How to Develop a Mindset for Math

There are several intermediate steps which connect Urnary and Roman.

visually highlight precounted intervals. This creates a new symbol out of 5. This system is well suited for incremental counting of potentially large numbers - such as counting days on a prison wall.

As the numbers grow large, a similar concept can be applied to group blocks of 50 or 500. Mayan, Egyptian, and other historical solutions exist.

We’re not limited to groups of 5, that’s just ( pun) handy. When playing Cricket (Darts) you’re using a base 3 symbology

While the Babylonians used a base 60. This number is usually chosen since it divides evenly in so many ways. This allows even partitioning among 2 to 6 people.

As the numbers grow larger we need groups of groups, and a new symbol for each. Roman numerals assigned a unique symbol for 1, 5, 10, 50, 100, 500, 1000.

This system can compactly represent very large numbers, in a very small space. But making the numbers follow a more uniform progression simplifies math. By standardizing on a 1,10,100,1000,… pattern all sorts of nice mathematical regularities pop out. Suddenly a mathematical operation such as multiplication that was difficult, could instead be calculated using a 10x10 lookup table. By choosing a binary system that table becomes 2x2.

I recall sitting through 3/4th of a semester of linear algebra. While studying for a test I finally had all the pieces in front of me, having that eureka moment. I’d wondered ‘why didn’t he just say that?!’ Having since done some teaching, I can appreciate that even when you know it, it can be very hard to convey. But I think too often we do fail to mention the forest while teaching the trees.

If I had ever previously encountered the Mobius Transformation, I suspect this youtube video might have been similarly insightful

Visual understanding, while not the only path, is often a key insight.

ps. loving the blog.

I have a hard time understanding why you would suggest negative numbers are not “real” (I understand the connotation). They are just as “real” a positive numbers (which are also contrivances of our imagination). I use the example of elevation for my math students as suggested by auferstehung (only I use the idea of a hole in the ground to illustrate the idea of negatives). And, to continue the analogy negative numbers are just as “real” as complex and imaginary numbers as well, with real-world applications and examples.

I think the best mindset is like you said think concretely, with concrete physical examples that you can sense and relate to. Everything in mathematics becomes abstract too soon in school mainly; when I’m doing trigonometry for real-world applications I’m not thinking what tangent really is which is a ratio of two sides but simply as a formular which I use and a button on my calculator. I think that you can’t make an mathematical concept abstract until you’ve grasped it’s physical real sense extremely well and then can progress to what if situations that aren’t present in the real world.

@Dorai: I agree that it takes a bit of time to view math as models and relationships. The why and how have a yin-yang relationship; each one feeds the other. Unfortunately, schools tend to teach the “how” and leave “why” as an exercise for the reader :). As you say, we learn a lot of math without realizing why it works and where else it can be used.

@auferstehung: Elevation is another example. I wanted to choose one where a negative number didn’t have a nice clean meaning.

@Mark: Thanks for the info! I’m glad you liked the article, I have quite the reading list now :slight_smile:

@Ken: I agree. Negative numbers are just as “real” (or better said, just as fake) as any kind of number (positive, fractions, imaginary). The goal was to explain that negative numbers are a figment of our imagination, which is easier to initially grasp than “counting numbers are a figment of our imagination”. I’ll be speaking about this more in upcoming articles, I hope to make it more clear.

@Pablo: Yes, I think there is a cycle of learning “how” and learning “why”. Unfortunately, many times we only learn the “how” (i.e., tangent just becomes a button on the calculator, as you say) vs. knowing “why” it has the properties it does. Thanks for the comment.

Interesting article, I’m glad you’ve pointed out some things here and that I’m not the only one feeling as you do. Here comes a rant. I am a college student in Calculus II (for the second time) and I often wonder “Who murdered math?”. It was my favorite and strongest subject throughout most of grade school, but in college I have developed a powerful hate for it. It has become “this is how you do it so now go home and do enough problems until you memorize it, then regurgitate it onto the exam.” There’s no soul to it anymore. Some of us aren’t interested in finding the area of the surface of a cylinder and don’t see ever needing to do so. Relate it to us, make it interesting. To me, if something needs to be vigorously memorized, it’s not being presented in a proper, meaningful way. Relationships should be drawn between concepts, like the relationship between Summation and Integration. I may be wrong, but it seems there is too much of an emphasis on breadth, not enough on depth. Please continue writing articles like this one. I doubt you’ll be able to make me a lover again, but maybe, just maybe, I’ll become less adverse to the subject.

