I could see using manipulatives for exponents. In fact, when I teach
exponents, one thing I do is we fold a piece of paper. We first fold
it in half and I ask how many rectangles did it make? We folded it
once and it made 2 rectangles. 2^1=2. Then we fold it in half again
2^2=4, then in half again, 2^3= 8, and so on. At the end, I ask, what
if we hadn't of folded it it at all? How many rectangles would we
have if we never folded it in half? 2^0=1.
How would you use manipulatives for PEMDAS? When I tech P E MD AS, I
let them know that P stands for Parentheses and all grouping symbols:
vinculum, absolute value symbols, parentheses, brackets,
etc.
Over the course of the class, we encounter several types of grouping
symbols.

Once P's and E's are simplified, I tell them"if we are walking along
the problem from left to right, will we pass division first or
multiplication first?" Same with addition/subtraction.
Maybe I'm going about it all wrong, but that's how I do it until I
find a better way.

On 6/22/09, Jo wrote:
> You could go a step further and ask how many teachers used
> really different methods of teaching exponents or other math
> concepts. Did they try manipulatives along with writing the
> information in various ways? What were the visuals like - just
> numbers written on the board 3 x 3 x 3 = 3^3? Some kids really
> need math to be hands on and visual, but there are few truly
> hands on math programs used in classrooms. And if they are, it
> is rarely by all teachers for different groups of students.
>
> So, I guess my point is, you CAN say it is the student if you
> expect that all should be able to learn by various explanations
> not different methods, but the reality is that not all students
> learn math that way and few teachers use manipulatives or good
> visuals after the first few years (if at all depending on the
> school). So, I would be more pressed to say it is the TEACHING
> (not saying bad, uncaring teacher - just not enough training in
> a system not designed to teach all students) not the student in
> many cases. But what is seen by the time the kids reach middle
> to upper grades is that they are woefully confused and many have
> given up trying to understand because it is futile and they are
> blamed for their failure (not trying hard enough).
>
> On 6/22/09, algie2 wrote:
>> Going out on a limb here, but I think we as teachers place far
>> too much blame on teaching methods when trying to figure out
>> why true learning hasn't taken place.
>>
>> I had a couple of really low high school classes this year.
>> They were all taught Order of Ops with P E MD AS. They all had
>> trouble with it. We spent tons of time on not just the what
>> but the how and why of P E MD AS. They were bored beyond
>> belief. We've been doing this since middle school! When do we
>> get to Algebra! But they still had trouble. As a class they
>> couldn't consistently evaluate an arithmetic expression
>> accurately. Some were very defensive about it -- but I did it
>> this way...(hmm, what is it about "order" that you don't get).
>> There were a few in this group that simply could *not* get the
>> hang of exponents no matter how many times (and ways)it was
>> explained and broken down for them.
>>
>> When we did get to algebra, they had mountains of trouble
>> "undoing" Order of Ops. No, you can't isolate 3x by subtracting
>> 3 -- let's relook at P E MD AS. (What, again? But we've been
>> doing this since middle school...)
>>
>> While at the beginning of the year I may have thought their
>> prior teachers did a poor job of teaching Ord of Ops, by the
>> end of the year I realized that it didn't much matter -- these
>> kids just were not going to master it. Meanwhile, other
>> students (college prep or honors level) were taught the acronym
>> the very same way by the very same prior teachers and fully
>> mastered Order of Ops. The variable is the students.
>>
>>

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