In my post Arithmetic and Algebra, I wrote that it’s how you solve a problem that tells whether you are doing algebra or arithmetic, not the problem itself. Here’s a description of algebraic thinking that I think teachers in elementary school mathematics might find useful especially when they are teaching about numbers and number operations:
Algebraic thinking is about generalising arithmetic operations and operating on unknown quantities. It involves recognising and analysing patterns and developing generalisations about these patterns.(NZCER)
I find the description clear, concise, can easily be committed to memory and can form part of teachers everyday discussion with a little effort. The keywords that should be remembered are patterns and generalizations. You can’t actually separate one word from the other. If you see a pattern, you can’t but make some generalizations. It’s human nature. If you make generalizations it must from the patterns that you recognize.
Patterns about what?
In the description of algebraic thinking above it says patterns about arithmetic operations (add, subtract, etc) or relationships between numbers for example in equations. Of course it could also involve patterns in shapes, colors, positions in sequences. In short, you also use algebraic thinking in geometry.
Here’s my other favorite description of algebraic thinking:
Algebraic thinking involves the construction and representation of patterns and regularities, deliberate generalization, and most important, active exploration and conjecture. (Kaput, NCTM, 1993).
It is similar to the first but added representing patterns and regularities observed and active exploration as important processes. Without these, generating cases needed for making conjectures/generalizations and verifying them would be difficult.
Click algebraic thinking to see the collection of articles and lessons in this blog about this topic. You may also want to check on these book for other lessons..