Month: September 2009

Aiming for Understanding…

On Thursday, in Algebra 2, I aimed for understanding… and came up short. It was one of those lessons I went in excited to teach, because I knew I could get my kids to “get” it. And then, I didn’t. It wasn’t a bad lesson, and I don’t think my kids are the worse for it. But it just wasn’t that killer lesson I had hoped for.

The constraints:

I had 50 minutes to teach absolute value inequalities. I definitely had to get through “less than” absolute value inequalities (like |2x-1| \leq 15, but I wanted to at least introduce “greater than” absolute value inequalities (like |-x+15| \geq 2).

The plan:

1. Start off lesson with a fundamental question raised by an image (found at 360).

3463662579_a02378447c

2. Generate a basic understanding of what a “less than” absolute value inequality is, using a simple question to illustrate why there are many solutions. We would then fill in a number line from our solutions and then talk specifically about why our answer is not [-12,2], but actually (-13,3) (for the question below).

Picture 1

3. Point out the basic geometric interpretation, and tie that back to the initial question we wanted to answer.

4. Show how we can use this geometric interpretation, and a compound inequality, to help us come up with a method to solve these sorts of inequalities. (Work backwards.)

5. Formalize our method of solution.

6. Practice, practice, practice.

The outcome:

I only was able to cover “less than” absolute value inequalities, and I doubt that my students have a good understanding of why our solution method works. I do think my students will be able to follow the procedure though.

Where exactly did I fail? I failed in part 3 and part 4 of my plan. For part 3, I should have found a better way to explain the geometric meaning. My students didn’t “get” it totally. You can see part 4 of my plan executed here on my SMARTBoard slide:

Picture 2

As you can see, I tried to start from the compound inequalities and work our way to the absolute value inequality. At the end, there is this “ta da” moment which was actually more like “ta WHA?” They didn’t get what I was trying to show them.

And I don’t blame them.

A huge part of me doesn’t want to teach something without proving — or at least deriving by example — why something works. I feel like a fraud, like I’m teaching ’em magic instead of math, when I teach a method of solution first and then show where it comes from. But in this case, it would have gone over so much better if I had shown the method of solution, and then after practicing it a few times, took a moment to look at our work backwards to see why it worked. Talking about what each step means — algebraically and geometrically — backwards might have clarified things a bit.

I could have also designed the lesson in a totally different way. I could have worked off of our understanding of absolute value equations (e.g. the equation |2x-1|=5). Then we could have had a great discussion on how to find solutions to |2x-1|<5, focusing on why we use open dots at the solutions to the equality, and why we shade inbetween those dots. Now that I’m thinking about it, maybe I should remember to try this out next year.

If you want to see my entire SMARTBoard for the lesson, look below the jump.

(more…)

My Favorite Rant From Today

Yesterday my calculetes took their first assessment — a algebra boot camp to help us prepare for limits. We focused on rational functions, piecewise functions, and basic function transformations (focusing on exponential and logarithmic functions). I haven’t graded their assessments. You know what? I don’t know why I’m talking about that at all. This post isn’t about that. It’s about…

… what came next … today …

Each year — this is my third year doing this particular rant — I am always surprised that we can go weeks (this year, we went 2 weeks, last year it was 6 weeks) before I ask my kids:

WHAT IS CALCULUS?

We do all this algebraic review, students are settled into class and into their routines, and then POW. I hit ’em with this question.

And now I can say for three years straight, I have been met with totally silence. Followed by a student saying “the thing after pre-calculus.”

Commence rant:

“Why are you here? What is this course you signed up to take? What’s the purpose of this course? Why have you been working your whole time in high school for this? I mean, you took a course last year called pre-calculus. And yet, here we are, already well into our year in a course called calculus, and NO ONE KNOWS WHAT THE HECK IT IS!

Seriously? SERIOUSLY?

Well, good. Let me tell you what it is.

