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:
- 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.
- Then I gave them 2 minutes to share ideas in groups.
- 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.
- 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!!
The Dead Puppy Theorem and Its Corollaries
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?
Calculus Standards 2011-2012: Feedback Requested
I’ve been toying around with my learning objectives for Standards Based Grading in Calculus for three years now, and I want to get some other people to weigh in on what I have. Please, take a look, tell me what you think!
- I love the first person language, which is an idea I think I stole from @kellyoshea.
- The physics modelers all have crazy acronyms for their standards like CVPM and UBFPM and ERMAHGERD. These seemed confusing to me at first, but then I thought that students might really benefit from this. The standards aren’t organized around chapter numbers, or something else arbitrary, but rather BIG DEEP IDEAS (models!). I wanted to do something similar for Calculus, so I organized mine around Local Linearity, Slope Functions, Proportional Rates and Accumulating Change (with short, simply worded descriptions in the document below). I don’t know how well this worked last year, but one goal for me is to try to always relate the standards back to their big ideas.
- I didn’t do the standards like this fully in order, and this year I am totally changing the order. But just to give you an idea of how I did things, I did all the IP and LL (limits) standards, then SF.a through SF.g (basic derivatives), then PR.a (optimization), then SF.h through SF.n (graph sketching), then PR.b-PR.h (exponential functions), then SF.o/PR.i (implicit and related rated), then all the AC standards. It was a bit confusing to go back and forth, but organizing the standards like that made it make so much more sense to me. Tell me what you think about that…
- I struggle with how general/specific to make the standards, and how to include both calculation and interpretation into the standards. Sometimes I split the two, sometimes I kept them together. This is the hardest thing for me!
Anyway, any thoughts are necessary! These are my standards from last year, the second time I taught Calculus.
Calculus Standards 2011-2012
Teaching Beliefs in Poster Form
Day one of the school year is rapidly approaching so I’m continuing my mad scramble to outfit my classroom (complicated by the fact that I’m trying to stock up here in the US because random things are very difficult to find in Jordan). My classroom walls were pretty bare last year, so I decided to make some posters and get them printed in large format (18″ by 24″). I wanted some of them to reflect core aspects of my teaching philosophy so that I can more easily continue great conversations that start in the beginning of the year and seem to peter out. Here is what I got printed:
(I’m counting this as #made4math for this upcoming Monday – I’ll be on a plane then!)
Memorizing vs. Understanding
One of the big themes in my classroom is the difference between understanding and memorizing. I don’t need to rant about this now, but I think that students just are not aware that when they cram algorithms in place of problem solving, they aren’t really learning. Getting them to understand the difference between understanding and memorizing is one of the most important metacognitive lessons of my math class. I thought it would be nice to have a visual reminder I could point to with a sort of classroom meme.
To explain, this is a picture of my family dog Whiskey. Whiskey is awesome at learning how to do all sorts of tricks. I think he could easily be a commercial dog. But when he messes up, he has trouble getting back on track, because he has no idea what he is actually doing. He has memorized what gets him a treat (like how students perform mathematical algorithms for grades) but has no deeper level understanding to be able to tinker with the process or apply his knowledge to a new situation. My favorite Whiskey-isms:
- BANG, YOU’RE DEAD: When my mom puts her gun finger out, Whiskey responds by sticking his paws in the air innocently. Then, my mom yells “bang!” and Whiskey awkwardly flops to the floor, flips over and plays dead. He’s really good at this, but if, for some reason, he messes us up, he just tries to throw all the steps of the trick at my mom until she gives him a treat (like students who just try to write stuff on a test for points). I show my students the video in this post, which shows Whiskey messing up, to illustrate the problem with troubleshooting a process you don’t understand.
