Four Big Ideas in Algebra
In his controversial post criticizing high school algebra, Grant Wiggins issued a challenge to his readers:
Can you identify 4 big ideas in algebra, ideas that not only provide a powerful set of intellectual priorities for the course but that have rich connections to other fields?
Yes, I can name four big ideas in algebra. Here they are.
1) Algebraic Structure
Maybe Algebra the course isn’t well-defined, but algebra the mathematical object is. An algebra is essentially a set of objects that can be both added and multiplied, with the two operations fitting together via the distributive property. The existence, and interplay, of these operations endow a rich and powerful structure on the set of objects: an algebraic structure.
Lots of sets of objects inherently possess this algebraic structure: numbers, both familiar and unfamiliar; matrices; transformations of the plane and space; polynomials; functions. By exploring the structure in familiar realms, like the integers and real numbers, we learn to understand, appreciate, and exploit the structure elsewhere.
2) Binary Relations
So we can add and multiply these objects, but how do the objects relate to each other? When are they the same? When are they different? What kinds of different are there? Equality and inequality are two basic, but extremely powerful, binary relations on objects that are studied in algebra.
The question “When are these two things equal?” may be the most frequently asked question in mathematics. By understanding how the relation of equality works within the algebraic structure (of numbers, polynomials, functions), we develop ways to answer that question.
And specific kinds of inequality—like greater than or less than—impose even greater structure on our systems. The more structure we have and understand, the more power we have to model and solve problems.
3) The Cartesian Plane
Cross-over techniques—those that allow us to re-imagine a problem in an entirely different, but equivalent way—are some of the most important tools in mathematics. The Cartesian plane may be the most powerful cross-over technique of all, and it’s another big idea in a high school algebra class.
The Cartesian plane is an arena where algebra and geometry seamlessly interact. Here, we can transform purely geometric problems into purely algebraic problems, and vice versa. We can turn questions of congruence, similarity, and intersection into questions about numbers, symbols, and equations. And we can explore the geometric interpretations of algebraic objects as well, giving us another window into their properties and behavior.
While the idea of function is more set-theoretic than algebraic, functions play a significant and relevant role in a typical high school algebra course. The primary role of functions is in modeling various kinds of mathematical relationships and expressing those relationships in different ways. But functions themselves possess an algebraic structure, so the algebraic rules we develop for integers and real numbers also apply to functions. Functions are fundamental objects in advanced mathematics, and knowledge of algebra allows students to arrive in this world with some experience and understanding of the structure of that world.
In addition to these four big ideas, formal abstraction is an important theme of the course and a key feature of all of the above topics. Moving from the algebra of the integers to that of polynomials; expanding equality of numbers to more general notions of equivalence; moving ideas back and forth between the algebraic and geometric realms—these are all examples of the power of thinking abstractly and leveraging the inherent structure of logical systems to solve problems and create representations. To me, this is what algebra is all about.