Math Is Fun Forum

algorithmicCreativity

Member

Registered: 2008-09-16

Posts: 1

"Learn by Doing" - what's it mean and is it the only way?

I am a computer science and math student right now. I am aiming to do a lot of the learning on my own, (not in a college/school).

I plan on taking the CLEP Exam, info in the link. It is essentially the same as an AP Exam or a Credit by Exam for CALC1 in college.

http://www.collegeboard.com/student/testing/clep/ex_calc.html 

I want to learn math to see if I enjoy it for its own sake, and for being generally more logical/smarter (mental exercise), and for complimenting work I may do in the sciences like physics and computer engineering/programming.

So, from what I understand, a lot of lower level Math, in classes like CALC1,2,3, Linear Algebra, and Differential Equations, is all about algorithms.

Anyway, what does it mean to learn by doing? How much of the learning is conceptual? How much is of it, is pretty much robotic rule following?

My main problem in learning Math, is getting stuck in cognitive loops trying to make sense of the conceptual/theoretical discussion presented prior to the exercises in every section.

But then there is this approach called "Learn by Doing", which says you basically do robotic exercises and drills and memorization, and then you learn math that way. Look at the examples before the exercises, and then just apply those algorithms to the exercises enough times so that you know the rules by heart.

I read a book on the topic,
http://www.amazon.com/Equation-Excellence-Make-Child-Excel/dp/0980144604
. It basically argued that drilling, memorization, and other "cranking"/"grinding" learning behaviors are critical to developing math skill.

He also said struggling through problems, (preferably with a mentor/tutor), is the best way to learn deeply, and develop your mind.

More importantly, and to the point, what is the best way to prepare for this CLEP CALC1 exam?

Also, the other notion I've been getting, is that only through struggling through problems do you really gain a deeper understanding. The memorization and drilling is important, but they are useful as supports for the problem solving which is kind of where the light bulb goes off, and the understanding clicks in your mind.

The critical subtext to this, which has really been bothering me, is that you also need someone who has seen the problem type you are working on, and can interface with you and let you struggle, make sure you are adequately struggling, and offer as limited assistance as is necessary for you to discover the solution on  your own.

Now, I don't know if this subtext was intended, but the analogy of doing math to doing a bench press lift in the weight room, really solidified this problem for me. And this is a problem, because I don't have access to someone who knows Math really well who can work with me one on one, and who knows me well enough to interface with my psychology to kind of train me and make sure I am working hard.

If the analogy didn't make sense, it goes like this. Most people don't use math in their work. However doing math makes you smarter and more generally intelligent. Similarly, few people become serious body builders, but many people play sports. Just as you do math to enhance your general intelligence for various non-explicitly mathematical applications, you do the bench press (or running (which doesn't need a spotter, weakness in the analogy)) to help your general athleticism in various sports, although in no sport, are you expected to actually lay down and do the exact bench press behavior.

The critical part of this analogy which is making me nervous is that, as Arnold Schwarzenger said in his movie Pumping Iron, paraphrased, "It is those last few reptitions that separate the great from the champions". And those last few reps are almost always ones which require spotting. Spotting means you need someone there, who can interface with you, and make you sweat.

Similarly, in math, you need someone who knows how to do the problems, and who is willing to interface with you, and make sure you work your hardest, because as the analogy/learning-theory goes, you learn the best by struggling through the problems.

I hope this hasn't been too much to read, but it is driving me nuts.

mikau

Member

Registered: 2005-08-22

Posts: 1,504

Re: "Learn by Doing" - what's it mean and is it the only way?

well i took the time to read it, for your sake!

I feel I should point out that math was invented entirely through the ingenuity of people, which means every topic was originally conceived in someone's mind through logic/reasoning/intuition, not by having someone show it to them. Therefore it is possible to understand it without having someone who can 'interface with you', you can come to understand it just through reason alone, thats where math comes from.

As for this statement "So, from what I understand, a lot of lower level Math, in classes like CALC1,2,3, Linear Algebra, and Differential Equations, is all about algorithms."

if by 'algorithm', you mean like a monotonous routine, a step by step process to solve a problem, that claim is absolutely absurd. Many problems in all of those subjects can only be solved through ingenuity. You are given problems you have never seen before, and you  have to use not only your knowledge, but also your wits. Let me tell you, many of the problems you will face in those courses are TOUGH and they give you no direct algorithm to solve them. This doesn't mean mean you can't solve it yourself, it means you actually have to think about it for a while.

The goal of these courses is to teach you not only how to solve common, and ordinary problems that have straightforward solutions, but also how to combine knowledge and experience, with your own ingenuity/creativity to find your way through problems you've never seen before.

Any mathematician that cannot solve a problem he's never seen before without help, would be absolutely worthless.

As for studying, if you want to take these courses in school, that works fine! You have a book, you have a teacher, you will have homework problems that are challenging and not just boring algorithms to follow. And yes, you will have them in Calc 1,2,3, Linear, and Differential Equations.

If you want to study these on your own, then buy a textbook, read the chapters, try to understand, and do the problems! The problems are designed to build up your skill, and let me tell you, they work! If you get stuck, checkout alternate explanations of the topic online/other textbooks, or ask questions here.   

I actually studied math up to and including calculus, entirely on my own before college, just by reading books, and doing the problems. I believe a really advanced member here, Jane Fairfax, learns only through self study as well. Let me tell you from experience, it works, in fact I personally think it works better than formal schooling, but its important to have official degrees if you're hoping to find a job!

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A logarithm is just a misspelled algorithm.

integer

Member

Registered: 2008-02-21

Posts: 79

Re: "Learn by Doing" - what's it mean and is it the only way?

what does it mean to learn by doing?

this approach called "Learn by Doing",  ...  you basically do ... the exercises enough times so that you know the rules

It may only require someone (usually a teacher/professor) to state the problem and a normally expected solution for a person to understand the concept.  However, there are some students that cannot grasp some complex problems.  Often an instructor will give an incomplete statement of the problem and a solution.  The student will then misunderstand why what occurs.  If you are adept at grasping (that is you can wrap your head around the idea) the basics then you wil require only a few exercises.  However, there are some who will struggle there entire lives doing exercises and will never completely understand the fundamentals of the terminology.

the other notion I've been getting, is that only through struggling through problems do you really gain a deeper understanding.

You can bench press a two-pound weight all want,  but that is unlikely to prepare you for championship bench pressing.  You will get better by doing more difficult problems than the ones you have previously done.   You may gain an extremely deep understanding by doing easy examples.  But most likely intricate problems will provide multiple doors through which you can gain entrance into the area where the light bulb flashes bright.

Memorization/drills are important.  They explain the best/correct method to bench press a small weight, so that when you want to bench press a hundred kilograms you have the knowledge to do it in a safe and correct manner.

If you do not know what you are doing, it is unlikely that anyone else will either.