[Insert discussion of the “tangent problem” and the “area problem”]

It’s the study of the very small to learn about the very big — to learn about things you never knew you could know. Like the basis for much of physics. Given just a little information about something moving — a roller coaster, for example — and a knowledge of calculus, what can’t you do? And let’s talk about how little you actually know about space. What figures do you know the areas of?

[Commence students calling out things like “square” and “circle” — of course followed by figures they know the volumes of.]

How sad and pathetic is this? You can only find the areas of silly, putrid little pretty shapes. What about the real world? What about this shape [commence drawing of crazy shape on board] or the volume of this figure [commence drawing of crazy volume on board]. I mean, seriously, think about it. Look at a sphere. What’s the volume? \frac{4}{3}\pi r^3. Fine. Great.

[slight pause, building suspence]

WHERE THE HECK DOES THAT COME FROM? I mean, really? You have no idea. It just popped out of nowhere and you never questioned it. In this course, you’ll know, not just accept, that it is true.

Your lives are about to be changed. [1]”

[1] Okay, so I didn’t say this last sentence, but dang, I wish I had. You know kids, I don’t know if it got through to them. But I love doing this each year. It never fails to shock me how it is that these kids work so hard so they can take this vague thing called “calculus” — they’ve even taken a course called “precalculus” which, if the name were accurate, was meant to prepare them for this course — and they come in not knowing anything about it. What other course could you be over 2 weeks in, and ask the students what the course is about, and they won’t be able to answer?

Solicitation for Math Club Advice

This year our Math Club members are really intent on training for the AMC. They want someone in our school to break a score of 100 to move onwards to take the AIME.

Here’s the deal.

I want to help the leaders of math club find a way to do this. I don’t know how. We only meet for 25 minutes a week.

Is anyone out there a leader of a math club, that “trains” students for these types of contests? How do you do it? Literally, I’m asking for how you structure a meeting, and what kids are doing, and what you are doing during that meeting.

Also, if you as a math club adviser have any websites or books that you find invaluable, that would also be of great help.

I assume one of the important websites is Art of Problem Solving. After a ton of digging, you find that on that site is a list of AMC problems of years past, and solutions. What else ya got?

PS. One of my favorite math competitions from when I was in high school was the USAMTS. It’s a mail in math contest with 4 rounds, and amazingly wonderfully frustratingly challenging problems. So if you don’t know about it, and you have a super talented math star in your school, I’d check it out and (if you like it) share.

Hey Mr. Shah!! (Reprise)

So I got that really wonderful email in my inbox last week, so of course I emailed this student back. I was insanely curious why s/he was taking Calculus in college. Most of the kids in my classes aren’t really interested in pursuing a science/engineering/math degree. So I asked why.

Hey Mr. Shah,
I’m glad my email could contribute to a better day I feel like that’s a very nice accomplishment.  As for the why and how – I decided I had such a good time learning Calculus last year that I should continue taking it in college.  I’m taking [course#] which is a basic Calc course […].  I’m actually really glad that I took Calc last year because sometimes the way my professor explains things makes concepts harder than they really are.  We learned about continuity today and I was happy to see I remembered basically all of it.  I think what we did last year was really good and I think you did a good job explaining things and making sure we really knew what we were doing.  The addition of the youtube videos and things like using our fingers as slope meters helps make things more visual too.  The sad thing about being in a college level class is it’s bigger and the professor goes much faster so you don’t always have those “a-ha” moments (or the time to appreciate them) that we had last year.  I also told [other student from the class] about your email and he said he really misses our old class too and that he’s taking Calculus this year as well and it hardly compares.

Um, at least TWO of the students in that section of 7 are taking calculus in college? Really? SERIOUSLY? ZOMG! I kind of can’t stop smiling. Can I think of any better thank you than that?

No, no I really can’t.

Commence swelled chest.

(That feeling is actually fighting with my feelings of inadequacy and failure which I’m feeling now at the start of this year. It’s a strange place to be in. Like “I must have been good last year, so why am I doing not so hot this year?”

