## Volume in Calculus: Conceptualizing before Formalizing

One of our PD sessions in the past was about how to support students with learning differences. One of the points that the presenter made was that most pedagogical tools that you would use the better serve these students are great tools to reach all learners. This struck me especially because I teach almost entirely students for whom English is their second language, and sometimes when I do something specifically to help students with the language of mathematics I come to larger and more general pedagogical understandings.

For example, this past week, I introduced solids of known cross-section in AP Calculus in a way that I thought would ease my students understanding of the tricky language involved in the problems, but what I ended up doing was really effectively let them develop their own conception of how these solids are formed and THEN interpret the AP problem language and integral notation in those terms. Conceptualize and then add mathematical formality to their own conceptual framework.

Here’s how it worked. I put 4 of these solids out around the room:

1. First, I gave them 1-2 minutes to SILENTLY write down in bullet points how they would describe to someone else how the solid was formed.
2. Then I gave them 2 minutes to share ideas in groups.
3. Then I cold called on 7 or 8 students via a deck small cards with their names on them (which is by far my new favorite teaching tool). After I called on some students, I called for volunteers with any other ideas.
4. LAST, I asked them to flip to the back of the paper and read the actual description.

During the “share” part, students said some of the craziest, random stuff, but most of the important parts of the description were said by various students. When it came time for them to read the description, at first they were like “whoa” because the language is still a bit daunting. But after a minute or so of close reading, they connected everything in that description with things that they themselves had said. So when it was time to do the actual integral, the intermediate notation I use made 100% sense:

So general pedagogical moral of the story? Letting students conceptualize something on their own before bringing in mathematical language and notation makes it more likely that the notation will aid in their understanding rather than provide another hurdle in learning.

A teaching fellow (a first year teacher) was observing my class (and has been observing periodically throughout the year). Afterwards, she remarked that she felt this was one of the most effective 10 minutes of the year, and I agree! And I think 10 minutes on this (instead of just 1 minute reading the question) will save lots of time in the future. Next week, I hope to try the same strategy with solids of revolution!!

## Whiteboard Experiments: Modified Mistake Game

I have used Mistake Game a lot in class. Students write up the solution to problems on whiteboards and purposely make a mistake in the solution. Then they present their solutions to each other, presenting their mistake like they meant to do it. Then, students ask thoughtful questions to try to find the mistake.

This works great with topics that are conceptually rich, but less so in topics that are more mechanical, where mistakes tend to be a bit harder to see and are less rich to talk about, like implicit differentiation for example. I did a modification of the Mistake Game that worked really well for this:

1. In groups of 2-3, students write solutions to a problem on a large whiteboard. After checking their correct answer with me, they go back through and make a mistake in their solution.
2. Students then flip over the sheet I gave them that had their answer and write what there mistake is, kind of like a mini answer key.
3. Groups then rotate around the room and try to find the mistake in the solutions in front of them. Once they find the mistake and check their answer with what the group wrote, they move on to the next board.

I wanted to train them in the art of looking over a solution and checking its correctness, and I think that this did that well. Compared to the mistake game, I felt like more students were active at any moment, more students could carefully follow complicated work, and it took much less time (20 minutes as opposed to 40)… but we also didn’t have the great mathematical discussions that we normally have during mistake game. I guess it really depends on the topic at hand which version is more appropriate, so I’m definitely going to keep this one in my teaching toolbox.

## Whiteboard Experiments: Practice & Reflect

I tried something new in class this week that I think solves a few random problems:

1. Sometimes, when working on whiteboards, one student hogs the marker and does a lot of the work (and thus the learning)
2. With whiteboard work, students don’t have anything in their notebooks to study later
3. When we practice things like derivatives in our notebooks, I feel like their notes become almost useless because it is a mess of 15-20 examples.

________________________________________________

## Practice and Reflect

I put the whiteboards out on the desk and left them there the whole period. We were learning the derivatives of exponential and logarithmic functions of bases other than e so I wanted to go back and forth between the whiteboards and their notebooks a few times.

After teaching them a derivative rule as a whole class, I gave them a sheet of 15-20 problems that definitely got more difficult as they went along. I gave them 12 minutes to practice (I put a timer on), and they worked on the problems with partners on the whiteboards, which gave them a chance to discuss, erase mistakes and see problems in large format with different colors.

Then, I asked them to put the markers away and open up their notebooks and gave them 3 minutes to reflect. I told them they could do whatever they want with this – copy down a few problems that were tricky, write down some things that they want to remember, write down steps for the problem. With this, I feel like their notes were a bit more focused and useful. I also felt like the whole routine was efficient, in that it kept a vast, vast majority of the students moving and engaged. I’ll definitely try this again.

