This isn't directly related to teaching, but I bet I can find something relevant to say about it.
As of the past two weeks, I've been taking a free, online college course called "Principles of Economics for Scientists." I'd somehow managed to never take any sort of Econ course before, and this one is a great response to my hesitation about the subject relying too much on discussion of human nature and the sociology aspect of things. (While I do respect both of those approaches as valid, neither of them would have helped me make intuitive sense of this new subject.) As it happens, the "for Scientists" part of the course title roughly translates to, "for people who know and understand Calculus", which squarely includes me.
(Ooh, I figured out the connection to teaching and learning!)
In this class (at least so far), the professor is using Calculus as the lens through which he is teaching the fundamentals of economics. Therefore, for the intended population of students, there is a basic level of math needed in order to comprehend the material of the course. We are not doing math just for math's sake; we are using it as a tool for something else. Without a conceptual and practical understanding of calculus, the professor would have to take a more convoluted approach to teaching the exact same topics, probably resulting in students appreciating the material less.
This is equivalent to the pedagogical idea of "learning to read vs. reading to learn," the transition that all Language Arts teachers encounter in their respective grade levels. Once students can physically read a text out loud, what do they get out of it? How will the skill of reading translate into a truly functional literacy?
These same questions can be asked of math: "learning to do math vs. doing math to learn." Many secondary math classes spend all of their time teaching and learning sequences of math skills that build upon each other, with the promise that students "will need to know this later on." But for students that have no desire to continue in math or science classes, that later time never comes. I see myself as a perpetual student of math, but I've realized that while some math is universally necessary in participating in adult life, and other math is elegant and interesting in its own right, some topics are useful not just for the sake of doing math, but rather as a tool for understanding other topics, spread far and wide across the human quest for knowledge.
Every math teacher gets asked the question, "Why do we have to know this? When am I ever going to use this?" In response, think further than just the upcoming test, and even further than the next few courses in the school's sequence. Math is pervasive, and we need to convince our students that there is a bigger picture to the subject, that stretches far beyond the classroom.

Life update, since the last post:
- Completed my Math major.
- Started and finished my student teaching (MS and HS), and my Secondary Education major.
- Graduated from college =)
- Became a certified Secondary Mathematics teacher in two states so far, working on the third.
- Taught my two summer classes for the 2nd year, and got asked back for my 3rd 2013 summer.
- Currently working as a full-time math tutor at a HS in urban Massachusetts, and job-searching for next year.

Hoo boy!

What's been on my mind lately:
- Do I want to start my classroom teaching career in an urban, more difficult-to-manage school, or in an "easier" suburban environment?
- How will I measure the success of my students mid-year and at year-end?
- What can I do to increase my students' retention of math content from week to week, other than just in basic skills?
- What else can I do this school year to improve myself as an educator and thinker, and make myself a more marketable teacher candidate?

One day down, only 18 more to go!

Note to self: Leave enough time to get coffee before classes. Also, bring water and wear comfy shoes.
So, my first day of classes was rather surprising. The younger class, which I was most worried about, is remarkably easy to manage, even though the kids are antsy at the end of the day. The older class, which I thought could be a little more open, is a mash-up of all different kids and grades and personalities and motivations (or lack thereof). This will be a challenge.
What I imagined for this first class: All of the kids show up within two or three minutes of the official start time. There is paper and a pencil ready on each desk. I have a 3x3 magic square drawn on the board, and ask the kids if anyone knows what's special about it. Some kids get it quickly, and after a minute I give a hint to the remaining kids. After a few more minutes, everyone had gotten it. I hand out a homemade worksheet with partially-filled magic squares, answer a few classwide questions, and let kids work on it in their small groups. Much inter-student discussion ensues. Once most have finished, I draw a table on the board to flesh out some patterns that showed up in the squares. For the last part of class, I hand out mini challenges to each group ("Make your own magic square that...") and let them work together on it. If someone finishes a challenge and doesn't want to start a new one right before class ends, I give them a puzzle photocopied from MathMania. Done.
What actually happened: Kids show up after lunch. One of the first girls in is not actually on the attendance list, so she sits and mopes until our counselor gets here and can take her to the office to see what class she's actually signed up for. I show the class the magic square on the board, and ask if they can tell what's special about it. Some kids know already. I give the hint and some scaffolding, and a few more kids get it. I, along with student volunteers, flesh out what the trick is, and a few more kids say they get it. Our last student to show up walks in, sits down at the very back table, and I tell her to move up to one of the tables with other people at it. Since there are about 12 kids who've already mostly figured out what's going on, I hand out the worksheet to everyone. The girl who came in late has no idea what's going on with the math, so I ask the girls at her table to try explaining how to fill them out. After a few minutes, I realize that one of the problems I put down doesn't actually work, so I tell everyone to cross out that square. After looking at a few finished papers, I clarify that each sub-square must have a different number, that zero isn't one of them, and that squares can have different magic numbers. A few more kids fix and finish their worksheets, or just get frustrated after trying for a while. There's about 10 minutes left in class, so I give kids the options of either trying one of the challenges, or doing the other puzzle. Most kids chose the puzzle, but a few attempt the challenges and come up with some good work. Kids say they don't get what's the goal in the puzzle, even though there are directions written on the page. Class ends.
What I could do differently:

• Hand out the photocopied puzzles at the start of class to keep students occupied while everyone gets to class, and to suggest some math techniques. (The puzzles will be related thematically to the actual lesson.)
• Have all necessary supplies on the desks before students come in, along with extra paper. 
• Up on the board or individually, give a set time for students to share some techniques and tips for the day's activity, to use as reference at all of the tables.
• Designate a bathroom/water break 15 minutes into class. (It is right after lunch, after all.) 
• Have one calculator for each small group, so that students who are slower at arithmetic can still get something out of it. (This is for elementary and not secondary students, so I'm not totally contradicting previous opinions >_<.)
• Check and double-check any papers I hand out.
• Use the worksheets as scaffolding, with directions, hints, and prompts included. 
• Figure out how to integrate the kids that are signed up for my class because their parents don't want them to fall behind over the summer, which may or may not be the same kids that show a lot of learned helplessness. Yeah, Differentiated Instruction!
• Commandeer another counselor (already done for tomorrow). 

I'm going to try to implement as many of these changes as I can for tomorrow's lesson. Hopefully, I'll be able to convince my kids that I do actually know what I'm doing, and that this will be a positive environment for exploring math. First days are always rough.

Strangest shopping list ever.

Things I have bought for my classes so far:

• Posterboard (for cutting up into DIY rulers)
• Small plastic cups
• Pack of plastic rulers
• Dry-erase markers
• Snack-size Ziploc baggies
• Wagon-wheel pasta
• Silver star stickers
• Mini post-its 

Also, my mom discovered about a dozen issues of Puzzlemania and Mathmania books (anyone remember those?) in our attic, along with a number of other math puzzle books. So many things...

How I (Will Have) Spent My Summer Vacation

The next few months are going to be very different from what I've gotten used to as a college student. Just as soon as I got the hang of being a decent student, I end up teaching instead. It's all for the better, anyway; according to my professors, I haven't actually learned any useful teaching skills yet. I have to get those on the job. Well, here's that there job.
For the next four weeks, I will be teaching two elementary school-aged math classes at a community summer program (where I previously worked as a counselor, and was a camper before that >_< ). My first class each day is "Math Fun and Games," where I'll try to convince students entering 3rd-6th grade that there's more to math than worksheets and drills by encouraging them to look for patterns and think creatively. The class for kids entering 1st-3rd grades is called "Number Sense," and I hope to give the younger students an idea of what different physical quantities look like by learning how to measure and record.
Classes start on Monday, and I honestly have no idea what to expect:

• The kids are outside of my typical age range for teaching or tutoring.
• This is my first time in charge of an actual class.
• I've come up with the lesson plans myself, and don't know how good they are.
• I don't yet know how old the kids that signed up are, or why they're signing up.

I'm pretty sure I find out part of that last one tomorrow. It's only really an issue for the younger class, because there's such a wide range in what I can do with a 5 y.o. fresh out of kindergarten, and a 7 y.o. who just finished 2nd grade. (All made more annoying by the fact that NJ doesn't have statewide math standards, nor does my school district across schools.) There will be a lot of modifying to do after the first day or two of class, but I feel like I'll manage to not screw up my campers too badly the first time I teach this. 

