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FULL DISCLOSURE: I am not a professional, amateur, or even wannabe poker player.[fn1]

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This post is about moving away from the transactional model of classroom discourse, where the teacher asks a question, a student provides an answer, the teacher validates or corrects the answer, then asks another question. Repeat ad nauseum until the students either graduate or drop out.

For extra disengagement, you can spice up the teacher-student transactions by eliminating wait time, cutting students off, shaming wrong responses, fishing and fishing and fishing for that reallllly specific answer, and best of all, flopping out undirected questions that lie dead in the classroom like beached whales.

We’ve all had moments (or lessons, or weeks, or years) like that. Even once I thought I was great at facilitating class discussions, I screwed this stuff up over and over again. Videotaping your own lessons is a great experience, humbling but hopefully not so humbling that you immediately quit the profession. Now, that wasn’t entirely my experience, but even though corporations may be people now, but I’m glad as hell that class discussions aren’t, because in my career I have murdered them over and over again and there is taped evidence that shows it.

So, the point is moving from a discussion model that is transactional to one that is interactional. Instead of TSTSTST, it’s TSSSSSSSSSSST (pronounced “tsssssst!”), where the teacher kicks things off, gets students responding to each other’s explanations, and only contributes again when it’s strategically necessary. Now, this is not to say the teacher has no role in the discussion — if you step out entirely, you may not like the results you get, especially early in the year.

But your role is facilitator, not arbiter. To use a comic book metaphor, you are Charles Xavier, not Judge Dredd. Keep people on track, set norms for respectful interaction, make sure everyone’s on the same team, but after that, let Storm and Wolverine and Nightcrawler take over, because they should be driving the story.

“BUT MY KIDS WON’T TALK!”

Of course they won’t! For their entire formal schooling career, they’ve largely been taught not to — sit still, be quiet, let the teacher talk. No matter how awesome and heartfelt your Day One speech is, it isn’t enough to wipe out a decade of indoctrination in the hidden curriculum. Talking to each other productively about a subject universally regarded as a barometer for “smartness” makes them vulnerable, and being an adolescent is all about trying to hide vulnerability. The good news is that this is actually fairly easy to do!

The bad news is, you’re gonna have to make some sacrifices.

Specifically, you gotta go buy some note cards:

Because here’s the problem with most classroom discussions. A small percentage of the kids raise their hands. The teacher is then faced with calling on one of that group (which leads to their thoughts becoming overrepresented and usually garnering resentment from other students), or cold-calling a student without their hand up (which tends to garner resentment from that student towards you). Or, worse, you just standing up front waiting for your beached whale of an undirected question to stand up and start dancing (protip: it usually doesn’t.)

Hands raised supports transaction. I need an answer, you provide it. I judge your answer, we move on. Cold-calling students is, according to at least Alfie Kohn, “fundamentally disrespectful“[fn2]. So what’s a teacher to do?

The answer is simple. You need a transparently random[fn3] method for calling on students in class. Hence, notecards. Each class has a color, each card has a student name. Whenever you are leading a discussion, have the set of cards in your hand. Ask your question, give wait time, then glance down at the card and say: “Well, what do you think, [name of student on card]?”

Now, here’s where the poker comes in. Keep your face completely neutral, no matter how brilliant or idiotic the response. Make eye contact, nod or purse your lips like you’re seriously considering what they said, then flip to the next card and say: “Do you agree with [name of student who just responded], [name of student on next card]?” Repeat as needed: neutral face, eye contact + nod, move to the next student. Once things have been well-discussed, make any clarifications or summaries that are important and move on.

This does many things, all at once:

**1) It lowers the stakes** — they can see you just literally pulled their card, so there’s no “why’d you pick me?” stigma, and kids with hands raised don’t wonder “why didn’t I get picked?”. If a kid isn’t prepared to answer, move on for a student or two, then circle back and ask whether those responses helped.

**2) It makes your calling patterns more equitable** — it eliminates all your unconscious assumptions about which students will provide useful responses

**3) It lessens the friction of student-to-student interaction** — you are explicitly asking them to respond to each other, so it’s not weird for them to say “I disagree” — and they can see that you didn’t pick “the smart kid” to shoot down their response, another toxic pattern that teachers fall into

**4) It at least doubles the rate of student participation** — because every question is receives at least two (and usually more) responses

**5) It encourages students to pay attention** — provided you shuffle the deck regularly (tho not too regularly as I like to get as wide a range of students to respond as I can)

**6) It forces students to listen to each other** — instead of relying on instant teacher feedback, they need to understand each other’s arguments and apply their own reasoning

**7) It reduces your decision fatigue** — you make a thousand decisions every day — don’t waste your brainpower on who to call on for every single question you ask

Now, this doesn’t mean you shouldn’t allow students to raise hands or volunteer participation. It doesn’t mean you can’t cold-call students — you still can! For best results, just look down at whatever card happens to be on top before you say their name — they’ll never know!

Respectful discourse doesn’t just happen. Tell students what you’re doing and why you’re doing it.

-“Raise your hands if you want to say something, but don’t get mad if I don’t call on you, because I want to make sure we all have a chance to talk.”

-“I’m not going to tell you right or wrong right away, because it’s important for everyone to think about this and make their own decision.”

-“If your card comes up, you don’t need to respond right then. I do expect you to be able to comment on the responses other people give.”

