If anyone is still out there reading this blog after I took a long hiatus, please indulge me with a response. How do you define algebra?

I just came from a great PD led by the lovely Kara Imm. She brought research concerning the lack of “algebraic thinking” in K-6 math, where arithmetic is taught almost exclusively and algebra is the step that follows for “those who are ready.” (I think some high school teachers have this mindset about Calculus. Students need to learn Geo/AlgII/PreCalc really well “to be ready” to start Calculus.)

So if (as research shows) it would be better to teach arithmetic and algebraic thinking from the start, can we first define algebra?

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Algebra = the logic of combining two or more objects through operations.

The first algebra we learn is on the set of integers with operations like +, -, x, divide. Then we expand to include new objects like fractions and decimals. Then we add in “variables” to the objects. Then we can expand into more operations like squaring, finding roots, trig functions, etc.

So, in my head (and by this definition) arithmetic

isalgebra. It’s just that the set of elements is not as expansive as we usually see in “Algebra I.”I think I’d still keep some separation between a problem that is strictly arithmetic vs. one that is more algebraic. The point I’m getting at is that we can do both. Arithmetic in first grade would be solving:

3+5=

Algebraic thinking in first grade would then be extending this to:

3+?=8

or even better (to emphasize the idea of equality):

3+5+2=8+?

Here is a possibility from Wikipedia:

The key points in that definition are “variables representing numbers” and “manipulating statements based on these variables”. It is certainly possible to do this form an early age—the Singapore Primary Math series introduces boxes for unknown numbers in one of the very early books (grade 1? grade 2?). Manipulation of equations with boxes comes somewhat later, but much earlier than in most US curricula.

Two thoughts:

1. In an equation like 3+x=10, is x really a variable? Can it vary?

2. See my reply to the previous comment with examples of taking the concepts of algebra (like equality) into elementary grades with problems like:

3+5+2=8+?

I teach high school Algebra and a lot of my students come to me with no idea how to think algebraically. I am constantly seeing them “guess & check” to find an answer instead of simply writing and solving a simple equation. In the most simple terms, I tell the kids algebra is the math of solving for an unknown variable.

See my reply to the previous comment…

does unknown = variable? Should we be more careful of our wording?

Strategic arithmetic

my favorite response yet!

I recommend this book linked below. If I remember correctly, he defines algebra as three distinct ideas around 1. equations (find unknown that makes this true), 2. identities (show that this is always true), and 3. functions (how one unknown changes as another changes.) (I know that you know what those words mean, was typing it out to get it straight in my own head.)

Name Your Link

Thanks, Kate! It’s a great book that I’ve read most of (and my dept has used for some discussions!)

I forget where I read this, where it entered my mind, but I just say algebra is taking math from the declarative, “This plus this equals that,” to the interrogative, “This plus *what* equals that?”