How to study math as an INTP

[dr froyd]

Whatup, people!

Recently I embarked on a graduate degree in mathematics, and this forced me once again to look into the issue of learning mathematics as an INTP. Looking back at my experiences as an undergraduate in mathematics, I believe the main reason that many INTPs struggle with math is this:

An INTP learns mathematics in a different way than most people. When people try to teach you mathematics, they are usually forced to reduce highly complex concepts into simplified, isolated facts and techniques. The problem for an INTP like me, is that I cannot understand something merely as a piece of isolated fact. I have to understand the whole system around it. I think this is one of those things that make an INTP feel stupid. While everyone seems to "get it" very quickly, you are clueless until you have studied the thing for a long time. So my theory is this: it's not that INTP's learn slower, or in a less efficient way or whatever, it's just that they have a completely different way of understanding a concept. While most people can quickly accept a series of statements and not think too much about how they fit together, an INTP needs to know where they came from, how they fit into a larger framework, how things might change if a parameter is altered, and so on.

What this implies is that as an INTP, you are simply forced to spend more time than others on learning the material. Again, not because of some intellectual deficiency, but because the way you understand a concept, or the way you learn a subject in an optimal way. An INTP should slowly and calmly analyze the concept at hand in a deep and broad way, slowly making a logical map of the concept. This might very well take you outside the bounds of the standard curriculum. It takes time, but it is an intrinsically pleasurable experience for an INTP, as opposed to trying to digest a collection of isolated concepts.

I view this as a positive thing. I have met many students who are highly proficient at solving exam problems, but are clueless once you take them slightly outside the domain of systematized, predictable exam-problem solving. That is the price one pays for learning in an way optimized for exams and not for understanding.

Why I think all this is true? Well, it is basically looking at the different mathematics courses I have taken, and looking at my experience taking them, in light of Meyers-Briggs theory. The most awful experiences in academia was trying to learn quite simple things but which didn't seem to belong to a larger logical system. The absolute most pleasurable experience was writing a thesis, where I had the opportunity to spend 5 months thinking about one single concept and exploring it in an autonomous way. I think it is easy to see how this ties into the INTP temperament.

 

[Architect]

Right ideas but wrong angle I think.

Here's the deal, yes we need to see the forest before we see the trees, and most education is designed for the majority of people who get to the forest (if they ever get there) via the trees - the details. However here's the problem, in many or most complex fields you can't get the big picture until you sufficiently understand enough specific details.

Example: Quantum Mechanics. I'd estimate that 90%+ of what you read about QM is wrong. It's written by people who jumped to the big picture without taking all the hard math and classes. They don't know the real story and so get it all wrong. In this case you have to work through years of maths to differential equations, understand the experimental basis, then study and solve problems. And those problems are by necessity the simplest. Anything harder than that can't be solved in closed form, you have to find a numerical solution.

Putting it all together, 'all' an INTP need do is to make sure and mentally take stock of the big picture whenever they're learning something. I suspect otherwise the educational system isn't broken. These big topics are taught in pieces by necessity.

 

[Polaris]

Putting it all together, 'all' an INTP need do is to make sure and mentally take stock of the big picture whenever they're learning something. I suspect otherwise the educational system isn't broken. These big topics are taught in pieces by necessity.

This is something I only realised when I re-entered university as an older individual. I had by that time understood the crucial role math plays in all scientific fields, and could thus accept with no hesitation the necessity of learning the topic in pieces.

At sixteen, I absolutely rebelled against what I perceived as the useless part of learning, because I did not realise education by necessity has to cover the topics in this manner to cover all areas of potential interest in students due to lack of resources, funding, etc.

I really wish there was a way of introducing students to topics that would encourage an appreciation for why they are taught that way.

My best friend is an ENTJ school principal and a teacher of hard sciences. She is currently doing a study on her pupils that describes their individual ways of learning.

She pointed out to me that there were the 'nerds', who just 'got it', or accepted the abstract concepts and sat there smirking while the others struggled with trying to understand the application of maths to reality.

So my friend introduced a system of learning that put the mathematical concepts into real contexts, and the rest of the students were suddenly interested and enthusiastic about maths -- and the result was that they started outperforming her so-called nerds.

I just thought that was interesting.

 

[Analyzer]

I always found it interesting that math is now taught first without teaching it's origins and logic. In the classical tradition of learning you would learn things like Euclid's Elements and the concepts related to the theorems, along with logic to have a more holistic understanding. Applied math was a separate area or concentration for those interested in doing technical or engineering work.

