I have two calculus classes, and one of them is deafeningly quiet. I know or have taught many of these kids before, and they are just a shy and reticent bunch. Enthusiasm and a lighthearted atmosphere has always worked in the past, as has groupwork, but not with these kids.

I normally can break through this, but so far it’s still a little weird. Today, though, I had a glimmer of hope. Just a glimmer, but that was enough encouragement!

Today, we were talking about average and instantaneous rate of change, and how these ideas relate to the slope of the secant and tangent lines. And of course I wanted to relate this to position/time graphs.

I wanted my kids to see that having a position of an object was so… powerful. I always assumed that my kids in years past understood the awesomeness that came out of understanding that knowing the position at all times tells you the velocity at all times. That knowing information about the position of an object was giving us information about the motion of the object.

Maybe to us this is obvious. It is an idea only truly half formed for them. They have an intuition about it, sure, but that’s about it. I really wanted to drive home the idea that we could see so much from a position versus time graph.

So I showed them this graph [1]

and I ask them where the object is at various times.

Then I say, let’s actually act this out.

I start banging my hand on the desk nearest to me, BANG BANG BANG BANG… I hold up my other hand in a fist 6 inches above the desk, and say “it starts out at 6 inches.”

As I did this, all my students joined in by banging on their desks. (I didn’t ask them to.) They all put their hands up. They all joined in. It was a true glimmer of life! of community!

For three BANGS we held our fist still, and then after three bangs, we moved it down to the desk and held it there for another bang, when all hands shot in the air and then stayed there for an extra bang.

We acted it out. And then I said: let me tell you what makes calculus so powerful. It allows us to look at this graph which gives us just the position an object — that’s all it gives us — and it lets us understand the MOTION of the object. What we have here is information about where the object is, but we now can find out how the object is moving. We could actually act out the MOTION of the object from that.  It seems so obvious, but the connection is so deep.

Then we constructed the velocity-time graph. And I pointed to the [4,5] interval and said “oh, negative. what does that mean?” (Our fist was going down.) And I pointed to the [5,6] interval and said “oh, positive. what does that mean?” (Our fist was going up.) And I pointed to the [6,7] interval and said “oh, zero, what does that mean?” (Our fist wasn’t moving. It was high in the air, but standing still.)

I’m building the concepts here. We do the math later.

[1] Pre-emptive footnote: Yes, we had a discussion about how this graph could not actually be representing something in the world, because of the sharp edges.

One of my pushes this year is to get my Algebra II students to write math better. Last year I put “explain this” problems on a few exams and wasn’t so impressed with their responses. This year I am teaching my kids to write responses.

On their first assessment, I put a question similar to one we talked about in class:

Explain to someone who doesn’t know a lot about math why you can never find an which would make .

The responses were disappointing across the board. There were bits and pieces of gems, but nothing complete. Not a single student was able to construct a well-written response. Things I received included:

• The other side of the equation is negative, leaving no possible solution to the problem.
• You can never find x because the answer is negative and an absolute value problem with a negative after the equal sign is not possible.

So what I did was type up the following document and passed it out a few days after the assessment:

We talked about the vagueness of the responses, the use of pronouns like “it” and making references to “the other side of the equation,” and most crucial, the lack of reference in almost every solution to the original equation. How can you answer a question about an equation without even talking about the equation?

My favorite moment of the discussion this generated was when one student raised her hand and critiqued her own solution, and then said: “I wrote this and don’t even know what I meant.”

On the next assessment, without telling them I was going to do this, I threw the exact same question down. It was on. I saw my kids reread their responses after they wrote them, and really pay attention to their writing. Let me tell you: it all paid off. On this second round, most students got full marks. (On the first assessment, almost no one got full marks, or close to it, for that matter.)

Here are some random smatterings of their thoughtful answers:

• You could never find an to make the absolute value equation above true because you would have to subtract -5 from -6, which still gives you a negative number. . An absolute value equation cannot equal a negative number because absolute value is the distance from zero and is always positive [my correction: or zero].
• In this absolute value equation there is no solution because any number in the absolute value has to be 0 or a positive number. And if you subtract 5 from 0 or a positive number, there is no possible way that can equal -6. So there is no solution to this equation.
• An absolute value of anything can never be equal to a negative number, since it expresses a distance. When this equation is simplified, it becomes . If the ‘-1′ were replaced with a positive number, you could find the answer [for] . But since it is a negative, you already know that is impossible.

I am continuing to ask them to express themselves through writing. On that same assessment where I asked them to repeat the absolute value problem, I also asked the following two questions, to which I got some really nice writups.