## An Anchor Problem for Riemann Sums

I like to start most new units in Calculus with an “anchor problem,” a common sense, every day problem that motivates new techniques and serves as a base that you can constantly refer back to. Some that I have used in the past, to varying degrees of success, are Infection for Inflection, Your Speedometer and the Intermediate Value Theorem, and Predicting Stock Prices with Differentials.

For Riemann Sums, and integration in general, I use the question that really inspired integration in the first place: how do you find the area of an irregular shape? I tell my students:

You work at the glass company. You are given the task of replacing all the glass on the front of this beautiful building, the Duxford Aviation Museum. How much glass do you need? All we know is that the building is 90 m long and 18.5 m tall in the very center.

(This task was partly inspired by this post from Shawn at ThinkThankThunk).

(isn’t this building beautiful??)

I have it printed on two sheets of printer paper for every group (so big enough to draw on and mark), and I give them 10 minutes to come up with an estimate. Every group writes it on a piece of paper, and then I put it in an envelope. About a week and a half later when we learn the definite integral, we calculate the actual area (using a parabola fitted to the top of the building) and the winner gets…. well nothing. But I announce it at least?

Most students struggle a bit at first and then eventually just start to try something. Some students try some sort of bizarre modified equation for the area of a circle (which I always find really interesting), some turn it into triangles, but most use the maybe-not-that-subtle hint that the window is broken up into square panes.

Right after they are finished making their predictions, we discuss. I ask them what their strategies were and how they could have made their predictions more accurate. I try to get them to come up a couple of points (that sounds manipulative):

• ### The smaller our shapes are and the more of them there are, the more accurate our estimate would be. In fact, if we could use infinitesimally small shapes, we could be perfectly accurate.

I think that this activity really shows them how difficult the problem that we are trying to solve is, and primes them to know why we set up Riemann Sums the way that we do, but to be unsatisfied with this solution to the grand area problem. Prepped and primed for Riemann Sums, but with some foresight to know where we are going.

## Whiteboarding Mode: Simultaneous Show and Tell

Side note: Simultaneous Show and Tell is a terrible name for this whiteboarding mode (because it kind of sounds like a lot of whiteboarding). Forgive me, I cannot think of anything better. So… propose a better name?

[update 11/25: Andrew in the comments suggested "Function Iron Chef" which is definitely the winner. That's what this whiteboarding mode is called now]

Students are in groups of two at a whiteboard with a VERY LARGE set of 3 X 3 axes drawn up on the board. They are sitting in a U shape so that if everyone put up their boards, every student could theoretically see everyone else’s. I put up a prompt like this:

Draw a function such that…

• $\lim_{x \to -2}=3$
• $f(-2)$ does not exist
• $\lim_{x \to 1}$ does not exist
• and $f(1)=-3$.

I put the timer on. Students are given a few minutes to draw a function (any function, lots of correct answers!) that fit the prompts. Then, at the end of the time, everyone puts their markers down and puts their board up. We spend a minute silently looking around the boards to look at everyone else’s work. Then, after a minute is up I allow the students to ask questions of each other (i.e. not just say “THAT ONE IS WRONG”). If they don’t ask questions about some that are suspect (or some that are totally correct), I will ask questions at the end to talk about specific boards. We then do 5 or 6 other rounds like this.

POSITIVES: We have done this so far with limits, continuity vs. differentiability and will do it in a few weeks with graph sketching – I think that making them do things the other way around, making them create (instead of just identifying limits or whether a function is continuous) really forces them to think harder. I also like this because when students have to show their work to their classmates, they often put a little bit more focus into making sure they are proud of what they have (and just about every student is engaged in the process, especially if you make them switch markers). I also love times to showcase mistakes as part of the learning process - we try to be as open and supportive as possible in correcting the boards. Lastly, having a discussion in a math class is always a really nice change of pace.

ISSUES: Students can get a little crazy during the discussion process and some can phrase things negatively. Not all students are good at following along verbally when discussing, and will wait for others to point out mistakes in the board. A few times the whole thing has taken a long time with all the transitions, but it has gotten better every time. I’m not sure how the weak students feel about this activity (having their work showcased and critiqued). Also, I’m not sure that this type of activity would be great for anything but a topic where the students already have some fluency and mastery.

## Tuesday, November 13, 2012

I am a math teacher at a boarding high school right outside of Amman Jordan. This is a day in my life.

## Tuesday, November 13, 2012

7:00 – Wakeup. The nice thing about living literally 2 minutes (walking) away from classes is that my wake up time is a little later than everyone else’s. But as I walk out of my apartment, a student grabs me to unlock the storage room – the downside of living in such close proximity?

