Category Archives: Teaching
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:
- 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.
- 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.
- 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.
I tried something new in class this week that I think solves a few random problems:
- Sometimes, when working on whiteboards, one student hogs the marker and does a lot of the work (and thus the learning)
- With whiteboard work, students don’t have anything in their notebooks to study later
- 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.
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).
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):
We took an irregular shape that has no simple geometric area equation and turned it into a shape that does have a simple geometric equation.
We split up a larger shape into a bunch of smaller shapes to be able to do this.
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.
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…
- does not exist
- does not exist
- and .
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.
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?
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:
- What was a speed that you are 100% sure that you must have gone in the time in between? Why?
- 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:
- What was a price that you are 100% sure that the iPhone must have had in the time in between? Why?
- 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?
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 so , 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 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 , 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 , and then once, in groups, on a huge sheet of butcher paper for . 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.”
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.
- My #180blog, Bowmanimal180
- A bundle of 36 #180blogs that you can subscribe to from Frank Noschese @fnoschese who has been the one to popularize this idea
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
My school uses Moodle as our platform for sharing course materials. I used it for two years, but it was just way too clunky for me – editing everything takes thrice as many clicks as it should. So last year I decided to upgrade to a Google Site for my class. I just redesigned it for the upcoming school year and it looks really pretty, so I wanted to share it:
Here’s the new site, which doesn’t have any content yet, but will aim to do the same thing as my old course website for the same class (if you want to see what content I put on there). On my site students…
- Check their homework assignments (I do not write homework on the board, just announce that there is homework)
- Check for upcoming tests and quizzes – I write which standards are included, so then students…
- Read the text of standards for the course
- Access materials to study for specific standards, whether for the first quiz, or for a reassessment
- Find daily materials including images, problems and links for students who take notes on the computer
- Leave anonymous feedback for me
- Check their grades (I make internet reports with EasyGrade Pro, host them in my Public Dropbox folder and link from this website)
- Access the grading scale for standards
- Request a Reassessment, through a Google Form, which goes to a spreadsheet I can see
- Access a virtual whiteboard through Scribblar where we can interact virtually after hours, or where students can interact with each other (experiment this year)
- Fold their laundry (there’s an app for that)
As you can tell, I use it for a lot of different things in my class, all aimed to INCREASE student accountability, which is why I spent time to make it look how I wanted it to. Some tweeps enjoyed the look (and the fact that I have some goofy elements in my Reassessment Request form, check them out), and they were wondering if I could post a template for the site, so I did! When you are making your Google site, if you are at the “Manage Site” interface, click on Themes at the bottom of the sidebar and then click on the Browse More Themes button at the top right.
Theme name: Math Class Portal – @bowmanimal
The pretty banner wont be there, but other than that everything else should. The only thing that you really have to change besides adding your own content are the forms embedded in the Reassessment Request page and the Anonymous Feedback page. Those are both Google Forms. You can either link your own existing forms … OR I posted templates of those in the Google Doc Templates which you can modify and use.
- Go to docs.google.com/templates
- Search for “Bowman Dickson“
- One is named “Anonymous Feedback Template” and one is named “Reassessment Request Template“
- Just click on “Use this template” to make your own. In the document, if you go to Form –> Edit Form, you can make your revisions.
- Then link the forms from the website to the one you created instead of the ones from my template.
PS. The background of my website is an origami crease pattern… if you do some origami and then unfold it so the paper is flat, you can color the creases and the folds different colors to reflect the 3D structure. Beautiful!