Reflections and Transformations

In my town, school starts this week. My children have new backpacks and lunchboxes and shiny pencil cases with 5 newly sharpened pencils in each. The New Blogger Initiative is filled with stories of first day jitters and school year goals. This is a hard time for me.

Ask me what I do, and I'll tell you that I am a math teacher. I have taught in urban, rural, and suburban schools. Unfortunately, life handed me a pink slip last year. It happens. Budgets get cut and new jobs are not stable jobs. But even without a classroom, I am still a math teacher. You are what you are. At the risk of sounding vainglorious, I know I am a good teacher. Not REALLY good, but working towards it.

So this past year I took the pink slip as an opportunity to reflect, learn, write, grow, and move into a new era. I started this blog (and thank the Initiative for kicking my butt into keeping it up). I took a very close and critical look at lots of stuff in my filing cabinet. I have made a little money by offering some of these things for sale. I know the mathtwitterblogosphere is a sharing culture. I have lots to share, but I have mixed emotions about sharing everything. So I may not be as avuncular as Sam Shah, but I hope to create a helpful space here on my blog.

Here's my first day syllabus. I've used it for a lot of years and it probably needs an update, but I still like it.
syllabus

I use this same format for all my classes, with tweaks to supply lists and calculator guidelines etc. The editable Word file is here, although it doesn't translate very well in Word (I use Publisher mostly, but no one else seems to). You'll have fun making it your own. I do.

Now, before you start feeling sorry for me and sending me job listings, I'm being picky. I know what it's like to be me as a teacher. It's a 50+ hour/week commitment, with lots of worry and stress. I'm at a stage in my life where I don't wish to handle too many other external stressors. 10 minutes is about my limit on commuting time. Besides the extra time, I just like teaching close to home, in my own community, where I run into kids on the street and their parents at the grocery store. Some people don't want this, but I do.

So in the meantime, I have a job. I edit math curricula at a huge publishing operation. I spend lots of time thinking intensely about tiny details, which is a wonderful contrast to teaching - where you have teeny amounts of time to maneuver a plethora of calamities. In the past several months I have been able to deepen my appreciation for:

• The pervasive misunderstanding of the difference between the terms inverse and opposite.
• The devastating impact of intermediate rounding.
• The art of posing just the right question to provoke intrigue and deepen student understanding.
• The subtle mathematical properties of okra.

Okay, maybe not that last one, but the point is that even without actually being in the classroom, I still find myself improving as a teacher. I see that there is a long and fascinating road both before and behind me. There are things to share and things to learn. Luckily, the company continues to be great, and just keeps getting better. Thanks for YOUR contributions to this fabulous community.

And there it is, my reflections on the transformations of my past year. Hardly Hemingway-esque, but veracious nonetheless.

My Backwards Approach to Inverse Functions

Joe "Math Guy" was one of the first lessons I ever created. I drew this comic strip 'hook' for a sample class that I taught on inverse functions during a job interview. Years later, it's still one of my favorite lessons to teach.

One problem with algebra is that there is often a disconnect between the meaning/understanding and the computations/doing. We try our darndest to bridge the gap between the two, but I find that the meaning often gets muddied by cumbersome symbolic computations. For me, I like the way inverse functions lend themselves to the meaning first, and symbolic abstraction second. And when I do it well, a beautiful aha moment can occur.

Step1: Start Simple.

• Functions are a series (composition) of one or more actions (functions) that maps one object onto another (as long as each input is related to only one output). For example, "Take something, add two and then multiply by 5," is a function. [It's also important to note that symbolic notation can differ in representations of the same function: like 5x + 10 and 5(x + 2). Why?]
• Inverse functions are a series of reverse actions that undo the actions of a function. So, "Divide by 5 and then subtract 2," would be the inverse of the above function.
• A function and its inverse, when composed together (in either order), always 'do nothing'.

