6 Essential Questions in Algebra

A year ago, I began this blog with the goal of uncovering some satisfying essential questions in algebra. These were to be questions that addressed the fundamental essence of algebra, while also being able to extend beyond a single discipline... and of course, they needed to be intriguing to both my students and to myself. A few months later I wrote that one of the qualities of a REALLY good math teacher is having a 'second set of objectives that go beyond the mastery of today's content.' A reader challenged me to identify these objectives, which I slyly avoided. But this week, in honor of my one year blogoversary, I present six essential questions or 'higher objectives' for my algebra classes. It's a start. In the spirit of UbD, expositions are voiced in the language of enduring understandings.

How is algebraic thinking different from arithmetic thinking?

It is my hope that my students will understand that algebra is a language of abstraction, where patterns are generalized and symbols are used to represent unknown or variable quantities. Arithmetic involves counting and manipulation of quantities where algebra relies more heavily on reasoning and generalizing the patterns that are observed from arithmetic procedures. It is my ultimate hope that they come to appreciate the power and utility of generalization.

What makes one solution better than another?

I would like my students to understand that numerical accuracy is only one piece of a good solution. The measure of a comprehensive and satisfying solution involves a subtle balance of precision, clarity, thoroughness, efficiency, reproducibility, and elegance (yes, elegance). I want my students to be masters of the well-crafted solution.

How do I know when a result is reasonable?

I want my students to understand that in math, as in life, context is supreme. There is no 'reasonable' or 'unreasonable' without an understanding of context. I hope that they can refine the skills to analyze and dissect problems that are both concrete and abstract, applied and generalized. I want them to develop habits of inquiry, estimation, and refinement. Ultimately, I hope that they will improve their sense of wisdom.

Do I really have to memorize all these rules and definitions?

Students will understand that mathematics is a language of precision. Without explicit foundations (axioms and properties) and precise definitions, reason gives way to chaos. On the other hand, they should understand that many perceived 'rules' in mathematics are simply shorthand ways to recall a train of logical reasoning (like formulas and theorems). It is my hope that they will appreciate precision but also understand the value of reason over recall.

Isn't there an easier way?

Without destroying their fragile spirits, I want my students to appreciate the benefits of struggle. I want them to realize that insight and higher knowledge are gained by approaching a problem from different angles and with multiple methods and representations. I want them to understand that knowledge about how mathematics works is on a higher echelon than the solution to a particular problem. In my ideal classroom, the students will understand how to spark their inner intrigue in order to move themselves beyond answers to seek connections, generalizations, and justifications.

Do I really need to know this stuff?

By sheer repetition and example, my students will know that the practical applications of algebraic thinking are numerous, especially in the rapidly changing fields of science, engineering, and technology. Beyond these undeniably important applications, they will know that confirmed correlations have been made between success in algebra and improved socioeconomic status. But ultimately, I hope that they will understand that the beauty and intrigue of mathematics is vast, and the limit of its power to improve the quality of their lives is unknown. I want them to glimpse infinity.

What's the Big Idea with Algebra 2?

Lately, I've been following some of the conversation around the big ideas in an advanced algebra/pre-calculus course. The Global Math Department* hosted an interesting panel discussion around this topic a couple of weeks ago. I appreciated the thoughtfulness and complementary ideas of the presenters (John Burk, Dan Goldner, Michael Pershan, and Paul Salomon), and especially the thoughts behind proof and 'the well-crafted solution.' Without entirely reaching a consensus, the focus of the discussions tended to lean towards prediction as the overarching theme for algebra ii. The reasoning was thoughtful and grounded, but this theme did not satisfy me. While I can certainly see it, I also think that prediction is the theme for statistics. Can Algebra 2 and Statistics have the same theme? They can, I suppose, but it is not satisfying enough.

Some of the new bloggers from the New Blogger Initiative also tackled this topic last week.
gooberspeaks got me thinking about the focus on families of functions and David Price included ideas about varying ways of representing functions and modifying their behavior. Kyle Eck has a strong bent towards applications which resonates with the GMD theme of predictions. And all these ideas muddled around in my brain for a long time before emerging as a single construct that currently satiates my desire for deeper inspection.

Algebra 2 is all about: generalizing patterns of behavior in bivariate relationships.

But that's my academic's definition. In the UbD-influenced language of a high school classroom, I'd say that Algebra 2 asks these questions:

• How can we communicate the behavior of a relationship between two ideas?
• Are there rules of behavior that apply to all relationships?
• Why is it important to be able to generalize patterns of behavior?