- OUR NEIGHBOR’S NEWSPAPER: Whiskey brings in the paper every morning for my parents, and loves life every time he does it. When he brings it in, he will chomp down on that paper until my parents give him a treat. The only problem with this has been that while on walks, he will sometimes see a neighbor’s newspaper and apply the same logic. He grabs it and the sprints home (however far away) and then wont let go until he gets a treat. I am going to use this example to talk about how you can misuse processes if you don’t understand them and try to apply them to different situations (like canceling out added terms in a rational expression).
If Whiskey could just ask “Why?” he could avoid these errors in his tricks! Good thing my students are capable of doing that.
The Growth Mindset
The Growth Mindset, a brainchild of Psychologist Carol Dweck, has become one of the lynch pins of my teaching philosophy. The philosophy espouses that believing intelligence is fixed hinders learning – “smart” kids will be scared to take risks and fail, and “dumb” kids will not see real results in their learning because they are comparing themselves to others instead of themselves. The growth mindset puts the emphasis on hard work leading to real learning, and normalizes (no, necessitates) mistakes as part of the learning process. I really like this image (which I did not make), despite how small it is, and I could see a student reading through this one day at the beginning or end of class. Even if not, I made the headers “Fixed” and “Growth” bigger so at the very least it will be a reminder to me while I teach about this important idea!
In the first week, I will give a survey that will get us talking about the Growth Mindset, as I did last year, but I hope to do a better job of continuing that conversation this time.
Math is Magical
I love the completely-not-subtle message that is so subtly expressed in this poster (which I also did not make). Yes, Math certainly is magical. In surveys, students always cite my enthusiasm for the subject as something that makes the class better, so I have embraced that as part of my teaching philosophy.
Foundations are Important
Okay, maybe this isn’t a huge part of my teaching philosophy, but I made this poster last year on regular printer paper and I really liked it, so I wanted a more durable poster. I like it especially because it worked as both a comforting thing for my Calculus students and an inspirational thing for the Algebra students. If you want a copy of your own, I have this image split into nine regular pieces of paper that you can print out and glue onto a poster.
Project Implementation Reflections and Questions
I really enjoyed doing final projects with the kids this year (which may be patently obvious considering that this is my 6th post on the topic). It’s such a fun way to end the year, seeing them get excited about doing something interesting with Calculus and coming up with ideas about math that I never would have even dreamed of.
But projects can also be very frustrating, and hard to implement. Here are the things I struggled with this year. I’d love any feedback or tips.
- Since these projects were very open-ended, some students felt a bit lost, and I struggled a lot with how much guidance to give and similarly, how much to let them struggle. I just find it so hard when we have such a short amount of time to see them getting thrown off in a crazy direction, especially if it’s going to lead them to a lot of useless work. I tried so hard to “be less helpful” but I just couldn’t resist sometimes! Part of me feels like I am stealing a bit of a learning opportunity from them and part of me feels like I am just advising them to help guide their crazy teenage thought process. Also, some students just started working on their projects without really knowing why they were doing what they were doing (they just wanted to do “something about optimization”). I wanted to help them do something for their idea without turning it into my idea, but I’m not sure how well I did at that.
THOUGHTS: I think that I am going to try to have them submit proposals next year where they present some sort of thesis, or a guiding question they are going to answer in their project. This might get them to plan out their project a bit better before starting, give me a chance to give good feedback and also give them an overall question which will really guide their whole project.
- One thing that I was continually frustrated throughout the week in class that I gave them to work on the projects was that students did not work very efficiently, leaving much of it for the end. Part of it was that they just had so much time in class, but part of it was that I have no idea how to help them structure their own project to use class time well. I had tons of students show up without materials to work on their projects, and even some who would sit there and do nothing telling me that they were just going to finish at home.
THOUGHTS: I wanted to do a midpoint deadline of some sort, but because all the projects were so different, it seemed really weird to me to organize something like that. I might try having them make a schedule in the beginning of the project, but I’m not sure how to help them stick to that, or if that is even worth all the work that it would be.
