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.

How Intermediate Rounding Took 20 Years Off My Life

Recently I found myself in a situation where intermediate rounding seemed inevitable, and so I sat there wondering, “Is there some kind of rule that would help me to discern an appropriate amount of rounding that is acceptable in the middle of a problem, so to not impact the final answer?” For example, if I need my final answer to be correct to the nearest whole number, would intermediate rounding to the nearest thousandth have an impact on the results of my final answer?

Potentially, it only takes 0.01 error to impact a final value rounded to the nearest whole number. That is, 2.49 would round down to 2, but 2.50 would round up to 3. Rounding intermediately to the nearest thousandth only introduces a maximum error of 0.0005  (say, from rounding 10.2745 up to 10.275 or rounding 5.25749999… down to 5.257).

Clearly, I could see that the answer to my conundrum would be a definitive “It depends.” Of course, it would depend on what happened in my problem between the intermediate rounding and the final answer.

As it turns out, there are lots of fascinating intricacies that play out in the solution of this problem. It's almost too embarrassing to admit just how much brain real estate I have dedicated to thinking about this.  But here’s one particular aspect that struck me hard.

If I am introducing an error of 0.0005 and then multiply this value by some factor, then my error would also be multiplied by this same factor. OK, so in this particular scenario, a factor of 20 would be sufficient to potentially impact the final whole number value.

What if I square the value? My instinct says that the error would also be squared, which would lead to an insignificant impact on my scenario. But my instinct is wrong. The reality is that the resulting error relies entirely on the initial value. For example, a value of 256.0235 that was rounded up to 256.024 and then squared would be off by more than 0.25, clearly enough to make a significant impact. And a larger number, like 10,000.0005 that gets rounded up to 10,000.001 and then squared would be off by more than 10.

BAM! I find myself in the body of an awkward teenager, struggling with the most famous algebraic misconception:

You see, I haven't made this mistake in years, but yet am amazed to find that the inner instinct still remains. I'm not sure what this means exactly, but at the very least it sheds some light on my teaching and perceptions of student understanding. Too often this particular misconception gets blamed on a misapplication of the Distributive Property.

What if, instead of insisting that "exponents do not distribute," or "the Distributive Property does not apply here," I allowed students to explore their misconceptions and discover that the Distributive Property does indeed apply? What if we embraced this instinct and used it to delve more deeply into quantities as factors?


What if I finally realized that even if they remember the rules and get this problem right every time it appears in symbolic form, that maybe, just maybe they still don't quite understand what it means?

What if.

I think I feel a performance task coming on.

Shifting Roles

What's the difference between a classroom teacher and a textbook author?

See, I know that you are all set for some snide punch line about curriculum writers, but actually, my motivation for this post is in defense of the 'other side.'

I should start by saying that the reason for my idleness since March is that I started a new job as a mainstream editor of secondary math resources. While this is not my first experience in the professional publishing industry, this past year has been a fascinating mental shift from classroom teaching, to independent writing, and now to my current role as a cog in a great big machine.

In my current project, I have spent over 300 hours pouring over every detail of a teacher's edition for a new pre-algebra program. The program includes animated media clips for daily lesson hooks, scripted teacher presentations that include digital slides/screens, daily student workshop routines, solutions and coaching prompts for anticipated student shortcomings, journal prompts, formative and summative assessments, digital math tools for presentation or student use, online homework with integrated media supports, and on and on.

Is it perfect? Certainly not. But boy, is it comprehensive.

My reason for saying all this is not for promotion of either this particular curriculum or company (who shall both remain nameless), but rather a two-fold defense for the corporate model of educational publishing:

• First, take a closer look at the current mountain of resources in your department's resource closet. Find one that jives with you and really delve into all that it has to offer. You'll probably be surprised. Cut and paste tactics are not really the best for continuity, so stick with one. And if your closet is old and full of terrible resources that you hate, don't dismiss the entire industry. There ARE good curriculum packages out there. Look carefully and critically, make the district see your case for investment in new resources, and the payoff is huge. Imagine the joy of not needing to create everything from scratch.
• Second, I LOVE TeachersPayTeachers.com. I have my own storefront, and I love browsing the beautiful, creative, and inexpensive resources that other teachers have posted there. But, shame on you Paul Edelman for the slogan "Teachers Pay Teachers, not big corporations... it's about time." There are some things that large teams of authors, editors, artists, programmers, and analysts can do better than individual teachers. Honestly, I think current, integrated, and comprehensive curriculum programs fall into that category. Yes, there are faults, and yes, there is room for competition from the classroom perspective, but I think teachers will be better equipped to survive as partners to big corporations, rather than opponents. As an inexpensive and creative way to supplement an existing program, BRAVO! As a protest against corporations, not so much.

