Wednesday, January 28, 2015

Grading Stories: "Cheese Weight" and Thusforthwith

One thing I love about the internet is being able to share stories and moments from everyday life. Here are a couple about something I'm sure other academics will be able to relate to: grading stories.

Cheese Weight

My alma mater enforced mathematical writing guidelines and the use of $\LaTeX$ very strongly. Yet some people, notably non-majors, chose to ignore those guidelines completely and complain when points were taken off for writing. Some people handed in scratch work done in pen on graph paper in consistently gigantic writing. Some people *coughEigenpetercough* printed out the questions in $\LaTeX$... two problems to one page, in landscape form... then did them out by hand in tiny writing. Some people *coughalsoEigenpetercough* did the homework in $\LaTeX$ but omitted large amounts of information to fit every proof-based problem on one side of one page.

Then there are the people with just plain bad handwriting. While grading with a friend, I encountered a homework that exemplified this while grading a core class; apparently, one of the people in the class was secretly a chicken tied to a Ouija board. Here's how it went down.

Me: Hey, do you have any idea what these two words are?

Friend: ..........

Me: It looks like it says "cheese weight".

Friend: It does, but that doesn't have anything to do with the problem.

Me: Can you tell from context?

Friend: .... no.... (to another person) Hey, do you know what this says?

Someone else: ..... looks like "cheese weight"?

Friend: How about you?

Yet another person: I have no idea.

Me: Well, "cheese weight" it is then.

And that's how someone got their work back with "what's a cheese weight?" written as a comment.

Runner-up for best handwriting-related mishap goes to the person who tried to write "I used Professor X's code," but botched the last two letters in "code" in a way that evoked, erm, Little Professor X.



As a fan of both analysis and silly things, I can't help but enjoy when they're combined. This story is about a friend who perfected this combination.

My friend, at the time, was taking the same real analysis course I was grading, so I mentioned to him how funny it was when people used archaic connecting words: "thusly", "wither" and the like. From there we started trying to come up with the most ridiculous word. Thenceforth! Thuswith! Whencehence!

So of course every homework I got from this friend had at least one made-up connecting word (despite being typed up quite nicely). This continued without incident, until one day:

Me: This is hilarious! I'm worried about you slipping up and doing it on the test, though.

Him: Why not?

Me: Well... the professor might notice, and you might get docked some points...

Him: Hmm...

Which obviously culminated in him PUTTING FAKE WORDS ON THE ANALYSIS TEST.

And guess what?




Readers, do you have any grading stories? Let me know if anyone tries to pull off using fake connecting words---not everyone may be as lucky!

Monday, January 26, 2015

Adventures in Linear Algebra with the Prismatoy

As the nature of the first few posts here should somewhat suggest, my fiancé and I spend a whole lot of time talking to each other about math. He needs a nom. Let's call him Eigenpeter.

The latest installment of "Peter finds an interesting idea, spends 1 hour worth of whiteboard lecture on representation theory to his algebra-phobic lover and makes a Mathematica toy in 15 minutes" is brought to you by Prismatoy, a cube that can be collapsed into a parallelpiped:

Basically, we wandered into a puzzle store where he picked one of these up and didn't put it down. (We did pay before leaving!)

I like this because, when restricted to any one of the 6 faces, it gives a visualization of the linear transformation
$$\left[\begin{array}{cc}1 & \cos\theta\\ 0 & \sin\theta\end{array}\right]$$
(up to transformations, scaling and unitary operations) with $0<\theta\leq \frac{\tau}{4}$* being the acute angle in the final configuration. You could derive this quickly at home by imagining one face as a unit square, then exploiting some basic trig to find that the transformation maps (0,1) to ($\cos\theta$,$\sin\theta$), (1,1) to ($1+\cos\theta$, $\sin\theta$), and leaves the bottom side of the square unchanged. The above then follows from knowing how the transformation acts on the standard basis vectors for $\mathbb{R}^2$.

