Setting: Algebra 1 Class. Unit: Operations on Radicals. At this point we have done adding, subtracting, multiplying, and distributing radicals.
Today I get a who student writes:
Then another student says, “You can add those?” I have been pushing really really hard order of operations so I go through the process:
The answer to almost every question in my Algebra 1 class is either order of operations or commutative property.
We teach students about “like terms” do we often show them why we can perform operations on like terms?
a brand new line. 
Is it because square root of 5 is a “funny” looking number? Students seem to have issues with anything but whole numbers, and possibly integers as long as the student isn’t too picky about paying attention to the sign.
Have them do whole number operations and they’re stellar. Throw in a fraction, or in this case a radical, and they act like you’ve asked them to prove string theory.
Is it because there’s more information “packed” into a radical or fraction? Show me the number 5 and I know what you mean. This is a quantity and I get it. Show me 2/5 and now I’ve got what looks like two whole numbers but I need to think about what each of them mean and how they’re related and I might have to change the top or bottom number for some reason, something I don’t have to do with whole numbers, and now my head hurts.
Show me square root of 5 and now there’s this funny checkmark around my number. Is it telling me the number is correct? It looks like 5 but it isn’t and I know it means some crazy looking number with lots of decimals and I’m feeling overwhelmed. Don’t even get me started with the number 35 with a checkmark over the 5.
I’m probably overthinking it, but it does interest me how rules break down so easily when certain numbers are involved, which may or may not be what was happening here.
I do like how you’re able to generalize to a few concepts – order of operations and commutative property – to help them work through it. That’s very powerful.
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I’m not sure how much my explaining helped, but at times like these I’d remind my algebra students that the biggest mistakes they can make are those when they forget some things are as easy as counting. I’d show them a sequence like this:
3 dogs + 7 dogs = 10 dogs
3 cats + 7 cats = 10 cats
3 fifths + 7 fifths = 10 fifths
3/5 + 7/5 = 10/5
3 fives + 7 fives = 10 fives
3x + 7x = 10x
By this point they’ve usually caught on and they can see how it’s just a matter of counting up 3 things and 7 things. This is perhaps not my greatest pedagogical moment, but this problem does highlight one of my favorite concepts from mathematics education research: the “dual nature” of processes vs. objects, written about by Anna Sfard in a brilliant 1991 paper. (A version of which can be found on her site at https://www.msu.edu/~sfard/Dual%20nature1.pdf.) In this particular problem, some students see the square root of 5 as a process — something to calculate before proceeding — instead of an object. In my examples, starting with dogs and cats, I’m just trying to emphasize the counting of objects. But it’s not always that simple, and Sfard does an excellent job describing why she thinks the transition from process to object is so difficult for many students. If you want to read Sfard’s paper and discuss it, I’d be happy to have an excuse to read it again!
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I teach the radical stuff as objects, too… that they can treat root 5 like an “x”. Last semester I had several students wondering why, when they put the quadratic formula into the calculator, it did different things with radicals than with whole numbers. What ?? “with square roots, you just put the numbers next to each other but with whole numbers you have to multiply.” It was news to them that when they “put the numbers next to each other” that the calculator was multiplying them.
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