Tuesday, July 31, 2012

Planning Calc I, Continued

I've changed the Google doc for calculus planning to be editable by anyone without permission, and made another one for my personal copy. If you'd like to play with it, please do.

A few people suggested resources including John Golden, who introduced me to his colleague, Matt Boelkins. Matt is putting together a free calculus text currently titled Active Calculus. Chapter One is online (pdf link at bottom of page), and the rest of Calc I is available by emailing him. His text does what I want to do, weaving the limits work into the derivatives, as needed. I plan to have my students use at least some of his work. Thank you, Matt! (I've listed about 8 free or inexpensive texts in the google doc, toward the bottom.)

I went through all my calculus bookmarks and added them to the google doc. I've also included many of the links in Sam's Virtual Filing Cabinet. (What a fabulous resource! Thanks, Sam!)

Course Organization
My course outline was broken into 9 units at first, which seemed like too many. Active Calculus (AC) has 4 chapters, and I liked how the ideas are organized. So I streamlined my units, partly reflecting AC's organization. Now I have 5:
  1. Slopes, Rates of Changes, Tangents (Understanding the Derivative in AC), in which we'll consider limits in an informal way.
  2. Exploring the Derivative (Computing Derivatives in AC), which includes power rule, product and quotient rule, etc. I'm saving much of the limit study for the 2nd unit, so the first focuses on the more important big idea. AC has limits in chapter 1.
  3. Exploring the World with Derivatives (Using Derivatives in AC). I want to weave the 'derivative rules' and use of derivatives together some, so will veer from AC's cleanly defined chapters.
  4. Area & Anti-derivatives (The Definite Integral in AC). I don't want my language to give away the deep connection of the fundamental theorem. I want the students to uncover that as much as possible.
  5. Volume, which I love as a grand finale.

Students need to get stronger with their algebra skills. Sam has some great algebra bootcamp ideas. For my college students, I'll move some of it from classtime to homework, and will put much of it in 'problem sets', I think. I'm hoping that challenging problems will get them to think deeper about things they've learned superficially before.

First Unit, Approaching the Derivative
This is a rough draft, mostly listing activities. I've included a practice test below, which gives a clearer idea of the topics.

I'm not exactly sure how things will play out, since this is so different from how I've done it before. I still need to find sources of good homework problems. I'll be going over the first chapter in Active Calculus over the next week to see what to use from that.

I'm thinking the test will have 4 sections. (If students need to re-take, they can do that by section, in my office.)

I make a Unit Sheet for the students for each unit, giving a tentative schedule, homework, and practice test. Our classes begin on August 20, so my unit sheet will be complete by then. Let me know if you'd like a copy. I'm going to make a shared dropbox folder for all my course handouts. I'll post again when that's set up.

Awaiting your ideas, advice, and questions...

Sunday, July 29, 2012

Planning Calculus I (Community College)

The last time I taught Calc I was before blogging changed my life. (It was spring of 2007, over 5 years ago.) I'm trying to plan out the semester, and I have questions for any other calculus teachers.

I don't like starting with all the details about limits - way too technical. (Nor do I want to start with weeks of review. Sam has some great ideas about quick reviews when needed.) I want to start with the beauty of the derivative idea. But I'll need materials to do that. (My blogging comrades will come to my aid there, all their lovely work is recorded in my google bookmarks.)

How do I approach the limits then? I've started thinking about how to bring them in at different levels, as we need them.

I'm gathering all my thoughts on a google doc. You can see my work, or you can email me (mathanthologyeditor on gmail) for edit access. I'd love to work together with others who are trying to move in the same direction.

Right now I'm working on getting all of my 50-some google bookmarks for calculus-related posts into my outline of the course. When I'm done, I'll make a version for others that focuses on the links.

My personal system is a modified SBG, I guess. I give tests on whole units, that are broken into sections for the major topics. Students can re-test on one section / topic in my office. Until now I've used percent grades, but I may put those in my gradebook and not on students' tests. Just let them know what went wrong and whether they've shown their mastery yet or not.

Right now I'm thinking  9 units. I'm not sure whether I'd test after every one. Maybe only after the 3rd, 5th, and 8th? Hmm...

Part I. Slopes and rates of change
  • Exploring Slopes of curvy lines / Rate of change
  • Playing with Limits (lighter treatment than most of the textbook stuff at this point)
  • Exploring Derivatives (the 'rules')
  • A more precise study of Limits
  • Exploring Further with Derivatives (more 'rules', applications, graphing - this one's huge)
Part II. Areas 
  • Limits and Area
  • Anti-derivatives (or "What's the big idea?")
  • Exploring Integration
  • Volume
I would love to collaborate. Please join me if you'll be teaching calculus. 

Sunday, July 22, 2012

Book: The Psychology of Learning Mathematics, Skemp

Two years ago, I linked to Gary Davis' post which quotes in full Skemp's paper, Relational Understanding and Instrumental Understanding (first published in Mathematics Teaching). Last summer and fall, a number of bloggers referenced the article. I wrote my own post about teaching for 'understanding' in June, and decided at that time to buy Skemp's book,  The Psychology of Learning Mathematics (1971).

