Monthly Archives: November 2011
After I put it up a student asked me if I drew it. Yes, yes I did. Because before I was a math teacher I was a Norwegian expressionist painter who died in 1944.
By the way, I just photoshopped this and split it up onto 9 pages, then used the Math Department color printer to print it off (oops). If you want the files, here they are in a compressed folder (nine 8×10 images and the original full file).
There are a lot of times in Calculus where stuff that you want to be true is (like the limit of a sum is the sum of the limits) but then there are a lot of other times where sadly your intuition just doesn’t work (like the derivative of a product is NOT the product of the derivatives). This is a random little thing, but I’ve been doing “Wannabe Math” in thought bubbles on the board to distinguish between rules we know that work and ones that we don’t know are solid:
I want to encourage students to try things (so encourage them to use their intuition to explore) but give them tools to be able to test to see if they are correct. I did this for the product/quotient rules and then used it to derive the chain rule with the class – with the Wannabe Math on one side and the actual math on one side, we could easily see what was “missing” and come up with the whole derivative of the inside part of the rule.
One of the mantras in my Calculus (non-AP) classroom is that the hardest part of Calculus is Algebra. Seriously. I find it really depressing because often when I am correcting tests and quizzes, they will have the Calculus PERFECTLY correct and then just royally mess up the algebra. This year, the students in this class have incredibly weak backgrounds and I’m not sure why (well, I do have an idea that I will keep to myself…), but I do know is that many are really smart kids who just have serious holes in their math education. I’m really struggling to find a balance between teaching Calculus and remediation of basic math, which forcing me to go really slow and snip out wonderful projecty-type applications that really make the class what it is. It’s a balance between “Teach them where they are” and “Teach what I am supposed to be teaching”
I’m posting about this as a call for help to try to fix the crazy problems that I am having. I think the biggest problem is that the notation is so abstracted that they don’t have a basic feel for what you can and can’t do when manipulating expressions. I’d say the problems fall into two main categories…
NOT RESPECTING THE TRIG FUNCTIONS AS OPERATIONS:
GENERAL ALGEBRA CRAZINESS, MOSTLY HAVING TO DO WITH PARENTHESES:
In all six of the examples I picked above, the Calculus is EXACTLY correct, but it just gets completely and unfortunately ruined. I think that I will show them this when we go back to school on Sunday:
My AP class literally laughed out loud when I put that up on the board and I hope they have the same reaction and maybe help them realize that I have the same reaction when I see them manipulate some of the algebraic expressions the way that they do.
Any ideas on how I can do an Algebra boot camp of sorts to remediate some of these misconceptions? Are these normal Algebra mistakes for a senior in high school? Do other people have similar problems?
Getting five mini-whiteboards was a real game changer for my classroom. It has completely changed the way that I go about skills-based instruction (which is what a large chunk of the first term of Calculus ends up being) and has added so many new tools to my instruction toolbox. I think that too often in the past I relied on variations of the learn-practice-apply model, and the “practice” part not only always seemed to make class drag, but never really felt effective. Well, doing practice on the whiteboards with some sort of extra little component to make it interesting has proven to be not only interesting (I’ve never seen students more engaged while practicing skill-based math), but far more effective at teaching skills. The only downside to the whiteboards is that they don’t have anything in their notebooks to study at home, but I’m brainstorming ways to fix that (mostly posting pictures of the whiteboards on our course website or not caring because they can always find examples of solved problems in our textbook – it’s the practicing part that matters).
Here have been my favorite things to do so far with the whiteboards and a few ideas for experiments that I want to try in the future. I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire.
1. UTILIZING DIFFERENT COLOR MARKERS
One of the great parts about whiteboards is that you can get get students to use different colors like you do on the board up front to get them to focus on different things. For example, above is a whiteboarding exercise I did with the Chain Rule. Students were in groups of threes – for each problem, one person had to rewrite the problem in different colors to indicate which was the outside and which was the inside function, the next person had to differentiate it still using the colors to point out where each part of the new expression came from, and then the last person had to rewrite the expression in a simplified form. This was perfect because the hardest parts of the chain rule are recognizing when you need, seeing inside vs. outside and then seeing where the parts of the new expression come from.
