Adventures in Math Land

Posted on January 23, 2015

In which Heath reviews Algebra, 3 months before his 30th birthday.

As you may or may not know, I am currently enrolled in a master’s program for Computational Linguistics at the University of Zurich. It’s a lot of fun. I’m already a semester in, and Mrs. has already pointed out that I’ve learned an incredible amount of nerd stuff.

And as I look to the future, I’m starting to think about what direction my career is going to take. Long story short, with a background in processing words, I’d be pretty well set up to do some sort of data mining (getting structured information from massive amounts of raw data).

I’m also pretty interested in Machine Learning, because I feel like I could teach a computer to do some pretty cool stuff.

“Today I would like an omlette Nicoise, Computer.”

“I’ve told you, Heath, my name is Reginald.”

~shakes fist~ “Asimov curse your self-awareness!”

But to do that sort of thing, you really need some math, like linear algebra stuff. Well to do that, you need calculus. And right now I have about a month off from school. So I started poking around the online courses, and started a Single-Variable Calculus course. Well within 30 seconds I hit a nested fraction.

And so around Tuesday of this week, I found myself signing up for the Khan Academy series on Algebra. Not even Algebra 1, but “Foundations of Algebra”. At this point in my life, it has been at least 12 years since I had done a prime factorization.

But I cruised through the basics. And I found myself being transported to Mrs. Darby’s class, where she explained you can’t compare apples and oranges. I remembered my mother trying to explain to me that she also had problems with negative numbers, and you had to think of it like a boat with two engines, each facing each other. A part of my brain that hadn’t been accessed in a very long time came to life.

I cruised through the basics. Many of the things took a little getting used to, and some things I had used in college (also freelancing and kubb), like functions and ratios.

I can safely say that I have a good grasp of the fundamentals. But why is it that I kept messing up problems? I would approach a problem, work it out, plug in what I was absolutely confident was the correct answer and then swear as it said that I was wrong.

“Oh look, Heath, you forgot to transcribe the negative sign, you idiot.”

In short, a stupid mental mistake. The same sort of thing that plagued me in middle through high school. I can’t tell you how many problems I worked through (then and now) where I had done every step correctly, but I had written a number incorrectly or forgotten a negative sign or read the problem wrong.

Again, I can see various higher authorities standing over me saying the same thing: “Heath, you need to show your work and take your time.”

Is this a solution? Can simple mixups like that be fixed? In Khan Academy I can’t tell you how many times I would get to the fifth problem (in a lot of these things you need to get five in a row) and mess it up, despite doing everything correctly.

It’s obviously a problem that needs solving. If I want to continue learning math, I can’t be doing this sort of thing.

So I took my teachers’ advice, and started being as explicit as possible. My rule, for about a day, was that I was not allowed to make mental leaps. I had to write everything down.

And it didn’t help. If anything, it made things worse. And it was starting to get frustrating, which is bad. You don’t want to get frustrated doing math, because you’ll start making more mental errors.

But I kept at it. And yesterday evening, I found that I was making very few silly errors. What had happened?

Well, teachers of math-class past, I was skipping steps, among other things.

The first thing I started doing, was making sure I was solving the right problem. Double check that I’ve read the problem correctly, and that I’ve transcribed the problem correctly to my scrap paper. I’ve had trouble with this at grad school as well. When I’ve double checked my exams, I’ve had to quickly re-answer some problems, simply because I didn’t read the question all the way through.

The solution is to read a question twice, and to double check that I’ve written the problem correctly. Advice I’ve received probably a hundred times, and everyone that gave it too me was right.

However the advice to be as explicit as possible was bad, at least for me, for several reasons.

First, and again, this does not apply to everyone out there, the more steps there are, the more chances I have to mess things up. The more often that I have to simply move a negative number from one step to another, the more likely that negative sign is to disappear into the ether.

Second, shuffling information around without manipulating it is boring and takes a long time. Despite being the child of the ’80s, I have a pretty decent attention span. But seriously, the more I have to do repetitive tasks, the more bored I am. And, more importantly, my brain is on autopilot. I’m not actually thinking about anything. A dangerous combination. That’s where errors come from.

And finally, the more steps I add in, the farther away from the original problem that I am, both in my head and on the paper, which makes it harder for me to do a quick check.

I found that the more steps I could do in my head, the fewer silly mistakes I made.

So, sorry Mom, Dad, Mrs. Darby, Mrs. Dix and Mr. Chubb, but shortcuts, at least as far as basic algebra is concerned, are my saving grace.

My point isn’t so much to gloat. I’m going over Algebra (and eventually Calculus), and I’m having a pretty good time. My point is that sometimes you have to go against prevailing wisdom. I’ll leave with a short anecdote from WWII.

When bombers came back from their runs, they would be looked over by a team of analysts, who would jot down where the planes had taken the most bullets/damage, crunch the numbers and see where the average bomber took the most flak. Then they’d armor up the remaining bombers in those areas.

On the surface this seems like pretty good advice. But it’s actually the opposite of what you want to do. If a bomber can return a hole, it means that it can remain flying with a hole in that place. It’s the places that don’t have holes that need the most armor.

I’m trying to find a source on that anecdote, and I’m coming up short. But I feel like the logic is sound. I want to say that I got it from ‘The Fog of War’, but I don’t think that’s the case. You should watch that documentary by the way.

I’ve learned a lot, going back to school at this age, and I’m not just speaking about knowledge. I’ve learned so much about my brain, and the best way to learn. And here are two (of many) lessons I’ve learned:

1) Make sure you understand what the question is asking.

2) Know the right time to take a shortcut.

Photo credit to stuartpilbrow at Flickr [CC BY-SA 2.0], via Wikimedia Commons


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One Response to “Adventures in Math Land”

  1. Yer Old Man on January 29th, 2015 11:10 pm


    Well written and well said, and I am (surprisingly) not taking any offense whatsoever. I remain a staunch advocate of step-showing, but if it works for you to skip here and there, then skip like the wind.

    The only caution that I offer is that you should take the time to do similar experiments as the level of difficulty increases and your intuition and mental math don’t quite carry you as far.

    Also, I have no earthly idea how programming works these days, but back in the dark ages of algorithmic languages, it was impossible — fatal, really — to skip even the teensiest step. The mental discipline it gave me in high school helped me with the couple of programming courses I took.

    The bottom line is, if you understand the problem and get the answer right, I don’t care if you use dice, prayer, or frog entrails to get it. Do what works for you.

    Thanks for another good article.


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