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!

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.

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:

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!