EXPERIMENT I WANT TO TRY: Mistake Marker. I want to have the students solve problems and then whenever they make a mistake, instead of just erasing part and fixing it, they write over their mistake with the mistake marker color (or make a note with the mistake marker if it’s not possible to write something in that color) and then continue the problem in a new line below with their original color. Then, at the end we can collect as a class the most common mistakes that are made when doing a complicated problem like, say, the Quotient Rule (not complicated you say? Then you have never taught students Calculus who have a terribly weak Algebra background).
2. THE MISTAKE GAME
I know I have mentioned this like twelve times already, but I absolutely love it. This is stolen from Kelly (read her description here), but the basic idea is that groups present solutions to semi-complicated/involved problems on whiteboards, but while presenting their solution, purposely make a mistake (and not an silly arithmetic mistake like , real hardcore-misconception-style mistake). Then, they present their work to the other students in the class, trying to sell their mistake as having been made for real. Then other students ask thoughtful questions about the presenting group’s solution to try to help everyone find the mistake. This is always great with a quick class followup at the end collecting the most common mistakes.
When I have wanted to show how a topic from my AP class could be applied to many different situations, I have done some sort of rotation so that students could be exposed to a wide variety of problems (without taking the time in class to do ALL of them). The first exercise with this is a simple gallery walk – each group of 2-3 students solves a problem individually. Then, when everyone is done, the groups rotate around to each station, taking a few minutes at each one. This is far more successful if you give them specific tasks, like “First, check if their answer makes sense, then see how they set up the limit definition of the derivative” or something concrete like that. The second exercise is when each group starts doing a problem, and then the groups rotate after about 5 minutes and they pick up where the last group left off (someone on Twitter gave me this idea, sorry, I forget who!). Then, they rotate every 4 to 5 minutes until all of the problems are completed. I tried to emphasize that while doing this, you must show your work neatly and clearly (an important skill for all math, but especially an AP test) so that the next group can quickly see what has been done to solve the problem and what still needs to be done. The thing that I liked about this was the meta cognitive mapping out of problem solving, though I don’t think I left enough time for students to really think about each one.
EXPERIMENT I WANT TO TRY: Rotate Marker. I didn’t love the rotating problems mid-problem solving, mostly because it stopped students in the midst of great problem solving, but I think one thing I am going to try is having the students rotate within their group who is writing. So a group would be solving a problem and every 2 minutes, the next person in the group would become the writer. This would ensure that all the students in a group are engaged in the problem solving process and that they are all talking math with each other.
One of my many goals this year is to step back and focus more getting students to figure out how to go about problem solving. One thing I did that I really liked was I photocopied a few pages from the book and cut out like 50 functions for each group that all required a variety of differentiation rules. Then the students made categories on their whiteboards and sorted the functions based on which differentiation rule they needed. It was a really interesting process, especially because many of the functions needed more than one rule. I really enjoyed seeing how students solved this – most just made a bunch of different categories (like Quotient & Chain, Product & Chain, Product & Quotient & Chain), but one group made a crazy complicated Venn Diagram and another made a table kind of like one of those that shows the distances between cities (so like Quotient, Product and Chain both across the top and down the side) then placing the functions at the intersections of the rules they needed. It only took 15 minutes, but after learning so many differentiation rules, I think it was great to give them a chance to step back and figure out what types of rules they needed to use and where. The next day on the quiz, I saw tons of students circle parts of functions and write “Product” and “Chain,” which is something I have never seen them do before. To me, this is a wonderful problem solving strategy that was explicitly identified and strengthened by a quick activity.
Overall, I just think the added presence of the whiteboards has given my classroom a much more dynamic feel. When I asked for feedback from my students about how class was going these were my two favorites:
I like how we change up the routine. We do not sit and do the same thing over and over again, its changes up and keeps me interested.
One positive thing is the different types of work you give us because it is not all the same thing it is diverse so it keeps things interesting.
I guess it was a good indicator to me that switching up the routine for switching-up-the-routine’s sake is not a bad thing. Knowing some basics about how the human brain works, keeping the kids from sinking into a comfortably numb routine will certainly make everything a little bit stickier.
And in case you forgot in all of my blathering, I’ll repeat my plea from above… I would love any and all comments about different non-topic specific modes of instruction that you use with the whiteboards to expand my repertoire!