## Wednesday, February 21, 2007

### Mental?

I am impressed by lots of people who manage to write regularly. I supposed I havne't quite learned how to manage my time. That's why my posts are short. Not that that's a bad thing.

I am still observing. Yesterday and today I saw something that seems recurrent, so I have to comment/ask. In a process of solving a problem my teacher gets to:

 $\frac{5}{\sqrt{10}$

She writes

 $\frac{5}{\sqrt{10}}=\frac{5}{\sqrt{10}}\times \frac{\sqrt{10}}{\sqrt{10}}=\frac{5\sqrt{10}}{\sqrt{100}}=\frac{5\sqrt{10}}{10}=\frac{\sqrt{10}}{2}$

Or, let's have even simpler example:

 $6\times(\frac{50}{3})$

She proceeds to multiply 6 and 50, and then divide by 3. I saw identical process in the first classroom that I observed. In fact, kids (9 grade, I believe, possibly even 10) there couldn't do things like -4-2 or 7*3 without a calculator (the latter we figured out, and then two minutes later we had 8*3 in a problem and the student couldn't do it even after I reminded her that she knew 7*3). The teacher there said "Their mental math skills are terrible"?! I wouldn't call that mental math. Should I? Regardless of what we may call it though, I was terrified. I still am.

Back to my examples, I can understand using the long process when teaching it, but this was not an introductory lesson. Why do the teachers not use the "shortcuts"? Is it out of some consideration for the students? Do they think the students can't handle it? Or they'd be confused? Or is there some other explanation?

Lsquared said...

How non-introductory is the math course? When I first started teaching college algebra (in college) I used the shortcuts. It seemed like I was always backing up because someone in the class couldn't follow the shortcuts, so I started doing all the details, all the time (mostly) when it's a class below calculus.

e said...

These were highschool algebra (not sure which number, 3/4? maybe) and geometry. In my mind they are advanced enough. Or they should be.

Mr. Carlin said...

I've found so far that many, many students have a tough time with fractions and with operations involving radicals. I think that many teachers blame what they say is a de-emphasis of "the basics" in elementary and middle school - no requirement (or minimal requirements) to learn times tables, long division, operations with fractions.

I like to "use shortcuts" and/or "skip steps" sometimes, and if/when I get blank stares, fill in the gaps and say what I did, and encourage students to start making those mental "leaps" (because for many of them, they are leaps).

Horrifying? Perhaps. But it's reality for many students, and we need to work with / work around these issues.

How would you define "mental math" ?

e said...

Mike,

I don't think we need to work "around" these issues. Can't we just teach them better? I know, I know.

As for mental math, I guess I don't actually have a definition. Maybe "doing basic operations quickly and being able to apply some of the rules in your head". But this requires you to know certain things, like -4-2, or 3*7, or the distributive property of multiplication.

jonathan said...

Where is the harm? Students who are weaker will benefit from seeing absolutely every step, and most kids get a reminder of what they skip in their own work (unless the teacher demands that the students include every line; that would be different)

e said...

Well, these are the lines I've heard from students upon multiplying fractions in a different way:
"You can do that?"
and
"This (student's way) is easier."
But see, it's not. It's much easier, and more efficient (and not to say that in some examples it'd be even more accurate), to divide 6 by 3 and to multiply 2 and 100, than to multiply 6 and 50, and then divide 300 by 3. Ok, so this wasn't a very involved problem, but you can imagine examples where it would be significantly easier, more efficient and accurate. Now, is there harm in doing it the way my teacher did it? No, BUT I think it is a silent endorsement of the clunkier method. Should we be happy that they know how to do it? Yes. Should we strive that they know more? Yes.
Also, to be honest, this type of a problem is a grind, and I'd like them not to get bogged down in performing long, involved calculations when they can avoid it.

jonathan said...

Then the complaint is that the teacher never demonstrates shortcuts? That would be different.

I teach and encourage short-cuts, but am conservative in my board work.

Again, I see no harm. And on the contrary, the value of using the shortcut is diminished if I have to double back to explain, or worse, if I just leave kids behind.

e said...

I agree with:
the value of using the shortcut is diminished if I have to double back to explain, or worse, if I just leave kids behind.
wholeheartedly. I guess I was being one of those annoying people who complain about what they think is a problem, but don't offer a solution. Sorry. I wish I had a solution.