Archives for category: Precalculus

My classroom is a game that my students play. I set the rules by how I allow them to succeed or fail in my class. If I’ve done it right, then the rules I set should motivate genuine learning and reflect that knowledge in the form of a ‘grade’. –Daniel Schneider

I’ve proposed a session for #TMC13 on “A map of problem-based class designs”.

Last year in pre-calculus, all my students had passed algebra 2, but some had passed with flying colors while others had passed with a D- (and a D- in geometry before that, and in algebra 1 before that). Everyone needed a different course. So, the rules were:

  • Problems spanned a range of difficulty from pre-alg to pre-calc.
  • You got credit for solving and presenting a solution, if you convinced the class you were correct.
  • You got credit for writing up what you presented and revising until I accepted it.
  • You couldn’t present something someone else solved, or write up someone else’s solution.


  • Almost everyone participated wholeheartedly and solved problems on their own that were challenging for them.
  • They all got better at writing.
  • Not much pre-calc was learned, as even the strongest mathematicians dropped in and out of the main precalc problem sequence in favor of brainteasers.
  • Presentations were a little desultory, since most students were totally unfamiliar with the problem the current student was presenting.

This year, we reorganized sections so that all pre-calc students had done reasonably well in algebra 2 last year. To get everyone learning the pre-calc content, I reorganized the course. The new rules:

  • Problems almost entirely required grappling with and mastering precalc concepts.
  • You got credit for solving a problem and presenting it.
  • You got credit for writing up someone else’s solution—but half as much as presenting an original solution.


  • People are learning pre-calc; some of them are mastering it.
  • They are all getting better at writing.
  • Presentations are better than last year and the discussions are more interesting.
  • Some students have earned good grades by reporting their classmates results, without ever solving a problem on their own.

Compare these examples to Exeter’s nightly problem sets, or College Preparatory Math’s groupwork, or any other problem-based course. All make different choices about what the problems are, who solves what on what schedule, and how assessment is done. I’m convinced the rules defining a problem-based course are the main factor in determining what students actually do, and therefore, what they learn. I’d like to discuss the feasible and infeasible regions in this choice space. What choices do you go back and forth about, and why? What results are you trying to drive, or avoid? What choices are you excited about? Worried about? Why? If “it depends on the kids”, what exactly about them does it depend on, and how?

I’d love to start hearing people’s questions and answers about this now. We can summarize and continue the discussion in Philadelphia.


I kind of feel like running up and down right now.

-A student, successful after 3 days of trying to prove the equivalence of the unit circle and right triangle definitions of sine.

Last year I taught one section of an all-writing-and-presenting precalc course. I was spending 5+ minutes reacting to each student paper and students were turning them in at 3-6 per day. So, manageable … but I would get 2-3 days behind, then save them for the weekend, then not get to them on the weekend … pretty soon I was buried.

The superstar ELA coach at our school helped me collect and combine the comments I was writing most often, and the things I wished I were seeing, into a rubric. We then ran off 20 copies and sat down with 20 papers. I had a highlighter and a pen and a stapler. She had a stopwatch. On your marks … get set …

The first batch of 10 took about 10 minutes. The next batch of 10 took 8 minutes. I just highlighted away, with no further annotation except sometimes to elaborate one of the highlighted remarks or to write a quick note that didn’t get covered by the rubric. After a year of practice and tweaking the rubric I now am down near 30 seconds for many papers, and average less than a minute. The feedback for students is much better than when I was writing it all out.

The time savings aren’t just in highlighting vs. writing. It’s also that the rubric focuses my attention. Title. Intro-does it work? Can I follow the explanation? Is it correct? Bam, bam, bam down the page. It also prevents me doing the work for them. “Solution is not correct” is all they need to know. If I write out a correct one, they don’t read it, and if they do, they don’t learn. Better that they find the error, or if they can’t, that they come find me.

The rubric I’m starting with this year is the product of having used and tweaked it for hundreds of papers last year. You can find it around page 4 of this document. But if you want to make your own from scratch, start grading papers freehand and keeping track of what you’re looking for, what you’re always writing. After a few dozen papers you’ll have a good draft of what you need.

This post was inspired by a twitter conversation between @mpershan and @cheesemonkeysf—I know the latter shares my enthusiasm. If we can save even one life, it will have been worth it.

Last year in precalc, everyone presented. (If you solved a problem, you had to present before you could turn in a paper, and the only way to earn a grade was to turn in papers. Publish or perish, baby.) But the audience often sucked. The crowd would wait politely without really trying to follow along, then clap politely and go back to what they were doing. Of course, that didn’t make anyone very eager to present.

So this year in Algebra II and Precalc we’re trying something devised by two of my colleagues and me, based on stuff we learned about feedback from Mylène and Patrick.

