Monday, May 16, 2011

What is = ?

I still remember the horror I felt at my first exposure to simultaneous linear equations. You know, this sort of thing:

2x + y = 7
3x - y = 8

I couldn't believe my ears when the teacher suggested we should just add the two equations together. How could this be? I thought it would violate the laws of God and man alike. It seemed completely arbitrary and inexcusable. It was a real "turn the giraffe upside down" moment for me.

Over time, I came to realize that it's OK to add the equations together because each equation represents two equal values. If 2x + y = 7, you can add the same value to both sides of the equation and the equals sign will still be true. In this case, the value we will add to both sides of the equation is contained by the second equation, 3x - y = 8. The equals sign means that 3x - y is the same value as 8, so we can 3x - y to one side of the first equation and 8 to the other side of the first equation, and the equals sign will still be true.

The power of the equals sign is a tricky concept. I recently showed my older daughter how to simplify one side of an equation, and she asked, "does this mean I have to do something to the other side?" I said no, because I hadn't changed the value of the expression that I simplified, so the equals sign still held true. I can tell she will need more work on this point.

This is where curricula like Trailblazers make a fundamental mistake. For some obscure reason, they explicitly discourage writing mathematical notation, in favor of either "mental math" or writing paragraphs explaining how you reached the solution. But it is essential that students practice writing the symbolic language of mathematics, which was developed over centuries with the exact purpose of expressing mathematical ideas in the most succinct, clear, and concise way possible.


  1. Interesting, FedUpMom. I've always liked this kind of math problem, and I've found that I do use this concept once in a while in everyday life. But until this moment I never thought about why you can get away with just adding two equations together to get an answer. Cool!

  2. We have a real problem with math education. Conrad Wolfram is way better at explaining it than I am... but, ultimately - this rests on the fact that math and computation (arithmetic) are not the same thing. Kids that rock at math may or may not excel at arithmetic. And, that frankly, we don't really do arithmetic in modern life (that's why we have computers). But, we do math like crazy.

    As Conrad suggests - we need to turn math education completely on its head and worry about teaching math - not computation.

  3. I don't agree. What I see is that kids can't even compute, and if they can't compute they can't hope to do the higher-order stuff.

    We don't need another big idea. We need to buckle down and just teach math. Singapore Math is an excellent curriculum. We don't need to re-invent the wheel.

  4. I am using Singapore with my oldest (homeschooling) and that is better than the public school fare.

    But, as a professor of biology that teaches experimental design and statistics, I would still argue that computation without covering mathematics principles makes a boring curriculum with no modern context. Watch the podcast that I refer to in my previous comment for a better explanation of the problem, and a solution.

  5. OK, I agree with your basic point, I think. Yes, it is possible to approach math in a mindless follow-the-steps, memorize-the-procedure, get-the-answer way. I would call this the "bad traditional method." Yes, it's much better to work towards a deeper understanding.

    My frustration comes from seeing the new-fangled curricula like Trailblazers at work. These curricula were designed in reaction to the bad traditional method, but in this case the cure has proven to be worse than the disease.

    I'd be very happy to see Singapore Math in our local schools. It would be a huge improvement over the nonsense they're using now.

  6. I would add that I am immensely skeptical about anyone with a revolutionary idea about teaching math. The chances that this could be successfully implemented by the schools are slim to none. On the other hand, a basically familiar approach like Singapore Math seems like a real-world possibility.

  7. When I was learning how to do simultaneous equations, the first thing the teacher did was prove to us that you could add equations together by adding simple equations such as 1 + 1 = 2 and 2 + 2 = 4 to get 3 + 3 = 6. I'm pretty sure everyone understood, because when the teacher asked, "Now, the sum of those two equations makes an equation that still works, right?" everyone just nodded and muttered, "Why wouldn't it work?"

    If every teacher took that 30 seconds to explain, there probably won't be as many students who don't understand math.

  8. Hienuri -- That's a smart way of teaching it. It's so long ago now, I don't know -- maybe my teachers did do that. I might have been looking out the window . . .

  9. The point that you are making is a different one. You are precisely correct that the typical K-12 teacher has such a limited understanding of the basic concepts that they fully embrace plug and chug - because they can do that. Before anyone get up in arsm with me, please know I agree that there are some really amazing instructors out there. But, I would argue that they are not the norm.

    I am greatly dismayed at some of the education-school math and science courses that I have examined. Many of these courses ask their college students (teachers in training) only to try the exercises that they'll teach - but, not to demonstrate understanding of the basic underpinnings.

    How could they shift to this alternative curriculum if they don't understand it? When I volunteer in the elementary school - it is not at all uncommon for me to hear them denigrate math, giggle at their lack of understanding, or thank me for covering science material that they don't fully get (in elementary school!).

    The other issue of getting and keeping bright, energetic and qualified teachers into our schools and keeping them - way too big to tackle in one blog.

    My poor second grader has seen nothing but worksheets, day in, day out all year.

    "How was your day today?"

    "Well, we did four worksheets for morning work, two for math, and one for science today".

    "Did you do anything else?"

    "Um, gym?"

  10. It would be great if every K-12 math teacher understood math concepts at a very high level *and* could communicate them to kids. But the higher you raise that expectation, the more likely it is that there won't be enough people meeting that description who are willing to choose K-12 teaching as a career. What then? I'm not sure how much of a difference ed schools can make in greatly expanding that pool of people.

    I don't know what the answer is. I do think that teaching experience helps, though, so it would be a good idea to make teaching a profession that people would want to stay in for a long time.

  11. Maybe the Ed.schools could raise their admissions requirements a bit and still attract candidates. In my state the requirement is apparently that the candidate be able to fog glass. Supposedly my state has the highest standards in the country but I can't believe that that's true. My kid is getting worksheets day in and day out, too. At least I've put a stop to it at home.

  12. Considering that 100% of the teachers I've encountered have attended education school and about 50% of them can't teach, I'm not too concerned about how raising expectations affects teaching schools because that system is utterly broken.