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.


Texts
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.)


Activities
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.

• Graphing the tangent
• Axes Exercise (math activity for group introductions)
• "What is the meaning of slope?"
• Graphing Stories (dog chasing ball video, guy on hill video, Another useful take...) 
• History Webquest
• Describing graphs game (graph on board, one partner looks and describes to other partner, who draws, helps students see the need for precision in language, for limits)
• What is pi? (Circumference and area; although Fawn's students are in 6th grade and mine are college students, I think these activities will work for us) 
• Limits: 60/x 
• more coming...

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...

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. 

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.

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?

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.

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?

More:

• The math behind a beautiful building
• "The hard bigotry of poverty" (re school 'reform')
• Learning time with Scaredy Squirrel
• The calculus behind an eyebrow
• Surface area for the kindergarten crowd
• A liar and truthteller puzzle (where the number involved can't be determined)
• Yoshimoto cube models (check out the videos at the bottom of the page)
• Letters to a new teacher (Bowman also got this going, amazing! I haven't had time to write one yet, but maybe in a few weeks, when I'm thinking about my courses...)
• Getting students to ask more detailed questions
• Teaching kids computer programming by having them 'program' each other to dance
• Inventions Festival
• Pairs of cities with the same population
• Getting better at teaching
• Show that the blue and red parabolas intersect at right angles
• Ten hours with Erica (in which Rebecca changes Erica's math life)
• Stephen Lazar's 2011-2012 Teaching Portfolio (I'd like to do this some summer)

_____
* Busking is a British term meaning performing for donations.

A Mathematical Story by Gary Davis

He's posting Matrix Algebra, one chapter a day, at Republic of Mathematics. Chapter one tickled me with all its math references, but the main character (in that chapter) is not particularly likeable. The main characters in chapters 2, 4, and 6 are much more fun to meet.

I'm looking forward to finding out what happens next.

Math Circle Institute - Day 1

I swam at 7 am, in the Knute Rockne Pool. Ate breakfast in the most amazing college dining hall I've ever experienced. And made my way over to our daily meeting place. In prior years, we walked all those calories off getting between far-flung buildings. Not this year. Easy on the feet, hard on the belly.

The Kaplans started us off with this problem: A light ray leaves one side of a mirrored square. What happens. We discussed the physics of it, and tried to get away from that, back to math. We talked about modeling this as a billiard table instead. We kinda sorta proved that angle of incidence = angle of reflection. We never worked together on the question that pulled me: How many line segments will be made by the path of the light, before it gets back to its starting point? I figured something out, but I'm not sure yet why it would be true, so it could be wrong. It was a great problem to pull people in at lots of different levels - I think.

Then we talked about relating math circle stuff to classroom teaching, since most of us attending are teachers. Then lunch.

In the afternoon, the kids came. Bob, Ellen, Tatiana, Anand, Amanda, and Rodi did math circles with the kids, and we all watched. Afterward we discussed what went right and what went wrong. Plans for tomorrow, then dinner.

There's a lounge in our dorm. We met in the lounge, and Bob ran a session on sizes of infinity. Those of us who knew the 'answers' managed to keep our mouths shut. I showed off my manuscript to a few people. It's bedtime, but I wanted to do what Fawn did and offer you some notes, so here they are.

Why do social networks matter in teaching and learning?

Alec Couros asked this question on his blog, and is looking for lots of answers. Here's mine:

Hi Alec,

No time for video. I barely know how, and I'm on vacation. But the networking I do online has changed my life, and I'd love to share.


How the Internet Made Me a Mathematician and an Editor

I've been teaching math at college level for over twenty years (mostly here).
I have a son who was in a 'freeschool' (defunct now)
And I wanted to help out with math there.
So I started reading Living Math Forum, to think about how to teach kids.

On Living Math Forum
I learned about Out of the Labyrinth: Setting Mathematics Free,
by Bob and Ellen Kaplan.
I read it and loved it.
Their website, themathcircle.org,
Mentioned their summer Math Circle Teacher Training Institute.
I went. It was amazing.
I met Kate, Jesse, Ben, Amanda, and a few others (who don't do the online thing).

After following Kate's wonderful blog for a while, I decided to start my own.
I also started connecting locally with math circle folks, like Josh Zucker who runs the Julia Robinson Mathematics Festival,
And Paul Zeitz, who wrote The Art and Craft of Problem Solving.

I also began a monthly math gathering in my home,
Which was videotaped by Jeremy Stuart and Roy Robles
As part of a film on homeschooling they're producing.

If I think of a mathematical problem I'd like to work on,
I know all these people have my back.
When I get stuck,
They'll be there.

I've solved a problem I was stuck on for decades
(And got my solution process published),
And then a problem inspired by the game of Spot It.
(Each pair of cards has exactly one matching picture -
How do they do that?!)
I've explored Pythagorean triples.
I've built a stellated dodecahedron out of Polydrons.

And then there's our book,
Playing With Math:
Stories from Math Circles, Homeschoolers, and Passionate Teachers.
I've gathered 35 authors together to share their stories
(Which I mostly found through my internet connections).
Due out this fall.

None of this could have happened even ten years ago.
Math is in my life in a way it never could have been before the internet.
I saw that when I taught linear algebra this year -
I had a blast teaching some hard material,
And I think my students had fun learning it.