7:30 – Breakfast. Sometimes, I eat 19 meals a week in our dining hall, which saves a ton of time and money (Why only 19 you ask?  Well, when you wake up at noon on the weekend, there’s really only time for 2 meals). Tuesday is bagel day which is my absolute favorite (ah the small pleasures in life)! This morning, a student asks me to tie their bow tie for them, which is actually a fairly common occurrence. I have to say, bringing the bow tie to our school has been one of my proudest accomplishments.

8:05 – Class starts. Except that I have two prep periods in the morning on Tuesdays, which makes life kind of nice. This year, because I am head of one of the dormitories, I only teach 3 classes, which makes for tons of prep time during the day (but lots of stuff to do in the evening). This morning, I made tests for my non-AP Calculus class and began to cobble together review materials for my AP class for our upcoming final.

10:45 – 12:20 – Back to back to back classes. I have three 45-minute classes in row, switching between non-AP Calculus and AP Calculus. Normally I find only having 5 minutes between classes stressful and exhausting, but today was pretty relaxing as my AP class was working hard on a packet of Related Rates problems, and my non-AP class was reviewing for a test the next day.

12:25 – Advisee Lunch. Two days a week, we eat lunch with our advisees (and every other day is formal, rotating assigned seating lunch – I have duty for one of those days). My advisees are four freshman and two sophomores from the US, Saudi Arabia, Jordan and Nigeria. They are an awesome group of kids, and a real breath of fresh air from the jaded older students (who are the only ones I normally interact with). I really love spending time with them, mostly because I feel like some of the things they say should be published in a book.

1:05– Class meeting. I’m associated with the twelfth graders so I trudge into the Lecture Hall with the senior class. I feel like my week is really filled with meetings. We have school meeting 3 times a week for 5 minutes and once a week for 45 minutes, class meeting once a week for 45 minutes and advisor meeting once a week for 45 minutes. Today, the class gave announcements and then watched a TED talk.

1:55 – One more prep period. That’s right, 3 prep periods in one day… I used this one to make reassessments for my Standards Based Grading system. Right now, I’m averaging almost exactly half of my students reassessing every day (I only teach 45 total, but still… making 2 standard checks each for 22 kids every day is ridiculous and takes forever).  Luckily this is an end-of-the-term-my-parents-will-see-my-grade-soon phenomenon.

2:45– Arabic Class. Three times a week I take Arabic class, which they offer to the ex-pat faculty (a little less than half of our faculty is ex-pat, and about 15% of our student body is non-Arab). I love these classes. It is fun to be a student again, and we learn a lot. I’m in the most advanced level, so we usually just sit around and talk in Arabic for 45 minutes about really random things. Last year, I took class with the students too – I took 9th grade Arabic – which was quite a trip. It’s funny to me that teachers are really the worst students. We don’t do homework, we’re always late for class, we forget about tests etc etc. Bust at least we’re enthusiastic?

3:35 – Reassessments. 23 students reassessed today, crammed into our math classroom, which fits about 18 comfortably. I find these times so stressful – I sit up front and correct their reassessments when they are done, but a line starts to build up and then I feel like students who are still taking reassessments take advantage of my attention being diverted to cheat. It’s frustrating and stressful, but I’m not really willing to give up the learning opportunities for many just because some people are complete jerks.

4:45 – Faculty vs. Student Swim Meet. Normally we have co-curriculars in the afternoon from 4:45-6. I advise the newspaper, and we meet once a week (which is an incredibly light load for co-curriculars at my school). But the co-curricular season ended last week, so this week we had a faculty vs. student swim meet! One of the boys in my dorm talked so much smack to me the night before, it was unbelievable… and then I completely crushed him in the water, muhahaha. Overall, it was very fun event, and one that must be repeated because we ended up losing to the students 75-72.

6:30 – Dinner. Again, my meal occurred at the dining hall. The food wasn’t very good, but I put up with it to avoid shopping, cooking and cleaning. Sometimes, I just don’t want to see students at night and get frustrated being there in the thick of it, but other times it’s kind of fun to be eating dinner at the table next door to some of your Calculus buds (I’m sure that’s how they think of me). This is when my day usually ends unless I have duty…

8:00 – Meeting with a student. One day a week and one weekend a month, I do evening duty in the dorm from 7:45 pm until 11:15 pm. Those days are long, and not much gets done during the duty time so you have to really plan well to get your work done. But even though tonight is not my duty, two students needed to schedule a makeup quiz so we did it at night. I was feeling sick because I have a sinus infection, so while the student sat at my kitchen table doing the quiz, I was lying on my couch with my hood over my head listening to RadioLab. My student must have thought I was nuts, but I guess that’s what they get for invading my house during chill time. Another student came at 9:00, so I didn’t really get time to myself until around 9:30.

