is 2x/2y = 2*x/2*y?

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michel paul

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Jan 26, 2013, 12:12:28 AM1/26/13
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According to what we teach, they should be, right?

Well, check out these Wolfram Alpha pages:



Is Mathematica wrong? 

I think this would fit very well in a discussion about what we mean by mathematical 'rigor'. Lots of kids have been marked 'wrong' in the past by teachers considering themselves 'rigorous'. But I'm starting to see that as a mistake. The world the kids will be entering requires the ability to think more flexibly. The picture that gets painted for them is that mathematics is a set of rigid rules, but in the real world, it's actually quite flexible. It is exact, yes, and precise. But it's not anal. 

-- Michel

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"What I cannot create, I do not understand."

- Richard Feynman
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"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
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"What I cannot create, I do not understand."

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"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
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Maria Droujkova

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Jan 26, 2013, 7:35:16 AM1/26/13
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On Sat, Jan 26, 2013 at 12:12 AM, michel paul <python...@gmail.com> wrote:
According to what we teach, they should be, right?

Well, check out these Wolfram Alpha pages:



Is Mathematica wrong? 

I think this would fit very well in a discussion about what we mean by mathematical 'rigor'. Lots of kids have been marked 'wrong' in the past by teachers considering themselves 'rigorous'. 

I literally cried in frustration and sadness reading protocols of some Piaget experiments. The whole setup of making kids misunderstand the task by refusing to discuss what they meant, how they understood what you meant, etc. seems so nasty! Then in the eighties all those studies started to come out "debunking" this or that claim of children's developmental limitations. People still publish such studies now. I am doing a lit review for a grant at the moment, so I am finding a good crop of them.

"Guess what, we repeated such-n-such Piagetian task! But we asked kids what they meant, and provided a bit of scaffolding for their representations! And now they can do math five years younger than their Piagetian developmental stage would make you believe!"

Gee, really?

Cheers,
Dr. Maria Droujkova

Olivier Leguay

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Jan 26, 2013, 8:59:07 AM1/26/13
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With other codes:


OL



2013/1/26 Maria Droujkova <drou...@gmail.com>

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Linda Fahlberg-Stojanovska

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Jan 26, 2013, 9:02:56 AM1/26/13
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i tell my kids that "order of operations" is just a convention like "right of way" and that after the accident, the only people who will win are the lawyers.
1. when two cars come from opposite sides, who has the right of way?
i add the following:
2. what is the derivative of sin3x? i was always getting cos9x until i told them that from now on we ALL had to write sin(3x).
3. what does sin^2x mean? and why can we write that garbage and complain when sinx/x=sin?

when we didn't have technology maybe it was important to save ink. now we must all save ourselves (and our kiddies).  we mathematicians must stop being righteous about things like "order of operations" and shortcuts and just make ourselves clear.

warm regards linda
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Bradford Hansen-Smith

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Jan 26, 2013, 10:00:37 AM1/26/13
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You are correct in stating that "rigor" is a quality mind and not the specifics of what we are taught. Mind is flexible and struggles when the brain has been taught ridge ways of thinking through habitual connections about static ideas.

Brad


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David Chandler

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Jan 26, 2013, 2:03:21 PM1/26/13
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I hadn't heard about this issue of revisiting Piaget.  Do you have a web site where your work on this issue is posted?
--David Chandler


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Maria Droujkova

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Jan 26, 2013, 2:28:15 PM1/26/13
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On Sat, Jan 26, 2013 at 2:03 PM, David Chandler <david...@gmail.com> wrote:
I hadn't heard about this issue of revisiting Piaget.  Do you have a web site where your work on this issue is posted?
--David Chandler

I don't have anything systematic written about it. When I next run into studies that are good examples of interventions that don't teach math but increase math success by better communication, I will post them for you.

Cheers,
MariaD


michel paul

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Jan 26, 2013, 3:15:02 PM1/26/13
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On Sat, Jan 26, 2013 at 6:02 AM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:
 
we mathematicians must stop being righteous about things like "order of operations" and shortcuts and just make ourselves clear.

Yes!