 

[Polaris]

I always found it interesting that math is now taught first without teaching it's origins and logic. In the classical tradition of learning you would learn things like Euclid's Elements and the concepts related to the theorems, along with logic to have a more holistic understanding. Applied math was a separate area or concentration for those interested in doing technical or engineering work.

Exactly.

If they had introduced those concepts in maths as well as focusing on the non-engineering side of physics, I would have had more interest in the topics.

 

[Architect]

I always found it interesting that math is now taught first without teaching it's origins and logic. In the classical tradition of learning you would learn things like Euclid's Elements and the concepts related to the theorems, along with logic to have a more holistic understanding.

Why? Because there's too much to do already. You get that approach with a Classical education. An INTP friend of mine got a Classical education (studied Greek, Euclid and even the "Humors" for gods sake) and so got that holistic sense. In a regular program there isn't remotely enough time.

I got math up through graduate level Physics studies, and didn't study a damn thing I use in real life. I could have used way more linear algebra. I don't presently use any of the calculus, but I did have a stint with a lot of Tensor calculus - but again there was no Tensor math in my background (I had to pick it up myself). Oh and statistics, I took a semester, but it didn't get into any Bayes theorem that I use now.

The job of school is to teach you how to get through projects and do things you don't want to do, in addition to a smattering of what you might need in your profession. It's up to you to figure out the rest.

 

[dr froyd]

Right ideas but wrong angle I think.

Here's the deal, yes we need to see the forest before we see the trees, and most education is designed for the majority of people who get to the forest (if they ever get there) via the trees - the details. However here's the problem, in many or most complex fields you can't get the big picture until you sufficiently understand enough specific details.

Example: Quantum Mechanics. I'd estimate that 90%+ of what you read about QM is wrong. It's written by people who jumped to the big picture without taking all the hard math and classes. They don't know the real story and so get it all wrong. In this case you have to work through years of maths to differential equations, understand the experimental basis, then study and solve problems. And those problems are by necessity the simplest. Anything harder than that can't be solved in closed form, you have to find a numerical solution.

Putting it all together, 'all' an INTP need do is to make sure and mentally take stock of the big picture whenever they're learning something. I suspect otherwise the educational system isn't broken. These big topics are taught in pieces by necessity.

Totally agree. I didn't mean to critique the educational system per se (although I could say a lot about that topic), it was more about the personal learning style of an INTP.

 

[dr froyd]

...
My best friend is an ENTJ school principal and a teacher of hard sciences. She is currently doing a study on her pupils that describes their individual ways of learning.

She pointed out to me that there were the 'nerds', who just 'got it', or accepted the abstract concepts and sat there smirking while the others struggled with trying to understand the application of maths to reality.

So my friend introduced a system of learning that put the mathematical concepts into real contexts, and the rest of the students were suddenly interested and enthusiastic about maths -- and the result was that they started outperforming her so-called nerds.

I just thought that was interesting.

As counter-intuitive as it is, that strategy will probably be detrimental to the pupils' understanding of mathematics in the long run. I am currently reading about applications of "behaviourist" techniques like these for motivating people to learn subjects. Turns out that when you imply that a subject is only a means to some other goal, as opposed to learning the subject just for the subject's sake, the intrinsic interest in that subject drops dramatically.

Besides, 99.9% of all mathematics have no real direct application in the real world. For example, in order to compute the price of a financial derivative you need math, but most of that math is based on purely abstract concepts which takes several years to learn (probability theory, measure theory, newtonian calculus, stochastic calculus etc)

 

[Analyzer]

The job of school is to teach you how to get through projects and do things you don't want to do, in addition to a smattering of what you might need in your profession. It's up to you to figure out the rest.

For a scientific school yeah but I wouldn't consider the goal of education is to merely learn instrumental knowledge. Perhaps this why the whole system and approach to education has gone down the drain. We went from teaching Latin and Greek to now having Common Core. If you don't understand why certain things are and it's history you lack a fundamental aspect of being educated. Einstein's theories are just a modern interpretation(using Newtonian mechanics) of Parmenides. Maybe we just live in a society that values having trained people versus people who actually understand the foundational ideas of knowledge, so it makes sense.

 

[Stardust]

I just recently found out that I am an INTP. This type makes the most sense to me in terms of functions, as well as interests. I have also recently embarked on a long journey towards biochemistry and/or theoretical physics (I haven't decided yet as I still have plenty of time to).

Everything you stated dr froyd makes complete sense to me. I have been told, multiples times (within the last two weeks) that I think about math differently than my peers. I was told that I needed to have it broken down and explained differently than what the conventional way teaches. So your post resonates with me very much.