Eventually, the notecards are more of a crutch than a necessity. Students will adapt to the discourse norms you’ve created and you can be more relaxed about how the discussion goes. But even on Day 179, I find them to be useful. I have them fill out the notecards on Day One and include stuff like parent info, their hobbies/interests, class schedules, etc. so it’s a one-stop shop for necessary info about that student. I use them to note when a student needs to make up work, or I need to have a conversation with them so I can’t forget. I use them for discussions, board work, setting up groups, pretty much everything, to the point where if I did not have my set of notecards, I don’t know how I would run a classroom.

Try it out. If they don’t help transform your classroom, at least you’re only out five bucks. Hit me up at @ejexpress and I’ll mail you a refund.

[fn1] I had a group of friends that played obsessively in college and I had to stop hanging out with them because poker is the second-worst social game[fn4] ever. Here are the possible outcomes:

1) You win, which means your friends are losing money, which makes you feel guilty, because they really need that money to fuel their expensive craft beer habits.

2) You lose, which means you are losing money, which means you won’t have enough money to fuel your own expensive craft beer habit.

Now maybe that’s not your situation but honestly, if you spend hours sitting around a table with non-friends drinking cheap beer, you’ve already lost more than you should allow poker to take from you.

[fn2] Don’t worry, he’s wrong.

[fn3] pseudo-random is close enough, you pedants

[fn4] Worst one is Monopoly, for all the reasons.

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**1) “You can’t divide by zero.” Explain why not, (even though, of course, you can multiply by zero.)**

You can divide by zero. To quote Rachel McAnallen, “You can do anything you want — YOU’RE the one with the pencil!” The problem is, there’s no way (assuming standard rules for arithmetic) to divide by zero in a way that gives a unique and logical answer. Division is the inverse operation to multiplication. Dividing a number by 2 gives an answer that can be multiplied by 2 to get the original number. So dividing by zero should give us an answer that can be multiplied by zero to give the original number. If I’m trying to divide 10 by 0, I should get an answer that can then be multiplied by 0 to get 10. However, since anything by multiplied by 0 gives 0, there is no real- (or complex-) value that we can multiply by zero to get 10. So the solution is undefined, which means we don’t have a way to describe that value (though if we wanted to create a new number system, we could come up with one — but it’s unlikely our new system would be as useful).

A nice concrete analogy helps. Imagine a table with eight 12-pack cans of soda. If we pick up 0 of those 12-packs and put them in our fridge, how many cans of soda are in the fridge? This models 0 groups of 12 cans, or 0 x 12, just like putting all eight in the fridge models 8 x 12, or 96 cans. So multiplication’s good.

Now open one of the 12-packs. Split the cans up into four groups and put one of those groups in the fridge. That models 12 cans divided into 4 groups, which gives 3 cans per group, so 3 cans in the fridge. (notice how this also helps us model dividing by 3/4… for exploration later). But if you try to split the 12 cans up into 0 groups, it’s very unclear what you should be doing. You can’t define the number of cans in each “group” because any number of cans in 0 groups leads to 0 cans, not 12. And soda cans, as matter, can neither be created nor destroyed.

**2) “Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.**

Okay, this one’s boring. The = sign means that the value of the expression on the left of the sign is equivalent to the value on the right hand sign. When we solve a complicated equation like 3x + 4 – 2x = 6x + 8, all we are doing is changing that original equation (which looks tricky) to a set of “simpler” equivalent statements, using the axioms of equality and various properties of arithmetic. So the LHS can be rewritten as 3x – 2x + 4 via the commutative property, which is equivalent to x + 4 via the distributive property, so the whole equation is equivalent to x + 4 = 6x + 8, and we can proceed, adding -x and -8 to each side (properties of additive equality) to obtain -4 = 6x, and finally divide each side by 6 (division property of equality) to get x = -4/6.

At each step, we replaced or previous statement with a new statement, but by following the rules of algebra, we were careful to make sure each new statement was equivalent to the previous statement.

**3) You are told to “invert and multiply” to solve division problems with fractions. But why does it work? Prove it.**

I’m gonna assume a less rigorous definition of “prove” here, but the simple answer is that multiplication and division are inverse operations, so dividing by a number is the same as multiplying by its inverse, which via the inverse property, can be found by “inverting” the number (changing 3 to 1/3, changing 3/4 to 4/3, etc.). That’s rather unsatisfying.

Think of the 12-pack of soda. Multiplying by 1/4 means taking one-fourth of the cans. That’s 3 cans. Dividing by 4 can mean two things: Either splitting the 12 cans into four groups and counting the number of cans in one group OR splitting the 12 cans into groups of four and counting the number of groups. You can see that these processes give equivalent answers. But how can we model 12 divided by something like 2/5? How can I split something up into 2/5 groups? Well, soda’s a poor metaphor here, but I’ll keep going with it, because I’m thirsty.

Let’s first divide 12 by 1/2. We can think of this as splitting the 12 cans into 1/2 groups and counting the number of cans in one group OR splitting the 12 cans into groups of 1/2 cans and counting the number of groups. The second seems more intuitive, because I can’t make sense of a 1/2 group easily, but I can make sense of a 1/2 soda can. So I pour each soda into two glasses and count the number of groups of 1/2 soda can (number of groups of 1 glass). There should be 24 glasses, or 24 groups of 1/2 soda cans, so 12 ÷ 1/2 = 24.