After all, believe it or not, both teachers and corporations are interested in the academic success of our students. Just imagine where we could take those kids in a consolidated effort.

A Note to My Former Self

Dear Me,

You're young, smart, ambitious, and about to embark on a wonderful adventure as a teacher. I know you are packed full of information and education about how to be the best possible teacher you can be... the enthusiasm, creativity, and energy are glowing around you. But I have a little bit of advice for you that isn't so academic. Maybe you will listen to your future self.

1. Don't reinvent the wheel. Yes, your ideas are wonderful, engaging, and creative... but you will burn yourself out faster than a candle in a vacuum if you try to recreate the whole curriculum. A healthy dose of trust and humility will take you far. Try to focus on building one or two creative ideas a month. Over the years, you will have plenty of time to build a terrific repertoire that makes you proud. In the meantime, look around - an abundance  of wonderful resources are just waiting for you to utilize them.
2. Share and share alike. Some people are not so good at sharing the ideas and resources that they create... maybe out of fear of criticism, or perhaps a lack of confidence. Luckily, this has never been your trouble. Your enthusiasm for getting your ideas out there will take you far. But don't forget the other side of the coin: invest time in listening to other ideas (even if you disagree at first) and don't be afraid to ask others to share with you. People want to help you; let them. Just treat your peers with respect and appreciation and you will be amazed by the wonders of reaping the benefits of someone else's experiences.
3. Reserve judgement as much as possible. I know that there are plenty of people who appear to be slackers, grumps, nay-sayers, users, and just plain jerks. As a nose-to-the-grindstone bundle of creative energy, it is so easy to criticize and see faults. Someday soon you will experience more of life's challenges: difficult or unreasonable students/parents, demanding administration, mountains of grading, illness, 24-hour infants, family management, ailing parents, crushing debt, and numerous untold emergency situations. You will see how easy it is to get bogged down and worn out, and then you will wish you could go back in time and just give those people a hug.
4. Enjoy your peers. Go out for beers on a Friday afternoon. Invite them over for a planning party at your place. Go to conferences together. Meet their families. And don't hide in your classroom at lunchtime... eat lunch together! These people are your best resource and safety net for retaining sanity in this job. Treat these relationships with the utmost respect, and don't forget to invite the grumps too. Some of them will surprise you. I promise.
5. Be diligent about keeping a diary. A key ingredient of personal improvement and professional development is self-reflection. Time spent on revisiting the day's (or week's) successes and failures is time well spent, and the rewards are even greater if this reflection is shared in a community of peers: like a blog. Just remember that the feedback you elicit will reflect the tone of your comments, so if you want constructive and uplifting feedback, dish out the same.
6. Make organization a top priority. One year, you will need to move out of state, and you will be inspired to go on an organizational blitz so that you can share your legacy with the friends/peers you leave behind. The fruits of this blitz will be wonderful paper and digital archives. I have relied upon these archives more times than you can imagine... and have regretted the multitudes of resources that have since gotten lost in piles (real and virtual). Think of me, your future self, as your very best friend - making my life easier will be rewarded handsomely.
7. Use your summertime wisely. You will be exhausted when the school year ends. I know you will have worked hard for 80 hours a week (or more) all year long and you will not be able to think of anyone who deserves a 2 month vacation more than you. I'm here to tell you that it gets easier... but do you know what really makes the difference in time demands? Preparation and organization. Give yourself ONE month of vacation. Really, it's enough... because your future self will thank you for the time spent cleaning up last year's mess and creating thoughtful, reusable plans for the future.

Most of all: Keep your head up. Breathe deeply. And don't let the turkeys get you down. Lots of beautiful people and wonderful experiences are coming your way. Don't forget to enjoy the ride.

Lots of love,
Your future self

Just for Fun: Problem 3

It's 6am and thirty lockers stand in a long lovely row, closed and neat, ready for the opening of another school day.

• The first student arrives and, what the heck, opens every locker! 
• The second student arrives and changes the position of every second locker (i.e. lockers 2, 4, 6, 8, etc all got closed).
• The third student arrives and changes the position of every third locker (from open to closed or vise versa)
• The fourth student changes the position of every fourth locker...
• This continues until the 30th student arrives and changes the position of locker #30.

What is the final configuration of the lockers after all 30 students have passed by?