Peter noted that the volume of this structure is given by the area of the base times the height, which, in this case, is the determinant of the linear transformation that takes it from cube form to its parallelpiped shape. To demonstrate this, we name the three vectors along the given three sides $\vec{a}$, $\vec{b}$ and $\vec{c}$:

The area of the base is given by $\lvert\vec{a} \times \vec{b}\rvert$---one can see this because
which corresponds exactly to the area in the first picture, except the vectors are no longer of unit length. Now recall that $\vec{a}\times\vec{b}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$. Taking the dot product $(\vec{a} \times \vec{b})\cdot \vec{c}$ only takes into account the component of $\vec{c}$ that is parallel to $\vec{a}\times\vec{b}$---in other words, perpendicular to both $\vec{a}$ and $\vec{b}$---in other words, the height of the parallelpiped! Hence taking the magnitude of this quantity gives us base times height, which is volume.

But wait, there's more! The quantity $\left\lvert(\vec{a} \times \vec{b})\cdot \vec{c}\right\rvert$ can be written as
$$(\vec{a} \times \vec{b})\cdot \vec{c}=\sum_{i=1}^3\left(\sum_{j=1}^3\sum_{k=1}^3 \epsilon_{ijk}a_jb_k\right)c_i$$
where (in case the reader hasn't seen it before) the Levi-Civita symbol $\epsilon_{ijk}$ essentially acts as the 'opposite' of the Kroenecker $\delta$ function, i.e.
$$\epsilon_{ijk}=\left\{\begin{array}{ll}1 & i=1, j=2, k=3;\ i=3, j=1, k=2;\ i=2, j=3, k=1\\ 0 & i=k=j\\ -1 & \textrm{else}\end{array}\right..$$
Now imagine taking the determinant of the matrix
$$\left[\begin{array}{ccc} \lvert & \lvert & \lvert\\ \vec{c} & \vec{a} & \vec{b}\\ \lvert & \lvert & \lvert\end{array}\right].$$
I won't put the algebra all out here, but calculating the determinant according to the definition and rearranging it will give the previous nested sum. This technique can also be used to prove that
$$(\vec{a} \times \vec{b})\cdot \vec{c}=(\vec{b} \times \vec{c})\cdot \vec{a}=(\vec{c} \times \vec{a})\cdot \vec{b}.$$
Yay! It is now evident that
$$V_{ppiped} = \left\lvert(\vec{a} \times \vec{b})\cdot \vec{c}\right\rvert = \textrm{det}[\vec{c}\  \vec{a}\ \vec{b}].$$
More generally, the determinant of a matrix is a factor indicating what change in volume (or area, or the appropriate dimensional quality) it produces.

EPILOGUE: He spent most of Saturday trying to sketch the manifold of possible shape configurations of this object, then trying to determine whether the set of operations on the toy in SL(3) was a group, then made several demonstrations of these and similar phenomena in Mathematica. However, he did not do the dishes. I cut the lecture for brevity.


Wednesday, January 21, 2015

5 New High-Level Math Jokes

We here at the Analysisters enjoy puns, and are happy to contribute to the already-gigantic list of math jokes every once in a while. Here are some we've come up with over the years!

1. A topologist walks into $\bar A$. It's closed.

2. (3 variants) Q: If chocolate is a Hilbert space and peanut butter is its dual, why can every element in peanut butter be written as an inner product $\langle y,x \rangle$, where $y$, $x$ are chocolates and $x$ is uniquely fixed?
        A: Reese's Representation Theorem.

      Q: Why is peanut butter the adjoint of chocolate? Why is chocolate the adjoint of peanut butter?
      A: Reese's Representation Theorem.
      (Thanks, Paul!)

      Q: What's a mathematician's favorite candy?
      A: Riesz's Pieces.

