We ask students to simplify expressions in various arenas: fractions, order of operations, radicals, and no doubt other topics I’m not recalling right now. What is the purpose of this? When is it appropriate? How much is too much? In my in-depth analysis of the Common Core Standards for Mathematics (CCSSM), I wrote approvingly of one feature of those Standards:
The word “simplify” occurs nowhere in the CCSSM. Understanding the concept of equivalent algebraic expressions is an important part of symbol sense, which can facilitate understanding and communication. But the mindless simplification of countless expressions with no rhyme or reason has been one of the ways we turn kids off to algebra and to mathematics in general.
In today’s post, I’d like to elaborate and clarify.
I can think of two valid reasons to simplify expressions.
- Communication: if one student’s solution to a word problem is 6/8 and another’s is 75/100, putting their answers in lowest terms is one way for them to find out that they agree. Likewise with √2/2 vs. 1/√2.
- Effective symbol manipulation: it is obviously easier to work with 2x + 4 than with
2y + x – 3 + x – y + 7 – y.
Clearly, some simplification can be useful. However, it should be handled with care.
- I stand by my statement above: lengthy “simplify” drills are counterproductive, because in general it is not really possible to get students interested in them.
- This is even more so if the expressions are artificially complicated solely to create opportunities for students to make mistakes. Those mistakes are often about lack of focus rather than lack of understanding. (See the example Rachel Chou shared at the end of her guest post.)
- Crucially, students should not be penalized for correct answers that have not been “simplified”. That is just perverse.
Here are some arguments for questioning the traditional obsession with simplifying:
- Sometimes 75/100 or 45/60 is more informative than 3/4 (for example if you’re dealing with percents, or time).
- Sometimes there is a good reason to complicate an expression (for example when completing the square).
- Sometimes what is “simple” depends on the context (for example, factored form vs. standard polynomial form).
… and so on.
So what do I suggest? Here are some guidelines that make sense to me:
- Discuss equivalent expressions in context, when it is actually relevant for communication and/or further work on a problem. What form of this expression is the best one to use?
- If your students need practice, keep it simple (heh). Choose examples that highlight important ideas: how to handle a minus in front of parentheses, what part of an expression the exponent refers to, and so on.
- Keep in mind that given the broad availability of computer algebra systems, the central goal for students is not computational speed and accuracy. The central goal is understanding.
Instead of an ironclad requirement to simplify everything in sight, we should take the time to think about each specific situation. Students will get better at recognizing equivalent expressions, and learn to use them strategically, if we prioritize the underlying concepts and encourage reflection and discussion.
— Henri
PS: I raised some of these points in a related post: Commitments