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I had just finished looking over the exams of almost 300 students. Nothing bothered me more than the silly mistakes of not knowing the order of operations. They would lose points for, say, not distributing a negative sign before combining terms, or (as unreal as this may sound), having parentheses in the wrong place or missing completely.
Then it dawned on me... why I love RPN/RPL so much. Every time I used one of the HP RPN/RPL calculators, it was implicit that I needed to know the order of operations. I had just been so used to the order of operations that it never occurred to me that RPN/RPL would be hard to figure out. If college students were required to use RPN, I think they would most certainly have a good understanding of the order of operations. You really cannot use RPN/RPL machines well unless you have a good understanding of the order of operations.
What's sad is that college kids are still lacking the necessary comprehension of basic mathematical principles, and these algebraicentry calculators are certainly not helping.
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Hi Han. You might find this document regarding common math errors useful:
http://www.math.vanderbilt.edu/~schectex/commerrs/
Regards,
John
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Thanks! This is a great link... and I even learned a few things as far as teaching goes. It's easy to fall into a habit and take for granted the smallest of details (referring to the article about differentiating x^k and requiring k be any CONSTANT).
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A quotation from the treatise on common errors:
"... Here is an example from Ian Morrison: What is –3^2 ? Many students think that the expression means (–3)^2, and so they arrive at an answer of 9. But that is wrong. The convention among mathematicians is to perform the exponentiation before the minus sign, and so –3^2 is correctly interpreted as –(3^2), which yields –9. ..."
Didn't we beat tat subject to death in the Forum a month or so ago? I haven't been able to find it. My recollection is that we didn't come up with the same answer.
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The thread discussiing the 3^2 problem is "TI84 Plus really that clumsy?" near the top of Archive 16:
Early in the thread the 2^2 problem was introduced:
"Re: TI84 plus really that clumsy??
Message #3 Posted by John Smitherman on 4 Sept 2006, 3:11 p.m.,
in response to message #1 by Hal
Hi Hal. Also, tell your son to be careful with the TI8x series as it interprets 2^2 as (2^2) instead of (2)^2.
Regards,
John
Re: TI84 plus really that clumsy??
Message #4 Posted by Valentin Albillo on 4 Sept 2006, 3:22 p.m.,
in response to message #3 by John Smitherman
Hi, John:
John posted:
" Also, tell your son to be careful with the TI8x series as it interprets 2^2 as (2^2) [...]
As it should. That's the correct way of interpreting 2^2, and the HP71B does exactly the same, returning 4. If you want (2)^2, you should write it that way, parentheses and all.
If in doubt, check any math books or articles and look for terms such as a2 , you'll easily find them aplenty. Do you really think the author actually intends you, the reader, to interpret that term as (a)2 instead ? Unary minus has lower precedence than exponentiation in standard math writing.
Best regard from V."
And the thread goes on and on from there explaining how a user can get different answers depending on how the problem is entered into different machines. What I think that all really says is that RPN really isn't the solution to reducing math errors by students. Neither is AOS, EOS, RPL, BASIC, FORTRAN, or whatever. First, the student has to know and understand the mathematics conventions. Then, the student has to understand the conventions for the machine he is using. Then, he has to be careful.
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There was an earlier discussion of the 3^2 problem in the thread "RPN vs AOS vs ALGEBRAIC ENTRY" in Archive 16:
Message #1 Posted by Andy Morales on 12 June 2006, 3:38 p.m.
I took the following equation for flight MACH no from a CORVUS 500 RPN Calc manual (ala HP45)
SQRT 5X(((((400/661.5)^2)X (.2) 1)X (29.96/15)+1)^.286)1) ANSWER .82 (.8232688 using a TI 35 Galaxy Solar calc with AOS) I started with inermost parenthesis (400/661.5) then ^2 and continued. I tried using the parenthesis keys from left to right and got weird answers so i just used solved the equation the old fashioned way. I took Algebra I in 1961 so i learned to solve equations starting with innermost parenthesis. Oh i got the same asnwer with the following hp calculators also. HP21, 15C in addition to TI 30, SR 50, 50A, TI SR56, T8C and the infamous TI59. I even tried using a relic Sinclair Cambridge Alebraic calc. Got correct answer. I do not know the equation entry procedure for the newer TI/HP calcs with parenthesis control. I know one needs to enter powers and squeare roots before the numbers. That is not how i learned math. Who in the hell solves a problem by trying to square a number before entering it. Also why in the hell do the newer machines give an INCORRECT number if a NEGATIVE number is squared? I was taught that a NEGATIVE times a NEGATIVE = A Positive. Try that on a newer algebraic machine and you get a Negative 9 when squaring a 3. Unless You use parenthesis. Seems the newer machines do a weird CONE HEAD type of math. Ahhh, its probably only me. Iam a SLIDE RULE Babyboomer from the sexual revolution that learned to do math on a HP35, then a HP21, 25C, 15 and then began using AOS TI Calcs. They all gave same answers ecept the new generation of CALCS (HP 39g,48gII, 49g, ti 83....) that required a whole new way of entering equations.