Now let’s look at 12 ÷ 2/5. Again, we can split 12 cans into 2/5 groups and count the number of cans (confusing!) OR we can split 12 cans into groups of size 2/5 cans, then count the number of groups (maybe confusing, but more doable). First we make our groups of size 2/5 by pouring each soda into 5 glasses, then combining two glasses together. For the first step, we wind up with 12 x 5 = 60 glasses. Once we combine, we are left with 30 glasses. This is 30 groups of a 2/5 soda can each, so 12 ÷ 2/5 = 30. Notice this could be modeled by multiplying by 5, then dividing by 2, which is 12 x 5/2. Neat!

**4) Place these numbers in order of largest to smallest: .00156, 1/60, .0015, .001, .002**

1/60, .002, .00156, .0015, .001

Trivial check for place value understanding. Would be more interesting with 1/600 as the second option, which would make it the same order of magnitude as the others.

[.002, 1/600, .00156, .0015, .001]

**5) “Multiplication is just repeated addition.” Explain why this statement is false, giving examples.**

Multiplying by whole numbers works as repeated addition, but how do you explain multiplying by 1/4? Multiplication is better understood as stretching, scaling, or intensifying. Keith Devlin gave a great example of a volume knob. Twisting the knob cannot be described as repeated addition, but is a clear example of a multiplicative relationship.

This often causes misconceptions when teacher talk about multiplying by 1 as well. “Now, multiplying 6 by 3 is adding 6 to itself 3 times,” says the teacher, and the poor student sees 6 x 1 on a later problem and writes “6 + 6” for 6 added to itself once. It’s much better to talk about something of length 6 stretched to 3 times its length, so that students more easily recognize 6 x 1 as an object “stretched” to its full length, 6 x 5/4 as something stretched to be 25% longer than its original length, and 6 x 2/3 as something shrunk to 2/3 of its original length.

Repeated addition can be modeled by multiplication, but multiplication is not confined to repeated addition, as examples using rational values quickly demonstrate.

**6) A catering company rents out tables for big parties. 8 people can sit around a table. A school is giving a party for parents, siblings, students and teachers. The guest list totals 243. How many tables should the school rent?**

Did students learn how to round in a context? Or do they see 30.375 and round that to 30, every time? You need an extra table to seat the 3-person “overflow.” (My favorite student response to a question like this is — “You only need 30, because there’s no such thing as .375 of a person.” What’s that misconception?)

**7) Most teachers assign final grades by using the mathematical mean (the “average”) to determine them. Give at least 2 reasons why the mean may not be the best measure of achievement by explaining what the mean hides.**

SBGers should own this question. Here are my two reasons.

1) Mean is sensitive to outliers (mostly, zeroes).

Assume a classroom where every grade has equal weight. Here are two sets of grades:

Student A: 100, 100, 100, 0, 100

Student B: 81, 80, 80, 80, 80

Student B has a higher grade than Student A. Do you think Student B has more mastery of the material? Do you think Student B is better prepared for the next math course?

Kids fail classes all the time due to compliance rather than competence, because it’s easier to give a 0 than track the student down and find out what they really know.

2) Mean is NOT sensitive to time.

Another set of grades, in sequence:

Student A: 25, 25, 50, 100, 100

Student B: 100, 100, 50, 25, 25

Same questions: Who has better mastery? Who will be better prepared for the next course?

Using a simple mean has a disparate impact on students who start “lower” than other students. Even if they do everything right and learn the material as well as anyone else, those low early grades will drag them down. Student A and Student B have the same grade, yet Student A seems like a star student and a great improvement story, while Student B is in a total tailspin. Thanks, mean!

**8) Construct a mathematical equation that describes the mathematical relationship between feet and yards. HINT: all you need as parts of the equation are F, Y, =, and 3.**

F = 3Y, but dear lord this is SO HARD if you try to jump straight to the equation without thinking. Yet another nod to the CME project for having a process that starts with numerical “guesses” that are then refined and generalized.

**9) As you know, PEMDAS is shorthand for the order of operations for evaluating complex expressions (Parentheses, then Exponents, etc.). The order of operations is a convention. X(A + B) = XA + XB is the distributive property. It is a law. What is the difference between a convention and a law, then? Give another example of each.**

Conventions are how we doing things that are largely irrelevant to the core of our mathematical system. Laws are important basic properties of our number system which have far-reaching ramifications if changed.

If we changed PEMDAS, all that would happen is that notation would be slightly different. For example, if I was going to model taking 5 cans of soda from one pack, and 3 cans of soda from another pack, then doubling the number of cans, I would use the expression 2(5 + 3). If we changed PEMDAS to PEASMD, I would model it with 2*5 + 3, as the parentheses would no longer be necessary. We’d write equations and expressions in different but equivalent ways if PEMDAS was changed, but it wouldn’t change the nature of our number system. However, if X(A + B) wasn’t XA + XB, we’d be in a whole different world. We’d be talking about some system that was NOT the real numbers. (Similar to how when we change the commutative property so that X*A ≠ A*X, we talk about matrices or other objects, not numbers).

Another convention is “simplified form.” Why is 1/2 the preferred way to write 5/10? We could make a convention that all fractions be written with denominator 10. It’d be the same math, just different optics (for example, 2/3 would be written as 6.66…./10). Why do we rationalize radicals instead of leaving √(2)/2 as 1/√2? We could easily switch that convention, with no far-reaching ramifications.