3. What did the analyst have for dinner?

4. Your mama isn't Lebesgue integrable because she doesn't vanish at infinity!

5. Eight mathematicians walk into a diner. The first one says they're not hungry, and orders nothing. The next three order a beer, a hamburger, and french fries, respectively. The next three order a burger and fries, fries and a beer, and a burger and a beer, respectively. The last one orders a burger, fries, and a beer.

"I'm sorry, I can't fill your order," she says.

"Why is that?"

"This is only a partial order."

BONUS: Not necessarily in joke format, but I refuse to refer to
$$F_n = \frac{1}{\sqrt{5}}(\phi^n - \bar{\phi}^{n})$$
as anything except "that formula that shoots water up your butt".

Monday, January 19, 2015

MIT Mystery Hunt 2015: Let's Hear it for Random Hall!

(Note: Links may temporarily not work without a login. I will update them as the situation changes.)

Readers, I have to admit that my wonderful fiancé has a life-altering addiction.

To puzzles.

He wakes up in the morning and does five crosswords, usually in five minutes each; is never caught without a copy of The Enigma; plays puzzle games when he comes home; and, over dinner, tells me about the latest puzzles he solved. He also participates in a minimum of 10 puzzlehunts a year with his MIT puzzlehunting buddies. I generally stay out of his way for the online ones, but there's no physically avoiding the MIT Mystery Hunt. So off I was dragged.

Let me admit: I was expecting to play a supporting role, making sure my dude gets enough food and sleep. I was not expecting to personally crack several puzzles, including 2 metas. Guess what happened.

As everyone on my team agreed, this year's puzzlehunt was particularly well-constructed! This year's theme was 20,000 Leagues Under the Sea and was presented by One Fish, Two Fish, Random Fish, Blue Fish. The short format of many of the puzzles made it possible for individuals to solve puzzles alone, which is a great confidence boost for newer puzzlers such as myself; and the way in which the puzzles were released (solving puzzles gave more Deep, which revealed puzzles hiding in the ocean) made it possible to work around roadblocks, eliminating the frustration of being stuck on everything. Feeling Bluefin was probably the team favorite, and a lot of people liked Nautilus's Duplicated Quest as well. I could not believe there was a Dresden Codak puzzle* released so early---we love Aaron Diaz here! Mad props!

There are always a couple puzzles that require special knowledge, which is great if you spent the last 5 years listening to showtunes instead of doing puzzles. I definitely enjoyed the auditory (directly or otherwise) puzzles such as the theater one*, Nina and Topsy-Turvy. Someone in Random Hall has pipes! The best 'esoteric knowledge' moment came when one of the puzzles required reading a diving chart*---apparently one of the members of our team was an experienced diver all along.

My fiancé, being a long-time language puzzle master, enjoyed everything from Flat Containers to The Curse of The Atlantean's Tomb, both of which he made me help with. Representative Characters (math!) was also a nice surprise (math!) because it required understanding his field (math!) in order to solve. Although our team concentrated on solving earlier metas rather than the Atlantean puzzles towards the end, he also glanced at Practice in Theory (physics!) and enthused at me about it for an hour.

Other than Feeling Bluefin, the puzzles with the cutest premises were Follow the Bees!, Montages and MIT Mystery Hunt. So adorable.

Our team enjoyed all of the physical puzzles (even the meta!). We had people solving cubes, picking locks, cracking the gelt puzzle so we could eat it; I decoded the knitted square, and one girl with INFINITE PATIENCE sat on the floor for hours putting together the paper jigsaw that had scared off everyone else. Seriously, I was amazed by her persistence.

This was a very different experience from last year's hunt: for me, personally, most of the change in experience quality was due to being on a smaller team. Fewer people means having more opportunity to get an 'aha!' moment and a higher fun-to-automation ratio. The large number of easy puzzles (School of Fish round) also made the hunt continuously accessible to new puzzlers, but I found myself avoiding them in preference of harder puzzles as time went on.