Fun- and preparation-filled weekend ahead. I'll keep all y'all posted. 

(Looking at past posts, I'm good about writing at the start of a "semester". I'm going to break that habit this summer.)

What my semester looks like

Sorry it's been quite a while since my last post. Classes started at the end of August, and life has been pretty crazy.

On that same note, my schedule will give me a good number of future post topics; I presume that's the occupational hazard of completing a college (double) major in what the blog is supposed to be about. My classes for this semester are as follows:

1. Multi-Variable Calculus
2. Linear Algebra (both for my Math major)
3. Adolescent Development
4. Computers, Teaching, and Math Visualization
5. Secondary Education Practicum
6. Jewish Science Fiction.

I know the last course seems a bit out of place, but it's a break from classes that I specifically need for my majors. Anyway...

The first five courses are fitting together surprisingly well. The straight-up math courses certainly teach the stuff they're supposed to, but it gives me some opportunity to play with ideas from the other three classes. And then topics covered in one of the education classes give me something to focus on from the other two.

An example of what's been on my mind:

In Adolescent Development, the 2nd chapter we covered was cognitive development. I forget which researcher it was, but someone talked about the stage at which kids are able to make leaps from the concrete to the abstract. In terms of implications on math education, that would be when students are able to understand the concept of a variable. So, depending on whether the student is an early- or late-maturer, he may get lost in class if the teacher introduces variables before he can really get it.

I witnessed a possible instance of this first-hand as I observed a classroom through my practicum: 7th grade Advanced Pre-Algebra. The lesson of the day was the Properties of Addition (Associative, Commutative, Distributive, etc.). The teacher talked about grouping like terms, and why students couldn't simplify the term "x + x^2". He asked a student why it wasn't possible, and the student replied that it was because they "don't know what x is." This sounds right, but it could be that the student thinks that "x" has a different value at both places in the the expression, which it doesn't. From a practical viewpoint, is there a good way of diagnosing cognitive misconceptions, or whether or not the student has reached the level of understanding in order to get what a variable is in this context?

So today in my Math Technologies class, we got into a discussion about whether or not technology should be used as a crutch to help students that are lagging still get something out of a lesson that may be over their head. The consensus was that if a student has a hard time carrying out the algorithm to compute a problem, they may still be able to benefit by viewing the problem more conceptually using a graphing calculator or Geometer's Sketchpad. In the above example, is there some way that a certain technology could be used effectively by the teacher to help the student visualize the logic of "grouping like terms" before they understand how variables are used?

And then when I don't feel like thinking about (relatively) little kids anymore, I can apply the same mindset of a teacher to my own math classes. Even once we reach college, the same things still apply. I find it interesting to contrast the teaching styles of both of my current math professors, mostly to find some justification in really not liking, at all, how one of them teaches. (The fact that I can't understand his accent half the time doesn't help, either.) A review book will be desperately needed for that class.

On that note, I should be studying. Any feedback would be enjoyed.

How's about those positive role models?

My friend showed me this video of Patricia Heaton at Guest Week on "Who Wants to Be a Millionaire". (Skip to about 3:00.)

It isn't often that I yell at videos, but this is ridiculous. As soon as anything with numbers showed up on the screen, she completely shut down even before looking at the question. This is a grown woman freaking out about middle school math. A kid would look at the video and say, "She can't do math and got to be famous. I can do that too!" Never mind the fact that when Regis walked her through it, she was perfectly capable of solving the problem.

A defeatist attitude is lethal to learning math or any other subject. Unfortunately, mathematics often gets the short end of the stick in situations like these; that's what credit cards are for, right? It's a lot easier to get away with never doing simple, mental calculations than to get through life unable to write a cohesive paragraph. Who cares that you can't balance your own checkbook, as long as you can write thank-you notes to your generous donors. 

Does anyone know of a celebrity who likes doing math in public, and is proud of it? Who knows, maybe Patricia is just stuck in a junior high mindset and doesn't want to be called out as a nerd.