 

[QuickTwist]

All I have to say is Si goes far with INTP and maths. Combine this with their abilities to see tons of possibilities and thinking things through and its easy to see why INTPs can be really good at getting a really good intuitive grasp of the rules for maths.

 

[scenefinale]

http://betterexplained.com/articles/developing-your-intuition-for-math/

 

[r4ch3l]

Why? Because there's too much to do already. You get that approach with a Classical education. An INTP friend of mine got a Classical education (studied Greek, Euclid and even the "Humors" for gods sake) and so got that holistic sense. In a regular program there isn't remotely enough time.

I understand what you are saying but I think that if you integrate this sort of "Math History" (like Art History!) in the younger years it can help give a sense of the bigger picture and help inform kids of the reasons why they should put the work into learning the details and the not-so-fun parts. Ralph Abraham has Amazon.com: Bolts from the Blue: Art, Mathematics, and Cultural Evolution (9780983051770): Ralph Herman Abraham: Books@@AMEPARAM@@http://ecx.images-amazon.com/images/I/41zTAR9ZPQL.@@AMEPARAM@@41zTAR9ZPQL that was written with the premise that people understand math more easily if they know the cultural context of the math they are being taught. He also champions the use of graphic representations in math education which I am a huuuuge fan of. Just these two things would have really helped me to integrate and retain my education and to actually care enough to bother to put in the effort when things got hard along the way. Here, he says it better:

This book is motivated to show the role of artists and intellectuals in the evolution of mathematics, and to promote Thompson’s scheme for world cultural history. But its chief concern is the perilous situation of mathematics in our contemporary society. For there is now a pandemic of math anxiety. And believing as I do that our future is doomed without the intellectual and cognitive support of a healthy and vigorous mathematical culture, this situation demands attention.

After more than fifty years of teaching math in universities in several countries, I have developed the conviction that every person is born with a substantial talent for math which is subsequently destroyed in our schools by a faulty pedagogy that has become traditional during the last century or so. Over the years, I have identified three major flaws. This book is intended to help remedy them.

Flaw #1: No graphics.
The first flaw came to my attention around 1974, when computer graphics first arrived at my university. After creating computer graphic software for research in chaos theory, then a new branch of mathematics, we adapted the hardware and software to support our lower division math courses: calculus, linear algebra, differential equations, and so on. With support from the State of California, these efforts evolved into a major program called the Visual Math Project. Computer graphic illustrations and animations were piped into classrooms using television cables.

We discovered that many students were saved from math anxiety and became successful students of mathematics. Some were so enthusiastic that they became programmers in our project, developing software to teach others what they had learned. Mathematicians communicate among themselves by coordinating multiple intelligences: verbal, graphic, and symbolic. For example, one cannot learn math without graphics. As our school math programs present math without adequate graphics, learning is handicapped. Students fail to learn, and then are persuaded that it is their own fault, which it is not.

Flaw #2: No history
The second flaw came to me around 1987. A book on chaos theory, in which I was quoted extensively, became very popular. Journalists called me to ask what the fuss was all about, leading me to write a book, Chaos, Gaia, Eros, on the historical context and philosophical significance of chaos theory. Meanwhile, a new course on the history of mathematics was instituted at my university. My colleagues, knowing that was writing a book on the history of chaos theory, offered me this course, and I taught it annually for a decade or more. A friend, Rupert Sheldrake, persuaded me that people learn things better if they are presented in historical order. In the case of mathematics, this is the opposite of the usual, logical order. Armed with this idea, and with my new knowledge of the history of math, I gradually changed all my teaching to a historically based style. In a historically based program, topics are presented in historical order, so that cognitive prerequisites are available when needed. This avoids the most common obstacles that prevent students from grasping new mathematical concepts.

Flaw #3: No integration.
The historical sequence is crucial, yet not enough. The whole historical sequence should be integrated with the cultural context in which it evolved, providing meaning and motivation for students. I am convinced that these three flaws are major causes of difficulty that students have in learning math in our school system today. I found fewer failures in my courses after adopting all three remedies — visual representation, historical sequence, and cultural
integration — in my courses.

The big test.
The weaknesses in our school math programs today are commonplace, and widely proclaimed. The remedies usually proposed — standardized multiple-choice testing and coaching, word problems, short-question drill and kill, and so on — will do more harm than good. Rather, we advocate graphics, history and integration.