9:30 – Colbert. Daily Show and Colbert come on at 9 and 9:30, which is awesome. I try to watch one every night. It’s sad, but it’s one of the best ways of keeping up with American pop culture.

10:00 – Finish prepping. I didn’t finish my test earlier, so I spent about 45 minutes putting the finishing touches and sending it off to our copy dude who prints our copies for us (amazing luxury).

11:00 – Read. I always read before I go to bed, every night, no matter how late. Right now I’m reading One Hundred Years of Solitude, which I’m liking enough, but is going really slow.

11:30 – G’night. I’m pumped because this is on the slightly early side for me.

One of the best things about being a teacher is that whenever you had a bad day you get a chance to do it all again better, but one of the frustrating things is that whenever you have a good day, it’s almost like there’s no time to stop and celebrate your victory. Moving forward, onward and upward… a new day begins.

## The Dead Puppy Theorem and Its Corollaries

To preface, I normally celebrate mistakes in my classroom as a part of the learning process. But there are some things that really speed the progress of my receding hairline, a large of percentage of which involve bad algebra. I saw this from @lustomatical and thought YES. This is what I need to get my kids to stop distributing powers over terms that are added, and “canceling” things willy nilly, and not respecting the trig functions as operations. Let’s concentrate on the Calculus! So I made these posters for my classroom:

Enjoy. I know my students will, and it will actually give us a funny and memorable way to talk about and avoid these common algebra mistakes.

The other thing that I showed them today to get them to stop just playing around with letters while doing Algebra is the following, which I believe I picked up at a summer workshop:

They literally laughed out loud at this. I said (in a funny, not mean and not sarcastic way), “You think that’s funny?!?!? This is the kind of stuff you guys do on quizzes. When I am correcting your work I sit and laugh and laugh and laugh at the crazy things that you do! No more crazy algebra!”

How do we stop/prevent crazy algebra mistakes besides carefully and repeatedly addressing them when they happen? Any ideas?

## Teaching Through Concrete Examples: The Intermediate Value Theorem

I’ve been getting pretty into cognitive science lately. I realize some of it is useless, and a lot of the rest of it is made up of kind of common sense things once you really think about it, but regardless, I have found it so helpful to put scientific names and research to intuitions I have in the classroom. One of the ideas that I have really liked (from Daniel T Willingham’s Why Don’t Students Like School?) is that we learn everything by connecting it to things we already know, and much of what we already know is concrete. Thus, the more you can teach through concrete examples, the more likely students are to learn the material.

### EXAMPLE: Speed, iPhone prices and the Intermediate Value Theorem

This year, while teaching the Intermediate Value Theorem in AP Calculus, I did not start with the theorem itself, as I always find that language so intimidating for what is actually a simple idea. Instead I started with this:

I showed them a video of a speedometer that cuts out for about 10 seconds in the middle (ah, you’re dizzy and you pass out for a second at the wheel!). Before the cut out spot, the car was going 60 mph, and after it was going 100 mph. I then asked the to tell me:

1. What was a speed that you are 100% sure that you must have gone in the time in between? Why?
2. What was a speed that you could have gone in the time between, but you aren’t 100% sure? Why?

We talked about this for a few minutes, letting the students argue a bit about their thoughts and came to an agreement as a class. Then I put up a new picture that showed the original iPhone prices at some intervals. It started at $599, a few months later was$399, and then two years later was \$99. Then I asked very similar questions:

1. What was a price that you are 100% sure that the iPhone must have had in the time in between? Why?
2. What was a price that the iPhone might have had in the time between, but you aren’t 100% sure? Why?

Again, I let them argue for a bit and discuss. After we had settled on answers, I asked what was different about the situation, keeping in mind that we had already discussed continuity in the class, but I had never mentioned this in this situation. Students said wonderful things like “To get from one price to another, the iPhone doesn’t have to pass through the other prices” and “Prices can can jump whereas speeds can’t” and I let them continue to do that until one student finally realized “Speed is continuous, whereas price is not!”

Prepped with the ideas of theorem, we took the speed situation and translated it into a mathematical theorem before looking at the actual Intermediate Value Theorem. It took about 10-15 minutes of class, which was well worth having a strong conceptual understanding of the theorem. Students still struggled mightily with proving anything with the theorem (as they have in proving anything mathematically both this year and in previous years – any advice there?) but the conceptual development of the idea was not only quicker, but I think stickier.

