Trevor Klee, Tutor

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Learning New Material

There are roughly three stages you can be at when learning new material. You can be at the stage where you’re just learning how to do the problem, at the stage where you’re learning how the problem works, and at the stage where you’re learning how to expand beyond the problem. The parallel, here, could be something like the process to become a competitive chess player. You learn how to play the game, then you learn the strategies of the game, then you learn how to combine the strategies to become an effective player.

Before you be confident that you know new material, you have to have completed stage 3. If it’s just a slightly different version of something you already know, you can learn it in a day. Bigger topics will take more time.

Stage 1: How to do the problem

If you’re at this stage, you are approaching this material for the first time. The key is then to put in the legwork to be able to replicate, independently, a solution to the problem. Once you can do that, take notes in your “permanent” notebook as to how to do the problem. You’re going to want to review this later. The end result is that whenever you are presented with a problem with an identical set up, you are able to find an answer.

Stage 2: How the problem works

Once you have an idea of what process reliably gives you an answer to the problem, you should begin asking yourself if there’s another way you could have solved that problem. Or, more importantly, if you had an incorrect intuition as to how to solve the problem, why your intuition was incorrect, and how it could have been correct. This will require some serious work on your scratch paper as you work your way through the dead ends. The end result is that if someone asked you why you solved a problem in a certain way, you are able to give a coherent answer

(One technique you might want to use here is the Rubber Ducky technique: place a rubber ducky (or stuffed animal) on the edge of your desk. Explain to him/her/it how to solve the problem, so that they could do it themselves. Anticipate any questions they’d have. Sound ridiculous? It is, but it’s also a real technique used by coders: look up “Rubber duck debugging” on Wikipedia. If you can explain what you’re doing to a Rubber Duck, you’re done with step 2.)

Stage 3: Going beyond the problem

If you want to get a 700+ score on the GMAT, you have to be prepared for problems to come in completely unfamiliar forms. You need to be able to recognize the concepts behind the question in whatever form they come in, and address them appropriately. The end result is that you can answer any problem based on the material you learned, even if the problem form is completely new.

Putting it into practice

These steps might be difficult to grasp in the abstract, so let’s take a look at an example.

I’m trying to learn simple Algebra. I have my equation:

3 + 5x = 25

My first attempt was to simplify the equation to:

3 + x = 5

So then I was able to say that:

x = 2

Unfortunately, I looked at the solutions, and this wasn’t the right answer. Apparently, I made a mistake somewhere in my simplification, because of something called “The Distributive Property”. So I went through my stages to try to learn this unfamiliar concept.

Stage 1:

In my textbook, I learned the correct way to do the problem was to subtract 3 from both sides, then divide by 5. I took notes in my notebook. I now feel confident not only doing this problem, but solving this problem as well:

4y + 6 = 24

4y = 18

y= 4.5

Stage 2:

I’m still not sure why my initial effort failed. It seemed reasonable, even though I now know it’s incorrect. I decided to try to make sense of it by working backwards from my new problem, using my new knowledge of the distributive property, as well as balancing both sides of an equation.

y = 4.5

y + 6 = 10.5

4(y+6) = 42

4y + 24 = 42

Aha! That makes sense. Order of operations means that I would have had to multiply everything by 4, which means that when I tried to divide my variable by 4, I also needed to divide every constant by 4. I now know I can also solve the problem like this:

4y + 6 = 24

y + 1.5 = 6

y = 4.5

Stage 3:

This exploration of the distributive property makes me wonder if multiplication and division is the only thing this applies to. My order of operations goes PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction). It makes me think that the same sort of concepts must apply to exponents as well. Let’s check it out. First I’m going to try the way I would have done it before.

(5x)^2 = 125

x^2 = 25

x = 5

Plugging that back into the original equation, it doesn’t work. This means that I was right: it’s the same concept. I need to take care of my exponents first, before division.

(5x)^2 = 125

25x^2 = 125

x^2 = 5

x= √5

Plugging that into my original equation, it works!

Just to make sure I understand this concept, I’m going to be on the lookout for questions which deal with the distributive property. Any I come across that I don’t understand, I’m going to take note of, so I can go through my stages with them later. I’ll keep adding to my knowledge and understanding of this concept. Eventually, I’ll become confident that I won’t run into any questions based on this concept that I don’t understand. Then, I can officially say that I know this concept.

How to review

Reviewing is essentially an abbreviated version of the above process. The idea behind reviewing is that you follow the same steps as you did before (which you’ll be able to do because of your detailed notes), but you don’t have to struggle through the same learning process as you did before.

The difficult thing about reviewing is it’s easy to skip steps, and it becomes rote. You look from step to step, saying to yourself: “Yup, I know that,” or you just skip straight from the problem to the solution. Unfortunately, by that point, you’re just wasting time.

While reviewing, you can’t fall into that trap. You need to make sure that you’re deepening the grooves that the initial learning made in your mind, so that you develop an intuitive understanding of the problems

It’s much like learning a language. When you first learn a language, you have to think about each word you use, and what order it should go in. If you just learn stock phrases and repeat them, like “where is the bathroom”, you will never progress beyond that stage. You don’t understand the language any better than you did before the stock phrase, and you can’t speak it any better than anyone else who has access to Google Translate.

However, if you force yourself to recall words, use them in new situations, and go through conversational patterns, you will develop an intuitive understanding of the language. More than that, you’ll develop an ear for the language. You will understand when things sound wrong, or when they sound right. You’ll notice when something is technically correct, but inefficient or just strange.

If you can get to that point with the GMAT, you’ll be golden. If you can look at a new problem, recognize it as a pattern, then instantly start writing a solution, you’re a GMAT master. But you won’t get to that point unless you deepen your learning grooves. You need to not only love the solutions and the answers, but the processes by which you get there.

So, in order to do that, you need to go over the same problems over and over again, especially if you got them wrong in the first place. It’s easy to get into the habit of constantly looking for new problems, because the old ones seem boring, or it doesn’t seem like there’s any more to get out of them. But that’s not how you learn. You learn by repetition. It’s like lifting weights: if you want to get stronger, you need to focus on a few lifts. You can’t be doing new things every time you get to the gym, or else you’ll never improve on any lifts, and you’ll never get stronger.