I'm old enough to remember that I was never taught PEMDAS. (Or maybe I'm old enough to be losing my memory.) Order of operations, sure, but the understanding I somehow developed as a kid was that products were 'tight' and sums were 'loose', similar to a chemical bond vs. a physical collection. So when I simplified an expression, I just perceived sums and products as different types of things. I didn't consciously follow a set of rules.

Just yesterday I had this very conversation with another teacher, which is what prompted the post, and she said that not only was she taught PEMDAS, she likes it and still uses it to this day and isn't gonna quit. I thought, "Really? Does she really scan an expression from left to right consciously applying PEMDAS? Really?" Funny thing, she considers herself a 'traditional' teacher, and the irony is, these acronyms like PEMDAS, SOH CAH TOA, FOIL, etc.are NOT traditional! I'm curious about when we started using them. I didn't see them until I began teaching, and at the time I thought, "How stupid." There was a long break between the time I graduated high school and when I started teaching. I went to college and then wandered around India and stuff like that. So when I again saw secondary math, the culture had shifted.

Linda Fahlberg-Stojanovska

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Jan 26, 2013, 4:11:55 PM1/26/13
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It’s odd you say tight and loose. I don’t remember being taught that, but I do remember the “idea”.

I certainly don’t remember “pembas” or “rise before you run”, but I was so lucky. I had the “old” math in the US. The next year (in 1960) the New Math started…

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michel paul

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Jan 26, 2013, 6:38:33 PM1/26/13
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On Sat, Jan 26, 2013 at 1:11 PM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:

It’s odd you say tight and loose. I don’t remember being taught that, but I do remember the “idea”.

I certainly don’t remember “pembas” or “rise before you run”, but I was so lucky. I had the “old” math in the US. The next year (in 1960) the New Math started…


 Well, this is interesting, it's the New Math that I got I think in the 4th grade, and I loved it. I remember totally getting cardinality vs. ordinality, set theory, and all that stuff. I have a hunch that the reason the New Math tanked had more to do with culture and lack of teacher understanding than the math itself. My 4th grade teacher was not necessarily a mathematician, but she must have done something right. One experience I remember was her asking the class a question and then asking for a show of hands for different possible answers. I was the only one raising my hand on one of the possibilities. She said, "No one else agrees with you. Are you sure you want to stick with that?" I hesitated a little and rethought it, and then shot my hand back up and said, "Yes!" She used that to tell the class that just because no one else agrees with you doesn't mean you're wrong. The majority isn't necessarily correct! : ) That experience stayed with me my whole life. Now, I have no idea what the problem was. I just remember the experience of having thought it through and being willing to defend my conclusion. And I've tried to use that as much as possible. I'll ask a class who thinks this is the answer? And who thinks this is? We'll tally opinions and then I'll ask volunteers from each camp to defend their answer. There have been many occasions where I've had the opportunity to give the same lesson - the fact that everyone disagrees with you doesn't necessarily mean you're wrong! Of course, it also doesn't necessarily mean you're right, and sometimes that happens as well. But I think a really important message to give kids is persistence and digging into concepts and really thinking for themselves.

-- Michel

michel paul

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Jan 26, 2013, 6:43:48 PM1/26/13
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On Sat, Jan 26, 2013 at 11:28 AM, Maria Droujkova <drou...@gmail.com> wrote:
 
When I next run into studies that are good examples of interventions that don't teach math but increase math success by better communication, I will post them for you.

This would be so great. It's exciting to hear about this. Along with a focus on communication, I'm curious about the myth of practice, the myth that if you practice enough, if you do all your homework, understanding will magically occur. I think these are connected. Under the practice myth, if you do enough PEMDAS, you will magically come to understand simplifying expressions. However, fluency is not the same thing as understanding, though they are often mistaken for each other. It's creating the expression in the first place that is actually most important.

-- Michel

kirby urner

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Jan 26, 2013, 7:25:24 PM1/26/13
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relevant reading.

Quoting from 'Leaning J' by Roger Stokes:  http://www.jsoftware.com/help/learning/contents.htm

1.5 Parentheses

An expression can contain parentheses, with the usual meaning; what is inside parentheses is, in effect, a separate little computation.