- Similar to supporting them in organizing their time, I struggled helping them work well together with each other. I think group work like this is crucial in high school to learn how to structure time with someone else and communicate about a project, but the students were terrible at this. They would do things like not show up to class without telling their partner, even though they had all the materials. I even had to mediate an email war between two girls who were flipping out at each other about who was doing less for the project.
THOUGHTS: Maybe this isn’t something that I need to do something for, and maybe this is something they just have to learn by doing the project, but perhaps I could find ways to help them structure their roles in the project beforehand, or maybe just do more long-term projects like this over the course of the year.
- Last, I really want them to show off their work to each other, but I’m not sure how to make class presentations anything but the boring yawn fest that they tend to be. Students did some really cool things, but were really bad at explaining those things in a way that the class could understand. Also, it’s really hard to listen to two full class days of presentations, even for me, and it’s really hard for students to get anything out of the presentation when they are not really expected to engage in a meaningful way (not one of the presentations was interactive in any way).
THOUGHTS: I’m looking for some sort of other structure to make it more interesting. Maybe some sort of gallery walk type structure? And I also want some formal way to get those listening involved so that they really pay attention and learn – some sort of commenting system, or interactive component. It’s very hazy in my head, but this is something I am going to try to flesh out over the summer.
Any ideas would be greatly appreciated!
(Also, below is my rubric for grading these projects)
Calculus Final Project Spotlight: Packaging Consultants
And the last project I am going to detail…
A few students did a pretty standard, but well done optimization project investigating different can shapes to find which one is the most efficient (Sam profiled his kids doing a very similar project, I loved reading his students’ reflections on it!). Then they redesigned the cans to help companies lower cost. The reason that I am profiling this because it made me realize what students find interesting in this whole optimization nonsense – I brought in cans in the winter when we first learned optimization, and we did something similar, but we never talked about the issue that really got other students’ attention…. money! I had been focusing on the shapes, but I should have been focusing on money! (Seems like a super “duh” in retrospect, and it’s not anything original, but helpful to realize nonetheless).
The students did tons and tons of calculations, but what I really loved is that they compared the price of producing the current can that the company produces and the price of producing the ideal can. They looked up the price of aluminum and estimated (or looked up? I’m not sure here) how many cans per day a factory would produce. After a bunch of multiplication, they showed that tiny, tiny changes in the shape would result in savings in the hundreds of thousands of dollars range for a year (see red number below), which is super cool.
Also, they had a really nice framework for their project. They pretended they were a packaging consulting company and even came up with a logo and a name that combined their names. I thought that was great!
NEXT YEAR: I am going to frame my optimization unit much more in the way these students went about it. I feel like this is a complicated mini experiment in terms of #anyqs – the students found for me what the actual interesting question is. For me, the shapes of the cans themselves is interesting (especially that it ends up being such a beautiful ratio), but I think a lot of kids were really amazed at how a small change in the size of the cans can result in huge savings and led them to wonder why all cans aren’t shaped the same way. So, thanks for helping improve my curriculum, (now former) students!
Calculus Final Project Spotlight: Math of the Pilgrimage (Hajj)
A student’s mother is completing the Hajj this year, the pilgrimage that Muslims take to Mecca. This is one of the five pillars of Islam (along with prayer, fasting, charity and testifying that there is only one God). All physically and financially capable Muslims must carry out this pilgrimage at least once in their lifetime. This student based her whole project on the Hajj and calculated many different things about it. Specifically, she calculated:
- How long it would take to complete each part of the Hajj (once you get there, there are certain rituals during which the pilgrims walk to various places). She used aerial photographs and official information to measure the distance (around 40 km!) and then used an average person’s walking speed to estimate that each pilgrims walks for around 10 hours during the Hajj.
- How many people can be expected to attend the Hajj in the future given data from the past 10 years and assuming exponential growth. She used previous data and the basic exponential growth model to make predictions for the next 30 years.
