# mixedmath

Explorations in math and programming
David Lowry-Duda

First, a recent gem from MathStackExchange:

Task: Calculate $\displaystyle \sum_{i = 1}^{69} \sqrt{ \left( 1 + \frac{1}{i^2} + \frac{1}{(i+1)^2} \right) }$ as quickly as you can with pencil and paper only.

Yes, this is just another cute problem that turns out to have a very pleasant solution. Here's how this one goes. (If you're interested - try it out. There's really only a few ways to proceed at first - so give it a whirl and any idea that has any promise will probably be the only idea with promise).

Looking at $1 + \frac{1}{i^2} + \frac{1}{(1+i)^2}$, find a common denominator and add to get $\dfrac{i^4 + 2i^3 + 3i^2 + 2i + 1}{i^2(i+1)^2} = \dfrac{(i^2 + i + 1)^2}{i^2(i+1)^2}$. Aha - it's a perfect square, so we can take its square root, and now the calculation is very routine, almost.

The next clever idea is to say that $\dfrac{ (i^2 + i + 1)}{i(i+1)} = \dfrac{(i^2 + 2i + 1)}{i(i+1)} - \dfrac{i}{i(i+1) }$, which we can rewrite as $\dfrac{(i+1)^2}{i(i+1)} - \dfrac{1}{i+1} = 1 + \dfrac{1}{i} - \dfrac{1}{i+1}$. So it telescopes and behaves very, very nicely. In particular, we get $69 + 1 - \frac{1}{70}$.

With that little intro out of the way, I get into my main topics of the day. I've been reading a lot of different papers recently. The collection of journals that I have access to at Brown is a little different than the collection I used to get at Tech. And I mean this in two senses: firstly, there are literally different journals and databases to read from (the print collections are surprisingly comparable - I didn't realize how good of a math resource Tech's library really was). But in a second sense, the amount of math that I comprehend is greater, and the amount of time I'm willing to spend on a paper to develop the background is greater as well.

That aside, I revisited a topic that I used to think about all the time at the start of my undergraduate studies: math education. It turns out that there are journals dedicated solely to math education, see here for example. And almost all the journals are either on JSTOR or have open-access straight from Springerlink, which is great. I have no intention of becoming a high school teacher or anything, but I became interested as soon as I began to come across people with radically different high school experiences than I did.

My high school tried to protect its students, sometimes in ways that I didn't like. It was the sort of place that, in short, held me back in the following sense: they wouldn't let anyone take 'too hard' of a course-load for fear that they would overwork themselves and therefore fail, or do poorly, or overstress, in everything. In more direct terms, this meant that you had to petition to take 3 AP classes and had to really work to take 4. Absolutely no one was allowed to take more than 4 in one school year - so that many of my friends had to choose what science to take. Those of us who were willing all had sort of the same schedule in mind - if you did an art (band/choir/orchestra, usually), then in 10th grade you took AP Statistics, 11th AP Language, 12th AP Lit, AP Calc, AP (foreign language or Gov or European History or Econ), and an AP science - if no art, then you could take an additional AP science in 11th grade. At least, that's how it worked while I was around.

So the big decisions were always around the senior year. For me, I had to ask: should I take AP Chem or AP Physics? (I ended up taking Physics, which was great - it was the curiosity and intuition from mechanics that led to me becoming a mathematician now). Many of my friends asked the same sort of questions. And it was very annoying - I hate the idea that the school holds us back, ever. It also turned out that one of my classes was terrible. I was so annoyed that one of my four choices ended up being bad that I wrote an embarrassing letter (which I regret to this day).

In short, I felt slighted by the system, and I've considered the system ever since. One of the articles I read was about the general idea that the sciences taught in schools and even at entry-undergraduate level in college are fundamentally different in both motivation and skill set from the ideas held by scientists and those who progress those subjects. The interesting part about the article was the amount of feedback that the journal received - enough to merit multiple copies of letters back and forth to make it to the next printings of the journal.

