Archives for category: Precalculus

All year long, the most common “proof” students offer has been to do an example with numbers. On “show that the distance between (a,b) and the origin is \sqrt{a^2+b^2}, I’ve been getting: “I chose random numbers for a and b, so I used a=4 and b=7. Then I plugged in and got \sqrt{16+49}=\sqrt{55}.” No mention of the Pythagorean Theorem we’d been playing with for the past two days.

To my students—most students?—mathematics is “Substitute, then simplify.” It’s all math has ever been. I have been aware of this, but my response has been inadequate.

  • Item: I would put two numerical examples on the board, then a third using variables, crowing that “if you do the same thing with only letters, then you’re doing it for all possible numbers at once! Do it once or twice with numbers, then do the same thing with variables! This is called generalization!” Result: I had one student do a numerical example similar to the problem above, then go on to say “I wanted to solve it with variables, so I made 16=U and 49=V, so the answer is \sqrt{U+V}.”
  • Item: The idea of a derivation is entirely absent. Later in the year I juxtaposed three student examples of finding a distance to show them that the distance formula is simply the Pythagorean theorem. Students were flabbergasted. I got compliments on my lecture. They had never seen anything like it.
  • Item: After developing a meaning for the distance between a point and a line, I asked them, “What is the distance between (x_1, y_1) and y=d?” and gave them some time to work on it. They were totally stumped. “Ok, let’s draw a picture.” Blank stares. “What’s up?” “We don’t know what d is—you have to know what number to use!” Ok. Let’s graph y=2. Fine. How ’bout y=-4. Great. What should we do for y=d? Someone, quiet, tentative: “Draw a flat line?” (indicating horizontal with his hand). Does it matter where I put it? No. What about here on the x-axis? No, don’t do that, that makes it look like zero, and it might not be zero.

In my mind, once you do something with specific numbers, it’s just one more small step to do it again with symbols. After all, it’s the same thing. But that’s the rub: to them, it’s not the same thing!. Torigoe (Thanks, Mylène) has got me thinking about what might be going on.

It never occurred to me that carrying symbols through a problem is more demanding than substituting numbers into as many variables as possible, as soon as possible. By stopping up all those leaky variables with numbers, I can stem the flood of confusion behind a nice high dam of something with just one variable to solve for. The structural relationships among whatever I’m modeling (say, distance between objects on a plane) are eliminated, so I don’t have to think about them anymore.

Anymore? At all! This explains everything! It’s why my precalculus students can’t articulate relationships between abstract quantities. They’ve never had to! So they don’t know how.

At least not yet. The technique of “give one specific problem, then a second, then a general one” and hope they make the leap has not worked. So how can I build the capacity to reason with symbols?


We drew a circle and measured the diameter and radius. The diameter was 3 and radius was 1.5. The formula was C=2ΠR so we multiplied 2 x 3.14 x 1.5 and it equaled 9.42. Then we divided 9.42/3 (diameter) and it equaled 3.14. Which is C/D and was the info that was given to us. So we proved that circumference is C=2ΠR. -A student proof.

The proof I was expecting was something like “Since Π is defined as C/D, then C=ΠD, and since D=2R, then C=2ΠR.” What’s the feedback that will cause them to be concerned about thinking in circles? The best (only) idea I’ve come up with is: “If you had multiplied 2x7x1.5 to get 21, then divided C/D = 21/3 = 7, you would have shown that Π=7.” But I have ’till Jan 4th to find something better. Any suggestions?

Update on my precalculus experiment:

The good news is, I have 110 papers from 50 students.

The bad news is, I need to read and comment on them all.

Two missiles are approaching one another moving at 9,000 mph and 21,000 mph, respectively. How far apart are they 1 minute before they collide? 9,000/60 gives 150 miles per minute. 21000/60 gives 350 miles per minute. Put those together and you get 500 miles in two minutes. Because we want to get one minute, divide 500 by 2. The missiles are 250 miles apart before they collide. -A presentation by two students.

The class buys it. I express confusion: Isn’t there just one minute we’re talking about? Exactly! The whole class responds—they’re so pleased I finally get it—That’s why we have to divide by 2!

By the norms of the class, if the community buys the argument, the presenters can submit the paper. It still has to pass review, however, and I’m the reviewer.

The goal, of course, is to avoid the completely useless path of simply telling them the answer is 500 and they’re just wrong. As far as I can tell, it is precisely as obvious to them that the answer is 250 miles as it is to me that the answer is 500 miles. Their answer makes sense to them, and my answer doesn’t. That’s what I need to reverse.

So I offer my burden up to the blogosphere: What question can I ask my two junior colleagues that will help them see what I’m talking about? Or, as a distant second choice: what’s a clear way I can explain to them why it’s 500 miles and not 250?

C-block, precalc, planning to discuss conditional statements. The anticipatory question: “Complete the sentence any way you want: “If _________, then __________.” Expecting a mix of banality and hilarity from my seniors. This is what I get:

“If you work hard, you’ll see results.”
“If I do well in school, I can achieve my dreams.”
“If I persevere, I can overcome any obstacle.”

And half a dozen more in the same vein.

I don’t know what’s happening at my previously-underpeforming turnaround school but whatever it is, I want more.

After all my talk about Algebra 2, it has worked out that this year I’m teaching precalculus instead. So I’m trying a hybrid of my favorite models: IBL, based on the Moore method; the Math Circle; and, following Lockhart, the art class at my school.

I have 80 minutes a day. I put up definitions and problems, which are often theorems to prove. Students work on the ones that interest them, or if none appeal, they can try to tackle one of the classic puzzles I have in a binder in the corner. We start each day with presentations. (Once a student has presented a solution that meets with class approval, they can submit it for publication in the class journal.) Then I might talk for a few minutes on problem-solving strategies, or values like persistence, or just give them the next few problems. Then work time for 30 minutes or so. I wander around to help or encourage. Students at a dead end can recharge with a Rubik’s cube or soma blocks. We end with a short reflection on where people are and how they’re feeling.

So, Friday – day 2. Everyone’s working. A few are still a little freaked out, but everyone is working productively on one problem or another. The first presentations were all over the map, but they were an occasion to talk about taking time to get good at presenting, which opened up a conversation about what makes a presentation good. Still, twenty percent of the students are ready to draft their first articles. And it’s only day 2! A student calls me over. I see he is working on Problem 1. He looks up at me and says, “I had a breakthrough.”

I know just how he feels.