Even if it is not directly related to this article, I’d suggest also to have a look at Mathematics: A Very Short Introduction by Timothy Gowers,
http://www.amazon.com/gp/product/0192853619 .
Gowers has also a blog, http://gowers.wordpress.com/ , but he did not update it for a while.

@Me: my understsnding is that this post - at least for the time being - is more a “call to arms” than an essay. Kalid will correct me, but I have the impression that he is trying to find the best (ok, a quite good) approach to teaching math, but he does not have the Answer right now.

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@cariaso: Thanks for the history, that’s some great background info. Yes, even when insights are in your brain it can be tough to get them out in writing. I try to use diagrams, analogies, dialogues, anything that can help convey the topic. Different approaches click for different people.

@John: Thanks for dropping by, you aren’t the only one who feels that way. “Who murdered math?” is a great quote, I think that phrase epitomizes what happened to many people along the way. Lots of people I know liked math as a kid but hated it later in life. I’ll keep writing, and maybe we can relight that spark :).

@Mau: Thanks for the info! I have quite a reading list now, I’m excited to start into it. Yes, this was a “call to arms” of sorts (nice description) – I want to do an assault on the insight-free math we’ve come to expect, and I’m looking for battle strategies :).

John @ 41: (Hoping you check in here again)

I understand you perfectly. I hated calculus and differential equations was a nightmare.

My major (CS) required, in addition to the usual calc, diff-eq and other maths, a Discrete Mathematics class. Within the first week, I was seriously hooked. Followed up with a couple abstract algebra classes and mathematical logic. After those, I enjoyed the hell out of math again. Because I understood the model - those classes are all about the ‘why’, not so much about the ‘how.’

Taking them made the all rest (relatively) easy; linear programming, coding theory, complexity theory, differential geometry, you name it, it was fun. AND, not nearly as challenging as basic calculus had been for me. So much fun in fact, I spent a litle extra time to get a second BS in math! (I will admit to struggling a bit with the Stat series though, probably because it bored me)

An intro course into discrete math could be just the ticket to resurrect math for you. Sure worked for me. I recommend looking into it.

How to Develop a Mindset for Math…

Math has always been my worst subject in school. No matter what I try, it has always been trivial and hard for me to get into for some reason.

I stumbled upon a post over at BetterExplained.com

I’m looking forward to reading more.

I am a former physics student with a form of dyslexia who “hit the wall” in university math… the point at which no amount of effort, practice or explanation could reproduce the concepts that I enjoyed so much on paper.

Like I said, I’m looking forward to more… I miss enjoying math as a concept as opposed to the terror and fustration it brought to my university years.

Hi wookie, thanks for the comment. Yes, unfortunately math lives up to the stereotype of being painful, not because it actually is, but often because of our approach.

Teachers often forget that many math “discoveries” were stumbled upon when tinkering. Newton used calculus without a formal definition of derivatives and did just fine. The constant e was discovered when tinkering with interest rates, not someone declaring “we must know the value of lim n->inf (1 + 1/n)^n”! But people forget this, or don’t know.

I felt a similar frustration and will do my best to take a different approach to learning :slight_smile:

[…] I sometimes wish there was someone who wrote about stuff like this when I was struggling through my math classes. Heck, I’d have taken it when I was struggling through my stats classes for grad school. […]

[…] Our number system developed over time. We started counting on our fingers, moved to unary (lines in the sand), Roman Numerals (shortcuts for large numbers) and Arabic Numerals (the decimal system) with the invention of zero. […]

I think it was a wonderful article and very thought provoking too. I guess the writer is right when he says we need to be a but tougher mentally because we stop thinking very soon and give up. And definitely I agree that math should not be treated as a ‘plug and clug’ kind of a thing but rather learning the way it came about to be. I think this article should be read by more people here in my home country in Pakistan.

Thanks Mohammad! I’m glad you enjoyed the article, it was a very precarious situation for me as I was almost discouraged after enjoying math for so long.

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