Never-Ending Amazement

I started my spring semester's classes in the last few days, including two introductory algebra classes. It's possibly the best and most powerful start to a semester I've ever had; I got an extraordinarily good vibe from all my classes. So that's a good thing.

So here's a quick observation I've said aloud several times. For me, teaching college math is a continual exploration of the things people don't know. If I listen really carefully, I continually discover that the most basic, fundamental ideas imaginable are things that a lot of people in a community college simply never encountered. Whenever I start teaching a class and think "oh, christ, that's so basic it'll bore everyone to tears, skip over that quickly," I discover at some later point that a good portion of people have never heard of it in their life.

That's actually a good thing. It keeps me interested with this ongoing detective work I do to see exactly how far the unknowns go. And it provides the opportunity to share in the never-ending amazement, through the eyes of a student who never saw something really fundamental.

Here's one from this week (which I got to run in each of two introductory algebra classes). We're going to want to simplify expressions, with variables, even if we don't know what the variables are. ("Simplifying Algebra", I write on the board.) To do that, we can use a few tricks based on the overall global structure of numbers and their operations. I'm about to give 3 separate properties of numbers, here's the first. ("Commutative Property", I write on the board.)

Let's think about addition, say we take two numbers, like 4 and 5. 4 plus 5 is what? ("9", everyone says.) Now, if I reverse the order, and do 5 plus 4, I get what? ("9", everyone says.) Same number. Now, do you think that will work with any two numbers? With complete confidence, almost everyone in a room full of 25 people all say at once, "No, absolutely not."

So of course, I'm sort of thunder-struck by this response. Okay, I say, I gave you an example where it does work out, if you say "no" you need to give me an example where it doesn't work out. One student raises his hand and says "if one is positive and one negative". Okay I say, let's check 1 plus -8 ("-7"). Let's check -8 plus 1 ("-7"). So it does work out. Now do you think it works for any two numbers? At this point I get a split-vote, about half "yes" and half "no".

Okay, what else do you think it won't work for? One student raises her hand and says (and I bless her deeply for this), "if it's the same number". Uh, okay, let's check 5 plus 5 ("10"). And if I flip that around I again have 5 plus 5 ("10"). So now do you think it works for any two numbers?

At rather great length I finally get everyone agreeing "yes" to the Commutative Property of Addition. And it's obviously a point that no one in the class had ever realized before, that addition is perfectly symmetric for all types of numbers. You can sort of see a bit of a stunned look on some people's faces that they hadn't realized that before. Isn't that just incredibly amazing?

After that, we get to think about the Commutative Property of Multiplication (pointing out that commutativity does not work for subtraction or division), look at the Associative and Distributive Properties, do some simplifying exercises with association and distribution, and so on and so forth. But the thing I can't get over this week is that something so simple as flipping around an addition and automatically getting the same answer can, all by itself, be an enormous revelation if you listen closely enough.


  1. I am impressed that they were honest enough to keep saying 'no'. Usually a class can read the teacher, and will prefer to give the answer the teacher wants. (Great article on clever Hans, here.)

    It sounds to me like you've helped them to feel really safe. I'm so glad I read this, because I do fly past commutative and associative properties, assuming students will know those cold.

  2. "Usually a class can read the teacher, and will prefer to give the answer the teacher wants."

    Exactly. I wonder if something about the way you asked it made them think you were expecting a "no". I think it's good to do that sometimes -- use a tone that implies the wrong answer so that they have to think instead of just giving the answer "the teacher wants".

  3. As a tutor of mostly High School students, I often find that revisiting concepts introduced many years before is very worthwhile. Students may have passed the test back then, but they did not understand the concept in a fundamental way.

    I believe that revisiting key concepts months or years later in a conceptual (vs procedural) way, and helping students make similar/different connections (+ and * are commutative, - and / are not, etc.) is incredibly important to their confidence and interest in the subject.

    Thank you for sharing your experience!


  4. That's awesome. But I can't help wondering ... if you really weren't sure whether addition were commutative, wouldn't you be really anxious every time you went shopping? You'd be thinking, could it make a difference to how much I'm going to pay if I put the bread through the till first, or the milk?

  5. Gareth, they probably imagine that any commutativity problem would apply to "special" cases, and their shopping isn't special. Notice that their failed counter-examples were special cases.

    The reason why a lot of early testing was changed to ask questions about real stuff was that the same question asked in this way got better answers from low-achievers. They thought they couldn't do addition, but actually they could, they just didn't know it. The high-achievers breeze past the fluff, so it's Win-win unless your objective is to make students fail tests.

    89 + 47 looks like hard arithmetic, which the weak student falsely believes they can't do. But 89 cents in one pocket and 47 cents in the other, how much do you have? - is an everyday problem they know how to answer. Their approach may take a minute of silent counting when it ought (at say age 12) to be instant, but it's a lot better than "I don't know".

    We see this in other subjects too. If a confident student advances the opinion that Romeo's feelings for Juliet are no more sincere than his quickly forgotten love of Rosaline they may press on even if the rest of the class don't see this possibility (maybe they're all hopeless romantics like me). But a weaker student may give up, believing they got it wrong, when really it's not settled and you just wanted evidence that they've understood the material.

  6. I do think a lot of my students are so messed up over negative numbers, that they just assume nothing is predictable or consistent once negatives go into the mix. It's always the first proposed counterexample to commutativity.

    I must confess that there was a stage at a very young age when I could do numerical arithmetic in school just fine, but was completely helpless at adding Monopoly money (I remembering running from the room and pestering my dad over and over). For me, it was partly having just learned about timekeeping base-60 and assuming that "real world" stuff had to then always be different from "school numbers".

  7. @Gareth: as Delta points out, shopping is all positive numbers, who knows what crazy stuff happens when you try and bring in negative numbers. Do people worry about whether "savings" are applied as each item goes through the checkout, or all at the end? I bet some people do.

    @tialaramex: While I agree in general with your broader point, there _are_ valid reasons for asking "89 + 47". For example: assessing ability for abstract reasoning.

  8. I'm deeply troubled that this stuff isn't explained thoroughly, tested for, stragglers weeded out, and taken for an everyday common thing by grade 3 of elementary school. Alas, looking how my daughter is being taught "math" so far up to grade 3 of elementary school, I stopped being amazed at anything. It's a miracle I'm not drinking myself silly every time I see what they "teach" my daughter.

    1. You basically have to teach this yourself to your kids. Sorry, but public schools, and even private ones, are not going to do it.

      This is the one single thing that I wish more parents knew.