But change a law like 0*a = 0, and we’ve shook mathematics to its core. [EDIT: yes, this isn’t a “law”, but rather of consequence of identity, inverse, & distributive properties, but I still think it gets the point across]

**10) Why were imaginary numbers invented? [EXTRA CREDIT for 12thgraders: Why was the calculus invented?]**

Solving cubics with real solutions sometimes requires taking square roots of negative numbers. Later in the process, those square roots of negatives are re-squared, so you still wind up with real answers, so no harm no foul. It was only later that imaginary numbers started to be accepted as solutions themselves. After I post, I’m Googling this to brush up on my history.

**11) What’s the difference between an “accurate” answer and “an appropriately precise” answer? (HINT: when is the answer on your calculator inappropriate?)**

I assume this is about “exact” answers, like 2/3 or √2 as opposed to “approximate” answers like .667 or 1.414, the latter type of answer being inappropriate when the teacher says to give exact answers, and the former being inappropriate when the teacher says to give approximate answers (j/k, but not really.) The answer on your calculator is inappropriate when giving it reveals that you have no idea what you’re actually doing (“The intersection happens at x = 1E-19”, “-3 squared equals -9”); alternately, when you hit the wrong keys.

**12) “In geometry, we begin with undefined terms.” Here’s what’s odd, though: every Geometry textbook always draw points, lines, and planes in exactly the same familiar and obvious way – as if we CAN define them, at least visually. So: define “undefined term” and explain why it doesn’t mean that points and lines have to be drawn the way we draw them; nor does it mean, on the other hand, that math chaos ****will ensue if there are no definitions or familiar images for the basic elements.**

Undefined terms can’t be rigorously defined, but they still have properties. There’s no such thing as a point with a length, width, or radius, or a line with area. There are canonical ways of drawing these things that help illustrate these properties (even though the representation of a point does have a radius, and a line does have a height, the drawing suggest that they do not). However, though the drawing suggest these properties, the drawings do not define these properties. We could draw points as squares, or lines as caterpillars, and these representations might suggest the correct properties in similar ways, but we could also draw points as triceratops, and lines as tyrannosaurs without changing the underlying mathematics (Hilbert liked tables, chairs, beer mugs, but c’mon man! DINOSAURS!), but it would make the representations much less intuitive.

Back when I still taught, I made a prezi for this: you can find it here: http://prezi.com/b47mqoqoxtv1/what-is-a-point/

**13) “In geometry we assume many axioms.” What’s the difference between valid and goofy axioms – in other words, what gives us the right to assume the axioms we do in Euclidean geometry?**

Valid axioms lead to logical systems of geometry in which theorems can be proven, and ideally, in which the world can be modeled. For example, the various version of the parallel postulate all lead to internally consistent systems of geometry that model flat space, spherical space, and saddle-curved space. Whereas if we changed the “Between any two points, there exists exactly one line” to “Between any two points, there exists exactly 41 lines,” it quickly gets silly. How do we define those lines? What interesting theorems can be proven about this geometry? What things can we usefully model?

I do dislike the language “what gives us the right.” We’re here, we’ve got brains and pencils, we can assume whatever crazy axioms we want! Saying “To hell with it, what if THIS is true” is how we got non-Euclidean geometry in the first place.

You don’t need a “right” to do math, you just need motivation.

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So, once kids are pros at both GCF factoring and 4-term “grouping” factoring (which is just more GCF), it’s time to introduce what I previously would have started with… the Easy Trinomial.

These are those lovely expressions like x^{2} + 7x + 12 that are so, so easy to factor but be careful! for as with all teaching, there are pitfalls that you with your superior knowledge will glide over while your students plunge down to their depths, which are filled with spikes, and possibly snakes.

Now, start with a challenge. “Wow, I totally want to factor this, but I can’t find a greatest common factor. Who knows what to do?”

Hopefully no one, but if some bright young thing pipes up right away, clap a hand over their mouth and promise them 100 bonus points if they shut up for the rest of the period. Let the class flounder for a bit: if anyone throws out solutions after a respectable time, put them up on the board, and — I’m glossing over this but WOW it’s important enough for its own post — have students check proposed solutions by multiplying. Maybe the correct answer is up there, maybe not, who cares, because you’ve bribed all the kids who know a sure-fire procedure into silence (you remembered to do that, right? I wasn’t kidding) so anyone who has a correct answer usually can’t explain it in a satisfactory way.

“Wow, I know this class is awesome at factoring,” you say, bringing pride into it. “What’s making this problem harder than those problems you were doing yesterday?” If they solved it, put up another problem, more challenging factors, change the signs on them if they’re sharp, whatever. For illustrative purposes, I’m sticking with x^{2} + 7x + 12.

Once someone can nail it down –“There’s not enough terms!” is my favorite response — emphasize that they are doing some real mathematician work here. If you like to digress, tell the mathematician putting out the fire joke, or point to your corny SOLVE A SIMPLER PROBLEM poster, or pull up Polya’s wikipedia page and assign a bunch of research papers. Obviously not really (they’ll just mock his hair, which will make you mad and derail the whole lesson), but emphasize that this is how for-real mathematicians DO WORK. Being a mathematician isn’t awesome because you solve super-super-hard problems all the time; it’s awesome because you take super-super-hard problems and make them look easy.

Now take the reins. Split up the middle term and explain how you’re doing it. “Read the expression backwards. We need two numbers that multiply to give (12) and (add) to give (7).” Ostentatiously circle the + in front of the 12. If kids freak out about negative 12 vs. positive 12, tell them to chill. Get the kids to give you 3, and 4, then you rewrite the expression: x^{2} + 3x + 4x + 12. Then, and this is the most important part… STOP TALKING.

