From twitter:
Seeing this everywhere across all grades. Anyone have ideas? #MTBoS #mathchat
The replay speaks volumes — seems that person didn’t catch the mistake, either.
I remember going through this tendency, and while I’m not a typical learner, it’s reasonable to presume that others might make the mistake.
The square root operation is even more abstract than other operations. When we add, subtract, multiply, or divide, or raise to a power — we include the number doing the operation. Square root? Nope. I know I tended to keep rooting things — even without adding the confusion of having x squared under the radical.
It took me a while to drill into my brain that the square root of x squared was x, in the same way that dividing and multiply undid each other. That connection wasn’t facilitated. When I teach it I talk more about that.
Quick suggestions before I get off to work:
Put up the big ol’ pictures of powers from mathequalslove and point to ’em often. One side of that square is the “square root.” That works well for visual learners (not me ;)) . http://mathequalslove.blogspot.com/2016/07/posters-of-perfect-squares-and-perfect.html
Put the “two” in the radical where you’d put a 3 if it were a cube root. What’s with these math people? It’s okay to say “if its’ not there it’s one” because the one *is* a tad redundant but … this isn’t redundant, this is “secret code you have to know…”
For me, turning it verbal was necessary. I’d tell myself that “the square root of four is two,” and visualize the two being a “regular” two, without its little radical, and acting just like a normal number instead of having that funny hat.
… I’ll ponder ways to do conceptual frontloading on this one… this is exactly the kind of predictable error that lends itself to that…
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howardat58
May 22, 2017
Once again “check your work” is fairly futile. The mistakes are passed over, or seem correct.
What is needed is “check your solution by substitution in the original equation”, as in
x=root 17
x squared = 17
root 17 = root 289
oops
This approach was drummed into me years ago, but is “out of favour” (!) these days.
xiousgeonz
May 22, 2017
I think it may have been overused and education culture is all about swinging pendulums and panaceas.
I learned to “substitute back” when I got sick of seeing my “careless errors.” (I’d done it when told to… then… made it a habit…) That’s what one student I know used to place himself about 3 semesters ahead of his skills on the no-time-limit multiple-guess assessment test.
I think a big part of this is that many students don’t really understand powers, so understanding the inverse kinda implodes on itself.
howardat58
May 22, 2017
It’s even worse when we get “little 3 root 27 = 3”, or “2 to the half”
xiousgeonz
May 22, 2017
Funny, that wasn’t worse for me. Either I had the concept more solidly welded down, or shifting the symbols and including the little 3 & the 1/2 (“power on the top, root at the bottom”) made it more verbal and less visual.
Opus the Poet
May 23, 2017
Yep, too many radicals in the final equation. Missed the first step of x2 = 289
Opus the Poet
May 23, 2017
And it stripped my HTML out so let’s redo that as x^2 = 289 instead.
xiousgeonz
May 23, 2017
I also make the effort to correct students who say “x two” since I”m 89% sure it’s a meaningless utterance and not that their inner HTML has been stripped 🙂