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Barring very elementary arithmetic, which skills from elementary school are essential for understanding the world better?

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Uh... sharing? But seriously, in the subject you mention math, but then you exclude arithmetic. Are you looking for non-arithmetic math skills in elementary school? – The Chaz 2.0 Mar 9 '11 at 15:02
Sorry, I meant very basic stuff such as counting, or addition/subtraction; these are very fundamental things. Elementary school syllabus stretches till algebra, geometry, etc., – CMR Mar 9 '11 at 15:06
None. Archimedes, Newton etc those boys never went to school – TROLLKILLER Mar 9 '11 at 15:16
Agree with @kakemonsteret, that list of people who never bothered to finish school is huge... Am sure so is the list of homeschooled ones: Edison, one I can instantly remember. However, this question is regarding kids who are not Einsteins or Newtons – CMR Mar 9 '11 at 15:25
@user02138, please move to a place where I can get some answers. – CMR Mar 9 '11 at 15:28
up vote 3 down vote accepted

Let me point that some ideas in the direction of what you are looking for can be found, for example, in Polya's work on heuristics (How to solve it), or in Davis and Hersh (Mathematical experience). Maybe after some reflection and elaboration, but there they are!

Maybe I should say something more about this. Mathematics exist because problems exist, particularly problems that admit solutions by a certain kind of means we could call mathematical (I know this sounds circular, but what other name could we give them?).

I would say that the two most prominent skills are idealization and abstraction. Idealization because in order to find a solution one needs to understand the problem and that involves being able to separate its relevant features from the irrelevant ones. Abstraction because the techniques developed for a particular problem become of general applicability in many other, possibly unsuspected, situations.

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Not everyone will agree, but I'd say the most important adaptive traits that a child can learn in elementary school are Deductive Reasoning, Inductive Reasoning, Abductive Reasoning and Healthy Social Skill.

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I was about to answer 'probability and logic', but your answer covers them better. – Mitch Mar 9 '11 at 15:14
Why the downvote? – user02138 Mar 9 '11 at 23:27

Wow, can i say everything? But let me just mention three that might be the most important:

1) Any money related mathematics will be very useful for obvious reasons. It amazes me how many of my students cannot even intuitively tell whether their answer to a problem involving tax makes sense. As consumers, we should be able to see a price tag and know about how much we will have to pay including taxes. And of course, there are many other applications involving money.

2) Spatial awareness can be very important. Geometry (composite shapes, three dimensional shapes, etc.) helps teach the brain to literally view the physical world.

3) Logic and algebra teach us everyday problem solving. Word problems, although they are often a dreaded topic, are SO beneficial! It's being able to comprehend a problem, assess what needs to be done, and find a way to get the desired result, that will likely be the most significant take-way from elementary mathematics for a person in a non-math-related (or really in any) field.

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Definitely the Intercept theorem or the related concept of similarity. Knowing that when two objects are similar, all length have the same ratio $\lambda$, all areas have the same ratio $\lambda^2$, ... . This leads directly to the concept of dimensional analysis. Knowing that you can estimate almost everything in life (and also in mathematics, see, e.g., Street-Fighting Mathematics).

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For some reason, this reminds me of Number numbness by Douglas Hofstadter.… – CMR Mar 9 '11 at 19:54

A short answer would be Identifying and Explaining Patterns

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First, you have to LIKE math. I've had no trouble getting there with 4 and 5 year olds. Something like Fractal brocolli or Fractal Extreme is enough to catch some interest. Then the game of Go is simple to teach, as is mastering tic-tac-toe. Reading Martin Gardner's books will also get you far.

At that point, math won't be able to terrify you. With more effort, you'll be able to recognize types of terminology, and will be able to identify likely areas for a solution. Math won't be a foreign language anymore.

Later, when a group of hard math problems drops on you and your colleagues like a pack of ninjas, you won't be cowering in the corner. You'll be pointing to some -- "I can take on these guys." You won't necessary be able to solve them in your head, but you'll know where you can look up the revelant solving tools.

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I find that one of the great benefits of studying math is the cultivation of self-reflexivity. Knowing whether you know something. This, I think, is unique to math, since there is no subjectivity involved, either you know it or you don't.

So it is a great developmental tool to practice knowing when you are telling the truth to yourself (and others) or not. And seeing when further understanding is necessary.

So on the basis of this, I would suggest written and especially verbal presentations by students in some context to enrich that capacity. This will also enhance communication skills of articulation and expression.

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