Algebra for Elementary ?

Algebra is for big kids.
Students must understand the number system before working on algebra tasks.
Algebra is confusing…I don’t think I could teach it.

If any of these misconceptions spoke to you, I have a blog post for that.

Truth be told, a lot of us have a strong misconception of what algebra even is to begin with. Algebra is representation and relationships.
WAIT - that’s TWO of the mathematical process skills. That means 2 of the 5 skills are essentially all of algebra. While it’s true that algebra uses our number system, it’s not completely true that students need the number system to understand representation and relationships (algebra). The base ten number system is a socially created system to represent natural quantities we can store in our heads. It is something we introduce students to and then provide experiences to help them develop a deep understanding of it. However, algebra is a more”natural” form of math. By this I mean that it is not a socially created system (other than the vocabulary and written symbols). Making connections & relationships is what the human brain can excel at. It is how we are able to function at levels higher than other animals. And what students are doing when they engage in algebraic tasks is problem solving, relationship building, pattern making, persevering, communicating….see the value?

And we also know that the best way for students (people) to learn new information is through connection making, pathways, relationships, solving.

So why are we teaching the number system and then starting “algebra” with preteens that are frustrated and confused…when we could be using algebraic tasks to teach the early number system?

Let’s use some examples. For these examples you will need my favorite math manipulative - cuisenaire rods. Cuisenaire rods are amazing because they allow students to deal with representation and connections without numerals/digits. This means that the youngest of kids with no formal knowledge of what numerals look like can still engage in joining (adding), separating (subtracting), comparing (subtracting), copying (multiplying), partitioning (dividing), making (representing), solving (balancing), and trading (substituting).

ALL WITHOUT NUMBERS.

Because you see…as much as I love numbers, we have often let them get in our way of deep conceptual understanding of mathematical processes. We encourage our kids to count when they have little sense of quantity, we hand them sheets of number writing practice…but we don’t give them manipulatives. We do pages of fact sheets where students are plugging in answers next to an equal sign, and wonder why older students struggle with equation writing. So let’s use our rods to delve into algebra.

If you don’t have any, here’s essentially what they look like.

For our purposes as adults that feel comfortable equating things to numbers, here is how we can think of the rods.
Orange = 10
Blue = 9
Brown = 8
Black = 7
Green = 6
Yellow = 5
Pink = 4
Light green = 3
Red = 2
White = 1

First, let’s do some representing:
What are the ways we could make (represent) the orange block (10)?
blue + white (9+1)
brown + red (8+2)
black + light green (7+3)
green + pink (6+4)
yellow + yellow (5+5)
brown + white + white (8+1+1)
black + red + white (7+2+1)

Let’s balance equations by substituting:
Orange = Orange (10 = 10) but we cannot use orange - so trade the oranges for something else! (the mathematical term for this is substituting)
We could turn orange = orange into (blue + white) = (brown + red)
Essentially, we just turned 10=10 into (9+1)=(8+2), helping students discover the meaning of the equal sign, how to balance equations, and substitute.

Orange can also be made by two yellows. Yellow + yellow is additive thinking, “two yellows” is multiplicative thinking. So now any time we see “orange” we can think “two yellows” (in algebra we would put 2y)
So in this case, orange = orange could turn into (green + pink) = 2y or even 2y = g+p (2x5 = 6+4)

Let’s simplify equations:
(red+red+light green) + (white+white+yellow) but I need it to be easier…only one block on each side.
We can turn red+red+light green into black….and white+white+yellow into black…
Black=Black…much simpler!

The youngest of kids can do this…because they are just trading out blocks! You can even write down the colors in equation form so they can see the math they are doing. Even better…you can make it easier on yourself because writing out the color names takes a while…so represent the colors with a single letter! Like y for yellow and R for red.

Wait a second……

we are doing algebra - letters in math!?!

These are algebraic properties that the youngest of students should be dealing with. We cannot assume they need a strong sense of our whole number system before they start representing, balancing, substituting, and simplifying equations. In fact, students can build a strong sense of number through algebraic reasoning tasks. They are more intriguing (read - engaging), challenging, rewarding, and tied to every mathematical process.

If students can use numbers in equations, they are going to get to know number well. For example, they could spend weeks on adding numbers in isolation. Or they could spend weeks thinking of ways to make (represent) each number. Instead of giving a fact sheet of +2 problems, tell students to think of many ways to make 8, many ways to make 10, many ways to make 12. They are doing algebra…and learning number.

So what about upper elementary? The beauty is these students are more rooted in the number system (hopefully) and can now start applying these principles to the number system they are coming to terms with.
And you don’t have to keep them within 10. For example, third grade teachers don’t want students to just be able to create ten…right?

Orange doesn’t have to be ten. Orange can be anything.

Make the orange 100 and see what happens now. Make the orange block ONE…and dive into fractional thinking. The point is still that students are doing a lot of math (thinking, problem-solving, reasoning, making relationships, etc.) and doing it with the larger (or smaller!) numbers they need a better understanding of. What you use, you know better. Let them USE those numbers. Instead of asking for answers on the other side of that equal sign, ask students to represent, substitute, simplify, etc. Have them make 100 in 10 different ways. Give them 43 on one side and ask them how to get to 100 (43 + ? = 100). Give them a long string equation like 15+32+18=16+24+? and ask them to simplify it…(not even solve - just simplify first!)

Because algebra is a lot to do with equations, students do algebra all the time…
We teach them how to represent (make) addition, subtraction, multiplication, and division equations. We just don’t think of it that way. Even as adults, we don’t think of that equal sign as a balance, choosing instead to carry the misconception that equal sign means “answer”.

So what if…

Instead of focusing time and energy on students learning facts, how about we start encouraging them to build facts?

Instead of focusing time and energy on getting students to remember answers, we encourage them to create the problems?

Let’s infuse sense-making into every part of our day…and bring algebra down to our youngest students!

Happy teaching!