson learned.

In the second, I had written down my solution in the days before and had checked with a classmate and (yes) the internet to see if I was correct. Unfortunately, the odds were against me two-to-one as both sources agreed with each other but not with me. But I just couldn’t see how I could possibly be wrong! Confident that my errors were truths, I submitted my solution anyway, hoping there would be no consequences. But alas, points were taken off.

Honestly though, is a lower grade such a bad thing? I think not. In both cases, I learned exactly where my understanding of the material went awry. And that’s great! It means that my comprehension of the math is clearer now than it was before (and that the chances of passing my third qualifying exam have just increased. Woo!) And that’s precisely why I’m (still, heh…) in school.

So yes, contrary to what the comic above says, grades do exist in grad school, but – and this is what I think the comic is hinting at – they don’t matter. Your thesis committee members aren’t going to say, “Look, your defense was great, but we can’t grant you your PhD. Remember that one homework/midterm/final grade from three years ago?” (They may not use the word “great” either, but that’s another matter.) Of course, we students should still work hard and put in maximum effort! But the emphasis should not be on how well we perform, but rather how much we learn. Focus on the latter and the former will take care of itself. This is true in both graduate school and college, but the lack of emphasis on grades in grad school really brings it home. And personally, I’m very grateful for it because my brain is freed up to focus on other things like, I don’t know, learning math!

So to all my future imperfect homework scores out there: bring it on.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Tai-Danae Bradley*