If It's Not Scottish Calculus, It's Crap

In memory ay me matile Colin Maclaurin, here's a summary ay whit ah did in mah Calculus BC class teh day bfer yeserday (ok, I'm afraid that's all the Scottish I know).

From our work with geometric series, we knew that if |x| < 1


A complete side note...I'm using Text the World to embed all the mathy stuff (thanks Sue).  While I've found it to be a bit buggy and clunky, it supposedly will work in blogger, google docs, gmail, etc. so I'm excited about the possibility.  This is my first attempt to use this so let me know if you're having trouble reading anything.

Anyway, from an algebraic perspective, I think this is quite surprising.  Basically, we're saying that this infinite polynomial is equal to a nice "simple" rational expression.

So is this true for other functions?

I asked my students what functions would be especially interesting to look at.

is not a terribly interesting expression to write as a polynomial.  sin(x) on the other hand...that would be curious.

Here's what they came up with as a class:

• trig functions
• log functions
•  
•  

I then had the following dialogue:

Me: So is a particularly interesting function in calculus because...

Class: it's derivative is also

Me: Great.  Try and find a polynomial with this same property.

They broke into small groups and began exploring.  Every group quickly realized that a finite polynomial won't cut it.  They then began trying different infinite polynomials.  I gave a few groups some guidance around starting with and then trying to find the coefficients that would make the property of f(x)=f'(x) hold.  Different groups approached this in different ways.  Some differentiated to get and then set the coefficients of each power to be equal to each other.  One group did something similar with the integral instead of the derivative.  Another group did something similar to this, but in more of an informal way where they realized that as you repeatedly differentiate you'll end up with factorials in from of your coefficients and then played around with some different polynomials with factorials in them.

I didn't help at all.   To my students' credit, when I asked 15 minutes later if they wanted us to come back together and talk/get hints/share ideas.  I got a resounding no.  Woo hoo!


They ended up working independently for about 30 minutes.  Every group found a polynomial that worked. Some groups realized that any multiple of their solution would have the property that f(x)=f'(x).  Two groups went further and attempted to define some equality between and their polynomial.  One individual began working to find a polynomial with derivative characteristics to sin(x) ie f(x)=-f''(x).  Anyway, everyone independently came up with some version of


I was pretty excited about this.

Sure, we still have work to do in terms of formalization to define a more robust definition of equality (more than just having two functions with the same property).  It'll be interesting, though, to see if this helps students understand MacLaurin Series and Taylor Series.

Reason #792 I Love Mathematics

Is 80 degrees twice as warm as 40 degrees?  Discuss...




Ok, so obviously I wouldn't be posting this if there weren't some subtlety to this question.  I will say that three days ago I was pretty confident in my initial answer to this question.

Tracking Study

Probably most appropriate for twitter, but I'm still holding out on that end...

At the Escape from the Textbook conference I attended last Saturday, Jo Boaler referenced a large scale study in England on tracking.  A cursory search came up empty so I came here to let someone else do the work of finding it.  Ok, ok, not really.  Actually, I was hoping someone familiar with the study could point me in the right direction.

Thanks!

Escape from the Textbook Conference

On Saturday I'm attending what I hope to be a great conference at The Urban School of San Francisco: Escape from the Textbook with speakers Jo Boaler (What's Math Got to Do With It), Paul Zeitz (The Art and Craft of Problem Solving), and me (too many books to name). Ok, I'm not really one of the speakers but I will be leading an afternoon workshop that I am calling Habits of Mind: From Student-Posed Problems to Assessment. 

Here's the plan:

We will complete a high speed version of a project where students create, explore, and generalize visual counting patterns (pile patterns for example). Sprinkled in will be discussions of how mathematical habits of mind (aka problem solving, aka mathematical practice) can be developed and assessed. Come prepared to do some math. 

Basically, we're going to start with

and go from there...

Here's the current draft of my powerpoint (zzzz) presentation.  I've never done this before (mostly because I've never been finished mostly finished putting together a presentation this far in advance). Obviously, there are lots of pieces that need to be filled in (I kind of hate powerpoints that don't have pieces that need to be filled in) but I'm hoping that I can get some feedback beforehand, send participants here during, and continue the conversation afterwards.

Here are some additional resources related to the workshop.

Edges, Faces, and Vertices

Counting Project

Painting Cubes

Mathematical Habits of Mind

Supposedly there will be attempts to stream the conference for those of you that fail to scalp tickets between now and then.  We'll see how well this works, but stop by if you have some time (9-3PST).

Teaching via Games: The Introduction

I like games.  Board games, video games, sports, Bear/Ninja/Cowboy, you name it.  I've even managed to get to a place where I even enjoy games when I lose.  I know...mom and dad are so proud.  When I moved to California, one of my best craigslist furniture finds was a nice big coffee table that opens up to reveal a not-so-secret compartment.  Mine's filled with board games.  Below are some of the most recent games I've played and/or added to the bowels of the coffee table.  

There are lots of games that are explicitly and implicitly mathematical.  Some a great and some are terrible.


Math Mama Writes has introduced me to some of the great ones.  Math for Love has a great post cautioning us that any game labeled as a "fun math game" is like calling something you would ingest a food product--typically not a good sign.

If your goal is to teach and you just slap the word “fun” on at the end, you end up with garbage. On the other hand, if you’re focused on making a game that’s really good, well, then it is fun. Of course, whether it has any educational value is pretty dubious.

I've played mathy games for many years.  There are hundreds (never mind the hundreds more using nothing more than a deck of cards).  Here are 20 (and please add your own favorites in the comments):

I find these games to be fun first, and mathematical second.  I use some of these in my classroom and a few to complement or inspire content.  For example, there are some great counting questions that can be asked about Set:


How many cards are in the deck? 
How many sets are in the deck?
What's the greatest number of cards you can have without a set?