The following two questions build upon each other. The solution to part (a) will very much help you explain part (b).

(a) Explain why without using your exponent rules. Explain it to someone so they can understand it simply!

(b) Explain why is true. You can assume and   are positive integers. Explain it to someone so they can understand it simply!

I still have to do more work with this, but I just wanted to say: it is worth it to talk with your kids about writing. One 15/20 minute conversation has already yielded great dividends for me.

The second day was a disappointment. Of the four talks I went to, three of them were bad. If they were a smell, I would be passed out. So bad. I actually felt angered by two of them, because the description was so fascinating that I felt betrayed. Talks in sheeps clothing.

I feel bad listing the three terrible talks, so instead I thought I’d at least point to the one good talk:

#201: Linear Functions: Much More than y=mx+b

The major thesis of this talk was that we might want to invert our traditional way of teaching linear functions. We tend to teach:

1.
2. make  table of x-y values
3. plot
4. connect the points. oh my gosh! a line!

But students find the equation to be the abstract part. The numbers and working with them is the easy part. So the speaker provided some ways to say let’s END with the equation and have it make sense to the kids, rather than START with the equation.

What was nice is that he started with some easy problems — that I couldn’t use in my classes — but then went to more advanced and more interesting problems — including one that would be great for an independent research project for a kid, and one that just blew my mind relating Pick’s Theorem to… systems of equations. Seriously.

But what was great is that he focused on student learning, and eschewed ed jargon and talked about why he made his choices for each lesson, and what his students got out of it. It was sweetness.

UPDATE: Commenter “m” below has prompted me to flesh things out a bit more. The easy part is with Pick’s Theorem… the speakers said he stole his connection to systems of equations from somewhere else… I suspect here! (He also showed a second way to derive Pick’s Theorem, which I am too lazy to do here. I remember first learning about this theorem in high school and spending days trying to prove it. I did eventually prove it and proudly showed my writeup to my math teacher.)

As for motivating simple linear functions, he basically had students engage in pattern recognition and play around with numbers.

White blocks in the first picture? The second picture? The third picture? What about the 5th picture? The 27th picture? He also talked about relating the blocks to tables to graphs really explicitly, as well as making explicit the connection between the “slope” (I put that in quotations because the speaker hates the term slope – he thinks it obfuscates) and the pattern, and the “y-intercept” and the pattern. His thesis was actualized: being explicit and very visual, and having students start with numbers and then come up with the equation out of these numbers provided a more natural and more deep way of motivating linear functions.

So right now I am sitting in Hynes Convention Center – room 109. In case you aren’t in the know (for shame!), I am at the National Council for Teachers of Mathematics (NCTM) conference in Boston. I just finished Day 1. I spoke to a total of three strangers, one of them who I recognized (and who recognized me) from the Phillip Exeter conference from this past summer. I don’t do well with meeting new people, which is such a shame in such a math-teacher-rich environment. But hey, three isn’t bad.

The sessions I went to today were:

#14: Identifying and Remediating Misconceptions [about CAS/TI-Nspire and developing numerical intuition]
#46: Show me the Sign! [about using sign analysis effectively in 9th, 11th, and 12th grades]
#79: Helping Students Read Math [about how to teach students to read their textbooks]
#142: Discovering Trigonometry [on how Exeter uses problem solving to teach their courses, using trigonometry as the vehicle to talk about that]

This was my first NCTM conference. Let me put this one piece of information about me: I don’t like my time wasted, so I tend to be critical of speakers [1]. I expected to really appreciate one or two of the sessions, and politely sit through the others. I thought I’d be inspired maybe once or twice.

You can see where this is going. I really, really enjoyed all four sessions. The speakers were prepared, and focused – for the most part – on concrete things in the classroom. It wasn’t about giving us the most difficult but interesting mathematical problems to work on. In other words, it wasn’t about mathematics. It was about teaching mathematics. We talked about topics and skills we work with everyday, and the speakers spoke about their approaches. None of them were zealots, saying “you should do it my way because it is the best.” It was “this is what I do, this is why I do it, and maybe you can use bits and pieces of what you hear here in your own classrooms.” I appreciated that.

I don’t know if I will have time to post about each individual session, but I will hopefully post some interesting bits later. (I said that about things I learned at the Exeter conference this past summer, and never did, though. So I can’t promise.) But maybe if (when) I actually apply some of what I’m getting to the classroom, I’ll feel more inspired to write.

(FYI, if you feel like you just absolutely need to know more about one of the sessions I went to, throw that down in the comments. You know I can’t deny you.)

[1] Yes, yes, I know our kids feel the same way, and we should always keep this in mind when we enter a classroom.