If you’ve done the previous two days right and your kids aren’t defective, they will already be working. x(x + 3) + 4(x + 3) turns into (x + 3)(x + 4). Just like before! Switch the order of the 3x and the 4x and send them off again — same thing!

New problem, with the middle term negative. Hope you remembered to give them plenty of examples with negative GCFs!

Keep going with the final term negative. Ostentatiously circle that negative and say: “instead of adding to get to the middle term, what should we do?” Cheap, I know, but working memory is a thing and let’s not overtax it.

Now, doing it this way is harder than the shortcuts. But it’s not much harder, and splitting up the middle term and writing the two pieces explicitly in the expression is keeping the math real (and rigorous!) instead of having those numbers just magically appear in the parentheses.

Next day (or that same day, even) when they are ready, bust out the terms with the leading coefficient >1. All you need to do is get them to split up the middle term smarter, but multiplying the first coefficient to the last term (yes, I still call this a “rainbow” and I still draw a big arc from first to last and write the product at the top — I can’t give up all my little tricks). And then, instead of some big new scary thing, it’s the same exact factoring they’ve been doing since DAY ONE. Yeah, it’s harder in degree, but not harder in kind. Instead of this being the day where I had to drag kids along and lose a good bunch anyway and get the “When are we EVER gonna use this?” barrage, it’s like I’m barely in the room. I wander around, spiking questions back at them (“Is this right?” -Did you check it? “Not yet.” -Then check it!… “Okay, I checked it now.” -Did it work? “Yeah.” -What does that tell you? “I should move to the next problem?”) and focusing on those who missed class or can’t multiply single-digit numbers or require their cell phones to be surgically removed from their thumbs in order to do math.

Final day is Differences of Squares. Same drill, throw up x^{2} – 9 and let them try. They will! Because now they have a mental construct for factoring. It’s not just a random collection of tricks, it’s a coherent method that they are beginning to master. They will give you (x – 3)(x – 3) and be hilariously convinced that it works, but if you’ve built your “check your work” culture right (needs its own post), they will get shouted down. Ask “what’s the middle term?” and if they need a little more, rewrite it as x^{2} + 0x – 9, because this “special” factoring too is also the EXACT SAME method: Distributive property.

Surprise bonus is that you will see persistence from students like you’ve never seen before. Previously, their mental model was “I need to know the right trick,” and they would just stare at a blank page until I came around to pull a new trick out of the bag. Now, their mental model tells them “It’s the distributive property, same as always” and they have a vague surety that somehow, this can work. And my rate of saying: “Did you check for a GCF?” dropped precipitously (still way too high — don’t kid yourself, this ain’t magic) because they are checking for common factors all the time, every time. SOLID.

As for Sum or Difference of Cubes? Ha! Leave SODOC for the Algebra 2 teacher. Sure, you can write up x^{3} + 0x^{2} + 0x + 8 and have yourself a field day (it does work), but it’s much, much nicer to attack that in the context of finding zeros of polynomials once your kids can do polynomial division. By Algebra 2, kids have an actual reason to want to solve cubics, and they will be much more tolerant of memorizing shortcuts — derive it once and tell them to do it like that every time or spend some time with an index card. In Algebra 1, I’ll push the kids who really like factoring (if they can do x^{4} – 1 without help, they can handle it), but spare the rest of the class.

HOW I KNOW THIS WORKS: Two years after adopting this system, I had a colleague drop by my room and lay this on me: “Every year, I quiz the kids on factoring in September and split them into ‘Got it’ and ‘Needs Practice’ groups afterwards. Now I split them up beforehand based on whether they had you or someone else. It’s 95% accurate and it saves me a pile of grading. What the hell do you do with them?”

I had to say it ain’t anything I did. It’s all D. Propz.

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So it turns out the secret to teaching factoring is the distributive property. I mean, who knew? I’d thought I’d been emphasizing that all along, but turns out the kids didn’t. They focused on the tricks, memorized a bunch of crap, and then forgot it. But riding with D. Propz was far superior.

Teaching strat: Greatest common factor, same as before, including a notebook file that goes step by step through a problem like 8x^{3} – 6x^{2}, including a key slide where the terms are displayed as 2x^{2}*4x – 2x^{2}*3. I never, never, never write work out on slides because then I miss the opportunity to completely botch my algebra in front of the class and let kids see that it happens to the best of us, but THIS one I put on slides, step by freaking step because I do it on smartboard and on the second-to-last slide make sure the 2x^{2} tiles are grouped so that I can drag my finger and factor them out (more on this later). Lame, but whatever. Tablet-people, please just project Algebra Touch for that part.

After some practice on that, I went straight into grouping four-term and five-term expressions with a common common factor. Stuff like 6a^{3} + 15a^{2} + 4a + 10. Before, these intimidate kids solely because they had more terms and we did them last. And always, the ones who were absent that day had tons of trouble catching up. But now, they are easy — just pull out the GCFs of each pair — 3a^{2}(2a + 5) + 2(2a + 5). Next comes the hard part and before, I thought it was my job to teach a bunch of tricks so that kids wouldn’t have to deal with the hard part, but guess what? THEY NEED TO UNDERSTAND THE HARD PART.

So we spent a ton of time on this, Socratic questioning-stylo^{f1}:

Etc. etc. until you get to the key insight that 2a + 5 is, at heart, just a number.