That particular article was very careful to simply assert that the current paths of education in the sciences and the sciences themselves are different, as opposed to positing that any particular idea or method is above or better than any other. But of course, it's perhaps the most natural response. Should they be different? Why does one learn math or the sciences in school? For that matter, why does one learn history (also oblique and hard to answer, but something that I maintain is important for at least the reason that it was the only substitute I ever had for an ethics class in my primary and secondary education).

These are hard questions, and ones I'm not willing to directly address here at this time. But I will quickly note that in both Tech and Brown, I am stunned at how many people lack any sort of intuition for the four basic operations - (I once tutored someone who, upon being asked what 748 times 342 was, responded that it didn't even matter because "math was made up at that point. It's not like someone has sat down and counted that high." oof. That hurts. Let's not even talk about being able to add or subtract fractions. As a worker at the 'Math Resource Center,' I've learned that about a quarter of the time, helping people with their calculus classes is really a matter of helping these people manipulate fractions. So if the purpose of primary and secondary education is to get people to understand arithmetic operations and fractions, it's not doing so well. John Allen Paulos should write yet another book, perhaps (Innumeracy is a good read).

Should they be different? That is, is there much reason for the sciences and the education of the sciences to align in method and motivation? I'm not certain, but perhaps they shouldn't pretend to be the same. I only ever learned arithmetic, as opposed to math, throughout my primary and secondary education with 2 exceptions: geometry (which had a surprisingly large logic content for me, and introduced me to interesting ideas) and calculus. Calling it math is a disservice - as Paulos mentions in his books, the general negativity towards math allows people to claim innumeracy ("I'm not really a numbers person") with pride - no one would ever say that they weren't very good with letters. But reading is useful, or rather widely recognized to be useful and expressive.

I end by mentioning that I think it is more important to come across real ideas of science and math at an early age, say elementary school, then middle school. In in elementary and middle school, there really isn't much difference between the maths and the sciences, so I clump them together. But in my mind, the initial goals of science and math education should be to spark creativity and wonder, while English and reading courses stress critical thinking (somehow, math, science, and English all get the boring end of the stick while reading gets full hold over the realm of creativity - how backwards I must be).

But those 4th graders whose teacher guided them towards the bee research, that has now been published under the 4th graders' names - don't you think that their view of science will be a much happier and, ultimately, accurate? Exciting, collaborative, uncertain with a scientific method-based structure. But then again, perhaps the lesson that my friends and I learned from our own high school is the most relevant: if you want to do something, then don't let others stand in your way. A little motivation and discipline goes a long way.

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1. 2011-10-26 gowers

I enjoyed that problem. A small remark about the solution is that you can do part of it more neatly (and transparently I think). When you've got

$\frac{i^2+i+1)}{i(i+1)}$

you can just instantly spot that the numerator is 1 more than the denominator, so it equals $1+1/i(i+1)$. At that point we're on familiar territory.

2. 2011-10-28 Jimmy

I really like what you are saying, and am glad I found your website. Please keep it up.

3. 2011-10-31 davidlowryduda

Thanks Jimmy!

4. 2011-10-31 davidlowryduda

Re: the first comment from Gowers

I agree - that's much cleaner. Thank you for that.

5. 2011-11-20 Elektrische Zahnbuerste

Lovely sharp post. Never considered that it was that effortless. Praises to you!

6. 2011-11-20 Elektrische Zahnbuerste

7. 2011-12-17 TomF

David, I think that you may find this interesting. It's about math education. http://www.maa.org/devlin/LockhartsLament.pdf

8. 2011-12-19 Gigili

What a deceptively simple question!

9. 2011-12-19 davidlowryduda

Hey Tom! I know the lament, and it's a great read. In fact, there's a pretty good book written largely as an extension of the lament (http://www.amazon.com/dp/1934137170).

How have you been? I haven't heard from you in a while.

10. 2011-12-21 TomF

I've been good! I had to transfer for Tech to a school closer to home because my mom was sick. But, the school i transferred to wasn't a good fit for me at all. So i dropped out, and went to fish in alaska, and travel. That has pretty much been my life for the past year and a half.