Functions certainly play a large role here, because it's easier to generalize patterns when there are overt rules of behavior to follow. But just as importantly, we also look at conic sections and the elusive inverses of even polynomials and periodic functions, because these ideas give us essential insight about the comforting nature of functions that are both one-to-one and onto, and about the obstacles presented by relationships that are not.

Graphing also plays a large role, because it is a most excellent tool for alternate representations of bivariate relationships. Seeing patterns emerge in the shape of coordinate graphs can be enlightening long before symbolic manipulation clears a path through the brain... and I thank the math gods for that! I am wary though of too much graphical emphasis, for our well-loved coordinate system has obvious limits as our brains allow us to consider relationships with more variables.

And applications clearly play an important role too, especially in the attempt to answer that third question. But I hesitate to put applications at the forefront of an advanced algebra theme. I think that is perhaps better handled by a physics class. In algebra we are attempting to represent scenarios with a generalized pattern of behavior, and manipulate this generalization to highlight useful information. I think I agree with Paul Salomon in that proof and 'well-crafted solutions' may trump (but certainly not replace) applications in the hierarchy of an overarching theme in algebra.

To end, I'll just say that my desire to ask (and attempt to answer) the big questions is never entirely satiated, but I do so enjoy the conversations that emerge from them. I welcome your thoughts, criticisms, and further insights. The discourse is what makes being a mathematician so much fun.

*Megan Hayes-Golding, where have you been all my life? What a terrific thing the GMD is, and one of these Tuesday nights, I will not have bedtime routines or NBI deadlines to worry about and will be able to attend a session while it is actually happening! Thanks to you and all others who are making this happen.

Parenthetically Speaking

I imagine that you, like me, have taught, or retaught, or referred to parentheses in the traditional manner:

Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules). In an expression like , the part of the expression within the parentheses, , is evaluated first, and then this result is used in the rest of the expression. Nested parentheses work similarly, since parts of expressions within parentheses are also considered expressions. Parentheses are also used in this manner to clarify order of operations in confusing or abnormally large expressions. (from Wolfram Math World)

Wolfram goes on to define seven other mathematical uses for parentheses, including interval notation, ordered pairs (0, 5), binomial coefficients , set definitions , function notation , etc. With so many uses, it's perhaps no minor miracle when students are able to emerge with any working facility of parentheses at all!

Honestly, I feel for my students. Even to me, mathematical definitions can sometimes seem inconsistent and confusing. Like the difference between terms (things that are added) and factors (things that are multiplied).
 I can hardly keep my own head on straight to describe the number of terms (2) and factors (0) in the following expression:

(To be fair, the first term consists of two factors, each containing two terms each, and the second term has four factors.)


And then recently, I reviewed a prealgebra curriculum that described parentheses as symbols that tell us to "treat part of the expression as one quantity." (from onRamp to Algebra) The book goes on to further implore the teacher to forgo the order of operations description in lieu of the 'one quantity' idea.

I knew that... 


So WHY have I NEVER thought to describe it that way??? 

Parentheses are grouping symbols that tell us to treat the group as a single entity. Period. No confusion. 

A function input is a single argument.
An ordered pair is a single location.
An interval is a single, uninterrupted region.
A matrix is a single array.
A set is a single collection.
A binomial coefficient is a single combination.
An expression in parentheses is a single quantity.

For some reason, this simple statement of an idea that seems so obvious is completely enlightening to me. The Common Core lists "Look for and make use of structure," as one of its Standards for Mathematical Practice. To me, this 'single entity' idea is paramount to the mastery of this standard: the key to seeing structure in long complicated algebraic expressions.

I find a subtle beauty in tiny moments of enlightenment, even if it is only my own. It's got to rub off on someone!

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?

Calendar of Best Fit

All my life I've operated under the assumption that a leap year occurs once every four years as our way to account for calendar discrepancies... a way of handling the actual length of a year: 365.25 days.

Vicky Rauch enlightened me a little today. In reality, the length of the year is closer to 365.2421897 days, and that's if you are measuring the mean distance between equinoxes, which is probably a good idea so that seasons stay intact. With this figure in mind, it has been calculated that we only need to have 97 leap years every 400 years! So, that's what we do. We, in fact, skip the leap year at the turn of the century 3 out of every 4 times. So, 2000 WAS a leap year, but 2100, 2200, and 2300 will not be. Cool.

