The biggest reason why some people are better at math/science than others

Science
#1

The biggest reason why some people are better at math/science than others

Reading Text

Do you ever wonder why some people are able to read books or articles at an insanely fast pace? You might think that they were born that way, but that’s not the entire story. In reality, while genetics always plays an important role, it has been known since at least World War II that the disparity between slower readers and faster readers is more a result of poor reading technique rather than different inherent reading speed limits in our brains.

So what are you doing that slows your reading down so much? Well, it actually comes down to several smaller inefficiencies that you learned to cope with the difficulty of reading during early elementary school. Think of how a child learns to read. They will often use their finger to point at a single word. They will read one word at a time and often sound it out internally or externally. They will focus on comprehending words, putting the words together to form the sentences and then put the sentences together to comprehend a paragraph. Most adults actually keep doing this for the rest of their life in subtler ways.

You may not use your finger to follow each word, but your eyes are almost certainly focused on individual words as you work your way across the page. This fixation on individual words not only tires your eyes from moving thousands of times per page (because there are thousands of words your eye has to refocus on), but it is also extremely slow. Far slower than what your mind is truly capable of processing.

You may not use your actual voice to say the words you see verbally, but I’d bet that there is an internal voice in your mind that is saying the words as you read. This is called “subvocalization” and it probably slows your reading speed by at least 50%.

If you aren’t properly trained in reading, I bet that you find yourself having to reread the same words over and over again (called “regression”) in order to comprehend a paragraph. This is often because you are still focusing on individual words/sentences and trying to comprehend how they fit in the broader chapter/paragraph/book through a bottom-up approach.

If you don’t believe what I am saying, then just think about all of the shit you process effortlessly everyday with no issue. If I walk in the room and meet 6 people, I don’t usually feel the need to subvocalize their individual names, appearances, behaviors, etc. I just immediately recognize them, what they are doing and what they look like right off the bat. All of you do that too. Processing visual text is no different from processing the billions of other pieces of information that you process in your life. There are several books out there with drills that, if followed, will improve your reading speed by at least a factor of 2 or 3. I’ve experienced myself.

Here’s one thing you can do that will speed up your reading: instead of focusing your eyes on individual words, focus your eyes on the spaces between words as you read. Your brain will almost certainly process the text faster and you’ll be amazed at how much easier it feels to read sentences. People who master reading, only need to look at two or three spaces per line of text in order to comprehend it entirely. This ability can be achieved for many people through one or two months of doing relatively brief drills every day.

Math

Most people have the same fundamental problems with mathematics that they do with reading: they learned several inefficient ways of doing things as children to cope with the inherent difficulty of learning the language of mathematics.

Consider the following problem:

What is the sum of the first 100 natural numbers (1 + 2 + 3 + 4 + … + 99 + 100)?

Instinctively, most people will proceed to add the numbers individually. This is inefficient. A much better way to answer this question is to think geometrically.

1 + 2 + 3 + 4 + … + 99 + 100 can be seen as the iterative construction of two different triangles.

1 (1 ones)
11 (2 ones)
111 (3 ones)
1111 (4 ones)
11111 (5 ones)
…

and

111111111…1111 (100 ones)
111111111…111 (99 ones)
111111111…11 (98 ones)
111111111…1 (97 ones)
…

We can combine the two triangles to get 100 rows of 101 ones, since
1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
and so on.

So using the formula for the area of a square gives us 100 * 101; however, we don’t want to count the same numbers twice so we divide the the product by 2. So the sum of the first 100 numbers is (101*100) / 2 = 5050.

This idea of transforming difficult arithmetic problems into easier geometrical problems and transforming difficult geometrical problems into easier algebraic problems or transforming some problem into some other type of problem is one of the key differences that sets excellent math students from mediocre ones who have to study 40 hours a week to get an A or a B in calculus. Learning to find the links between different perspectives is something that is drilled into you just like learning how to speed read.

There are some people who have disabilities that make math harder for them, but most people who struggle in math do so because they did not have the proper guidance to move them out of the inefficient grade school habits that they learned as children in order to cope with those difficulties. There was a time and place for those inefficient methods, but the time for using them is long-gone once you get into algebra and geometry. Unfortunately, school systems around the world didn’t get the memo, so we live in a world that fears math.

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#2

I agree that thinking about things in a different way often yields better/deeper understanding and often more efficient approaches. But,

Actually, the better way is to think of it how Gauss discovered back when he was a young math student:

i.e. that every sequence of consecutive integers with an even number N of numbers in the sequence is also a set of N/2 matched pairs of numbers that add up to the same number. This yields the same result as the brute force addition approach and whatever you were doining with triangles above, but this algebraic solution is far easier to grok, express as a formula/equation, and apply in real-time especially programatically.

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#3

There isn’t a perfect way to solve any math problem. Each way you solve it just adds further understanding.

Geometric thinking is great, but starts to become untennable when you exceed 3 dimensions in a problem. Few people can think in 4 dimensions, let alone 5. This problem breaks down nicely into a 2D geometry problem, thankfully.