Everyone gets a slip of paper to fill out for each presentation:

The questions correspond to the full rubric for presentations (the rubric shows up around page 5).

The goal for the audience is to ask questions of the speaker in order to help the speaker score all YESses—without doing it for the speaker.

Everyone fills out yes/no sheets and turns them in to teacher. Teacher reviews quickly and that day or next gives general feedback to group about their sheets: “Didn’t seem to me that you guys could actually check the steps in Sam’s presentation, but you said you could.” or “Here are 5 samples of comments you wrote for Lianne. Which ones are specific and which are not?” Teacher gives sheets to speaker.

Any troubleshooting you wish to offer prior to launch will be greatly appreciated.

Problem 132. Write an essay describing what, if anything, you have learned from this course that will have a lasting impression on you (from Ted Mahavier’s trigonometry course).

[Three out of fifty students chose to respond to this problem. I wanted to share one of the responses, reprinted with permission below. -DG]

Struggles, Overcoming, Achieving

Completing this class has been a rollercoaster ride starting in September of 2011. I entered this class without any confidence in myself, in a way doubting that I would succeed, sad but true. I was extremely impatient, lost motivation quickly, and when I needed help, I just gave up. Just reading the word “pre-calculus” and seeing it printed on my schedule was quite intimidating. Not knowing what I was walking into was nerve-wracking. I had no idea of what it was.

In the beginning, this class was a tremendous struggle for me. Colleges paid very close attention to the first two semeseters of a student’s senior year. So I was trying my utmost to achieve nothing but good grades. The way class worked was completely conflicting with the type of learner I was. We had to solve problems practically on our own, PRESENT them to the entire class, and write them up as papers. Woah, well first off I am a very shy person, not only that but I am a visual learner, I learn best when the class lessons include lots of note taking, examples, pictures, etc. None of these were involved in the way this Pre-Calculus class was taught. I rarely ever reached the point of presenting, never mind writing up papers. Presentations were more difficult for me than people understand. It was a real personal issue, I had trouble overcoming. Let’s just say I wasn’t happy to be in this class.

I was getting tired of watching my peers constantly present and write up new problems, they just made it seem so easy. Then there was me, always struggling to simply figure out and solve one problem at a time. I think this is what sparked my interest to do better, and become more motivated in this class. See, the thing wth Dr. Goldner is that he likes for you to figure things out on your own, he wants you to think, and work hard for your answer/solution. I hated this. But I wanted to do better, I wanted to succeed. So I began working with mathematical puzzles, in the infamous black book. These had many pictures, diagrams, tools that were used to solve them, so they seemed quite easier to me. It took me three weeks to present my very first puzzle, but at least it was a start. Success with these puzzles led me to wander off to the orange book of straight math problems. When I was stuck and needed help, I was already hesitant to ask the teacher. So when I built up enough courage, and pushed my pride aside to ask for help, it was quite the inner conflict for me as Dr. Goldner refused to help me any further than explaining what was asked in a problem. Ultimately, his strategy forced me to work strenuously and diligently on my own. I accomplished more than I thought I would, and learned how to have patience when working with new things, how to be persistent and never give up on something I want, along with not letting my pride hold me back from things like asking for help.

In the end, everyone’s experience is different, and mine was a struggle before I finally became successful in my class. However, through it all I gained a lot of new skills. I learned how to have patience, to have persistence, and to ask for help when I need it. All these skills will be beneficial to me in the future and I am glad and proud of myself that I will depart from this exceptionally greater than the person I was when I entered.

So i times i is negative one. Good. Now, what’s the square root of -4?

I expected someone to shout out 2i and we’d move on. But no one did. A pause. Then guesses: “2?” “-2?” We had just—I mean 30 seconds earlier—discussed why neither 1 nor negative 1 could be the square root of negative one.

Decision time: I could show them, giving them this tool, and getting on to the problems I’d planned to introduce. Or, I could leave this as an open problem for them to work on, on the theory that moving on won’t have much value until they put this step together for themselves.

I decided to leave it open. That was Friday. Today, Monday, it’s still open. Most students have raised their eyebrows at it and turned away to work on other things. A few are still poking at it: “Mister-nothing works.” “Well, we know it can’t be positive, or negative, so it must have something to do with i.” “Oh, I have to use i?” “Well, try it.”

My internal dialogue on this is punishing:

I’d be an idiot to tell them—a few are perplexed, and they’ll enjoy getting it when they figure it out. The ones that aren’t working on it are more interested in other things, and what’s wrong with that?


I’m an idiot NOT to tell them. The reason most aren’t interested is because they have no fluency with this thing yet. They’re bogged down and need me to give them a push out of the mud so they can get going. And anyway, my job is to make sure they all know how to find the square root of negative numbers, not to let a few of them figure it out while the rest do whatever they want.

So, am I an idiot, or am I an idiot? And why do I love this job so much?

