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What comes to your mind when you hear the word variable?
Take a minute and write down or draw out what you are seeing in your mind's eye before you continue reading.
Whatever image presents in your mind-the letter x, an empty box instead of a number, or something else - is most likely how you are approaching the idea of a variable when teaching foundational algebraic concepts in the classroom. When I first started my teaching career in the early 2000's, I remember emphasizing the idea of "substituting a number for x" in order to find a solution. I proudly told students that a variable is just a letter that you choose in order to later replace with a number. While I had taken many algebra courses in my own educational career, this was the extent of my background knowledge of a variable. I was teaching my students exactly as I had learned without ever thinking of the possibility that there could be another definition or interpretation.
My previous concept image of thinking of a "variable simply as a symbol for which things can be substituted" is also the image that presents in the minds of the majority of elementary teachers around the country. And since this "'symbol for an element of a replacement set' conception of a variable seems so natural today...it is seldom questioned" and is what is taught in grades K-5 (Usiskin, p.9).
Thus, many students leave elementary school having this same concept image of "variable as unknown" as their only image when in fact the idea of a variable is more more complex in nature than that of an unknown waiting to be solved. The truth is that "the values a variable takes are not always numbers" and we are hindering the future success of our students when we do not explore this truth with them and instead lead to them to only believe in one general view of a variable (Usiskin, p.9).
Fortunately for me, I had the privilege of learning about the complexities of a variable through some readings assigned in a recent course. Almost two decades after beginning my career as a teacher of mathematics, I am finding out that "solving for x" is only a small part of what a variable can mean.
Now that this world has been opened for me, I want to share my knowledge with the Kid's Math Talk community.
Below is my thinking about how the three major conceptions of a variable present in the K-5 Operations and Algebraic Thinking standards. This is not to say that a variable can not take on another meaning instead a particular standard, but instead of what type of emphasis is placed on the variable use.
Below I've given you a brief synopsis of each kind of variable conception. The major sources referenced for this understanding are found at the bottom of this post.
1. Variables as specific unknowns
Examples: 5+__=11 4x=20 x-7=14
This conception is known as "simplify and solve" and is what is generally taught in elementary classrooms (Usiskin, p.13). There is typically one answer that can be solved by undoing the operations and typical language used by teachers to help students connect include: "replace", "substitute", and "Solve for x". While this is one use of a variable, it is important that teachers do not oversimplify the idea of a variable by stopping instruction with this conception.
2. Variables as varying quantities
Example: I have seven cups of juice to pour. I can use grape juice and apple
juice. How many cups of apple juice might I pour?
These types of problems present more ambiguity and generally lead to more problem posing on behalf of students. Usiskin refers to this conception as "variable as an argument" (p.14) and it is also argued that this conception should be a starting place for students in elementary school.
When thinking about how children are naturally curious and are always looking for patterns and relationships, starting with this conception makes sense. Early elementary teachers already are using this conception - perhaps without realizing it-when they ask students to explore combinations of 5, 10, and 20 (K.OA.A.5, 2.OA.B.2) for instance. Now with this new layer of understanding about how these standards lay the foundation for deep algebraic reasoning, teachers can approach these tasks with a new intensity and purpose.
An extension of this starting place is the use of both addend unknown word problems where the variable actually can vary and the student is looking for different combinations that make the equality true.
For more information about this concept, see my summary of the article Variations in Both Addend Unknown Problems (Slide 4).
3. Variables as generalized numbers
Examples of using variables as generalized numbers:
•Zero plus a number equals that number (0+a=a)
•When you subtract a number from itself, you get zero. (a-a=0)
Understanding a variable in this way helps students talk about algebraic patterns and relationships between numbers. The main takeaway from this conception is that even though there is a letter indicating a variable, we don't have any unknowns.
This view of generalized arithmetic is actually one of the key understandings of domain of algebra and should not be overlooked when planning for mathematics lessons.
Further reading and sources about the conceptions of the variable: