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[…] How to develop a mindset for math […]
[…] Seeing imaginary numbers as rotations gives us a new mindset to approach problems; the “plug and chug” formulas can make intuitive sense, even for a strange topic like complex numbers. Happy math. Posted January 2, 2008, under Math Tags: arithmetic, complex, conjugate, diagram, imaginary, Math, number, visual Related Posts: […]
One “Eureka!” moment I had came recently when studying matrices in school (for the second time (I hate the fact that Analysis is a repeat of advanced algebra)). I was wondering why the determinant of a matrix was ad-bc and not some other combination of the elements of the matrix. It took about 10 minutes of thinking about Cramer’s Rule before I understood exactly what the point of it was.
I happen to believe that teaching the history of mathematics along with the subject itself is probably the best approach to the subject. You really have an appreciation of why things are the way they are when you see how they came to be that way.
Seeing relationships really helps you understand math; it’s great being able to see the derivative as an “instant change” and the integral as a “cumulative change” - might make a lot of people less scared of calculus, and who’d oppose that?
Hi Zac, thanks for the comment! That note about Cramer’s rule and determinants is really interesting, I hadn’t thought about the reason for that format either.
Yes, having math & history go side-by-side really helps show why certain techniques developed, not just how. For me, it makes understanding much clearer and more interesting.
And on your calculus topic, that’s exactly it – I want to show that “scary” topics like calculus really aren’t that bad :). Thanks for the note.
I think you’ve got your representation of negative numbers all wrong. In your example, you said that if I gave my friend three cows, I had -3 cows. I disagree. I would explain things this way: I promised my friend 3 cows. Then when I get three cows, I will give them to my friend, bringing my total to 0 cows.
Representing “owed assets” is a more difficult problem. The best way is to separate your assets according to whether they are concrete or whether they are owed to you by someone else. Then you can add them together to get your total theoretical net worth.
Note: To get total net worth, you would also have to subtract the amount of money you owed to other people. I kinda forgot that part.
Hi John, thanks for the comment.
Yeah, the example I used may be more confusing than helpful. The idea was to show that negative numbers aren’t “real” – they have a certain relationship, and in our world, debt seems to model a similar relationship. So we use them to represent the bookkeeping of debt.
Hi Kalid, Thank you very much for this wonderful resource. I admire the way you explain things, especially math. Thanks again.
I am looking for a math book which can teach me things in a way similar to your posts. Can you think of any?
Thanks in advance.
Hi Prashanth, thanks for the comment, glad you are finding it useful :). Comment #33 (above) has a list of some books people have found useful.
I really admire the way Richard Feynman explains physics and other topics (videos here: http://vega.org.uk/video/subseries/8). I’m not sure of many mathematicians who explain things in his style.
[…] The idea that “close counts” is weird — shouldn’t math be precise? Math is a model to describe the world. Our equations don’t need to be razor-sharp if the universe and our instruments are fuzzy. […]
Great article. Hope the site gets converted to a print version.
My personal favorite for learning about ‘e’ is e:The Story of a Number- Eli Maor. For a good of general mathematics my current favorite is Mathematics: From the Birth of Numbers.
Thanks Sameer – I’d love to turn this into a book one day. Thanks for the references, I’ll add them to my (ever growing) list :).
Thanks or information
[…] How to Develop a Mindset for Math | BetterExplained (tags: learning education math) […]
Thanks Kalid. Your posts are excellent and contain some beautiful revelations.
Only one thing (there is always a but). Could you swap the es in the article above. Generally and on calculators in particular e represents 2.71828… and E x10^n.
That is “e is not a number” rather than “E is not just a number”.
And in the table and below. 1E3 vs 1.000E3 rather than 1e3 vs 1.000e3.
It may seem a trivial point but surely will confuse my students.
Thanks again, Peter
Hi Peter, that’s a great point about the "e"s, I hadn’t realized their potential for confusion. I’ll update the article & diagram.
I am really impressed. A link to your site is going straight onto the Aberdeen College VLE in the hope that it will help and inspire my engineering students.
A good site that has also inspired me:
http://jedidiah.stuff.gen.nz/wp/?page_id=10
@Peter: Thank you for the kind words! I hope your students enjoy it also. I’ll check out the article you linked.