But, here's a question, because surely even this system will not be entirely accurate: How long will it take before the calendar will be off by a full day, despite our leap year alterations? And for all you scientists out there, the tropical year (measurement based upon equinoxes) varies slightly from the sidereal year (measurement based upon earth's orbit in relation to fixed stars). How would the calendar change if it was based on the sidereal year (365.256363004 days)? And dare I even mention that the tropical year varies slowly as the years progress? The calendar is such an easy thing to take for granted!

Some fascinating classroom discussion and fabulous mathematics work might just ensue on Wednesday. Happy Leap Day!

Just for Fun: Problem 2

How do you find the center of the circumscribed sphere of any triangular pyramid (not necessarily regular)?

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!)

Just for Fun: Problem 1

Because math is fun, and sometimes I like to work on interesting problems. 

AND because I see so many UNinteresting problems.

Just For Fun, Problem #1:

On Being a REALLY Good Math Teacher

I started a grad school class this week. It's been a while, so on my way out the door, I grabbed an old notebook off my shelf and shoved it in my bag with my new textbooks. As it turns out, this was a fortuitous move, for the first half of this notebook was filled with journal entries from my days as a student teacher and a first year teacher. It's been a fascinating read.

One particular entry contained notes from a lecture I attended by John Benson, one of my highly admired mentors. His topic: "The difference between good teaching and really good teaching." He said that GOOD teachers:

1. Have a clear idea of what students know and can do,
2. Know what is necessary for success on a particular task,
3. Use a variety of instructional methods to reach many learning styles,
4. Are eager to spend extra time outside of class to answer questions,
5. Establish clear guidelines for student success and performance,
6. Hold students to high standards,
7. Account for individual differences,
8. Provide clear explanations of the concepts that students are expected to master, and
9. Continually preview and review.

Competent teachers may accomplish a satisfactory subset of these objectives, but do not provide the care and variety that is evident in the classroom of a good teacher. On the other hand, REALLY GOOD teachers:

1. Have a second set of objectives that go beyond the mastery of today's content,
2. Can seize teachable moments and move towards higher objectives,
3. Are able to recognize when students have lost interest and can seize the opportunity to teach something really interesting, and
4. Believe that these higher objectives are the really important part of their mission.

There's more in my notes, but I'll stop there because these lists still make an impact on me. The 'short' list for really good teachers is appealing to me. I believe in these four objectives and I like working to improve on them. They seem to be about passion and that makes me feel good. But as I read #1 a little more closely, I pause on the word 'second,' and realize that there is no hope for me as a really good teacher unless I can also move towards better mastery of the objectives of a good teacher... and that's a LONG and difficult list. I think John Benson is spot on when he suggests that really good teaching comes from a long list of grueling, difficult, AND passionate objectives. It's a package deal, and it's really hard. But the company is great.

Can You Help Me with an Algebra 1 Sequencing Project?

I created a checklist style review guide for my Algebra 1 students at the midpoint of the course several years ago. Over the years, I have reviewed and revised this study sheet multiple times… tweaking phrasing and sequencing, but also changing my mind again and again about what is (or is not) a basic skill in a beginning algebra class. Recently, I took the leap and created the second half: committing myself to a firm opinion about the essential nature of 55 basic algebra skills. The problem is that every time I pick it up, I find something else that I want to change. It is becoming an albatross for me, and so this is where I could use your help. Take a look and let me know what you think. Look at language, sequencing, design, etc. Did I miss something important or include something they already mastered (or are not ready for)? I really want it to be great, but have become overwhelmed in my solitude.

If you click on the image of the study guide, you will be directed to a site where you can upload it for free*. (*This project has been completed, and the free version is no longer available on the site.) As you read through this study guide, please bear in mind my goals/objectives for this type of algebra review guide:

• The individual skills are meant to highlight the essential tasks that an Algebra 1 student should have mastery of. I firmly believe that good algebra teaching revolves around problem solving and applying multiple skills to illustrate, communicate, generalize and verify solutions to problems. I have purposefully left off any topics that I consider to involve multiple skills and the critical thinking of deciding which methods are appropriate and useful. I lovingly refer to the chosen topics as our ‘bag of tricks,’ or the tools from which we pull from to solve problems.
• I have tried to include only topics that are learned in an Algebra 1 class (and not earlier) although there are a few that I have found to be so essential that they bear highlighting again (like order of operations, graphing points on coordinate plane, and properties of real numbers).
• I have attempted to order the skills according to my best sequence of instruction, but I have found that there are some that tend to bounce all over the place. Ratios, proportions, and cross products, for example, have felt comfortable to me in many different locations in the course. The same is true of datasets and statistical analysis. Bear in mind that this is meant to be a cumulative review and not necessarily a course outline. For example, I thought it best to group box plots with scatter plots on the study guide even though I don’t necessarily teach them at the same time.
• The code after the topic name is my newest attempt to align this sheet with the Common Core Standards. If you are a Common Core expert, I would greatly appreciate fact checking and additional input with this alignment.
• The last two columns are intentionally blank, to provide teachers the flexibility of aligning the guide with their class textbook and supplementary materials. I go back and forth on the usefulness of this.