It didn't hurt that everyone on our team was awesome and cracked jokes the entire time! Even my fiancé, who is normally reserved and academic, was dropping the sass left and right. Highlights include giving Ariel a bottle of hair dye when she asked for us to give her a soul (she's a ginger) and the phrase 'Chocolate Rain' being used to describe makin' it rain with gelt. Nevertheless, Puzzfeed has outdone us all.

If you're new to the puzzle scene, get on a smaller team (10-30 people) with some experienced puzzlers and some MIT/Boston residents. You can also look at any of the puzzles listed above for at least an entire year (and Random Fish plans to publish a fancy book of the School of Fish puzzles.) I'm not even going to preface that with 'if you want to'. GET ON A TEAM. IT IS FUN.

Now for some flats before bed...

*(not linked due to spoilers)

Wednesday, January 14, 2015

Why Learning Math is Important (General)

A common question I get from tutees is, why learn math at all? Many others on the internet have provided satisfactory answers to this question, so I'll try my best to come up with a couple new points.

(A side note to those who are highly educated in humanities, social sciences, etc.: I am not claiming that every one of these benefits is unique to mathematical learning. Those disciplines are useful for critical thinking as well!)

Problem-solving practice

If you don't go into a technical career, the odds that you'll have to use algebra or calculus every day are slim. However, no matter what you do in life, you will have to know how to effectively solve problems. It's unfortunately difficult to practice and develop good problem-solving skills on their own---that often comes with experience---but doing relatively simple math problems is a good substitute.

How would that work? When you start a new videogame, the game doesn't immediately drop you into the final boss fight; you start by doing the tutorial instead. This is what schools are hoping to accomplish by giving you simple problems to solve: they may be in a weird format, and you may not be sure how they connect with day-to-day life, but you're being given them because these number problems are some of the simplest problems that are possible to solve. Furthermore, it's not a bad thing that algebra problems are disconnected from your real life---if they were, then there would be much greater punishments for failure to solve them. I'm certain you'll agree a broken friendship or broken arm is much worse than losing a couple of test points!

To summarize: think of grade school math problems as the tutorial level to real-life problems, and their disconnect from regular life as a protective safety net.

Increased ability to communicate abstract ideas

Math can be seen as not only a tool, or a scientific discipline, but also a language. Understanding and communicating mathematical ideas requires a set of symbols and vocabulary that people wouldn't learn just by going through life. Ideas related to math do pop up from time to time, and it feels great when you know what the answer is, and how to explain it!

Here's an example: let's say you and some friends are trying to get to a frozen yogurt place a block away. We call your current location A and the location of the frozen yogurt place B. Also, let's call the frozen yogurt place Froyomorphism for the sake of puns. Your friends, whose favorite colors are purple, green and blue, suggest the following three paths on the map:

Now you are asked to choose the best path. Because you are good at judging distances, you know that the purple and green paths are the same length, but the blue path is much longer. Could you communicate this concept to your friends without using math? Without using the word 'sum' or 'length' or evoking a visual proof? Probably, but it would be much harder. The point is: even if you're right, you may end up taking the blue path if no one can explain to Blue why the other two paths are shorter.

If two people have a shared vocabulary that can be used to talk about abstract objects, they can exchange information about what essentially amounts to different lines of thought. This is how people get smarter and better at problem solving.

Protection against being exploited

Most people think they are smart. However, as I'm sure you've figured out by now, not everyone is. Several people/institutions/etc. have realized this and use people's lack of mathematical awareness to make a living. Gambling is a classic example: also see the Monty Hall Problem or Bertrand's Box Paradox for situations where common sense can be deceiving.

However, not only can ill-meaning people use your unwillingness to think about mathematics (and academic prospects in general) to separate you from your money, they can twist information to separate you from your ideals and beliefs as well. Most people like the idea of experiments being able to prove, disprove, support, or refute ideas, but don't want to dig through heavily written academic papers to find the point. This is where exploitative people come in. If they can bank on the audience being too busy or unable to read the source material, they can make their audience believe whatever they want---even if it comes at the expense of the audience! See Flaws and Fallacies in Statistical Thinking for tons of real-world examples; Stephen Campbell explains this better than a blog post ever could.