Summations are awesome for more advanced problems, too. The story of Gauss is pretty infamous, but you can go a step further with integrals when your numbers are no longer discrete.

Personally I think the biggest reason some people are better at math is simply practice. Math is like riding a bike; nobody is BORN with the understanding needed to solve problems. But, by solving problems with small variations you can become extremely quick at solving them. Sure, approach can make a difference; RPN is faster for calculators than a TI is, simply by reducing button presses, but on a fundamental level there is only a minor difference between expert calculators when using the two different devices.

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#4

First of all, your intuition for why the method works is wrong. This way generalizes to literally all finite arithmetic sequences of common difference 1 (not just sequences of even length). Second of all, my way is Gauss’s way but with a geometric explanation instead of a combinatoric one. In fact, if you read the article, you will see that the method I give is in the article you cite. They both yield the same algebraic solution [(n+1)(n)/2] to the same recurrence relation. They are therefore equally efficient.

#5

Obviously, it would be weird to suggest that there was a “perfect way” to solve a problem. Geometric thinking is great even outside of three dimensions; in fact, it often does a great job even in infinite dimensions since so much of plane and spherical geometry generalize to higher dimensions. A ton of Ronald Fisher’s core statistics proofs viewed ideas like the correlation coefficient as a comparison between two n-dimensional vectors of data. Differential geometry is used all of the time in high dimensional settings (like those settings used to investigate hamiltonian markov chain monte carlo methods) because thinking geometrically helps so damn much. Thinking geometrically isn’t just thinking in terms of plots.

Honestly, I don’t think we disagree here. I didn’t say that you had to use geometry for everything; the important idea is being able to change perspective and this is usually done through drilling. I explicitly say that practice is important, but I go further to describe what kind of features should arise from the practice you do. There is good practice and bad practice. You could practice all sorts of different problem by using a handful of slower methods that are more taxing to use for different problems. It is not obvious to people ahead of time whether their practice is good or bad. If you practice playing the guitar by playing different songs with the same chords, that’s far less effective practice than playing the same songs with different chords (think different interpretations of the same songs). Good math practice should allow you to look at one thing from many different perspectives so your life can actually be easier.

I also don’t think your analogy fits here. The difference between the naive method and the solution given above is literally the difference betweeen a constant time O(1) algorithm where you plug-in (n)(n+1)/2 for some n vs a O(n) [linear time] where you have to actually loop through the numbers. This is not a minor difference. That’s why Gauss’s story is still remembered.

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#6

You’re right, we don’t disagree. Breadth of practice is critical for learners.

As for my analogy, I’m not referring to the time complexity itself, but rather the ease in which some people are prone to thinking and their own limits. Even though I have a degree in physics, people are usually surprised to find that I can’t do mental math that requires 5+ digits to be stored at a time. Multiplying two digit numbers pushes my ability to do mental math. 13*46, for example, is pushing it for me. No issues grinding out a triple integral on paper, but adding up a grocery bill is taxing. (pun intended)

Some methods might be computationally more complex, but at the same time become easier for certain people to do. I do better with the geometry method for this problem and imagining pyramids of counters than I do playing the algebra side (using algebra to derive (n)(n+1)/2). However, this varies from person to person, I believe.

If you have a computer doing the number crunching for you rather than doing it yourself, finding the most optimal time complexity is key. I hope you are not doing monte carlo simulations by hand, but weirder things have certainly happened. As for in-your-head stuff, sometimes a less optimal method can be easier and faster.

Thanks for starting this topic, it’s a great discussion!

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#7

I picked up reading pretty quick. I was the 2nd-grade teacher’s pet, and she’d take me up to the 3rd-grade class to read out loud to show them how it could be done.

In college, I got a job with a tutoring group. Our manager was mad for speed-reading, and had us all use the machine to read faster. I gave it up pretty quickly, as it was only increasing my casual reading speed, and I couldn’t afford that many more books. It didn’t help my class reading much at all. My photographic memory isn’t 100%, so I need to take a good look at a page to be able to see it later.

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#8

So it is true that there are certain methods that just happen to click with people better than others for different use cases. I think it very much depends on what mental muscles you’ve flexed in life. There was one exchange student in undergrad who was insanely good at combinatoric and geometric puzzles because all he did growing was play various types of games where winning in a competition required you to find short cuts and read books about theories about strategy patterns. He could relate pretty much everything he learned, particularly in discrete math, to his passion directly. If you are a competitive deckbuilder, you have almost certainly come across the hypergeometric distribution, possibly without even realizing it. I’ve found that outside, non-math experiences do indeed have an direct impact on different kinds of math competency. These relative levels of competency are going to color how you approach new problems that you encounter. If you are really good at combinatorics, you might look at that summation problem and think in terms of pairs. If you are more geometric, you might look at that summation and derive the algebraic solution using triangles and squares. The relative competencies are going to correspond to habits, so this should be expected. Building more and more habits so that efficiency can be found is the true key to good practice imo.