I've started reading it, and it's great. If anyone would like to buy it and read it together, I'd love that. His discussion of concept formation (chapter 2) and schema (a mental structure, organizing concepts, chapter 3) helped me think about why things seem so easy once we've learned them, and how hard it is to really get back to the student's perspective, in which the concept hasn't formed yet, and understanding is a struggle. Chapter 5 talks about ten different functions of symbols. From that chapter:
Thinking is hard work. Once we have understood a mathematical process, it is a great advantage if we can run through it on subsequent occasions without having to repeat every time (even with greater fluency) the conceptual activities involved. If we are to make progress in mathematics it is, indeed, essential that the elementary processes become automatic, thus freeing our attention to concentrate on the new ideas which are being learnt - which, in their turn, must also become automatic.  ... In mathematics, this is done by detaching the symbols from their concepts, and manipulating them according to well-formed habits without attention to their meaning. (page 88)
I hope to post a more complete review of this fascinating book once I've finished. I wanted to post now in case anyone would like to read it before they start back to teaching. Let me know if you'd like to discuss it.

Saturday, July 21, 2012

San Diego in January, Joint Mathematics Meetings

Joint because it's the MAA (Mathematical Association of America) and the AMS (American Mathematical Society). Their annual meeting is huge. I went a few years back when it was right next door in SF. I went to lots of pedagogical talks, and to all the math circle events. I went with friends and had a blast.

It's hard to travel in January, but I'm going to do it this time. I lived in San Diego for a year, and I can attest that it's a beautiful city. The meetings are Wednesday, January 9 to Saturday, January 12.

I've been invited to help host the poetry session that will happen (most likely) on Friday evening. I hope we can make that a delightful experience. I'll also be attending every math circle event I possibly can. And if all goes well I may even be trying to sell a certain book.

I would love to meet some of my online friends there. I just spoke with a blogger I've admired for a long time. He wasn't planning to come, but said I'd gotten him to think twice. So I'm writing this post to get more of you to think twice. Let's meet up in San Diego in January. Who's coming?

Wednesday, July 18, 2012

Playing With Math, the Facebook Page

I've set up Playing With Math as a page on Facebook, to report little bits of progress on the book. Today I posted that:
This morning I worked on a chapter I hope to include from Rodi Steinig, who blogs at Talking Stick Learning Center. I'm taking bits from a number of her posts, and highlighting her mindfulness practice. I love what she does!
Of course Facebook includes a nice little picture of her last post, which I can't figure out how to easily reproduce here.

I'm going to try to post updates every few days there. Check it out if you'd like to follow the progress of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.

Tuesday, July 17, 2012

Lotsa Links: Tanton, Busking, Cat in Numberland, Spot It, ...

While I was on vacation I kept seeing fabulous stuff online. No time to write about it all, but I want to both share and store my many finds here.

Shecky reminded me of this set of 4 videos that James Tanton did.  I'm not sure what #2 means (watched the video a while back, forgot the details), but the rest sounds great to me.

This might be a good place to direct my students in the first week. Maybe I'll give them some choices, and ask them to tell me what they think of at least one website.

Mary O'Keeffe wrote earlier this summer about doing Guerrilla Math Circles at a local park. She did a Pascal's Triangle question, making a big triangle with sidewalk chalk. I like the idea of grabbing the attention of random people in public. This video on Maths Busking* looks like something similar.

Malke wrote a great post about her daughter's take on The Cat in Numberland, pointing to a fabulous review of it [pdf] I hadn't seen before. It goes into great depth on the pedagogical issues involved.

While I was at the Math Circle Teacher Training Institute (just last week? is that possible?!), I ran a math circle for the top level kids, mostly 12-16. I had them analyze Spot It. (Posts here, here, and here; can you see any progression?) There were 9 kids, in 3 groups of 3. Each group approached it differently, and one young man (call him Trevor) became absolutely obsessed with the problem. It was a delight to get to work with this group. Our institute participants were there to observe me, and I seduced them into thinking about the problem too. A few of them were able to multi-task well enough to give me some good feedback. One noticed that Trevor and I got into a conversation that excluded the other kids (oops!). But the other kids watched us for a few moments and then decided to get back to work on their own ideas! (Invisibility achieved, though indirectly.)

The adults, who were working on the floor in the back of the room, made a bigger dent into the problem, and spurred me to think more about whether cards with 5 or 7 pictures per card can be made with the process I came up with in January. (5's, yes. 7's, I'm not sure yet. This question has connections to something called finite field planes, which I intend to explore soon.) Bowman Dickson was in that group and wrote a great post about their thinking. He also wrote about the origami he learned to make.  (I meant to make some origami, but never found the time - dang it!)

[Note added on 11/30/12: Perhaps 5's cannot be made. The procedure Bowman's group used, which I had also used, does not work for 5. A student apparently made a deck using trial and error. I wonder if I can replicate that.]

Dan McKinnon seems to be thinking about number theory on his blog, mathrecreation:
[For] these kinds of sequences (generated by polynomials with Integer coefficients) - their terms are either always even, always odd, or alternate between even and odd values (e.g. you won't get a sequence that goes "even, even, odd, .." or some combination other than the three possibilities mentioned). Can you see how you can show that this is true using similar arguments to the ones used here for last-digits?


* Busking is a British term meaning performing for donations.
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