The only way to protect yourself against this is to be able to read and analyze scientific papers without needing someone to tell you what you mean. In many cases, this requires some knowledge of statistics (math), experimentalism versus mathematical modeling and the implications, or what conclusions can be drawn from the data presented (logic, which is part of math).

Not looking like a tool on the internet

If you've spent any amount of time on the internet, at all, you may have come across someone who is angry at their opponents for not understanding "logic" and "reason." You may have seen someone make a statement along the lines of "that doesn't make any logical sense" without noting what the error is (or invoking a fallacy incorrectly). You may have seen someone who is incapable of understanding that a smart person may disagree with them, and who concludes that if someone disagrees with them, that person is stupid.

Judging by the relatively low proportion of people with bachelor's degrees in mathematics or philosophy, it stands to reason that very few of these people have had real training in formal logic. Someone with completely illogical arguments would have no way of knowing so (i.e., a special case of the Dunning-Kruger Effect.) On the other hand, someone who has studied higher-level mathematics can recognize what is and is not logically consistent, which affects how they act in everyday life as well. This makes everyday life a lot---a lot!---easier. (I could go on for days about this; but that's a story for another blog post... or twelve.)

Yet embarrassing oneself is often caused by a lack of empathy---what about that? Math has no relation to that, unfortunately. [Sad face.]

Concluding remarks

Oh, and talking about math is awesome, and we have the best jokes.

(Check back in the future for specific examples of how some topics you may have seen are actually used by mathematicians and scientists!)

Monday, January 12, 2015

Vector Calculus... with Poles?!

A couple days ago, my partner and I were about to go to sleep, when I wondered out loud whether Stokes' Theorem and the Divergence Theorem would hold for functions that were analytic except at finitely many poles (it's what engaged couples do in bed!).

Since my fiancé is a physicist, he already knew the answer for the Divergence Theorem, and was happy to clue me in: for $f(r, \theta,\phi)$ in polar coordinates,
$$\iiint_\Omega \nabla \cdot f(r,\theta,\phi) \; d\Omega=\oint_{\partial \Omega} f(r,\theta,\phi)\;d\partial\Omega.$$
appears to break down when a pole occurs in the interior of $\Omega$. In order to demonstrate this, we consider the function $f(r,\theta,\phi)=\frac{1}{r^2}$, which has the classical divergence (in polar coordinates)
$$\nabla \cdot \frac{1}{r^2} = \frac{\partial }{\partial r}\frac{r^2}{r^2}=0,$$
forcing the left-hand side to be zero. Yet, because $\frac{1}{r^2}$ is constant on the surface of the sphere, the right-hand side evaluates to
$$\oint_{\partial \Omega}\frac{1}{r^2}\hat{r}\cdot\hat{r}\;dA=\frac{1}{r^2}\oint_{\partial \Omega}dA=4\pi,$$
which is nonzero! Hence the Divergence Theorem does not hold in this case... for classical forms of divergence.

Knowing my interest in analysis (see blog title, above), my fiancé clarified that when a function has poles, we redefine divergence in the sense of distributions so that Green's Theorem does hold. Apparently then
$$\nabla \cdot \frac{1}{r^2} = 4\pi\delta(r)$$
which I plan to check rigorously in the future (no money for books). It hadn't occurred to me yet that that distributions could also be used to generalize multivariate forms of the derivative, so this was an interesting way for the conversation to go.