An extended discussion of the 3^2 problem followed. It seems that a lot of confusion is generated by not being careful to recognize a difference between the way mathematics is written and the way calculating machines work.
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Hi, Palmer;
yeap, one can see that there are many considerations to be done in these cases. Not the ones concerning precedence, that I believe being the key to open the dicussion. Or lock it out, definitely.
I remember that some calculator manuals explicitly define their precedence for pending operations. Others may not explictly do so. In any way, when using parenthesis OR RPN, the user has the 'power' to change 'natural' precedence, as the expression he is computing demands. My question is: having found so many precedence 'rules' in computer languages, algbraic calculators and the like, I found myself in doubt about the actual 'natural' precedence to be followed. AFAIK, there is only one valid math precedence, so why not to strictly follow it? Many of these problems would never happen, I think. One math expression should generate the same resulting value whatever math tool we use, considering that with pencil and paper we would always get the same resulting value.
That´s the main reason I believe RPN as a keystroke feature helps developing math reasoning. As a programming tool I see it as the best portable FORTHlike implementation. Many contributors will point it differently, as I know some of then may find many negative points when programming in RPN. I do not see RPN as a programming language, in fact it is NOTATION ([N] stands for Notation in RPN), and I gladly accept FOCAL as a valid reference for the HP41 way of programming. I rememeber that in some previous posts of mine, a few years ago, I pointed Algrebra as a notation. Indeed, I failed to observe taht RPN is a notation as well. Math itself is what we are representing with these notations.
As Palmer states, I was also 'raised' in a time when math was taught differently. Does it classify me as outdated?
Cheers.
Luiz (Brazil)
Edited: 28 Sept 2006, 10:11 p.m.
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Hi, guys;
I logged at the university when I posted the previous message. Now I'm home and I cannot edit it, don't remember the password... Sorry!
This is what I should have written (with some typos already corrected).
Thanks.
__________________________________________
Hi, Palmer;
yeap, one can see that there are many considerations to be done in these cases. Not the ones concerning precedence, that I believe being the key to open the dicussion. Or lock it out, definitely.
I remember that some calculator manuals explicitly define their precedence for pending operations. Others may not explictly do so. In any way, when using parenthesis OR RPN, the user has the 'power' to change 'natural' precedence, as the expression he is computing demands. My question is: having found so many precedence 'rules' in computer languages, algebraic calculators and the like, I found myself in doubt about the actual 'natural' precedence to be followed. AFAIK, there is only one valid math precedence, so why not to strictly follow it? Many of these problems would never happen, I think. One math expression should generate the same resulting value whatever math tool we use, considering that with pencil and paper we would always get the same resulting value.
That´s the main reason I believe RPN as a keystroke feature helps developing math reasoning. As part of a programming tool I see it as the best portable FORTHlike implementation. Many contributors will point it differently, as I know some of them may find many negative points when programming with RPN. To be more precise, I do not see RPN as a programming language, in fact it is NOTATION ([N] stands for Notation in RPN), and I gladly accept FOCAL as a valid reference for the HP41 way of programming. I rememeber that in some previous posts of mine, a few years ago, I pointed Algebra as a notation. Indeed, I failed to observe that RPN is a notation as well. Math itself is what we are representing with these notations, and it surely is one single stuff to be represented by both of them.
As Palmer states, I was also 'raised' in a time when math was taught differently. Does it classify me as outdated?
Cheers.
Luiz (Brazil)
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Here's a decent reference describing the convention used for ordering math operations:
http://mathforum.org/dr.math/faq/faq.order.operations.html
Remember PEMDAS  Parentheses, Exponents, Multiplication, Division, Addition and Subtraction.
Regards,
John