Now, depending on your students’ tolerance for this type of thing (which I call “sustained thought”), you can go in a few directions. I like substitution (or “chunking” in EDC-lingo — buy their books, srsly).

First, round on the kid that said “A NUMBER” and is trying to grandstand about winning this latest round of “guess what teacher wants”

**Me**: “So if 2a + 5 is a just a number, what number is it?”

**Prize student, suddenly floundering**: “…. uh….”

**Me**: “Well, this is ALGEBRA, Jimmy^{f2}, what do we call numbers we don’t know?

**Every other student, in unison:** “CALL IT X!^{f3}”

And then it’s time to do some damn teaching. If I write x = 2a + 5, then let’s get fancy with the whiteboard markers here and….3a^{2}(2a + 5) + 2(2a + 5) becomes 3a^{2} x + 2x, which factors into x(3a^{2} + 2)

(NOTE: do NOT gloss over this part. It’s boring to write about, but make sure every kid is on board. x and 2a + 5 should both be written in the same special color while the rest of the expression should be flat black, the kids should help with the factoring, let them talk it through… do NOT screw this up on the home stretch by whiz-banging your way through)

Then bring it home….

**Me**: But what’s “x”?

**S, excited, in unison**: A NUMBER!!

**Me, winding them up**: WHAT number?

**S, super-confident, in unison**: WE DON’T KNOW!!!

** Me, once all the tears have subsided: **We don’t know *everything* about x, but we do know* something*

**S, timorous**: it’s 2a + 5

**Me**: then we get (2a + 5)(3a^{2} + 2) OH MY GOD YOU JUST FACTORED IT

Hard for kids to handle that cognitive load. Hard for them to know when to use that “chunking” move. I should get up on my pedestal and talk about crafting a unique cooperative learning experience that guides them to develop it as their own solution, but I would be BS ing cause I do that by Direct Instruction ALL. DAMN. DAY.

Now pull up the smartboard file, where I have cleverly written everything out slide-by-slide, just as before, again with the 2a + 5 common factors in the same color and grouped just like before. And I factor it out via gesture. Just. Like. Before.

Why can I do this? Because it’s JUST A NUMBER.

Minds get blown, high fives all around, and we spend the rest of class on whiteboards factoring big scary things like champs. Something with a negative factor gets put up as a challenge, kids talk it through, I push them by putting up pairs like (c + 2) and (2 + c) vs. (c – 2) and (2 – c) and we hash that all out. They go home, crank out some practice problems, and we come back again, stronger, and ready for the real challenge. HOLLA

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^{f1:}We always do Socratic questioning early in the year, and I always introduce it by getting kids to talk about Socrates b/c a few always know. Then I emphasize the point that Socrates asked the Athenian muckity-mucks questions that were so hard to answer that they made him drink poison. Full stop. You WISH you could be that annoying, kids.

So when I have to push them on stuff later in the year, I put on my Socrates hat (actually a toga) and remind them: “If you get annoyed, Socrates wins.” Lampshading it like that really increases their tolerance for being jerked around (or at least decreases my guilt, which is almost as good)

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^{f2}Feel free to substitute your student’s actual name here

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^{f3}“Oh wow, I see why he used** a** instead of **x** as his variable! To reduce the cognitive load on students who are inevitably trained to call everything **x**! Such brilliance!”

I know, I know, ultra-pro teaching going on here

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So that’s how I taught. And boy, did I think I was good at teaching factoring. Because when I was going through school, I was taught using the tried and true “figure it out” method. This was where the teacher put up a factoring problem — like 3×2 – 5x + 2 — put a couple parentheses below it ( )( ) — then said “Figure it out!” This process was then scaffolded by smacking the blackboard and saying “Figure it out!”, except louder, and then eventually jumping up on the desk, waving the hands wildly and yelling “FIGURE! IT! OUT!”…. because learning.

I’d share this story with my own students and it had the desired effect — it made them super-glad that they had me, extra-helpful teacher, with my rules for identifying types of problems and my efficient tricks for solving them. And all was good until I moved to a small school district, had kids for more than one consecutive year, and realized that none of my former Algebra 1 students could factor.

Sure, they all could factor during the factoring UNIT, and many could factor on the Algebra 1 exam, but in Geometry? In Algebra 2?

HAHAHAHAHAHAHAHAHA

Yeah, after review some of it would come back, there was always trouble with those sneaky two-step GCF problems (“stands for Gotta Check First,” I’d say, for the billionth time, debating the utility of jumping on my desk for emphasis), but what got them, every time, no matter how much we reviewed and no matter how many reminders and hints and prompts and blahs I gave were those damn trinomials with a leading coefficient greater than 1.

Because no matter how slick I could make the trick, the cognitive demand was too high for many of my students. They could remember the easy tricks, but when you started talking about rainbows, or cows jumping around, or any of the crazy “best ways” I heard from other teachers how to teach these problems, it was just a procedure and it was a procedure that did not make sense.

So I burned it all down. I threw out my beloved sequence: 1. GCF (Gotta Check First), 2. DOS (get it? “dos” means “two”, there are two terms, it’s the second type), 3. ET (easy trinomials), 4. HT (hard trinomials, or rainbows, or whatev) 5. Grouping (let’s put this last because it has more than three terms so it looks really hard), and 6. SODOC (eh, leave this for the Algebra 2 teacher, poor sucker, oh wait that’s gonna be me in two years).

And I threw out the “Field Guide to Factoring” we’d make every year, I threw out all the tricks that I used as a student and that I diligently had taught my kids, and I taught every factoring problem the same way. EVERY. SINGLE. ONE.

