Reports from the Front

So I’ve comfortably slid into the new year. I can’t say the transition has been all laughter and involves me skipping and trololo-ing. There’s an energy drain that comes when you have to be “on” all the time — and it’s no truer than at the start of the year. If you’re me, you’re on hyperdrive, being very purposeful in what you do and say, because you know that this is when you’re building your reputation with your class. (Just as kids are building their reputation with me, by what they do and say.) And that is the most important thing for me at the start of the year. I want them to see what I value, but enacting it.

Yesterday I had one of my favorite teaching moments. One of my students, who I happened to have taught in a previous year also, said when she arrived to class: “When I realized I had to leave for class I started singing I’m off to see the wizard, the wonderful wizard of Shah.” I heart my kids so much, because… well, I just do. They’re awesome. I promise if they come to me saying they don’t have a brain (“I am just not a math person”) that I will give them a brain (because isn’t that what wizards do?). I will also give them courage (pronounced coo-raj like the french) and confidence. At least I will try.

Our school mascot is the Pelican (ferocious! fierce! or not…) and I want to feel like this at the end of the year:

Honestly, I feel like I’ve been doing a pretty good job in some areas and a crusty job in others.

Rational Functions in Calculus

Example of a crusty job: In calculus I am teaching rational functions to prepare us for limits. I am really focusing on getting kids to understand why. In particular, I’ve been working on getting ’em to understand what a hole truly means, and what a horizontal asymptote means (and no, it is NOT a line a function gets closer and closer to but never touches) and why they might arise. The problem is that this sort of work is hard and takes time and my approach just wasn’t super effective. It was too me-centered, and I didn’t design a way for them to grapple and discover… instead I just kinda gave and explained, in the guise of student questioning.

Still I did get one amazing question which I have to type here so I can use this to provoke discussion and investigation in a class next year…

Why is it that holes appear at the x-value that makes the numerator and denominator of a rational function equal 0, but vertical asymptotes appear at the x-value that makes the numerator non-zero but the denominator 0?

And then to muck them up, after we come to some sort of understanding, I will ask a follow up question:

Graph f(x)=\frac{x}{x^2}. At x=0, you have f(0)=\frac{0}{0}. You’d expect that to be a hole, but … SHOCK! GASP! EGADS! … ’tis not. Explain.

(This came up in one of my classes, and it was precisely at that moment I realized how deep and complicated rational functions are, and how they are just blind algorithms to my kids. I hate that students use procedures and rules to memorize how to find x-intercepts, holes, horizontal asymptotes, etc… but that’s how we teach ’em so I shouldn’t expect any differently.)

I wonder if I asked students in AP Calculus BC to explain why f(x)=\frac{x^2-1}{x-1} has a hole at x=1, could they give a comprehensive answer that doesn’t rely on the fact that “a factor cancels from the top and bottom”? I’d bet not. That makes me sad. I don’t want to be sad.

This is good stuff. I could have introduced it and had my kids muck around with it in a more meaningful way.

The other hard thing that I’m finding, as I really really highlight why, is how much longer things take. I’m okay with it, because I’m not teaching to an AP exam. But it’s a change I have to get used to and honor, but that’s not going to be easy for me.

I have a couple great concept questions on tomorrow’s calculus assessment, so we’ll see if all our discussions about these things have actually made an impact on student learning.


Today in one of my Algebra II classes, I used an exit card. We briefly went over why — when working with inequalities — you “flip the direction of the sign” when you multiply or divide by a negative number. I waited a day or two, and then I put the following on an exit slip for them to fill out at the end of class:

I am unsurprised by what I got back. About 1/3 of the kids said “you only switch the direction of the inequality when you divide by a negative number, so matt is wrong.” Almost all of the rest said “when you divide or multiply by a negative number you switch the direction of the inequality.” Only two actually got close to a meaningful solution.

So why am I unsurprised? Because this kind of explanation is new for them. They really haven’t been asked — at least not on a regular basis — to justify their reasoning. It’s a procedure. They “think” they understand it, but when probed, they don’t. Also, more importantly, I’ve realized they have no idea what the word “why” means in math. They think stating the rule is the why. It’s become clear to me in the past year that they don’t know that when I ask them why, I am not asking them for the rule but for the reason for the rule.

The great thing is: this was formative assessment. Without it, I wouldn’t have known that about 1/3 of the kids didn’t even fully know the “rule” for inequalities. And that those kids don’t see that multiplying by -1 is the same as dividing by -1. I also wouldn’t be able to talk specifically with them about what why means in math, and what a comprehensive explanation might look like.

