A Few Ideas on Education

jonah

“Wow! 2500 hits! I really didn’t expect such an enthusiastic response to my first blog post! Thanks for all your enthusiasm!

Andy Colyer asked me the following question in the comments.

I’d love to get your thoughts about how to get children interested in math and physics. What the progress of the curriculum should be. Any text or material recommendations, etc. Did you do mathcounts?

I decided that maybe I could share my answer with all of you. (Don’t worry, an article answering Kosh’s question on the requests page will be up soon. This is just bonus content.) Before I go on, though, I should say that I have absolutely no training as an educator. I’ve tutored at the college level, and I’ve taught my 9-year-old cousin, but that’s about it, so take everything I say with a grain of salt.

I would really like to see a stronger emphasis on discrete math. I think that topics like boolean algebra and basic set theory are well within the grasp of a motivated middle school student. These topics are very important in higher-level math and require a very different way of thinking than algebra or precalculus.

I also think we underestimate the power of programming as a teaching tool. Even the simplest program relies on ideas that are important in higher mathematics. You need to know how to assign a value to a variable, how to perform basic algebra, how to perform boolean algebra, and how to think abstractly. If you do any kind of graphics, you need to know enough geometry to make the picture look right. Programming is also really fun to learn because your efforts are rewarded in a very immediate and tangible way.

I’d really like to see a joint programming/math class, where students learn some math and then apply it to make a cool program. Repeat ad infinitum until the school year is over.

Even if we ignore the benefits of programming as a math teaching tool, I think we’re entering an age when everyone should know a little programming. Computers are becoming as ubiquitous as pen-and-paper notebooks, but relatively few people know how a computer works on even the most basic level. Understanding the inner workings of computers and computer programs is very empowering.

Teaching math and physics should focus on teaching students to think critically. Math and science are all about thinking critically, after all. Both involve facts, of course, but the facts are less important than the deep understanding. Unfortunately, it’s very tempting to just teach students to memorize a problem-solving algorithm, rather than how to think about a problem, especially in typical subjects like algebra and precalculus. A classic example is the binomial equation, the solution to the equations of the form

    \[ax^2 + bx + c = 0.\]

I can still recite it from memory:

    \[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}.\]

I didn’t know why it worked, I just knew it worked. However, we can easily solve for the binomial equation using high school algebra:

First divide everything by a.

    \[0 = x^2 + \frac{b}{a}x + \frac{c}{a}\]

Add b^2/4a^2 to both sides.

    \[\left(\frac{b}{2a}\right)^2 = x^2 + \frac{b}{a}x +\left(\frac{b}{2a}\right)^2 + \frac{c}{a}\]

Recognize that \left(x+\frac{b}{2a}\right) = x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}.

    \[\left(\frac{b}{2a}\right)^2 =\left(x+\frac{b}{2a}\right)^2+\frac{c}{a}\]

Move c/a to the other side of the equation and take the square root of both sides.

    \[\pm\sqrt{\frac{b^2}{4a^2}-\frac{c}{a}} = x + \frac{b}{2a}\]

Solve for x.

    \[x = -\frac{b}{2a}\pm \sqrt{\frac{b^2}{4a^2}-\frac{c}{a}}\]

Rearrange the terms a little.

    \[x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

Voila! I never saw this derivation in high school, and I think everyone should.

Part of the reason I feel this way is because I was actually very bad at math in primary school. I never learned my multiplication tables because they required too much memorization. In high school, I thought math was boring, and I didn’t pay much attention in physics class. It turns out that what bored me was the approach of memorization. As a result, I didn’t do mathcounts (although I did do academic decathlon) and I didn’t really get excited about math or physics until college.

Some materials that I think would help teach kids programming:

  • Invent with Python is a free ebook that teaches kids to program by teaching them to write video games in Python. Python is a great first programming language because it’s very powerful, but it has very intuitive syntax. Python is also an interpreted language, so kids can experiment with programming concepts in real time.
  • The Raspberry Pi is a $35 computer. It comes with Linux installed, and it’s a great platform for kids to play around on.
  • Five Tools to Teach Kids Programming.