You have a student who is trying to solve a 3×3 magic square. She’s been trying for a week, and keeps getting it almost to work: usually she gets to 6 directions out of 8 that add to 15, the other two add to 12 and 18. She keeps wanting to give up and keeps asking for help. So far you’ve just told her to keep going, and she has, but her near-misses are starting to repeat themselves, and she’s getting bored.

What should I do you tell her to aid her investigation without taking it away from her? In other words, help her out, possibly making the problem simpler, without giving an answer or algorithm.

No deadline, but answers arriving before Monday will be more likely to have real-world impact.

It is 4:22pm on Wednesday, April 4th. An eccentric billionaire has gathered $5 billion in 500-dollar bills and proceeds to hand them out, one each second, without stopping. What is the exact date and time when the last bill is given away? -A puzzle I got from Mo Page

I’m flabbergasted. I have a number of students—maybe 10? 20?—who determine by division how many bills there are, then figure out by multiplying 60x60x24 how many bills are given away in a day. Fine. But then they start subtracting … after day 1 there are 9,913,600 bills left. After 2 days there are 9,827,200. Almost immediately many students lose interest, but there are a few arithmetic ox that start chugging through it (with calculators, to be sure). 9,740,800. 9,654,400. I watch in disbelief as the markerboards are filled in, line by line. 8,617,600. 8,533,000. After a while I can’t help myself. I casually mention that people sometimes use division to do repeated subtraction, and I countdown from 10 by 2’s and compare to 10/2. They are a little chagrined at not having thought of that, but they try it. Then they face confusion about handling the remainder.

I don’t believe math must be learned in a particular order. My seniors don’t need to model repeated subtraction with division in order to learn the basics of trigonometry. But, really? Should I really keep teaching precalc instead of throwing it out and teaching pre-algebra? I don’t even know how to teach pre-algebra! But I’ll learn … if that’s what’s needed.

What should be the primary goal for instruction of students who are placed at one level but who are missing huge chunks of what they were “supposed to have learned” years ago? I feel like I keep asking this same question in different ways on this blog. Instead of elegantly reasoning my way to a solution, I just keep doing what I know how to do, day after day, hopefully making progress, gobsmacked at the glacial slowness.

Brute force. Them and me both.

Student: Hey, Goldner, check this out! I started with multiples of 6:
18 + 24 + 30 + 36 + 42 = 150
Then I subtracted 2 from each one and it went down to the next 10!
14 + 22 + 28 + 34 + 40 = 140
And it keeps going!
12 + 20 + 26 + 32 + 38 = 130
10 + 18 + 24 + 30 + 36 = 120 … 

Me: That’s cool! Do you think it will keep working?
Student: Yeah! It’s a solid pattern!
Me: Why do you think it works? What’s going on?
Student, smiling: I don’t know!
Me: That’s awesome – see if you can figure out how it works.
Student, 15 minutes later: I don’t want this class to end!

Me neither. But you can always tell whether I’ve had enough sleep and a good breakfast on days like this. When I have, I’m thrilled to see someone get hooked by structure. When I haven’t, I despair because this marvelous encounter with distribution is happening to a senior who has been passed through Algebra II and is currently taking a course from me labeled “pre-calculus”.

I’m suffering from competing values.

I want my pre-calc students to make as much mathematics as they can. When they solve a problem, I want their excitement to carry them into the next problem.

I also want them to practice getting their ideas down on paper, with correct spelling and grammar. They are seniors, and their great ideas won’t get them very far if they can’t express them cogently. Their professors and employers will take incomplete, ungrammatical, mis-spelled, mis-punctuated, mis-capitalized sentences as evidence of ignorance.

The rules in class are that grades will be based only on finished papers. Finished means (among other things) a sensible problem statement, explanation and conclusion, with no grammar or spelling errors. (I secretly allow one error per paper.) I give lots of feedback on every draft, but no credit for unfinished work. Result: as the term ends, the relaxed pursuit of new problems has given way to a grumbling, resentful process of revision after revision. “We already have to do this in English and History. We understand having to write it up once, but why do we have to make perfect drafts? This is math class, not writing class!” They’re stressed: students with ten papers in progress have no credit yet. And I sympathize. I remember this from grad school. The fun part was figuring out the solution and a neat way to explain it. The long, boring, maddening part was the subsequent rounds of reviews, revisons revisions, rewriting.

So why am I doing this? I guess because my experience is that everything I start teaches me something, but only the things I finish move my life forward.

Is that the most important lesson at this stage of their lives? If it is, am I teaching it the right way? Turning a fun class into a tedious chore seems like a bad idea. But letting them skip off to college in the habit of turning in papers that would embarrass many 7th graders seems like a bad idea too.

Maybe I should ask them.

In the meantime, if you find any errors in this post, please let me know.