# Understanding The Beauty Of Math

May. 15, 2012
Sangsan is relatively enthusiastic about math, education, cool designs, and good films!

First of all, I understand that not everybody likes math.

I am currently tutoring kids at a primary school nearby my college campus. Today, in the 4th grade class I work with, I was assigned students who were doing a practice exam, preparing for the 60-minute state-wide assessment tomorrow. My job was to help this group of students get through the whole set of problems in two hours. The practice exam featured problems about time and date, approximation, and spending. I thought to myself that I would have loved the class if I was given these kind of problems in 4th grade.

However, despite their real-life simulation, the problems did not interest the kids. Everything else seemed fine; my students did the exam anyway since the teacher was sitting at the next table. At first my excuse was that it is normal that math is not their favorite subject.

As my observation continued, however, I found that they were all using exactly the same approaches in solving these problems. It should not be surprising since everyone including me did that in 4th grade too. We were told how to solve the problems and we just kept using the method. Do my students know what mathematical concepts are behind it though? I asked them. It was surprising that “No” was the answer. For example, given the date of the first Monday, students were asked to find the date of the third Wednesday in that month. They knew that they had to add multiples of seven but they did not know why. This is indeed sad.

I was trying to think of what went wrong but my students were finishing all the problems in such short a time. Why do we prefer speed so much in a test? I have come to accept that a short exam time does indeed turn their “getting interested” button off. Although generalizing my observation is risky, this is the problem that we are facing in our education system.

To get the work done quickly, my 4th graders simply put everything into formulas they had already memorized. Such methods give them right answers and save some time; therefore doing so does not seem wrong, at least in their thinking.

I believe that grounded understanding is fundamental to early math learning. So it may be way too early for the state to assume mastery in both speed and preciseness in fourth grade. My observation suggests to me that an inadequate proportion of short exam time to a number of problems in the exam prompts teachers to teach their students only exam-taking strategies and neglect mathematical constructs.

Students, knowing that they have to finish all problems in time, cannot help ignoring the importance of formulas’ origins. Thus they respond to this exam format by merely memorizing all strategies needed. It entails that they skip mathematical foundations directly ahead to the formulas that are fruitful for them in an exam. Such a jump intensifies the problem that students lack basic backgrounds when they learn more complicated materials because the only time they learn the essentials of mathematics is when they are in primary schools and the problem propagates as a class moves on.

The elaborate demonstrations are becoming less important and undervalued. When the question asked them to show all work, my students just wrote down the answers and skipped the rest of it. If to increase speed means to have something like formulas in mind, something in our current math curriculum is not right.

However inevitable in learning, memorization should not serve as a main component. Knowing formulas alone does not demonstrate students’ comprehension of the subject as when everyone uses the same method to solve the same problem without knowing its mathematical concepts. The recipe they have used is not derived from their own experiences. Memorization therefore must only be a by-product of learning processes, i.e. everything will come to one’s mind immediately as soon as one thoroughly understands the concepts.

Instead of exciting students, learning math is now submissive with teachers only giving and students just taking — since all we want to achieve is high quantitative academic performance. The beauty of math that they could find after numerous attempts at a problem is lost and will never be found. Students do not get to know how enjoyable (or painful) solving a math problem can be if they are programmed to receive inputs to produce outputs through given methods only. Weakening also is mathematical creativity. While handing formulas to kids in order to ace the tests in time, their boundless creativity is bounded. Why do they need to think if tools are given so easily?

Why must we do math in such a rush and neglect its beauty along the way? Solving a math problem requires logical thoughts that in early math should be traced step by step in an explanation of the problem. Revision is crucial. Once students practice such repeatedly, not only does revision give opportunities for the student to verify their process with their mathematical knowledge but it also allows teachers to know their students’ progress.

Short exam times, in contrast, give rise to multiple choice questions which are easier to be graded but incentivize students to memorize. As I already explained, we know how memorization undermines the perception of the mathematical beauty.

One may argue that 4th graders may not be proficient enough to critically understand or to question mathematical concepts in depth. What I say here, however, is not to have everyone grow up to become innovative mathematicians; rather, as well as in other subjects during early learning period, I want fourth graders to have a strong math foundation before they even make a judgement about whether they like math. We, as educators, need to properly nurture their attitude before it is too late because speed can be increased as long as students are grounded.

Or maybe, I just want everyone to enjoy math.

image – joanna wnuk

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• guest

this is awesome. i just finished a minor in math and everybody is always shocked to hear that. judge me all you want , peeps, but while you’re off hating life writing papers that barely give you any sort of emotional stimulus, i will be enjoying that amazing feeling you get when you finish a really hard problem. or i’ll be crying because math is the only subject i care enough about to cry over haaa. i’ll admit though, foundations of analysis was a bitch. but i simultaneously loved it and i now know how to prove the chain rule. so whatevs.

• guest

there’s so much joy in math that most people never see. finally solving a problem that you’ve been thinking about for hours is one of the most exhilarating experiences you can have. (of course, even after you get past high school math, these affirming moments are hidden among hours of late night complex analysis or algebraic topology problem sets, with endless rote computations and epsilons and estimates and identities…)

So true.
Now that I want to ‘understand’ math at 18 years old, I can do the work assigned if it’s on a step-by-step basis but I have trouble if it requires me to use previous concepts to infer solutions when formulas are not given to me.
I do enjoy math though. But even in high school, there is the problem that there is not enough time given for the reasoning.
I plan on teaching myself more of the important principles this summer, before college.
Might not be as good as learning them while young, but it never hurts to gain knowledge.