We quickly noted that for Stoke's Theorem in 2D, with $f(z)=p(z)+iq(z)$ and $z=x+iy$,
$$\int\int_\Omega \frac{\partial p}{\partial x} - \frac{\partial q}{\partial y} \;d\Omega=\int_{\partial \Omega} p\; dx - q\;dy$$
for the real part and
$$\int\int_\Omega \frac{\partial p}{\partial y} + \frac{\partial q}{\partial x} \;d\Omega=\int_{\partial \Omega} p\; dy + q\;dx$$
for the imaginary part. Assuming that $f(z)$ is analytic leads to one of the proofs of Cauchy's Integral Theorem, but, as differentiability is a condition for Stokes' Theorem, it is expected to break down under classical conditions for a complex function with finite poles. However, given my partner's earlier insight with the Divergence Theorem, I wouldn't be surprised if a distributional equivalent existed for Stokes' Theorem as well.

I'm certain there are a few texts that could clear up exactly how this happens rigorously... sounds like something fun to do in the future!

(Please excuse my slightly incorrect use of notation. Some symbols are not supported in MathJax.)

Wednesday, January 7, 2015

Funny Papers: Overly Expressive Lab Mouse

Scientists are people too, and they have a sense of humor. We don't claim to be a research humor blog (that's what Annals of Improbable Research is for.) However, sometimes we come across something that is so hilarious it has to be shared.

In this case, it's a silly figure in a medical research paper. I doubt the legality of posting an image from someone else's protected academic work, so here's the source (Figure 2) and a short description of what's happening in the figure:

  • The first image in the chain is a newly infected lab mouse with a darling smile on its little face.
  • Endotoxins produce IL-12, TNF and IFN-gamma in the mouse, contributing to shock. The mouse's smile has been reverted; it is now unhappy at its predicament.
  • The toxic shock leads to weight loss. Here, weight loss is manifested as someone copy-pasting an image of the sad mouse and shrinking it a bit.
  • The poor mouse dies. The authors have helpfully put 'x's in the eyes and put the mouse's little feet up to communicate to the reader that the mouse is, indeed, dead.
A shout-out to C.A. Biron and R.T. Gazzinelli for demonstrating that, while mathematicians have the best jokes, experimental biologists have us completely beat on black humor!

Monday, January 5, 2015

Math Test-taking Strategy

Math testing strategy varies a bit from the strategies that would normally be useful for classes requiring a lot of rote memorization. This is mostly because, in addition requiring a student to interpret and regurgitate information beforehand, math tests also involve a performance element that tests short-term problem solving ability. You may recognize this as an important skill to have! Before we list some specific ways to prepare, keep in mind that:
  • Mathematical ability can be improved. Some people have a tendency to give up on math if they aren't good at it immediately. However, if you spend more time on math than your classmates; whether it is going to math camp, working with a private tutor, thinking about outside problems, or figuring out how things work; you will get better faster. Nobody comes out of the womb being able to solve every type of math problem. It's like playing a musical instrument or learning to dance: the more practice, the better.
  • Getting an A in most classes requires a level of understanding that is not taught in the class. In American schools, an A grade is meant to signify that a student is going above and beyond what is required, even if it looks like all of the testing material is being taught in the class. There is often far more to the material! For example, a lot of students understand the general material being taught, but get slammed on small mistakes such as minus sign errors. Catching and being aware of these errors is something that the student must develop on their own, and it is hard to explicitly teach. Small things like this are often the difference between an A and a C.
Got that? Good! If you're studying for a test and aren't quite sure what to start with, there are several 'levels' of understanding the material, which, for your purposes, we'll express in three separate categories:
  • Basic understanding. You're at this level if you can read and understand everything in the textbook chapter, and know how to do the problems that aren't word problems.
  • Familiarity. You're at this level if you understand what the answers should 'look' like, why the answers 'look' that way, and how to fix something if you've made a mistake. To get to this level, you have to be observant and look for patterns in the work you're doing. Knowing where mistakes can arise in a certain type of problem is very powerful!
  • Creative application. If basic understanding is like knowing how to get home, and familiarity is knowing how to get home even if you've taken a wrong turn, creative application is like getting home by parkour. You're at this level if you understand the technique so well that even when it's not mentioned, you know when it has to be used. This type of understanding is the most important for word problems.
Let's use (scalar) multiplication as an example. Someone would have basic understanding if they could multiply 45x25 on paper (or in their head! Can you?). They're familiar with how multiplication works if they can explain why the answer cannot be 725, or 329670, or 2621. Lastly, they will have mastered multiplication as a concept if they can use it to solve problems such as 'how much do 45 vending-machine gumballs cost'? or use multiplicative identities to prove the exponent rules.