Tools for Teaching Discrete Math to Kids

Some Other Resources:

  • The QAMA calculator is a new calculator that asks users for a reasonable guess before it will return an answer to a problem. This might be a great tool to teach math at the elementary school level.
  • There’s a growing number of excellent educational youtube videos that I think are really engaging. A short list of them I watch is Minute Physics, The Idea Channel, Veritasium, Numberphile. There’s more, though. Many more.

EDIT: Andy Colyer suggested the following resources in the comments, and they look very good.

  • The Life of Fred series is a huge series of educational books, all of which look very good.
  •  Math from Three to Seven is the story of a professional mathematician’s experience teaching math to preschoolers in Moscow. It looks very interesting.

As a final thought, I follow Charlotte Bouckaert on Google+, and she often shares interesting articles about education. I think she’s probably much more qualified to comment on this topic than I.

Questions? Thoughts? Hatemail?

Now I’ve shared my thoughts, and I’d love to hear what you all think. If you have objections, suggestions, thoughts, or teaching materials, please share them in the comments.

6 thoughts on “A Few Ideas on Education”

  1. Someone on my facebook feed asked me for my thoughts on the ideas promoted by this video:

    http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

    For the most part, I think this school of thought is spot on.

    Conrad Wolfram is simply wrong when he says that by-hand algebra and calculus aren’t used in the real world. Real-world physicists and mathemeticians do by-hand calculations all the time. We do this because it helps guide our physical or mathematical intuition, because a computer can’t solve the problem yet, because it’s difficult to understand the meaning of the output of the computer without doing the calculation yourself, or because it eventually becomes faster to do the calculation in your head than to plug the problem into a computer.

    However, a lot of students won’t neeed this ability. Math and Physics are on the forefront of building new mathematical tools, as opposed to applying well established tools in novel ways (although they do that too). Perhaps students destined to be mathematicians or physicists can learn to do by-hand calculations, and everyone else can move on to computers.

    At the college level, I’ve benefited hugely from playing with and interacting with an equation or a physical system on a computer, and I’d love to see this transferred to primary schools. Wolfram also argues that we use programming as a way to teach students the algorithms that we typically force them to practice by hand. I couldn’t agree more. I remember taking introduction to chemistry in college. I was really frustrated that every problem on the homework was the same question wearing a different hat. By this point, I had a little bit of programming knowledge, so I just wrote a program that could solve every single problem on the sheet. If everyone had been taught to do this, we all would have wasted a lot less time.

    Perhaps a good way to go might be to teach the calculation method, go through some examples, and then for homework as the students to write a general solver. In my quadratic formula example, students would learn the method of completing the square used in the derivation, and then have to generalize to write a program that could solve the problem for them.

    Thoughts?

  2. Jonah – Thanks for your thoughts. You’ve sent me in a number of directions that are new and welcome. Thank you! I love the idea of introducing programming, and have downloaded Scratch to try out on my son. Will pursue the other suggestions as well.

    I’ll pass on that I’ve gotten some great ideas for working with my 5 year old from this book:

    http://www.amazon.com/Math-Three-Seven-Mathematical-Preschoolers/dp/082186873X/

    And my five year old has also enjoyed the Life of Fred series, which, at the elementary level (it goes up thru calc), introduces arithmetic in a gentle way–in the context of a five year old prodigy who teaches math at kittens univ.

    1. Andy–Thanks for your suggestions. I’ve added them to the main post. They look very good!

      I think I may have done you a disservice because I promoted teaching discrete math without offering any resources other than programming. Here are some more:
      http://www.infoagepub.com/products/Discrete-Mathematics-For-Teachers
      http://www.ehow.com/video_4974325_teaching-children-discrete-mathematics.html
      http://scholar.lib.vt.edu/theses/available/etd-04252007-140123/unrestricted/DM-Dissertation-Olgamary-May2007.pdf

      1. Jonah – Thanks. You may want to change the link to the Life of Fred books to the following:

        http://stanleyschmidt.com/FredGauss/index2.html

        It’s much less expensive to get the books from the author’s publisher than from the Amazon marketplace. Additionally, he has a first chapter for preview of each of his books.

        I’m not in any way affiliated with the author or the books, but have really enjoyed and benefited from them.

  3. Pingback: Some Thoughts on Education, Part 2The Physics Mill

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