Isn’t that better than starting with this?

## Let THEM Figure Out the Power Rule

I have been reading and enjoying (though not fully buying everything in) Daniel T Willingham’s book Why Students Don’t Like School: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. One of the ideas that I think is really useful in planning instruction is that humans are wired to enjoy learning – some scientists believe that the brain releases a little bit of dopamine every time we solve a problem. We actually physically get pleasure from solving problems.

As an example, check out these two word picture puzzles (figure out the common expression indicated by the words and their placement):

Which of the two puzzles did you enjoy more? If you’re anything like me, or most human beings, you didn’t really enjoy the one that had the answer right above it. Even if you didn’t figure out the other one, you probably at least thought about it more than the other one (though Willingham points out that the physical response only occurs when a person solves a problem). How often do we give the answers to the riddles first in math instruction?

Here is a rule and here are examples of every type of problem you will have to do with it, now do problems like those even though you kind of already know the answer.

### An example of posing math as a riddle instead:

It took a few days for students to learn the power rule this year, as opposed to me just writing $f(x)=x^{n}$ so $f'(x)=nx^{n-1}$, which takes about 10 seconds (if you talk while you are doing it and write very, very slowly, and have to erase something in the middle because you forgot what you were doing). Despite the time needed, I felt that the cognitive payoffs with the progression I used were great, and students really internalized the idea because THEY FIGURED IT OUT THEMSELVES. Figuring out the Power Rule is something that is totally in their reach, and I would have been robbing them of some learning pleasure had I just given them the power rule at the beginning.

PHASE 1: What is a derivative? We started just by drawing tangent lines to $f(x)=x^2$ at a bunch of points, estimating the slope and then making a table of values. I chose this function specifically because with the derivative of $f'(x)=2x$, it’s easy to see the pattern for the slopes in the numbers without graphing them (saving one level of abstraction). Yay, the slope at any point is just twice the x-value!

Then we did this two more times, once on a small sheet of paper for $\sin x$, and then once, in groups, on a huge sheet of butcher paper for $\frac{1}{3}x^{3}$. This was laborious and took a ton of time in class, but by the end I felt like students really understood well the idea of a derivative. More importantly, were ITCHING for an easier way to find it. They had all these great ideas that they were proposing, so it was easy to funnel their energy into the next phase.

PHASE 2: Finding the Rules. Then, I introduced the derivative tracer, a GeoGebra applet that does in seconds what they did in 15 minutes. I gave them a sheet of functions (below) for them to find the derivative of using the derivative tracer (kind of like collecting data in a typical canned high school science lab) and asked them to make conclusions about the derivatives that they found.

Though it was interesting to talk through with them the idea of a constant function’s derivative being 0, and a linear function’s derivative being a constant, the highlight of the lesson was seeing students figure out the power rule. When students got to that section, they seemed really proud that they could see the pattern. I had numerous students raise their hand to call me over to ask me if their idea worked, and then were so excited that it did that they immediately gave me a high-five. Students raised their hand so that I would come give them a high-five… in math class. I know the power rule is kind of easy, but I felt like they were so much more invested in the quest of learning mathematics because they figured something out themselves. Further, instead of trying to get ideas from my math notation, they had the ideas first and then I formalized it with math notation (though many students could do this no problem for themselves).

#### Long story short: The excitement in the room while the students were discovering something mathematical was palpable, even though that thing had been discovered many many many times before, including by their classmates sitting a few seats down. There was no “real world” motivation in this progression, no gimmicks – just the pure pleasure of mathematical discovery. So, to add to my ever lengthening list general goals for the year: I hope to avoid at all costs robbing students of the pleasure of figuring something out for themselves.

Here is my derivatives “lab” using the GeoGebra derivative tracer. Note that I’m not quite as adventurous as some and still want some structure in the classroom while “discovery” is happening. This is part of my controlling personality – tell me if you think this is too guided given my goals.

By the way, the answer to the other word picture puzzle is “mathematical induction.”

## Bowmanimal180

I’ve been a little quiet on the blog front for a couple of reasons, the main one being that the beginning of school is always crazy busy as you’re trying to claw back into work mode after a few months of leisure while at the same time attempting to teach students whom you don’t know and don’t know you or your routines and expectations…

But another reason I have been quiet here is that I have been steadily working on my #180blog. For those that haven’t heard of these, the idea is simple - I post a picture and a few sentences about every day of class for a whole year. I have found the process to be awesomely reflective, and I can see myself totally looking back at this in future school years.  Also, a bunch of other teachers are going the same thing, so it’s fun to get a picture of other people’s classrooms.