I don’t know what it is but thing have been so busy in the last two weeks that I don’t remember a time I left school before 7pm (most days 8)… and I continue doing work at home until 10 or 11. I honestly don’t know why. I’m not getting any more work done that I have previously. I’m just grading and lesson planning. Maybe it’s that all my free time at school is now being taken up by meetings with students. (Note to self: do you really want to encourage students to seek out help? [1])

It’s a little bit crazy. Actually totally wackadoodle crazy. I mean, anytime that I don’t read my “back twitter feed” (the tweets that happen when I’m in school or working), it’s just wackadoodle crazy. And I’ve had a couple of those days where I thought “Can it be winter vacay already, please, Mr. Calendar, because I seriously need to lie down and sleep for hours.”

Unfortunately, I won’t have time to do that for a while. Because today right after school I am getting on a bus to go to Boston for the NCTM conference. I don’t return until Sunday evening. And what do I have waiting for me on Monday morning? Yes, the substitute is administering exams in 3 of my 4 classes. So I totally win the prize by having tons of stuff on my plate when I return.

AWESOME.

Followed quickly by the end of the quarter, and comment writing time.

Please, sir, I’d just like some sleep.

[1] In case it wasn’t obvious, sarcasm. Obviously.

A couple o’ days ago, I posted a question about how to come up with a set of parametric equations equivalent to an implicit equation. It seemed to me like the general solution to this broad question would be like differential equations. There would be certain tools you could pull from the toolbox, once you saw what “kind” of equation you were dealing with. There isn’t a one-size-fits-all algorithm for solving differential equations (at least, not that I learned).

I got to thinking… didn’t I learn how to convert between parametric and implicit equations some time years ago? And, in fact, the answer was: yes. I took a class on algebraic geometry. The book we used (one of my favorite math textbooks when I was an undergrad) was:

The way this course was designed (it was officially a “seminar”) was that each day, two students would “teach” a section from the book to the rest of the class. We somehow made it through the whole book. It was a great experience, having to learn a section well enough to teach it the my classmates. The class was — however — a bit of a failure. The desks were in a row, people rarely asked questions, and no one engaged with each other. (Much like most of my college math classes, actually.) For something so student-based, it was strange that I didn’t make a single friend in that class. Plus, there was no “teaching” us how to teach well. Some some of us were great teachers, but most of us sucked. I can’t say what I was, really. I don’t remember. Regardless, I remember thinking: this textbook was incredible because I pretty much had to teach myself the subject. (I have to give major kudos to the instructor because he forced me to learn an entire course by reading a textbook.)

So upon reminiscing about this class and this book, I pulled it down from my bookshelf.

I am so dumb.

I will revise: I am so dumb now.

I look at the pages, and read theorems like Theorem 9 on page 241

“Two affine varieties and are isomorphic if and only if there is an isomorphism of coordinate rings which is the identity on constant functions.”

and see words like “Nullstellensatz,” and wonder how I ever got to the point where this stuff made sense, and how I got to the point where I see a bunch of gibberish now. Seriously, it’s disturbing. I mean, I don’t expect to be able to pick up a book I learned from years ago and know everything in it, but I do expect that it is in a language I can read.

I’ve figured it out. I am Charlie in Flowers for Algernon.

I don’t know how to feel about the loss of my mathematical mind, besides sad.

Maybe I’ll try teaching myself some math again, to either prove to myself I still have it somewhere in me, or to know that my brain has truly atrophied to a giant anti-intellectual morass.

Today, actually just an hour or so ago, another math teacher asked me if I knew a way to parametrize the following:

It is also known as the Folium of Descartes and looks like:

Her purposes was just trying to find a quick and easy way to graph it on the calculator. It was just a small, unimportant question. She didn’t need to know the answer, if it wasn’t easily doable. But to me, I needed to know! How do we find the parametric equations which define this? I just don’t know. The answer I found online is:

and

And it’s pretty easily verifiable when you work backwards with the parametrization.

Is there something I’m missing? Is there a method to working from implicitly defined 2D functions to parametrizations for them?

Last Thursday was Parent Night. Also fondly known as The Longest Day You Experience In The Entire Year. Yes, indeed, on this day I woke up at 6:30am and taught until 3:10pm, followed by an hour of tutoring and a little bit of working, followed by running out to grab an early dinner in the ‘hood with some colleagues, followed by Parent Night! Which concluded, for me anyway, at 10pm-ish. Note that there is no time to lesson plan for the next day. Which is why I worked backwards and planned my classes so each would be having tests on Friday. Genius?