- How large the current area around the Kaaba is (the holiest site of Islam around with the Hajj is based). She used GeoGebra and Google Earth software to measure the area.
- And how much the area will have to increase in future years to accommodate the extra pilgrims. Based on her predictions of the increase in the number of pilgrims, she mapped out how big the area around the Kaaba will have to be for the pilgrims to all have the same amount of area. She thought it was cool they they would have to restrict the number of pilgrims, or knock down highways in order to keep the area per person the same.
The math wasn’t perfect and there were some crazy assumptions made, but I absolutely loved this project. It was from someone who had told me in the beginning of the year that math wasn’t her thing, and it was really cool to see her get excited about the project because it applied to something really interesting. All the math was very well motivated and taken from a wide range of things that we did this year. Great stuff!
NEXT YEAR: I could see doing some sort of city planning project involving Google Earth that somehow involves population growth. It would be really cool to look at current rates on population increases in areas and see what that would mean for the physical space. I am so happy that a lot of these final projects have translated into great teaching ideas!
Calculus Final Project Spotlight: Twitter Followers Math
For their final project, one group decided to make a twitter account and track how many followers they gained over time. The account was called “UknowURatKings” (King’s is our school… so YOU KNOW YOU’RE AT KING’S for those who hate txtspeak). They tweeted inside jokes about the school that you would only really get if you were pert of our community. I was following them, which was good because they ventured into inappropriate territory once (it was a nice mini experiment in social networking with students!). Here was my favorite tweet of theirs:
They had predicted that the followers function would follow a logistic model. Using a few data points, they created a logistic model of their own: they thought they would max out at around 100 followers (the size of the senior class population on twitter plus some extras), they originally told 13 people, and after one day they had something like 40 people (unfortunately, I can’t find where they uploaded their project ahh!). Based on that they created their logistic model. Then, they tweeted furiously for about a week and recorded how many followers they had each day. At the end, they compared their results with their model…
They were way off. Though they had chosen the right model, the number of followers increased slower than they thought and maxed out around 60, not 100. My favorite part of their project was that they didn’t try to fudge their numbers or make the data fit their model – instead, they talked about their assumptions that may have been flawed, their tweeting behavior skewing the results, and inconsistencies in data collection. I ❤ data.
NEXT YEAR: I thought that this was a really fun and simple project, and it might be something that I try to do with my whole class when we study exponential models next year (I swear I could teach a whole term on just the logistic function). I think we could have an awesome discussion about modeling with all the different inconsistencies that will arise, and we could even add a competition component, to see who can get the most followers for their account under certain constraints… Too many ideas, too little time.
Calculus Final Project Spotlight: 3D Solid Modeling
**The next few posts are going to be spotlights of final projects that students did that I thought were cool or interesting and then a few reflections on doing final projects in general. I could picture doing a lot of my student’s projects as a whole class!**
If I had one more week in my non-AP Calculus class, we would study volumes of revolution. That’s probably the biggest weakness of my course right now, and I am trying to figure out a way to include that next year. A junior who is in my regular class and is taking AP next year was a bit lost when coming up with an idea, so he asked me for a topic that we do in AP but did not do in our class so he could be a bit prepared. I suggested volumes of revolution and after a lot more nudging and guidance and idea planting than I did for other students, we decided that a good project for him would be to recreate an interactive 3D model of a solid of revolution using GeoGebra and Winplot. (actually it works with solids of known cross section too).
Here’s how it works…
1. Upload a picture into GeoGebra (he chose a huge vase from the art room). Fit functions to the edges of the object on the part that will be revolved.
2. Recreate the same exact functions in Winplot (which has much better 3D capabilities than GeoGebra does).
3. Use Winplot’s revolving capabilities to revolve the surface around an axis (any axis!). And then, voila, you have a 3D model of your object that you can use the arrows on the keyboard to rotate in any direction. It actually ends up being really impressive – my student told me that he left the model up on his computer and every time he would turn it on he would rotate his vase a bit.
