Me: What does the square root of 5 mean?
Sophomore 1: A number times itself that equals 5.
M: So what’s 
S1: [Reaches for calculator, evaluates 
…
Me: What does the square root of 5 mean?
Sophomore 2: A number that if you times it by itself it equals 5.
M: So what’s this? [writing:
S2: [Helpless look, grin] 2? 2.5?
…
And three more today that are similar. When I hold their hand through 
Work in Pencil 
I am in tremendous suspense at the end of both anecdotes – what did you say next? What did they say?
What Ben said…
Also, this worked well for me, used in Algebra 2 when introducing cube roots and nth roots: http://function-of-time.blogspot.com.ar/2010/09/radical-comes-from-radix-which-means.html Of course I have no idea if it would help you in this case, just letting you know it’s there.
But I realize with sophomores, you may not be teaching them about what a radical means, maybe it just came up when using the Pythagorean theorem to solve problems, or something? With my particular sophomores, I’m struggling with their poor understanding of linear functions. It’s really annoying.
So far I’ve been running into this one-on-one, refusing to acknowledge that it’s an epidemic. In the one-on-one conversations, after the square-root-of-four example, there’s usually a long “Ooooh!” and maybe a little sheepishness. So it *seems* like they have it … but I’m skeptical on whether it’s really sticking.
@Kate – thanks for the reference. As for linears, I find that if I give linear problems, no one knows anything about linears, but as soon as I give a non-linear problem, they all have linear suggestions coming out of their ears.
Ditto that. Besides memorizing the visual pattern of root x * root x = x, it’s possible they’ve memorized the auditory pattern “a numb burr tie mz its elf that eek walls.”
If non-linears are the way to generate linear ideas, that’s great. What’s the (apparently) unrelated discussion that generates square root ideas? I can’t think of one right now but will keep thinking.
Is it possible that this is about spotty understanding of multiplication (which prevents people from seeing its relationship to roots)? Is it possible that this is about spotty understanding of division? If students are thinking about root 4 in terms of dividing it by things until one of them gives the same answer as the divisor, they could get stuck on their difficulty dividing 5 by anything and not be able to move ahead in their thinking process. Could this problem be a symptom of not having internalized the relationship between multiplication and exponents? I wonder what students would say if you asked “what is (root 5) squared” — would they give the same answers as they give in response to root 5 * root 5?
I just laughed when I read this because this happened to me such much in Algebra II. They were saying the words but could not connect them with anything/meaning. It’s all of us. It happens to all of us.
Sam
Yeah, reading Mylène and Sam’s responses, I think the way I’d like this to go in future is:
Me: What does the square root of 5 mean?
S: A number times itself that equals 5.
Me: Oh. So, what does that mean?
*lol* A new trick I’m experimenting with is “explain (concept x) without using (words y or z).” Sometimes that helps me understand what students are thinking… other times it helps students develop their own sense of what it feels like when you don’t understand something.