Hey Mr. Shah!!

In my inbox this morning:

Hey Mr. Shah!!
I just wanted to tell you that I’m taking Calculus this year and I have to say, it is just not as fun as it was last year.  My teacher doesn’t have fun pictures and she writes on a CHALK BOARD *gasp*.  I miss our class!  Hope you’re having an awesome year so far.

She’s not the only one who misses our class. This just made my week. Seriously.

Don’t Judge A Book By…

Today I decided to do my classic “tie with a polo shirt” look.

Photo 11 copy

Oh yeah, there was a hat.

One of the great things about my school — torn between being progressive and traditional, nurturing the whole child but with the looming vista of college admissions at the end — is that this tension is actually generative. It has its moments of maddening frustration, but it also allows for some pretty great things to happen.

Like what I wear to school everyday.

Let me explain. In my first week of teaching, I wore kakhis and button down shirts. (Which, by the way, I look awful in.) I had been told from everywhichway that your dress matters. That you need to dress older, to gain authority. I think even that horrible “First Days of School” bible gives the prescription of dressing in a suit everyday for guys, or something crazy like that. So I tried it out. And I noticed that some of my colleagues dress more relaxed, so each week I very consciously started dressing down. Button down shirts to polos. Kakhis to jeans. Dress shoes to grey sneakers. Polos to t-shirts. Grey sneakers to colorful Adidas.

No one commented. I hadn’t broken any norms.

Then, afterwards, I started adding to the outfits. A hat here. A scarf there. 5 pins one day. A cardigan with binder clips the next.

Why am I bringing this up? A couple reasons.

1. I was stopped on the street two times today by strangers because they needed to tell me “I found Waldo!” (One had alcohol on his breath.) And a few colleagues said something similar, and one said something about “Papa Smurf.” But he said it to me last week too, because I’ve been wearing this hat almost daily. It’s pretty kickin’, right? Okay, so I just thought it was neat and wanted to share.

2. I definitely think dressing up for teachers new to a school is important. Each school has different norms and they have to be carefully navigated early on. But I really, honestly think that people need to rethink this whole “you have to dress up to have authority” trope. Majorly. I get if putting on a suit makes you feel older, and that feeling gives you confidence, fine. But the suit does not make the man. I know I have control in the classroom. I can stand with a certain look and bring my class to silence in 3 seconds. My kids don’t take advantage of me. We have a good time. But I have control of the classroom.

And it’s not because of what I’m wearing. Wow, yeah, if that was a meterstick of anything, I would be an EPIC FAIL.

3. Most importantly, because of today, I was thinking again of why I dress the way I do. I’m not fashionable, I know. But I love my fashion choices. And it all started when I was in high school. My clothing was how I expressed myself. Badly, but it was. And in college, well, let’s just put it this way: one of my friends snuck in my closet and took out some of my signature pieces and was ME for Halloween freshman year, and definitely didn’t need to explain his costume to tons of people. (More impressive: my parents are from India, and his parents are from Korea. In other words, we don’t look at all alike.) Clothes are about individuality. And one benefit of being able to dress the way I do is that I get to express myself to my kids, and they see that it’s okay to express themselves in quirky, unfashionable ways. I don’t know if any of them take that away, that it’s okay to be yourself, by looking at me. But today, when I was pondering, I thought it might be nice if one of ’em did.

You know, teachers as role models, and all that stuff.

If you really want me to return full circle to the idea that the school is “torn between being progressive and traditional,” it is this: it is only because of this tension in my school am I allowed to express myself in this way, and still garner the respect from my students and from my colleagues that I think I have. 