With that one switch, I all of a sudden noticed something. Some of the “easier” problems under my old system became harder. But all of the “harder” problems became easier. Much easier. And I found that even when kids didn’t know how to complete a problem, they knew how to start one — without hints, without prompting, without any help but my standard stonewall: “Well, what do *you* think you should do here?”

And I felt dumb. Because what I was teaching was dumb. It was basic, basic, basic. No special teacher skills required, no flashy tricks, no special techniques. No secret knowledge. But, by gosh, it worked.

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So, what’s your reaction to this? I mean, #firstworldproblems, amiright? Kid got a taste of real life, got the smackdown! YEAH!

But seriously, this needs some unpacking, and the more I unpacked, the deeper I got in until I realized that I couldn’t solve this with a Facebook comment…. I needed to BLOG.

Because is this how far we’ve fallen in US education — To draw a favorable comparison for what we do to students, we’ve got to reach for kids in Nigeria swinging machetes for 14 hours a day? That’s not even a real comparison! No one in Nigeria is making those kids swing machetes to EDUCATE THEM. They are swinging machetes to PRODUCE FOOD. The two situations are not analogizable (trust me, that’s totally a word) — they are completely different contexts, with completely different social purposes. If your farmhand is whining about sitting on a tractor to harvest wheat, by all means, bring up the agricultural production system in Nigeria. But we’re supposed to be EDUCATING these kids, not using them as cheap labor!

That’s like saying: “You think your boss sucks because he yells at you all the time? If you were in solitary confinement, you’d pray for that kind of human contact.” And you know the boss in Nigeria is saying “Man, if you kids were in the Congo, you’d be using these machetes to save your families from the LRA. Feel lucky that the only thing you need to hack is cassava.”

People use false analogies all the time as weapons against teachers, and it’s important to call them out when it happens. My friend is a speech therapist who is awesome with kids, and totally on the side of the angels — she even disclaimered her post before I jumped in there, but I just couldn’t get it out of my head. We’ve all worked under administrators whose only response to teacher pushback was “Well, if you really cared about kids, you’d be willing to …..” followed by the most cockamamie educational scheme this side of a Dukes of Hazzard episode.

Let’s be clear: No one is opposing the testing regime in this country because they think America is an awful place to grow up, compared to the rest of the world. We are opposing it because we think it’s harming kids, and that this country can do better.

But here’s what really kills me: Go back to that conversation again, the last line, which goes “**Kid: (pause) Point taken.**”

Now, I’m guessing the way this “**(pause)**” was meant to be read was “spoiled American kid learns life lesson.” But there’s another way to read it, which is this: Someone told the kid that instead of being educated in the greatest city in the entire United States of America, he could instead be in Africa, doing hard physical labor in a field for 14 hours a day, and….

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…wait for it….

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THE KID HAD TO THINK ABOUT IT

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The topic for the talk was “Motivating Students through Assessment” and it was about — what else? — SBG.

My two biggest fears going into the talk were these:

1) no one would show up

2) EVERYONE would show up

Lucky for me, I hit #2, talking to a 30+ person room for each session. I also discovered 5 minutes before I was scheduled to go on that my laptop doesn’t connect to projectors (damn you Alienware and your sleek, sexy design!), but since I’m a genius (aka I listened to people at PCMI who are way more tech-savvy than me), I had Google-Doc’d it, so all was well.

The thesis of the talk, if it had one, was this:

“Many of the academically self-destructive behaviors we see in students are the product of the warped incentives created by traditional grading schemes.”

Seems like every teacher has stories about how the kids only care about points, or don’t care about learning, or don’t see the purpose in studying, or just grub for grades, and on and on, and if indeed this type of behavior is widespread, my inclination is to blame the system, not the individuals.

After all, if a student does not understand a concept, and you hand that student a 30-point worksheet due the next day on that same concept, with a test coming up later in the week, what’s the best plan for the student? If you said — work hard, get help, and use those problems to obtain mastery then DING DING DING you are a teacher because the actual best plan is to copy those answers off of someone who has already mastered the concept, get a guaranteed 30/30 and make a vague promise to yourself to learn the stuff before the test. After all, if it’s a concept that takes more than a day to learn, getting a low score on the worksheet and then failing the test definitely earns you an F, while locking in that 30/30 means you can blow off the test entirely and still probably pull a D. (especially if you’ve got a parent that will come in to the teacher conference — “Well, how can Sunshine’s test/quiz average be so low when s/he has SUCH GOOD GRADES on homework?”) And once that test is blown over, hey, you never need to worry about that material again because it ain’t coming up until the midterm.

In this situation, as in many others, we incentivize students to not learn and to not try, to either cheat or give up. Remember the Dunning-Kruger effect. Non-proficient students do not prepare well for tests because they do not KNOW they are non-proficient! We need to build up those meta-cognitive skills by giving them repeated attempts to demonstrate proficiency, coupled with timely and relevant feedback. Otherwise, how will they build those skills?

Anyway, this deserves a much more thought-out post, but I am headed out to teach some kids the only thing in life more fun than math — fencing.

Presentation is here, but be warned! I am not one of those presenters that uses PowerPoint as notecards, so it may be of little value. The Futurama characters come from the grading discussion I do with my students. The teachers were a lot more generous in grading!