Last year I put concept questions like this one on tests, but that was problematic. Kids usually did poorly on them, and they wouldn’t have a chance to really revise their response because their grade was fixed (I don’t do SBG in Algebra 2). So the feedback loop was stunted: kids saw their score on these kind of problems, they quickly read the comments, and never revisited it.

(I should also say that we did talk about these sorts of concept questions during the lessons too — they weren’t just sprung on them at the time of assessments.)

I’m in debate how to follow this up, after I have my in-class conversation with my kids. Right now I’m leaning towards making a graded take-home “paper” where students answer this question as comprehensively and clearly as they can. And if they want, after I comment on it, they can revise and resubmit. This closes the feedback loop. And I figure if I do this a few times early in the year, I’ll get dividends later on.


I always have my kids fill out an online google docs survey at the start of the year. It has logistics (e.g. do they have the book yet? what’s their graphing calculator’s serial number?), but it also asks them some questions about their thoughts on math, their hopes and fears, anything else they’d like me to know, whatever. It’s really useful because you get, with a few questions, exactly the things you need to know in order to start getting to know your kids as math learners (and as people, yadda yadda, blah blah).

In previous years I wrote special emails only to students who said things in their survey that I thought needed a response. Like a student sounding especially nervous about class, or who has a learning difference and wants me to know what sort of things work well for them. However, this year I decided to respond to all surveys. I have already done all my calculus students, and I hope this weekend to get my Algebra II students done too. It takes a surprisingly long time to do it, but I enjoy it. And I hope this is one of those small things that I do that shows these kids, who barely know me, that I care about them and that I’m going to be listening to what they say.


Tomorrow is the first calculus assessment. It’s only a 20 minute thing (I’ll let ’em have 30 minutes though…). Beforehand, I’m going to talk with ’em about integrity. I tend to overplan things, but I want this to be a more spontaneous discussion that revolves around the ideas of respect and trust. So in opposition to my own inclinations to overthink this, I’m going to wing it in the hopes that it will be more powerful that way. Then I’m going to start ’em on the test, and leave the room for about 10 minutes. (I won’t be far, because we’ve been having lots of firedrills.)

And yes, like last year, I’m going to continue to have my kids sign these integrity statements. (And I even have another teacher doing it also!)

It’s not that I think it will stop cheating. But I do think that talking and reminding them about it semi-frequently, they at least know that integrity means a lot to me.

With that, I’m done. I’ve almost finished our first full 5-day week of school. Huzzah!



  1. Your Front is about 90% like mine. Except I’m starting rational functions in precal rather than calculus and the problem that our school distract has scared me in trying to contact students via e-mail outside of class unless absolutely necessary.

    Trying to talk about “why” something is or motivate it in Alg2 this year is like pulling jawbones for me. Once in a while I’ve given up and told them the “rule” and they get so happy about that. “Mr. Petersen, why don’t you just tell us the steps?”

  2. You’re absolutely right when your say students think “why” means the rule rather than the reason for the rule. Like you, I was disappointed with how students answered these types of questions on tests. It requires a lot of modelling and discussion to break this perception of math as a set of rules to be recalled.

    For example, ask grade 8 students to explain why they think Austin is wrong when he says “5^6 divided by 5^2 is 5^3” and they’ll say something like “Austin should have subtracted because you subtract the exponents when dividing”.

    I haven’t given up on these types of questions since they give me valuable information about what my students really understand. I’ve had many students be successful on a quiz on integers by applying the rule “a negative and a positive make a negative”. It works, half the time anyways, when adding integers: (-6) + (+4) = (-2). By asking questions like “What should Matt say?” I can see when students do not have conceptual understanding – even if the answer is right.

    Also, I love the integrity statement. Usually when I leave the room it’s on a smaller scale. You know, a few students are writing a test afterschool and I’m dying for a Coke. 99 times out of 100, they don’t let you down. I also tell my students early in the year that when I run into graduates years later I never remember what percentage they got in my class. But I do remember if they were caught cheating.

    By the way, you might appreciate this:

    You’ll get laughs from some kids and others will be Googling trololo later. If nothing else, that’s got to make it a successful lesson.

  3. Sam

    A quick story – I hope to have the energy and focus for a more thoughtful reply to share soon. I am on dorm duty this weekend, so I should have time. Energy????

    So – last year (which was my first at my new school) I am talking early in the year with my Precalc Honors class about rational expressions and functions. The usual confusion over whether zero is bad in the numerator or denominator comes up. A relatively quick consensus on the denominator is arrived at by the group. At this point I select a student and say WHY are you not allowed to divide by zero. His response was – Because I was told that I can’t. I gave him credit for honesty but ended up having about a 10 minute conversation about the problems inherent in dividing by zero. I like that he was so direct, I was SAD that he was certainly not alone in thinking it is true because he was told it was true.