Sam

Calculus Fail

xHsHnSTczqfu5oadlECRl4aDo1_500

I’ve been beating myself up, and it’s only day 4 of school. It’s sad because I just want this year to be the most fantastic year ever, and I wanted it to start so positively. But I’m feeling sad about my classes. I am okay being a teacher centered teacher for my Algebra II class. I really am. We have a curriculum that we are following, and we don’t have too much time to dawdle. Also, the kids are younger, so I feel okay keeping them mostly reigned in. And my MV Calculus class is going to be relaxed, though more challenging to teach than last year, because there are only two students (gasp!). That is a nice combination of student and teacher directed.

However, my calculus classes are a different story. I don’t have a set curriculum, which allows me a lot of freedom. I want to make sure that these students leave understanding calculus. I want them to see what makes calculus cool. What makes math cool.

So I promised both sections of my calc kids on the first day that my goal was to make math understandable to them. And I secretly promised myself that day that I would make math more interesting than they’ve ever seen it before.

It’s day 4 of teaching, and I feel like I’m flopping already. My classrooms are depressing (no sunlight in one; loud sounds of recess floating through the window in the other). I haven’t made one interesting lesson or one group/partner activity. I’ve just been up at the stupid SmartBoard pointing, talking, asking questions, going over homework. We’re just reviewing. And honestly, I don’t even really know where the students are in terms of what they know and what they don’t know. I call on random people, I walk around when they’re working on problems practicing in class, and still: not much clue. That’s not good.

I want to feel okay letting go this year and shift from having a teacher centered classroom to having times when the room is student centered. Where I’m not the one talking for most of the class. And I feel if I talk about that goal here, it’ll force me to keep it in mind. And be slightly more accountable.

As Alison Blank (@pvnotp) said on Twitter: “Maybe just try to be student-centered a little more often – like set aside one class every two weeks where you switch it up.”

Baby steps. And I’m going to try that. Even if it means something as small as playing a review game with students, or having some sort of hard problem challenge we spend the whole period solving together (like I do sometimes in MV Calculus), or making a guided worksheet to lead students through a concept. I should also remember that I can mix things up by asking for different forms of homework, instead of book homework, worksheets, etc. I can ask students to write a letter to their math-illiterate uncle explaining a concept we’ve been working on in class, or create a quiz of their own, or write a formal solution to a challenge problem. I can have students each work on different problems and make a SmartBoard presentation of their solutions for the class — and grade their presentations. Or even have students research the practical applications of calculus.

My brain to itself: Okay, Mr. Shah, keep these things in mind as things you can do instead of traditional classwork and traditional homework. And you just came up with these in the last 5 minutes. Imagine what you could come up with if you gave yourself 10 minutes, or (egads!) 15 minutes?

So I’m going to try to experiment a little this year in calculus. Be slightly daring. Put my foot in the water.

Optimism! Glimmers of hope!

If you want to see why I’m so dejected at the moment, you can see my SmartBoard presentation for my calculus class.

Setup. We’re in Algebra Boot Camp and we’re learning about rational functions before we start on the Limits unit. Up to this point, these kids have reviewed holes and vertical asymptotes, and have just started thinking about the domain of rational functions.

It’s not that the SmartBoard is bad, exactly. I actually think it’s pretty well thought out and organized. But you can see what my class would look like, by looking through it. (FYI, this particular lesson on domain, x- and y-intercepts, horizontal asymptotes, and sign analyses takes more than a day to go through. It will take 2 days to teach and 1/2 a day to pull it all together.)

I know I shouldn’t beat myself up too much. It’s only day 4. But I am. I’ve just been in a bit of a teacher funk. I’ll get out of it. All I need is some kid to say that they’re actually learning something in my class, and that they’re excited about it. I’ll get that.

Important Note. I don’t mean this to be a pity party. I don’t want pity comments – please. I only posted this because this is a place for reflection, thoughts, emotions, whatever. An archive of how I’m feeling today, so I can look back later and see how I’m evolving.

However if you have ideas on activities/games that work for you, things that break “the teacher introduces an idea –> teacher asks questions to develop idea –> teacher goes through example applying idea –> teacher asks students to practice a few problems –> start over” cycle, I would love for you to throw those in the comments below.