Thank you for your assistance with this project. I will happily share the final results with you.

Algebra is Not a Four Letter Word

Despite the overwhelming evidence that a foundation in algebraic thinking is essential to a sound mathematics education, algebra continues to get a bad rap among the populous. Billy Connolly's Algebra rant (foul language warning) is hysterical, but sad because it spotlights the popular opinion that algebra is incomprehensible and useless.

I've been thinking about algebra's reputation a lot lately:

• What do we need to do to make algebra seem approachable and  useFUL for everyone? 
• How can we improve the way we rationalize algebra - so that our arguments are convincing and appealing to even the most jaded among us?
• What can I do in my own classroom, so that my students better understand both HOW to use algebra and WHAT algebra is useful for?

These questions have been driving my thoughts lately, most likely spurred by the heightened student frustration I have experienced as we switch gears from one semester to the next. There is nothing like a cumulative exam to dredge up student anxiety and feelings of hatred for the source of those anxieties!

What I Read

I know some of you have blogrolls as long as my arm. I don't know where you find the time. But, without further ado... my blogroll:

• For their overwhelming enthusiasm, extreme generosity of ideas and resources, and high quality of mathematics education, I like Dan Myer (dy/dan) and Kate Nowak (f(t)). Both blogs are friendly, helpful, and FULL of endless discussions about best practices for teaching math.
• I also like Sam Shah (Continuous Everywhere but Differentiable Nowhere), Vi Hart (Vi Hart - Blog),  and Christopher Danielson (Overthinking My Teaching) mostly because they remind me of the colleagues I've had who are wonderfully smarter than me - with all the quirkiness that comes with overactive synapses.
• Richard Byrne (Free Technology for Teachers) introduces me to something new every single day. I love/hate him for that!

These blogs pretty much satisfy my need for young, enthusiastic, creative, and innovative voices in the math education blogosphere. I'm sure there are more great ones out there, and of course I'd love to hear about them, but these have made me happy so far.

But there is another type of voice that I continue to search for - the voices of experienced, wise, veterans. I wouldn't mind reading a few more blogs in this category, but I currently recommend:

• John Benson (along with a young colleage, PJ Karafiol) who writes Angels of Reflection which is filled with tough lessons and ideas with teeth - all backed with demonstrated success and not theoretical success. This is immensely appealing to me.
• Vicky Rauch (aka Scipi @ Go Figue!) who writes from the sobering perspective of a veteran teacher in the midst of products from a failed secondary mathematics education. Her wisdom in the context of community college mathaphobes makes me consider the cost of too much innovation in math education.

A Versatile Blogger

I was recently awarded the 'Versatile Blogger' award. I know! When I read the announcement I was instantly overcome by a rush of pride and disbelief - "My wit, charm, and unique spin on teaching math are getting acknowledged after just a few short months of blogging! I didn't know I was so awesome!" I read on to the award description and requirements for acceptance:

• Thank the person who gave you this award. Include a link to their blog.
• Next, tell 7 things about yourself.
• Finally, select 15 blogs/bloggers that you’ve recently discovered or follow regularly. Award those 15 bloggers the Versatile Blogger Award.

POP!

According to my most casual research, the 'Versatile Blogger Award' appears to have originated this past September. According to blogpulse.com  there are about 182 million public blogs in existence today. I have three questions:

1. Assuming awardees all accept their award and complete the requirements in the space of one week, how long would it take to award ALL 182 million blogs? 
2. How many people have a blogroll long enough to discriminate 15 award winners from it?
3. Didn't you just award me a blogging award? If you don't already know 7 things about me, I probably should give the award back.

And my bubble has completely popped, leaving me with only one lingering question: "Why did it take so long - after 16 weeks of circling the globe, why was I not recognized sooner?" Mathematics can be seriously depressing sometimes.