Check.

For those without Parent Night, it involves, in short, parents arriving at 6:30pm and attemping to follow their child’s school schedule — spending 10 minutes in each class (and 5 minutes getting lost between classes). I think it’s a very good thing we do. As one of my colleagues who retired a couple years ago said: “You know, it helps the parents know there isn’t a crazy person watching their kid for 50 minutes each day.”

Check.

My first year, I was told the two tricks for the night:

1.) Do not like parents corner you to talk about their individual child. If it does happen, either say “I’d love to talk but I don’t have my gradebook in front of me. Can we set up a time to talk by phone or in person later?” or “I generally don’t talk about individual students tonight, but I’d be more than happy to sit down and talk with you sometime soon.”

2.) When you’re “teaching” your class, talk for the entire 10 minutes. If you go under, and parents start asking questions, the night can turn very quickly if you have one upset parent.

My third trick to surviving the night without going crazy:

3.) Accept that parents are going to be on their PDAs while you’re talking. It’s annoying, but not worth getting riled up about.

Of course, although I tried my hardest, I got drawn into 4 conversations about individual students. You know how parents are, so sneaky. They lull you into a sense of calmness, and then THWACK: “Mr. Shah, we really liked your presentation. I really liked calculus when I took it in college. Stu is really excited about your class. How is Stu doing? THWACK!”

If you wonder what I spend the 10 minutes doing in each class, I give a SmartBoard presentation. Of course, my presentation is 15 minutes if I’m talking fast, so I basically edit to 10 minutes based on the cues my parents give me (if they’re stoic, I skip over the jokes; if they look interested in the structure and content of the class, I speak more in depth about that).

My calculus presentation is below.

Of course, this year, I came home sick. I barely made it through Friday. And I slept all of Saturday and Sunday. Yes, even though I know germs get you sick and not Parents Nights, I blame you, Parents Night, I blame you.

In my most recent calculus classes, I wanted to show my kids their first “not nice” functions. After being introduced to how to find limits graphically (fancy way of saying: looking at the graph of a function) and numerically (fancy way of saying: using the graphing calculator’s TABLE function to guesstimate limits), I wanted to have them think about what they learned.

I had time to show one class that these methods aren’t foolproof — that the calculator can lie to you, and make you think a limit is 3 when it is in fact 3.004, or that it can’t graph things when numbers get too large or too small. So they have to be careful. And that we will be learning algebraic methods to do limits. But for now, they need to use their brains and wits.

So I divided them into groups of 2 and 3 and had them use whatever methods they wanted to find:

I made them each draw a sketch of the function, write down an appropriate table of values, make observations about the function, and then decide on an answer. (In one class, I had each group turn in their findings, and then I photocopied them and distributed them and had the class talk collectively about the results the next day. In the other class, we didn’t have time for this, and we just met up together as a group to talk.)

FYI, the graph is here.

It was great. Students were debating whether the craziness was a function of the calculator lying or if that actually was what the function looked like. They wondered if the limit was 0 or if it was “does not exist.” They noticed that the function starts to oscillate more and more rapidly as approaches 0. They noticed that it bounced between -1 and 1. It’s not an easy question to solve with this information.

When we came back as a group, we talked about their observations and conclusions, and documented them on the board — so everyone had the same notes. Then I said: “so… one of you said that the function is crossing the axis more and more as is getting closer and closer to 0. Can we be more exact? Where does the function cross the axis?”

Of course my students didn’t know exactly what to do. We got to the point where we knew we had to solve:

But then they were stuck. So I guided them through it.

I asked: “When is

We generated:

We then said:

We went through solving one of the equations for and saw that we needed the reciprocals…

We concluded:

I then asked: So what? Why did we do this? Don’t lose the forest for the trees…

Finally, we converted those numbers to decimal approximations

and saw that the zeros were getting more and more frequent as we approached 0. No matter how close we came to zero, we were still going to be bobbing up and down on the function. And crucially, we’ll be bobbing up and down between to .

We then talked about what a limit means again… what the value of a function is approaching as the value gets closer and closer to a number. Using that informal definition, I asked them if the value of the function was approaching some number as was approaching 0.

At this point, most of my kids had that “a hah” moment.

I am definitely doing this again next year, but perhaps more formalized. I might generate a list of good conceptual questions to walk them through this more systematically. One such question: “How many zeros are there in the interval (.5,1)? How about (.1,1)? How about (.01,1)? How about (.001,.1)? How about (0.0001,1)? And finally, how about (0,1)?” Another such question: “How do we know the function will bounce between and ?”