After I saw the success of this project, I suggested the same one to a few students in my AP class (who were required to do a much more low key, shorter version of a final project because of time restraints). They decided to recreate a bunch of sports equipment using the program, which I thought was a really cool idea! Their rotate-able objects:
NEXT YEAR: I made an instruction sheet for those AP kids because they had less time, but I’m glad I did because this was a really cool project and is something that I can see myself doing with a whole class next year. Here it is below. If you haven’t tried making any 3D models (not necessarily real objects) with Winplot, definitely try it out – it’s super cool!
And the Calculus Final Projects Begin
For the last week and half of school, my non-AP Calculus class is embarking on a free choice final project. The only requirements are that they must use some sort of Calculus, they must use a real artifact (data, a picture, a video, history etc), they must incorporate technology, and they must find a way to present it to their peers.I have been so excited to see their creative streaks and see some of them get really excited about this, especially because I am impressed that they are still energized two weeks away from their graduation.
Here are some of my favorite ideas. Note that some are not very sophisticated, but are interesting nonetheless and I have been supportive regardless, as I want to see them really carry out something that they feel is their own. I will report back on these after a week and a half when they are done.
- COMPETITIVE EATING RATES: A few students want to eat as many chicken wings as they can, but as they go, time when they finish each one. Then they are going to calculate the rate at which they are eating wings at a few points during the eating. Their prediction is that the more wings they eat, the slower they will eat them. I am hoping they will try to fit some sort of exponential function to the data (that might tell them their limit). They are going to compare their rates to that of an actual professional eater.
- ATTENDANCE TO THE HAJJ: The Hajj is the annual pilgrimage to Mecca that Muslims embark on once in their lifetime (or sometimes more). One student wants to look at aerial photographs of the Hajj to determine the area that the pilgrims fill up and compare the relative areas from different years to the relative levels of attendance. Then, she also wants to make functions for an old man, a young man and a woman doing the hajj that will give their position at any time given the size of the crowd in a given year.
- THE SPREAD OF SENIORITIS: A couple of students are collecting data from their friends about their GPA throughout the year to see how real senioritis is. Then, they are going to use the idea of differentials to expand on the data and predict students’ GPAs in future terms (college?) given their current slide.
- DESIGNING A GREENHOUSE: One girl wants to make a model of a curved-roof greenhouse and then use Calculus to find the amount of glass used and the volume. She also wants to do some sort of optimization exploration to see if the shape has to do with using the least amount of glass for the most sun exposure.
- CELEBRITY LAND AREA: One student is using Google Earth to find the area of various celebrity plots of land. Then he is going to compare the Google method to numerical methods (like Riemann sums and trapezoidal sums) and he is going to try to determine how Google’s mechanism for finding area works.
- INFECTION: A student has a game on her iPad where a disease is being spread around the world. I can’t remember if the object is to infect the world or to save it. Either way she is going to pick a few regions and track the spread of the disease through those regions to see if the curves are logistic, and to see how the curves of regions close to each other relate to each other.
- DERIVATIVE/ANTIDERIVATIVE CHECKERS: Two students are going to design a checkers board to practice derivatives and antiderivatives. The checkers will have derivatives on one side and antiderivatives on the other. When you jump a piece, you have to solve a derivative or antiderivative before you can capture the piece.
- GATSBY’S OPTIMAL PARTY: One student is going to design a prompt from Gatsby himself asking Calculus students to optimize his guest’s happiness at a party. I don’t know the details, but the sense I get is she is going to give Gatsby a limited budget and things that he could purchase for his party – I’m excited to see how this one turns out!
And there are lots of other great ideas too! I liked the ones above because they took one of my ideas for a prompt and totally made it their own, or just came up with something totally random that they wanted to do. I’m excited to hear how these turn out. I had a million other ideas too… here is the packet of ideas that I gave them to get them thinking.