    1. Yup. I get the same thing everytime I push a little bit. I realize how surface-y their knowledge of math is. I suppose I’ve seen it in previous years (I know this is only the first week) but I guess I’ve put blinders on. I’m now hyperaware of this. Now my big question is: now that I see it for what it is: what now?

  4. I’ve also been noticing that doing things right takes about twice as long. I’m really grappling with how to deal with that. My department values covering the whole curriculum above all.

    1. Mine does too, but probably not as much as yours because you have regents always looming. This is tough for me, because the more I see about my students and what they *truly* know, the more I think that we need to really overhaul how we think about math. I used to not think that.

      1. To Kate and Sam,
        Maybe it is just pie in the sky, but one of the ways I have fought against the calendar to buy time for the ‘WHY’ conversations is to seriously trim down the nonsense review time at the beginning of the year. Almost every test I’ve dealt with seems to disrespect our students when it comes to thinking that they can remember anything. So, I buy time by flying through that. Two things happen – (A) Kids seem to realize more quickly that they have to get serious and (B) I can buy time for conversation about why and I can review in a more focused and meaningful way later on when specific old skills are needed. Regardless of regents exams, etc. I think that we all feel responsible to the students (and their next teacher) to ‘cover’ the necessary amount of material. In a class like Algebra II, I do not want to spend much time reiterating lines when they’ll need that time dealing with logs and exponents.

  5. “The Rule is the Why”-is very true when it comes to describing the mindset of most learners . In Singapore our students “do” maths without even attempting to dissect the meanings behind various mathematical operations-they just accept things as they are. Year after year, I still find myself explaining to kids why differentiating x wrt x gives 1.

  6. @Sam I really like the personal integrity statement. One of my college professors has a cover sheet for exams that we sign. I’ve been wanting to that with my exams, but haven’t done so yet. I like the way you’ve worded it and asked them to speak with you if they can’t sign. I also really like that you have the mention of witnessing any unauthorized assistance. Good stuff.

    @CalcDave I have the same problems with students. They just want the rules. I’m trying something a bit different this year to see what happens. I’m giving in a little bit and just giving them all the rules; so I’ve started calculus with derivatives and all the rules right up front. Later on I’m going back to limits and then really going through the theorems, etc. I’m hoping that this way they don’t have that feeling that I’m holding back on them when I do limits and theorems but that it actually serves a purpose and explains why all of the rules work the way they do. If you’re interested I’ll let you know how it goes when I get to all the “why.”

  7. Kate really hit the nail on the head for me here. So much value is placed on ‘finishing’ the year’s curriculum. I love that I have a class this year that’s ‘off the grid’. It’s an 8th grade pre-algebra class of students who failed or nearly failed 7th grade math. The goal is that they get above a C in Algebra as freshman and don’t have to repeat it (as prior to this intervention, nearly all students in their boat did repeat it). I feel like I finally have the freedom to do math right…i.e. taking twice as long…

    And like Sam, the longer I teach, the longer I think our breadth of standards for each course is totally whack.

  8. @Sam I was also wondering if you would be willing to share your google doc form. I’d like to know what you ask and also save myself the trouble of creating one (let’s be honest here, I’m all about not doing the work myself if at all possible.)

    Related to this, one thing I realized this year is that I always ask students about themselves on one of the first couple days of school, but that I don’t know any of them at that point. So none of the information really means anything to me yet. I’ve decided that this year I’m going to have them fill out a questionnaire thing, but wait until a month or so into the school year. This way I actually know the kids a little bit and the information will stick with me a bit better. Since it’s been about a month and you posted about the google doc, it seems like a really good time to try it.

  9. Just curious what kinds of answers you were expecting from the “why” exit ticket, and what work could lead students to the understanding you were after. My personal opinion is that teaching the “switch when you multiply or divide by a negative” trick is just that — a temporary trick that then ends up either forgotten or memorized. To understand why requires a whole different direction of thinkin’… :)

  10. Hi Bowen,

    It’s a good question. I was actually expecting them to say “that’s because that’s what the rule is.” Which is what they did (I was not expecting any of them to not know that multiplying AND dividing by the negative number applied to the rule). I am now having discussions with my class about what constitutes a WHY in mathematics. I showed ’em various examples from their class, and we talked briefly about them. Next phase: having them write their own, revised, responses in comprehensive form. Then they will hand them in, I’ll give ’em feedback, and they will be allowed to revise again.


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