Funny thing about the monte carlo stuff. I actually remember the story of a physics professor way back in the day who generated random numbers and wrote them on a piece of paper he kept in his pocket for later use during class or when he had to use a pencil and paper. When he would come across a problem that required monte carlo methods, out came the paper.

#9

Interestingly enough, photographic memory is actually the most general case of what I was describing in the first section. The point of looking at the spaces between the words is to get your eyes to photograph several words at once rather than a single word at a time. People can then drill their way into skipping more and more spaces. Depending on how good they can get and how hard they drill, people can eventually get to looking at the middle of a line and comprehending the entirety of the line. People who have photographic memories in the sense that we typically mean it are those who extend this idea all the way to the page level through a mix of peripheral vision and other processes that are actually available to most people. I don’t know if it is possible for most people to be drilled into photographic memory (I’m certainly not even close), but I wouldn’t be shocked if one day we found a way to teach people to that level of reading. That would be great.

You definitely strike me as a heavy reader.

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#10

Interesting discussion, the takeaway for me is if I run across a time consuming math problem the quickest solution without memorization and out of the box thinking is to ask Kevin how to solve it.

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#11

We had to use Euler method once in my diffeq class by hand, calculating 50 or so future positions of a projectile. It was a lesson in why computers are so important, rather than a lesson in math!

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#12

Back in the dark ages, Dif E taught me that we were really learning how to use look-up tables…

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#13

Look-up table is the correct way to do Diff-Eq unless you are paid hourly.

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#14

There was a device called the controlled reader; it is a filmstrip projector modified to display only one line at a time. A mode adds a sliding window from left to right to “enforce” the pace of reading. It was probably some kind of experiment and was not around for very long.

I took an elective in high school called Number Sense. It included fast mental arithmetic techniques; we even went to compete in UIL contests.

#15

I can’t read this entire thing right now but I once had a boss, who when he read (silently, to himself) sounded out every word in his head. It took him ages to read anything.

My wife is a kindergarten teacher and tries to teach kids to recognize words by sight without having to spell or pronounce them in their heads. I know nothing about elementary education but this seems legit because it’s the way I’ve always done it.

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#16

I too, was in Number Sense in middle school and again in high school. It is actually why I chose to study math in college. My BS degree is in Math/Computer Science from Baylor. When I saw the math problem I immediately thought of a brute force Fortran program of about three lines.

I loved the Mathmatical approach, too though. .

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#17

Was subjected to this device in a required 1st semester college class called Developmental Reading Which how to speed read (Evelyn Woods Speed Reading courses were all the rage) but also how to study, test taking strategies, etc.

There the main strategy was to read blocks of words. The key parameter was to get comprehension and retention … these become very antagonistic as speed gets above a certain point. Also what works reading say 6 pages of a History textbook in a minute was very different than reading a page of calculus/physics/chemistry text loaded with formulae and having comprehension and retention.

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#18

One big challenge is that everything in advanced math is based on simpler math. There’s a lot of pressure from parents, and it’s also pretty demotivating for a student to be performing poorly at algebra, when the real issue is that they didn’t memorize the multiplication tables. It’s already challenging, but when you add up all the little frustrations it’s easy to find comfort in beliefs like “I am just bad at math” or “we will never use this math in the real world.”

For English, the biggest differentiator in reading ability is whether you read books for fun. (Duh.) Students who speak English at home also have an advantage.

The sub-vocalization stuff is just a meme – for technical subject matter, you’ll be tested on some key terms. The fastest way to study is to read through it multiple times, and in that context, comprehension isn’t defined the same way as it is on the WPM test. I wrote a thing about note-taking here.

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#19

We used to call that “banjo music”

#20

In my journey, we were taught addition in first grade, subtraction in second. At the beginning of third grade, we were given a sheet of multiplication “facts” and told that we will eventually learn them, but at first, we must memorize them. That was scary. We had help in the form of flash card drills both in class and as a recommendation that parents do that at home too.

We also had these cards with the facts printed on them and a hole beneath each one. Put the card over a plain sheet of paper and write the answers into those holes. When done, turn the card over and the correct answers were next to each hole for grading. I took the option to purchase one for use at home; my siblings also benefited from that.

My brother had difficulty memorizing the table and the recommendation for him was to say 1 times 1 is 1, etc, into a tape recorder and then repeatedly listen to that tape. With a voice for newsprint, I am glad I did not have to do that…

I agree with that to a point. My first years were in a non-English speaking home. Mom was learning it as we were, so we read together. She would watch television to learn to understand the spoken word since she did not go to school during the day like we did. I am very impressed with the progress she made on her own. Over the years, we transformed to an English-speaking household because we kids insisted on it.

Since that beginning, I have come to appreciate the advantages of growing up bilingual. Being accustomed to dealing with multiple spoken languages, I can switch between different programming languages and computer operating systems with ease.

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