Also, maybe next year, I’ll couple it with an analysis of the function:

The function behaves similarly (crosses the axis more and more rapidly as approaches 0), but the limit in this case is 0. You can see it in the graph easiest.

So if anyone out there is looking for something to spice limits up, you might want to really go in depth into these functions. They are often used as exemplars, but rarely investigated.

Disclaimer: This post is snarky, a little exaggerated, and in no way is meant to be anything more than me just hamming our meeting up.

We like having people talk to you. So says my school. Which is why they bring in consultants. And honestly, although it may sound like kvetching, well, it is. But it also has a larger point.

We have consultants for curriculum mapping, for drug and alcohol training, and just yesterday, for creating an “anti-bias” curriculum. I want to talk about yesterday.

My school is amazing in many ways, and one of the ways it excels is with its diversity work. Our diversity coordinator (yes, we have a full time person with that title) has been pushing for us to move to the last frontier: take what we’ve already done so well at and start applying it to the curriculum.

I can definitely get behind that.

So after school we heard a consultant tell us how we can do that. And throughout her whole talk, I did what I try never to do with the speaker in the room. I rolled my eyes. A lot. I might have been having eye seizures. Yes, it was that bad. Maybe worse than the consultant who told us to “imagine we were on a plane… but the plane wasn’t built… and we were building it while it was flying…” In that instance, we were getting an analogy for, well, maybe how to write terrible analogies. Who really knows? All anyone remembers from that consultant is the bad analogy, not what it was for. But that was then. Back to my eye rolling speaker.

So this speaker comes to us, in my opinion pretty unprepared [1] and with bad powerpoint skills, repeats “that’s why they pay me the big bucks,” and spews off either commonsense as Great Knowledge Being Imparted or pure tripe.

I know you’re just salivating now for an example of pure tripe. And it’s juicy. Her math example. How to make math anti-bias. (And, I should preface, that in the beginning of her talk, she mentioned the questions we may be having, like how to make polynomials anti-bias, and then later when asked what her answer is, she responded “I don’t even know what a polynomial is.”) Here it is:

Passive curriculum: “There are 200 miles between point A and point B. Driving at 15 miles per hour, how long will it take to get from point A to point B?”

Multicultural curriculum: “Jose lives 20 miles from school. Driving at 25 miles per hour, how long will it take Jose to get to school?”

Anti-bias curriculum: “Jose needs to get from his home to the far side of the city. If he goes the most direct route, how long will it take him? But Jose can’t go in certain parts of the city because they are dangerous. So how long will it take him if he goes an alternative route.”

No, seriously. SERIOUSLY. SERIOUSLY! I can’t tell you how dead serious I am. It was the most unbelievable thing I’ve ever heard.

Later, when pressed on it, the consultant admitted the problem was bad and then changed it to some problem with numbers of people on a bus and the probability that someone black would get out somewhere… wait that sounds wrong… by then, my head was expelling smoke and I couldn’t really focus…. but it was something like that.

Individually I talked with her and turned the screw by really giving her the lowdown of what math in a high school is, and how statistics easily lends itself to these sorts of social justice questions, but much of what we do doesn’t. And that I don’t see why we would want it to. It felt somewhat vindicating to hear her concede and say that the math teachers couldn’t really make an anti-bias curriculum.

She was — in my opinion — playing a giant joke on us. Giving us a parody of what an anti-bias curriculum might look like. You know, to get us to rebel and talk amongst ourselves about what we really think one would look like. Right? RIGHT? Hey, if that’s the goal, it worked. I have had three conversations about anti-biased curriculum since, all initiated by the “do you believe that consultant yesterday?” If that wasn’t the super double secret plan, then I can’t believe the school just spent goodness knows how many dollars to waste my time.

I know this is a rant. And I need it to be a rant. Because even though I love my school, sometimes I wonder what they are thinking. Knowing nothing about an anti-bias curriculum, I could have given a better talk. After my second year of teaching. That’s what infuriates me.

I know a lot of people I talked to were upset with the speaker. I don’t blame the school for working on this initiative. I love that we’re a school that actively works on diversity issues. The only thing I ask from the administration is that next time we meet, they agree with us that the consultant wasn’t up to snuff. Just hearing that would make me feel the administration understands us teachers and is on our side and we are working together.

My larger point: consultants are soooo overrated.

Signing off,
Sam

[1] Example: One of the go-to examples of how to create an anti-biased curriculum was a lesson designed around “To Kill A Mockingbird.” And the speaker said she hadn’t read it in years, and had cagey phrases like, “if I remember correctly…” If this is your paragon example, shouldn’t you really have read the book and know it pretty well?