Saturday, September 22, 2012

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.


  1. Algebra is one of the most important topic of mathematics.It has immense application in research field and almost all type of science subjects.

  2. Very nice blog and excellent ideas. I will stop by more frequently to get more ideas. Thanks.

  3. I recently became a maths teacher. but I think I have a few things to learn from your blog

  4. >Arithmetic involves counting and manipulation of quantities where algebra relies more heavily on reasoning and generalizing the patterns that are observed from arithmetic procedures.

    Sheer nonsense. Generalization should start with arithmetic, but because such misconceptions plague teachers, once students start algebra, they too get confused. And so the cycle of weak math understanding is perpetuated.

  5. Written with excellency. I liked it very specially :Do I really have to memorize all these rules and definitions?, and Isn't there an easier way?.

  6. Very articulate and to the point. Thanks for sharing.

  7. I am interested to hear your perspective on standardization.

    The company I work for Learn Bop ( is striving to adhere to the common core standards while empowering teachers to have better relationships with their students.

    Traditionally technology doesn't permit for solid personalized instruction because its very simply oriented and non-adaptive. From what I infer in your article here beyond simple arithmetic its very difficult to make procedural, non-linear math instruction or abstractions.

  8. I stumbled across your Blog while searching for Algebra 1 Essential Questions. I have been trying to come up with questions similar to these for weeks now. I think more teachers need to go through this exercise. I love the idea of a "second set" of objectives. Uncovering these is a difficult and delightful process. I think it's important for students to see this second set objectives....maybe more important than seeing the first set! This is why we teach math. To inspire students to ask these questions in their every day life and have practice coming up with reasonable, logical answers to them. Thank you!

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