As a child I remember being baffled as to why my teachers taught me to add by essentially just counting on my fingers, why multiplication tables were important if I could just use a calculator or memorize them as I actually implemented them instead of in practice, why long division was necessary now that we had calculators. I always got As until high school but was so unmotivated because I actually thought that math was just for stupid day to day problems like the ones in my homework and exams. How boring. Why would I want to study that? Had I learned what calculus actually does instead of it just being some mystical term to describe what kind of math "smart" people do in high school and college I would have been captivated. If modular arithmetic was introduced in 4th grade instead of that hellish year of long division I think I would have been able to understand division as a concept.

TL;DR: Word problems suck and totally misrepresent what math actually is. Learning the important details requires motivation to keep going. Now I'm a 28 year old taking calculus at night with 19 year olds because I finally have sufficient motivation and context.

 

[dark+matters]

Now I'm a 28 year old taking calculus at night with 19 year olds because I finally have sufficient motivation and context.

Me too! I might have to check out that book. (I plan on being in one class or another for the rest of my life, but does feel strange being ten years older than most of the people around me.)

 

[manishboy]

This resonates--

So my theory is this: it's not that INTP's learn slower, or in a less efficient way or whatever, it's just that they have a completely different way of understanding a concept.

I've recently had to internalize some math (a bit of diff. eq, lin. alg., stats., and abstract alg.) and have found that I cannot simply learn and apply techniques. I'm actually quite bad at blindly running an algorithm. Nor do I need a deep history and context (the big picture). What I need is an intuitive way to feel my way around the subject at hand. For example, learning to do numerical simulation of feedback systems before learning much about differential equations helped to give me an intuition of what that math is trying to do.

At some point, the intuition breaks down and I have to rely on technique. But having that basic intuition in the background is a framework around which to hang the specific methods, making it easier to select and use them.

Writing code is not the only way to build intuition, but it's easy, cheap, and more fun than working with clay.

 

[Trebuchet]

I think another factor is that there isn't just one math, and people are not equally good at all of them. I've known people who had a knack for topics like

Arithmetic and Cryptography (these seem to go together but that may be coincidence)
Algebra and Calculus
Geometry and Trigonometry
Topology
Set Theory
Formal Logic

People who are good at one aren't necessarily good at another, but it is easy to think you've reached your limit when you start a new class that just doesn't make sense at the start, and give up. I had to struggle with Arithmetic and Algebra initially but Geometry and Formal Logic were easy to me.

I agree that context and an understanding of why is critical for INTPs. I'm just offering another angle, that some things come easier than others but we call them all "math." (Or "maths" if you prefer.)

Word problems suck and totally misrepresent what math actually is.

Ha ha! I came across a real word problem one time.

Two trains left from several hundred miles apart (I was on the NB one) and the teens who were working as staff needed to go home at night. They stopped the NB and SB trains next to each other and the staff all swapped trains to go back where they came from. So they had to figure out where that would happen.

I didn't have to do the calculations since I was a passenger, but I was delighted to find a real example of this old classic.

 

[Direwolf]

Dont know about u guys but i absorb it through the skin

 

[Sockrates]

Read a book.

 

[Coolydudey]

With some classes I don't have an intuitive grasp on (like the Geometry course which has just got into Riemannian metrics and hyperbolic geometry) I build whatever understanding I can from the course and wait for my intuition to develop further in doing the questions. I feel this process has eventually led me to understand and grasp everything I've done so far very well, but then again I am at a leading university so it definitely helps a lot to have high-level teaching aimed not just at providing techniques but developing deeper insight and understanding in its own.

TLDR; my attitude has been be patient and the intuition comes from the problem sheets and general progression of the course.

 

[Vion]

Truth doesn't come from any secondary sources. One man's conclusion is another man's introduction. Start from the back of the book right at the index and slowly work your way back dumbfounded and confused like the day you were born into this world. "There is no substitution for knowledge." I now know W. Edwards Deming, thanks for the thread.

 

[Keep on Thinking]

Modern Math and Physics is full of mistakes:


http://milesmathis.com/death.html

 

[Keep on Thinking]

What you learn today at school is down to earth materialistic / applied math and physics (engineering) which is a good thing if you want a job. (Ergo it Works)

Much from the enlightment era: Descartes.


Modern Theoretical math and physics however...Einstein, Hawking, black holes, dark matter curved space, string theory etc. is full of mistakes

 

[Coolydudey]

Modern Math and Physics is full of mistakes:


http://milesmathis.com/death.html

I wasted 20 minutes of my life reading some of the stuff by this guy, because I was interested in what one earth he was on about. But in hindsight, I knew even before reading. His attack on calculus in particular is quite laughable. It seems really strange that someone so knowledgeable and well-written could make such basic mistakes. Oh well, that's my opinion at least. You probably wouldn't trust me cause I think maths is beautiful for its own sake.