@Peter: Thank you for that link, it was an excellent discussion similar to Lockhart’s Lament. I agree with the way English/Math is taught, with a focus on “grammar” vs. ideas.
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[…] Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is. […]
[…] Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is. […]
Wow! Love this site! I actually have been dreaming of doing something like this for a while, as I noticed in the comments for this and a few others many have found these in interesting moments of synchronicity!
Anyway, I completely agree with you. Our education system should be showing people how to apply what it teaches, thus the student gains interest in the subject and desires to learn, thus learning. Spitting brick wall rhetoric only annoys and encourages the student to tune out the teacher.
Keep up the excellent work and thank you very much for clarifying my understanding of many, many topics!
@Jasmine: Thanks for the kind words! I completely agree, right now much of math education consists of memorizing and plugging values into formulas. The mechanics of math has its place, but it isn’t real understanding (and can be boring/demoralizing).
Thanks again for the note.
Greetings,
Absolutely ignore comment 1! I can’t believe someone would start with that. Getting the right mindset is vital - a process I’ve been learning as part of my PhD, returning to maths after an extended break.
I’m in the process of trying to learn some (for me) hefty, complex stuff from Paul Krugman’s papers. once I’ve nailed it, I hope to follow your example and make it easier for others. E.g. I’m doing interactive visualisations to help -
http://www.personal.leeds.ac.uk/~gy06do/processing/ces_function/
(Though it needs more writing up!)
This stuff is fantastic - and really, really, not just for elementary school! Thanks. One line in particular I’ll stick on my wall:
“Don’t stop until it makes sense, or that mathematical gap will haunt you.”
I’ll do my best…!
Dan
[…] Mathematical Mindsets I evidentlly prefer a more philosophical mathematical mindset. Explore posts in the same categories: Mindsets […]
Maybe I’m missing the point here, but shouldn’t we say that the person who takes the cows has -3 cows? In that case, negative numbers certainly do make sense.
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[…] Intro: Mental math shortcuts, adding 1-100, how to learn math, understanding averages […]
This an eye opener, loads of a-ah moments. I wanted to purchase yet another maths book because of math’s abstract nature the ideas aren’t discuss only the equations are presented in text books. I decided not to purchase yet another math’s or novel until I came across this website by accident. Keep up the good job!!!
@Mike: Yes, that’s the way we’d have to look at it in our accounting system: I have 0 cows, and my friend has -3 cows. So nobody has any positive number of cows? The total # of cows in the world is 0 + (-3) = -3? 
[The point is that negative numbers are a mental construct that we try to apply to scenarios we encounter in the world].
@Anthony: Thanks for the kind words!
@Dan: Thanks for the comment! Really glad you are pursuing more learning, and I’m even more delighted that you want to share your knowledge with others.
I think that quote might be a little too cut-and-dry in hindsight, more that you can understand anything simply. There’s always a period of “haunting confusion” before understanding a topic – but we can always push through! Thanks again for writing.
You: Negative numbers are a great idea, but don’t inherently exist. It’s a label we apply to a concept.
Me: Numbers are a great idea, but don’t inherently exist. It’s a label we apply to a concept.
All numbers have the same amount of “inherent” existence or non-existence.
@Peter: Great point. Yes, all numbers can be considered an abstraction or a creation of the human mind.
There’s a lot of philosophy of mathematics that deals with this dilemma – one view is "“God created the integers, all the rest is the work of Man. (Leopold Kronecker)” [substitute Nature, Flying Spaghetti Monster, etc.].
Another view is that the universe is the result of a mathematical structure (http://discovermagazine.com/2008/jul/16-is-the-universe-actually-made-of-math), so in that case it’s math creating man :). It’s fun to think about.
hey kalid,
I have gone through the post and comments, and also read views of proponents of formalism, and I must say that i could pick up most of the points correctly but somehow I could not think in a free and intuitive way when i tried to learn maths on my own (using standard books though).
When i try to think of concepts, find relationships as in real world, just nothing to my mind. Can you guess me (assume that i have average IQ) whats wrong have I been doing and suggest me an intuitive way of conceptualizing that is faster and can also be applyied in real world
@hitendra: Finding the right mental model/analogy can be tricky. I usually try to look at the history / context of an idea to see where it was used & what it was useful for (calculus, for example, was primarily used around rates of motion, so intuitions can involve speed, acceleration, etc.). I try to find a few ways of looking at a concept and “circle around” to see if I can connect them. You might find this post useful:
http://betterexplained.com/articles/developing-your-intuition-for-math/
This is probably silly to all of you, and off topic. But I am in 5th grade and our teacher gave us a challenge question: What is the relationship with the numbers 10 and 1,000. He wants us to write/draw 3 relationships. I may be overthinking this, I can think of the obvious, but then the draw part is confusing. Any ideas?
Maths, its always a dream for me… Not to learn it but to get passing marks. My college teachers of Anna University so much spoiled the interest tat i almost run from the subject. Perhaps they themselves dont know the subject.
Im so much frustrated with maths that i have left my mechanical engineering degree since i was not able to clear the maths paper alone. Now i finally say or have to say, Maths Sucks and the teachers who teach them sucks more…