So, in a nutshell, if you understand the book and can do all the problems, you're still not guaranteed to do well on tests. Nooo! How can this be?

The primary problem is that very few books go into detail on how things work and how to recognize mistakes. Yet recognizing where mistakes happen and how to fix them is vital---not to mention that knowing how methods work is the only way to understand word problems! Here are a few helpful things you can do to improve your math testing ability:
  • Know exactly where your abilities are for each type of problem. Can you do mental multiplication? Can you do basic algebra problems? Can you prove that L^2 is complete? Being honest about where your weaknesses are makes it much easier to conquer them. Needing more practice on a specific type of problem isn't a bad thing, and you'll save time if you focus on only the hardest problems.
  • Develop 'sanity checks'. 'Sanity checks' are what I call pieces of information you remember to check whether you've made a mistake in the problem. Using the multiplication example above, a good example of a sanity check would be 'an odd number times an odd number cannot be an even number.' This helps build your familiarity with the material and could save you from losing tons of points.
  • Build the test. Even if you don't know which specific problems will be on the test, you can generally work out how many of each type of problem will be on the test. This will help you direct your attention towards whatever will win you the most points. For example, you might be okay with everything on the test except one very, very hard type of problem: would it be better to focus on figuring out the hard problems, or making sure you don't mess up on the moderate problems? This depends on how many of the hard problems will be on the test.
  • Talk with friends. We don't mean about videogames. There aren't a lot of ways to develop creative thinking for mathematics other than 'think about math a lot and try to come up with and solve math problems outside of school', but this is one of the more fun ones. Your friends may have some insight about math that hasn't occurred to you yet, or you may be able to solve a hard problem by working together. In any case, talking directly to people who know more than you will teach you a lot.
  • Look for patterns. If you don't want to talk to your friends about math, or have some pride about developing things on your own, remember that math is all about finding and exploiting patterns. Once you've found a pattern, try to figure out where it comes from. This line of thinking often leads to developing newer, faster ways of solving problems. See our mental math post for some simple examples of pattern exploitation.
  • Hire a private tutor. We said earlier that talking to people who know more than you will improve your skills very quickly: well, private tutors know a lot of math and they can teach you a lot about math. There's no shame in needing one! If someone is very good at the guitar and wants to become even better, they hire a private guitar teacher. The same is true for math. This is how you should see a private math tutor: someone who can help you improve very, very quickly and understand mathematics far beyond what you are taught in class.
These are all things you would do before the test happens (we've left out the obvious "read the textbook and do the problems" advice.) While the test is happening, try to:
  • Read the test beforehand and do the easiest problems first.
  • Temporarily improve performance by stretching, chewing gum, drinking caffeine, or listening to energetic music.
Now you should be focused and in the moment!

Lastly: even if progress seems slow sometimes, or you're having trouble catching up, don't feel hopeless. If you spend time thinking about how the methods work, you will improve.

Good luck, and happy testing!


Good day, internet!

We are (currently) two flat broke prospective scientists: one future physicist with no money and one mathematician with no money, who thought it might be a good idea to start a blog in order to spread the love. Here you'll find insight gained from teaching, stories about research, discussion of odd problems, jokes about LaTeX, and possibly an informal paper review once in a while.

We had intended on starting a free tutoring video series, but had to quit due to severe technical difficulties (see 'broke,' above). Boo! Perhaps that's what the future holds.