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Next slide is a relation with one domain going to twelve different ranges and I try my best to wheedle some student into playing, but they know something’s up. I go through each domain value, asking “Do you know what range I’m thinking of?” I usually get someone with a relation that has a bunch of one-to-one values and a single domain that goes to two range pairs. The student often justifies playing by saying the odds are in their favor, only to have me break character and pick that same value time and time again — “oooh, this time I was thinking of the 5, not the 2! You almost got me! Let’s keep playing!” After taking all that student’s money (again: not really), everyone is more wary. Each new slide prompts a burst of hurried strategizing, followed by a decision to play with me or stay away. The first slide that has multiple domains going to a single range value always elicits some good discussion, as students initially suspect a trap, but then figure out that they will rob me blind.

After they are okay on maps, we move on to ordered pairs, tables, points on a coordinate grid, continuous graphs (where brief mention is made of the Vertical Line Test), and eventually equations. Students eventually realize that in theory, they should be able to clean house on ANY equation.

I usually save the equation part for a few weeks later, because once students realize that the game is no fun with equations, we start playing backwards. Now I give them range and they have to determine which domain I used, and all of a sudden, we’re talking about which functions are invertible and which are not. We can push it further and put up equations that have values which “break” the game like radical or rational functions, and now we’re talking about restricted domains.

The advantage to this approach is that it lets students engage functions using their already well-honed intuitions about fair vs. unfair games. Since adopting this approach, students have a much better (if at times more casual) language for what makes a function a function (not that “oh, there’s a repeated x-value, so it’s not a function” BS), and are much more able to explain why a single “problematic” value is all it takes to lose function-hood, and why it’s totally okay to have multiple domains going to the same range value.

And it sticks. First year I did the lesson like this, I had a student mis-identify a function literally MONTHS later, and another kid chimed in with: “Nooooooo, he’s gonna take your money!”

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to:

Here it is, and I realize probably everybody already does this, but I didn’t, and it is SO MUCH better.

Start with one-variable equations: |x + 5| = 8

I do everything on a number line and make students interpret the meaning of this as “the distance from x to -5 is equal to 8.” We solve some problems just like that, talking through them on a number line, but when I go to show the algebra “trick”, I teach them to drop the absolute value and set it equal to 8, or drop the absolute value and set it equal to … (here it comes)…. (seriously this is awesome)…. the OPPOSITE of 8.

So when the right-hand side has two terms, like |x – 2| = 2x – 12, the very first example of that type we look at, everyone already knows that either x – 2 = 2x + 12 or x – 2 = -2x + 12 and when I called on a student to explain why, she was like: “Duuuhhh, it’s the OPPOSITE.” Didn’t have to mention distributive property or anything and 100% of the class was ahead of me.

Then when we get to inequalities, it’s all very natural that something like |x – 2| > 7 will split into one branch of x – 2 > 7, and the other branch should be x – 2 < -7 because duuuhh… it’s the OPPOSITE!

And now we’re on to absolute value in two variables, and after some good old table of values graphing to convince them that they really didn’t want to have to do that all the time, I went back and talked about splitting up the absolute value equations into 2 linear equations, and wow… did not feel like I was earning my paycheck.

“So when we look at the second branch of y = |x + 4| – 2, how do you think we should write that? Should it be y = -x + 4 – 2 or y = -x – 4 – 2 or y = -x – 4 + 2?”

Usually questions like this spark an intense debate, or at least a lot of students with that scared “I ain’t picking anything” look.

But this year, all I got was eyerolls as 25 kids all picked the middle option. WHY?

“Because x +4 is in the absolute value, so you just take the opposite.”

“And why not take the opposite of -2?”

“Well, it’s not in the absolute value, so it doesn’t represent a distance where the answer could be in either direction. You’re just subtracting 2, like it says.”

So, 3 days of absolute value and we hit equations, inequalities, graphing two variables with tables and linear equation, including writing out interpretations referring back to the number line, no solution & all real number solution situations, and everyone’s on board. So if anyone out there is still teaching absolute value without using THE OPPOSITE, come give it a try. The most remarkable thing about this profession is how those tiny, tiny changes can have such an impact on students.

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Then I divide the class into pairs, one as “the explainer” and one as “the questioner” and write a question on the board such as: “What is a rectangle?”

Here are the rules:

1) the explainer must make a good-faith attempt to explain the concept at hand

2) the questioner must ask a question designed to expose a flaw in the definition or to frustrate the explainer

3) Neither player can give up. No matter how good the explanation, the questioner must find something to question, and no matter how annoying the question, the explainer must try to answer it.

After a few minutes, I have the class vote on which role they thought did a better job, then we share some of the more frustrating questions that were asked. This is usually awesome, as the questions will range from the completely off-the-wall to the remarkably-apposite-to-the-lesson. “What is space?” “What is a point?” “What is a thing?” “What is a shape?”

We usually play four rounds, switching roles each time. The first two rounds I use actual defined objects (rectangle, circle, square, etc.), while the last two rounds I go with “What is a line?” and “What is a point?” The proportion of the class who thinks the explainer “won” goes sharply down in rounds 3 and 4.

I use this to talk about the four basic types of mathematical statement: Undefined Term, Definition, Axiom/Postulate, and Theorem, and show how definitions are based on undefined terms.

I played a game with a student where I asked: “What is a point?” and he said, “Don’t make me answer that, give me something else.” I let him pick what he wanted to explain, and showed how in a few questions, I could get him right back to “What is a point?”

Try this out with your classes — loads of fun, starts the year off well, and really gets students thinking about definitions in a more precise way.

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