First, thank you for this amazing blog and the insightful articles that make math so easy… ok maybe not so easy but easier than usual :D.
I really like the idea of the bow and arrow metaphor, but wanted to know how could I implement it into every day learning (high school, college etc.).

@Sh: Thanks :). Applying the bow/arrow metaphor is tricky, it’s more of a general attitude to take: make sure you build up your bow skills (intuition), and not spend too much time on arrows (memorizing facts). I have an article on building your intuition which might be helpful: http://betterexplained.com/articles/developing-your-intuition-for-math/

Yep, I agree, the bow/arrow concept is not just limited to math (math is a “fun” area to tackle because if we can make it understandable, or even enjoyable, think what’s possible for a subject which isn’t as generally reviled).

The more I think about it, the more I realize the value of truly understanding one bow and one arrow. Appreciate the note!

Hi psoe, thanks for the note! I don’t like to make clearcut distinctions like specialist vs. generalist, but I think someone who is really, really good at something has figured out a) how to practice b) how to work through difficulties c) what it feels like to “know” something. The so-called “generalist” might actually be really good at A, and ok at B-Z, but would still have that mastery experience with A. In more abstract terms, if you can get into the “flow” state with one endeavor, you’ll know what it’s like to be in the zone with a new one (and everyone has hit that “in the zone” feeling at some point).

It’s really hard for me to comment on education systems as a whole. In general, I think flexibility of options good, and the focus on lots of arrows (or not) is a combination of the attitudes and efforts of the teacher and student.

I honestly never loved maths in high school. Today I realize that what i learnt in high school was not maths at all. It was bihearted literature…just a pattern of alphabets and symbols memorized in a specific pattern(formulas).
Infact even students actually studying literature know more about their subject than what I or even the maths teachers knew about maths.
It was just recently that i actually started learning intutively.
And today I realize that physics…which i always loved and maths…which i never loved were actually one and the same.
I feel i am the only person in my vicinity who truly understands maths.
There was a time when i used to truly respect people who used to bring full marks in maths. However…no longer.
With Respect
Binnoy

Hi Kalid, enjoy your site immensely, keep adding more!
when you say learning how to learn is not learning more and more arrows, but mastering a single bow and a single arrow, does that imply that specialists are better at learning new things, i.e. they are also better generalists? And on a sidenote, american liberal education that teaches multiple things in college seems to be giving students just the arrows, and not allowing students to master one bow and one arrow, thus fail? (or maybe it’s doing well, what do you think?) erm… this puzzle has bugged me for a long time, so I’d like your opinion.

One way to REALLY improve intuition in math (and science) is to learn the history behind it. The history often provides the conceptual development of the subject that most textbooks skip. For example, ancient Babylonian tablets and Greek, Arabic, Hindu texts are a record of the “firsts” in math. Imagine developing a new topic by introducing these sources. It also makes the math more human - there’s the first person to develop it and there context to do so is such and such.

@Mark: I completely agree. e and natural log were discovered in the context of computing interest. Complex numbers weren’t fully utilized until the geometric interpretation came about. There are so many aspects where knowing the history makes you understand the thought process that went into the idea.

I personally really like the idea (use the computational power of computers to help teach math nowadays). A lot more “high-level” thinking (real world - delving in IDEAS themselves) can be done by students; as opposed to drowning solely in the nitty-gritty “low-level” thinking (symbol manipulation, hand calculations, but LACKING the big picture of the IDEA), typical of traditional math education. *(high-level and low-level used from a computer science perspective - e.g. C++ vs Assembly)

Also, it’s a perfectly feasible real-world solution - the question is whether the educational institutions will slowly begin to adapt this style of teaching (knowing how stubborn and resistant to change they tend to be.) From an individual (self-learner) standpoint though, it is CERTAINLY feasible, and a VERY exciting idea!

A great status-quo breaker - thanks again for a marvelous article, Kalid! (My thought is that your site, combined with Lockhart’s Lament AND Wolfram’s idea - would be a very powerful movement indeed!)

I hadn’t seen that video: “Conrad Wolfram: Teaching kids real math with computers”. He makes some good points. It reminds me of “A Mathematician’s Lament” by Paul Lockhart.

I agree that there is a HUGE gap between mathematics at school and mathematics at work. I also agree with him that there is too much emphasis on doing the same calculation for the hundredth time. I have experienced his idea of using a computer to make mathematics more interactive when I did a course on Operations Research. But I’m glad he acknowledged that computers could be used wrong.

The problem is, mathematics could be improved a lot(!)—even without computers. And we aren’t doing it! Instead we bombard students with theories, lemmas and so on. We teach them how a formula is derived; we might(!) explain what it physically means, but we rarely explain what it means mathematically. Which is what you do with this site. I have had very few teachers actually try explaining mathematics really well. I was fortunate to be good at it.

I like the fact that Conrad Wolfram takes an engineering approach: (1) take a real world question; (2) model it mathematically; (3) solve the mathematical problem; (4) check the solution against the real world. I also like that he only showed interactive programs and computer algebra systems.

It’s unfortunate that Conrad Wolfram didn’t talk about the history of mathematics. Because many of the new concepts in mathematics were introduced specifically to be able to solve real world problems.

@Stan: Thanks for that pointer, I had forgotten about that TED talk! Yes, I really agree with his approach – we need to learn how to use and manipulate the tools we have, to really get an intuition for them. Memorizing giant tables of integrals isn’t doing anybody some good. See, the funny thing is after getting that intuition you become more interested in learning the nitty gritty also (since you’re interested in the subject in general).

And yep, I think think a joint movement would be fun… everybody is putting together the pieces :).

@PL: Thanks for the comment – yep, a balance is needed. I err on the side of the minimum level of rigor needed to begin talking, then getting a deep intuition, then sprinkling in more rigor as needed.