Do students at Harvard, Yale, Princeton, Stanford, and MIT ever struggle with proof-based mathematics such as Linear Algebra or Abstract Algebra or Real Analysis or they always find these very easy?
I graduated from Stanford in 1964 as a mathematics major. I can assure you that I had plenty of difficulty with proof-based mathematics. I was in an honors calculus class and we used the two volume work, Differential and Integral Calculus, by Courant (2nd edition 1959 reprint). The epsilontics1 in this class were my first exposure to proof-based mathematics (except for a proof based plane geometry class in a public high school in Grants Pass, Oregon — in southwestern Oregon). I did get an A, both semesters but it was lots of hard work and not easy for me.
Then I took linear algebra. The text was by Hoffman and Kunze (1961 edition), not an easy book. I was ill-prepared for this and found the course extremely difficult. It was then that I realized that mathematics was not going to be an easy major.
Most of my friends who were in these same classes also found it not easy going.
Later, abstract algebra seemed much easier. Mostly we did group theory, if I remember correctly. The differential equations class was taught by a Hungarian refugee. His English was utterly incomprehensible. We all madly copied whatever he wrote on the blackboard. For the most part it was not proof-based. I remember that one of the recommended books was by Ince. However we did not follow any text.
Well, that was a lot of years ago. I have no idea how it is today.
Edit: I just remembered that I took a course in Complex Analysis (sometimes called functions of a complex variable, or just function theory). It was proof-based but there were also applications, e.g. hydrodynamics. I remember liking this course a lot and not finding the proofs too difficult.
Buddenhagen, James. “Do Students At Harvard, Yale, Princeton, Stanford, And MIT Ever Struggle With Proof-Based Mathematics Such As Linear Algebra Or Abstract …”. 2022. Quora. https://www.quora.com/Do-students-at-Harvard-Yale-Princeton-Stanford-and-MIT-ever-struggle-with-proof-based-mathematics-such-as-Linear-Algebra-or-Abstract-Algebra-or-Real-Analysis-or-they-always-find-these-very-easy.
How to Learn Math Fast and Easy: Tips and Tricks
Whenever a person asks a student or anyone about their favorite and toughest subject in school and college, then most of the time you might receive only one answer, toughest would be maths. But, there will be some people who believe that to learn math fast is not a difficult thing and in fact it is easier than it seems.
There are people who think that if someone find maths as easiest subject than anything, then those people definitely will be declared as the most intelligent people.
To overcome that thinking, a person needs to understand their capability of handling maths and then maths is going to be the easiest subject.
Maths is the only subject which can be solved by continuous practice.
“How To Learn Math Fast And Easy: Tips And Tricks – wisestep”. 2016. wisestep. https://content.wisestep.com/learn-math-fast-easy-tips-tricks/.
Why is math easy for some people and hard for others?
Oh this is a fabulous question, and one I have spent gazillion hours thinking about. As in almost all such questions, I think it is a combination of nature and nurture.
Lets get the nature part out of the way
The nature part is understandable, and is true for almost every subject or topic. For instance, if you showed me a sequence of numbers – 343, 243, 1001, I would probably think 7^3, 3 ^5, 7 * 11 * 13 – what wonderful numbers; while a great many might simply not think like that. Some brains are wired to be amused by numerals; while others are like Calvin and are meant for far greater things.
Nurture component is definitely worth exploring
Math is too often seen as a set of formulae to find things, and so the automatic questions become why these things need to be found and what in the wide world is the point of it all?
If learnt well, math is a fabulous tool to structure thought processes. Reducing math to a set of tricks is akin to reducing Picasso to a few brush-strokes. But this is precisely what we have done. Instead of unraveling the layers beneath which the beauty in math resides, we have decided to cloak it with further obscurity. This is why math gets broken into haves and have-nots so easily.
We take our kids and tell them that in order to subtract 29 from 83, we ‘borrow’ one from the tens place and it translates into ten in the units place and voila we have 54 in no time. Humans had a decimal system in place 5000 years ago, and struck upon a finer version of the system that accommodated positional ideas about 1200 years ago. So, for a period of 3800 years there was no carry over or borrowing in math. This was an era where there was no easy way to multiply XIV and XXIII, where there was no face value and place value, where a single digit number could be far higher than a 4-digit one. No wonder we had the dark ages. Some kids ‘get’ the idea behind carry-over, a great many only understand the process behind carry-over – therein lies the rub.
We are anxious to teach our kids
but are not keen enough to take them through the method of completion of squares. This is the issue with the pedagogical script. Sitting on top of this is the behavioural angle. A lot of math ends with some form of computation. And there are always a few who can do 114 * 538 before you can say hello; so a host of kids, already unsure about the fundamentals get intimidated to within an inch of their lives by these computational whizzes.
The one-word answer is simple – Intuition
Some kids learn math intuitively, while the vast majority dont get a chance to see this. What exactly is this intuition? Let me outline this with an example. Put a group of 25 six-year olds and ask them the following question, chances are most of them would get it right.
Qn: I have a big cake here with me. I can either cut into 3 equal pieces and give you one, or cut it into two equal pieces, take one of those two equal pieces cut that into two equal parts and give you one of those. Which would you prefer?
6-year olds get the idea that 1/3 is greater than half of 1/2. They might struggle to articulate it, but they get it. Chances are that by the time these same kids are 11 they would have learnt some trope like “If the numerators are same, then the fraction with the larger denominator is actually smaller” in some form of weird supplementary class and have the intuition strangled out of them. India is very good at these supplementary classes. We are masters at chewing off intuition, adding layers of short-cuts and getting our kids competition-ready. We have giant abacus and kumon farms and a thriving industry called ‘Vedic Mathematics’. I cannot describe what Vedic math does to intuition without violating Quora BNBR.
The very essence of math remains undisclosed to many
I am going to outline a series of questions, in increasing order of difficulty, to convey this point.
- Can you find the smallest 4-digit even natural number, sum of whose digits is 17?
- If sum and product of a set of natural numbers is odd, can we say that the number of numbers is also odd?
- Can there be a sample of 5 numbers such that their median is greater than their mean? To this set, can we add 2 numbers such that the median becomes lesser than the mean? Am I being mean?
- Can an irrational number raised to the power of an irrational number be rational?
- Can we have two non-congruent triangles T1 and T2 such that three angles and two sides of T1 are equal to three angles and two sides of T2?
These are the kind of questions that any kid likes thinking about, whether they claim to like math or not. We are depriving both the have-maths and have-nots by not including this element of intuition-building in our curriculum.
Balasubramanian, Rajesh. “Why Is Math Easy For Some People And Hard For Others?”. 2022. Quora. https://www.quora.com/Why-is-math-easy-for-some-people-and-hard-for-others.
References
“Not Sure If Calculus Test Was Easy Or It Just Seemed Easy Because I Have No Idea How To Do Math”. 2020. quickmeme. http://www.quickmeme.com/meme/3qydk1.
1 An approach to mathematical analysis using the epsilon-delta definition of a limit, i.e. with explicit estimation of error bounds, as opposed to using infinitesimals. “Epsilontics – Wiktionary”. 2022. en.wiktionary.org. https://en.wiktionary.org/wiki/epsilontics.
Mathematical Mysteries 