I really wish this talked about like Fibonacci, or the Golden Ratio, or patterns that are found in both math and nature and how those could be used to like, I don’t know, show people math isn’t all numbers they don’t understand or something.

• Kelly

I used to think like those fourth graders until I saw Aronofsky’s Pi a few years ago. There is a kind of beauty to math that doesn’t really get touched upon in the public education system for many reasons, mainly because of time and resources. I think apathy could possibly be reduced if we changed the way we teach math currently, but that’s not likely to happen any time soon. Plus, to most people, doing math for hours to figure out methods rather than being shown the formula is not only time-consuming, but terribly, terribly boring. You may not get the personal revelation and insight if you don’t spend hours of time on a single problem, but you also avoid the dullness of sitting there trying to reach that revelation. Which is why I never found math so intriguing until all of the interesting, crazy elements were dramatized in an easy-to-digest format which skipped all of the…having to do the actual math part. Guess I’ll just have to stick with the plebeians.
(Regardless, everyone should see Pi. That movie changed my views on math forever.)

I found beauty in Calculus. Too bad I’ve lost my touch since it was four years ago when I last had it as a subject. Though a literature major, I don’t dislike/loath/whatever the field of Mathematics. I actually think that it’s an untapped resource of knowledge that only a few people know how to harness. Why? It is because the population of learners are never given enough time to explore the undertow of its meaning. LOL at knowledge being lubricated for the mechanism of society.

• Bleachedpeace

As a high school math teacher this is totally what I try to get across to all of my students.  With my Algebra 1 and Algebra 2 students especially, if I can help them to see the joy in figuring something out on your own and understanding why a pattern comes together in the way that is does, these are the students that continually tell me how math is actually pretty fun.  It’s a lot of fun to look at a problem and try to figure out how to solve it on your own using only the combination of mathematical tools that you have discovered.  I want them to look for, see, and understand the beauty that exists in mathematics.

Science is faaar more beautiful.  Although where would science be without math?  Hmm.  I need to evaluate this further.

• Anonymous

TEAM SCIENCE. But also, math is fine too!

• http://raymondthimmes.com/ Raymond Thimmes

This is why I ended up at art school. And why I can’t even multiply in my head let alone figure out a problem without google.

• Privileged Misogynist

I tutor the remedial maths (pre-Algebra, Algebra I&II) at the school I’m attending and I see this all the time.  People are always asking me for the ‘steps’ to solve problems and I’m always trying to dissuade them from thinking that way.

As a tutor, it is never my place (as in, contractually) to have an opinion on how a professor taught them to do something.  What I do instead is just kind of talk out how I would go through the problem solving process:  what are we given, what are they asking for, what are some properties of numbers that we can use to get there from here.  I’ve found that I produce a lot more “a-ha!” moments by focusing problem solving rather than rote memorization.

I can also personally attest to how gratifying it is to have an insight on a problem you are working on after being up against a brick wall for a while.  I remember one particular trig problem that I stared at for ~45 minutes before my a-ha! moment happened.  Even if you don’t get it, if you have put the time in thinking about it then you know exactly what it is you don’t get and when someone finally explains it in a way you understand the feeling is almost as good.  I just had this happen to me a few weeks ago.  I, for whatever reason, just could not follow the derivation of something (some mumbo-jumbo about the multiplicative inverse of a complex-imaginary number) and when an old professor explained it to me a few days later it felt almost as gratifying as if I had derived it myself.

I think math helps to you be a better whatever else it is that you want to be.  Really, it’s all about creativity and abstract deduction.  Thinking.

What’s not to love?

• Jessica

math sux

Maybe I’m just completely f’ed up but I didn’t find beauty in math until I took statistics. I wish I could go back 100 years in time (you know, the first time I was in college) and major in stats. It’s so fluid and…artistic. I swear I saw it in color, where all other math is just black and white.

Funny, same for me. I hated Math all the way through high school, and when I went to college and heard Advanced Statistics were obligatory for my degree, I thought I was gonna die. It took me months of hard practice, but it the end it just clicked, and I started to enjoy it.

I wish I’d put more effort into Math in high school – practiced more, paid attention during class and remedial class – because I think once you understanding why you’re doing what you’re doing, it actually becomes fun. I

(my sixteen-year old self is rolling in her proverbial grave right now)

• mathemagician

Math is definitely one of the most beautiful things one can learn. Really advanced math can get ugly and difficult, but once you push through that obstacle, it really is beautiful and amazing. It’s just so truthful and precise. What saddens me the most is that introductory math courses are taught in a way such that the beauty is completely obscured. My introductory math courses didn’t convince me to major in math, even if they were easy and fun sometimes. It’s when I had to sit down and prove everything that I really enjoyed it.

There would have to be a lot of reform in math pedagogy to remedy this tragedy. I feel like after basic algebra and geometry, we can strive to introduce students to the basics of set theory, and then go from there — possibly delve into linear algebra and a more rigorous treatment of calculus (with epsilons!). If we are rigorous from the start, then maybe, just maybe we’ll learn to appreciate math on a deeper level. I’m also tired of hearing people say, “when am I going to need this?” What people don’t realize is that it’s not so much about being able to solve for x or y, but really, learning how to think critically and how to solve problems in a variety of ways.

Thank you so much for this. It is so reassuring to read this. If there are more people like you out there then there is still hope for our education systems (both India and US).

• Guy

Not sure how this wound up on TC, but I loved it. Have you read “A Mathematician’s Lament?”

• Guest

I completely agree… I also find with my students that they are not confident enough in their abilities…

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