Then we practice finding inverses of simple functions by first identifying the sequence of actions and reversing it. It's wonderfully intuitive and students 'get it' right away, just as long as Joe and I keep it relatively simple. Challenges at this point come in the form of four and five step functions, and not rational and quadratic curveballs.


Step 2: Complicate Things

Suddenly we find ourselves confronting rational functions and functions with multiple x's and our intuition begins to meet its match. At this point either I or someone in the class will throw up their hands and beg for a methodical way. I'll mention that one of my colleagues told me that I could just solve for x and that would be my inverse function. Dubious, but worth a shot. And so we try it, and yes it works. WHY??? Will that always work? What is going on?

Why is finding an inverse like solving an equation?

It is at this point that we talk about notation and graphs and all the algebraic aspects of inverse functions, keeping a tight grip on meaning: inverse functions 'undo' functions... no. 1 application for us right now? solving equations.

Have you noticed that we have not yet encountered any functions that don't have inverses? We do a lot of practice with functions that do have inverses before we even think about ones that don't.

Step 3: Complicate Things Again



So, now Joe finds himself confronted with two more functions and builds two more function machines. The problem is, Joe just cannot get back all of the numbers he threw into the original function! Why not?

What's wrong with these inverse machines? Is there any way we could tell in advance that these functions would have inadequate inverses? Is there any way to compensate for the missing values?




I purposely try to stay away from formal language at the beginning of this topic, but suddenly there is a lot of talk about inputs and outputs and mapping two inputs onto the same output. So the formal definitions come out, and lo and behold, they don't seem like jibberish.

If I'm lucky, something wonderful happens. They see a connection between this new topic and what they've been doing all along (solving equations). MAYBE they begin to appreciate the need for abstraction, formalization, and making compensations for small discrepancies.

And when that happens, my head rests peacefully on my pillow at night.

For Free or Not for Free?

For free or not for free: that is the question; 

Whether 'tis nobler in the mind to covet
The wealth and fame of outrageous fortunes,
Or to take arms against the seas of ignorance,
And by benevolence, end them? To donate: to cede;
No more; 

Honestly, I can see both sides:

• Providing original curricular resources openly and freely creates an atmosphere of collegiality and solidarity among teachers. Ideas are given more room to grow, although they are potentially less developed (which can often be a good thing). Plus, ideas can be more widely spread, since there is no cost involved.
• Offering original resources on a fee basis limits their influence to those that are willing to pay the price, but rewards the author for his time and creative genius. This incentive has the capacity to encourage greater care in the production, and can lead to higher quality and more thorough resources.

On a personal level, I believe in the benefits of benevolence, but I also obsess about perfection. After I have put together a lesson, activity, or unit for my students, I try it out. Sometimes it's great, sometimes, not so much. But then in the hours of afterthought and redesign, I try to address the quirks: design away the flaws, fill in the gaps, remove the bumps, and polish it up with some serious rationalizations. I have been known to spend an additional 17 hours on this revision process for a single lesson.

And you must be thinking, "Who has this kind of time?" 

I do. I suppose it's time for me to be transparent: I've been without a classroom since June. I've been embarrassed to admit it - afraid of a loss of credibility and upset by my role as victim of the down-turned economy. Nevertheless, here I am, hoping for a new position in September, and filling the time with lots of intense self-reflection and curricular revision. Sometimes I'm empowered to share my work freely, but lately I feel validated in asking a small fair price for my time and ideas.

And in the spirit of sharing, I'd like to open a forum for you and I to share some original resources. What is the best thing you created for your students? Link it up below, free or not. We'll let the submissions determine the mood of the masses. Add a couple things if you like, but please: 

• only post links to actual resources and not your general website or blog. 
• only middle/high school math products, like algebra, geometry, trig, calculus, stats, etc.
• free or cost items are both welcome. If you would like a nice recommendation for a marketplace to host your items, click here to join the TeachersPayTeachers community. You can give your things away or name your own price. It's a lovely community, and they could use some more good secondary math products.
• in the URL field, put the location of the actual product, and in the Name field, write a short description (subject and topic are good to know!)