I've used Mastermind lessons as a tool to develop the concept of proof with middle schoolers.  Backgammon is a great game to practice probability.  


Overall, though, many of these games present challenges in the classroom: they take too long to play or teach the rules, they require too much set up, they can only be played by a limited number of people, students can play without actually trying to implement any thoughtful strategy, and the big one: it is challenging to convince kids (and schools and parents) that playing these games is not just an extra, froofy, free time activity and a break from doing "real" math.  True, becoming better at chess will most likely not help you become better with fractions, but I've begun to think about how these above games could help develop mathematical thinking and habits of mind.

New Semester, New Job

While technically I still have the same job (and the job I signed up for) and I'm still at the same school, last week marked the beginning of a significant transition from teaching 6th grade to Calculus.  I've taught both before (sometimes back to back), but going from all sixth grade all the time to all seniors (and a few juniors) all the time has been more jarring than going from one to the other in the same day.


I miss my sixth graders a lot.  They were excited and eager and energetic and brave.  We'd built a pretty fabulous learning environment.  Yes, I still had kids who were struggling with the content, with my style, with the transition to a new school, etc, but I knew who those kids were and they were making real, measurable progress.


Now I start all over again (never mind the fact that if there had been a memorizing/remembering names course in high school and if this course had been a required course for graduation, I'd probably still be sitting on the other side of my teacher desk).


Don't get me wrong.  My calc students are friendly, studious, and smart (in both traditional and non-traditional definitions).  


One gave this great "proof without words" of why 1/3+1/9+1/27+...=1/2

  
See it?  Beautiful!

My 6th Grade Version of Algebra

This is probably better fitting for a twitter tweet twit, especially considering how delinquent I've recently been but couldn't pass up sharing a few questions/comments from my sixth graders today.

Ok, so maybe a little context to defend my choice of not being on twitter:

EDIT: Ok, maybe more than just a little context.  If this is too much to read, at least skip down to the end and look at the questions/comments.

My sixth graders just started talking about algebra.  I love this because, at least with the way I approach it, it's the first time that I don't have to do much unteaching (fractions, on the other hand...).

I began by asking them to come up with mathematical questions motivated by a 1x1, 1x2, and 1x3 strip of squares made from Construx.

We shared out.  The good: Lots of kids asked questions about counting the pieces and looking for patterns.  The bad: I was going in this direction regardless of the questions they asked.

Then in groups they played around with this Edges, Faces, and Vertices activity for a while, building models and looking for patterns so that they wouldn't have to build huge models.
The next day I asked them what they thought their new unit was on.  Most guessed geometry.  Others guessed graph theory and combinatorics.  No one used those words.  A few said algebra.  I told them all of these are reasonable choices, but we're going to make the choice to explore algebra.  I then talked ranted for a few minutes about some misconceptions of algebra:
1. It's something you do in 8th grade.
2. If you do it before 8th grade, you're ahead.
3. It's something you check off the list and don't do again after 8th grade.
4. It's math with letters.
5. It's something my smart friend did in 4th grade.
*6. It's something people use all the time in their jobs.

*There may be some disagreement with this misconception.  It'll have to wait for another post.

I then gave my algebra schpeel.    Talked a little history.  Talked about how the word comes from the Arabic al-jabr roughly meaning restoration written by a guy named al-Kwarismi (where the word algorithm came from) and how these ideas managed to avoid Europe for close to 1500 years.  Spent the few minutes to share the aside that while the mathematics of algebra didn't make it to Europe, the word did as barbers in the medieval times were called algebristas because in addition to cutting hair they were also bone setters. As an aside to my aside, they were also bloodletters which is where the red in the barber pole comes from.  Well, at least that's what the internet tells me.

Then we talked about three aspects of algebra:
1. Finding the secret number or numbers, ie solving algebraic equations.  Told them that the problem
3+?=7 was really not so different from x^2-5x-6=0 except that the latter was harder to find the secret number(s).
2. Describing and using patterns, ie writing algebraic expressions and using formulas.  For these types of problems, it's not about finding secret number(s). Instead, it's about describing relationships and making generalizations.  Hey, we've already done this when we described the number of edges, faces, and vertices in a 1xN strip.
3. Proving equality or inequality, ie manipulating expressions.  Hey, we've already done this when we showed that one students pattern for vertices of 2*(N+1) was the same as another student's pattern of 2*N+2.

I'm sure there are other great ways to think about the subject of algebra.  Feel free to share your own in the comments.  After making a few examples, I left them with 2 questions:

1. So we've already done some of #2 and #3.  What about #1?  Can you make up a problem in the context of edges, faces, and vertices that would use the first aspect of algebra?
2. Someone mentioned that 3+?=7 has a solution of 4.  Does it have any other solutions?  Can you create an algebraic equation that has more than one solution?

Mixed results with #1.  Some kids were right on it and came up with the expected "What size strip has 68 vertices?" along with some other good questions.  Others were pretty confused by the task.
Great results with #2:
0*N=0
(11-11)*N=0
1*N=N
N/N=1
7*N=E where E is even

There were more, and this second question prompted some fabulous questions/comments, the point of this post.  While I've tweaked some of the language to be more concise, these are all from the 11 year olds.

Can I make it so N has to be a whole number?
Can I make it so N has to be an even number?
Is 2+3=x an algebraic equation? (anyone have thoughts on this?)
In the equation 0*N=X, N can be anything but X has to be 0.
Can I make the operation a variable? (I had a hard time containing my excitement)
If variables are used to talk about a bunch of numbers at the same time, can we use variables to talk about a bunch of algebraic expressions at the same time? (more failed attempts to contain my excitement)