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.

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.

A Gift for You

One Christmas long ago, I was maybe 9 or 10, my grandparents  gave me a subscription to Games Magazine. While I cannot give this magazine full credit for turning me on to math (so many contributions there), it did go above and beyond in its ability to feed my brain and satiate my thirst for challenging and entertaining puzzles. It still does. What I especially love about this magazine is that there are plenty of puzzles that I cannot solve... and still plenty that I can.

When I met my first 'Paint By Number' logic puzzle, I remember the wonderful satisfaction of incremental achievements. At first, it seemed so impossible, but bit by bit, the little squares got filled in. Each success led to another inspiration and slowly a picture emerged.

And I remember the first time I introduced these puzzles to my students - 14 year old minority algebra students stamped as 'underachievers'. Sure, there were some that took one look and walked away, but I was surprised by the ones that gave it a try, and amazed at their tenacity in pursuit of success. I can still picture Nestor at a desk in my empty classroom for an hour after school!

So, my gift to you is this snowflake puzzle... for yourself and your students. If you click on the image, you will be redirected to a site where you can download it for free. I created the puzzle using Griddlers Deluxe software that I downloaded online. You can find more puzzles there and also make your own. Enjoy!

Happy Holidays!

Show your work!

One of my colleagues recently said to me "You know, nobody does proofs anymore." Seriously? I think that would be a catastrophic move in the wrong direction, and it reminds me of a nagging problem that I have avoided.

Since the beginning of my teaching career, there are a handful in every class: those students who, despite numerous pleas to "show work," refuse to write anything but the answer on the page. I give what I think is a thoughtful (and lengthy) practice set as homework, only to find that student who has successfully reduced it to a smattering of digits that fills up only a quarter of a notebook page!

"You need to show your work," I'll say.
"I did it in my head," he replies.
"You did what in your head?" I'm not letting him off, I don't care if he's a genius or not. "You need to write down what you did."

At this point, blank stares or shrugs are about all I ever get, but depending upon my mood, I might launch into one of the following attacks:

• "You need to know what you've done, so you can correct yourself if mistakes are made." (You're not as clever as you think.)
  
• "A good mathematician must show diligence in precision, and exercise good habits of mind." (You'll never amount to anything.)
  
• "It may seem easy now, but it won't always be and showing steps now will make the steps more natural in the future." (Pleeeease... I need to validate my existence.)
  
• "You only get partial (or no) credit. You need to show your work to get full credit." (You will do it because I SAID so!)

The truth is, I don't really understand this student. It shouldn't surprise you to learn that I never was this student. I was the neat girl, with the 5mm mechanical pencil, gum paste eraser, and tiny, Times New Roman handwriting. My geometry problem sets looked like architectural journals and my calculus tests got photocopied and passed around as examples. The aesthetic appeal alone of those neatly written problems in nicely aligned columns with balancing symbols and digits was enough motivation for me. Those other students have always frustrated me to no end!

So then comes the blow: the thing I've been forcing myself into these past months. I ask myself "WHY?" And I don't mean "Why do these kinds of students frustrate me?" I know the answer to that question. The real question is "Why is it important to show your work?"

That particular question has evaded me many times, probably because of a lingering fear that the answer might be "It isn't." Thankfully, I don't think I'll need to go there.

At the heart of the issue is the realization that there is a quintessential difference between an answer and a solution to a problem. A solution illustrates, generalizes, communicates, and verifies the results. An answer is just a number. In mathematics it is important to find the correct answer, sure, but more importantly I want my students to gain skills in writing accurate and convincing solutions. It's not enough to just know the answer, you need to be able to prove it beyond a doubt to yourself and others. Take this example:

Student A has successfully shown that 4 is AN answer to this problem (I took the liberty of writing what I believe is going on in Student A's head, because usually this version of the solution is simply a scrawled 4 on the page) Student B, on the other hand has successfully shown (and verified) that 4 is THE answer to this problem. The difference may seem subtle and unnecessary for a typical beginning algebra student, but it quickly becomes essential, like in the case where more than one solution exists or when extraneous solutions occur.

The result of treating 'work' as 'proof' could very well be an improvement in reasoning and logic in students across the board. Certainly it would make a difference to my geometry students when they are first confronted with conditional statements, laws of reasoning, and proofs. It seems that more and more, students (and people in general) are forgiven for their inability to write a logical and convincing argument.

So, is my colleague right? ARE we moving away from proof? Are YOU? Seriously, I want to know.