   (2+1)*(2+2)
12

Parentheses are not always needed, however. Consider the J expression: 3*2+1. Does it mean (3*2)+1, that is, 7, or does it mean 3*(2+1) that is, 9 ?

   3 * 2 + 1
9

In school mathematics we learn a convention, or rule, for writing expressions: multiplication is to be done before addition. The point of this rule is that it reduces the number of parentheses we need to write.

There is in J no rule such as multiplication before addition. We can always write parentheses if we need to. However, there is in J a parenthesis-saving rule, as the example of 3*2+1 above shows. The rule, is that, in the absence of parentheses, the right argument of an arithmetic function is everything to the right. Thus in the case of 3*2+1, the right argument of * is 2+1. Here is another example:

   1 + 3 % 4 
1.75

We can see that % is applied before +, that is, the rightmost function is applied first.

This "rightmost first" rule is different from, but plays the same role as, the common convention of "multiplication before addition". It is merely a convenience: you can ignore it and write parentheses instead. Its advantage is that there are, in J, many (something like 100) functions for computation with numbers and it would be out of the question to try to remember which function should be applied before which.

In this book, I will on occasion show you an expression having some parentheses which, by the "rightmost first" rule, would not be needed. The aim in doing this is to emphasize the structure of the expression, by setting off parts of it, so as to make it more readable.

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Phillip Kent

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Jan 26, 2013, 8:37:51 PM1/26/13
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I was taught BOMDAS, MDSB, SOH CAH TOA and others in middle grades in
the late 1970s in England. I don't know for sure but I think these have
been a staple of English maths education for a long time - I learnt
'Silly Old Hitler Caught A Headache Through Our Army' (and still say
this to myself!) which must have a fairly well-defined date of
introduction!

My take on Michel's first question is like this. I don't see that
knowing precedence rules helps your initial (and most likely incorrect)
interpretation of what Wolfram Alpha is doing. What happens for me is
that the tight/loose heuristic (as Michel calls it) overwhelms any use
of precedence; it dropped me into a conflicting situation which I was
able to recover from only by getting the heuristic out of the way.

The key issue I think about is the essential ambiguity of conventional
written maths notation. In the 80s-90s when maths education started to
embrace computer algebra/symbolic maths systems (Mathematica, Maple,
DERIVE etc) one of the huge pedagogical benefits was that things like
precedence rules became more than apparently authoritarian impositions,
they came to be seen as central parts of the systematic grammar of
mathematical expression, as instantiated in the software command
language.

I can't help wondering where that has gone since 20 years ago. I suppose
what happened is that software got more clever, and people got more lazy
- they would rather cobble together 'natural math' expressions using two
finger gestures (like an ape) on a tablet screen, instead of using a
keyboard to type grammatically correct symbolic expressions. (OK, I
know the cause lives on in computational maths using Python and all that
- but that is a discussion for a different thread.)

- Phillip
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Dr Phillip Kent, London, UK
mathematics education technology research
philli...@gmail.com mobile: 07950 952034
www.phillipkent.net
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"A picture had better be worth a thousand words,
it takes more bandwidth"

Juan

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Jan 26, 2013, 9:53:33 PM1/26/13
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Michel,

From the pages you provide the links to, it seems to me that WolframAlpha

is interpreting 2x/2y as (2x)/(2y) = x/y

while interpreting 2*x/2*y as (2)(x/2)(y) = xy

which is interesting (somewhere in between confusing and intuitive). Thank you for bringing that little nugget to my attention.

Unlike WolframAlpha, the TI-89 in 3D-graphing mode handles both expressions as xy

However, I fail to see a context where the word "wrong" in your question "Is Mathematica wrong?" would mean something.

Are we grading Mathematica? Under what set of criteria?
Handling the above expressions as they do is just the way they do it.
As far as I know, each computer language has its very own syntax rules, conventions, and exceptions; and programmers and users alike have to struggle on their own with those rules, navigating instruction manuals or internet forums, to get to the proficiency level they want to attain using the particular program of their choice.
So, in that very limited sense, you could say computer languages kind of "get to write their own rules," and people in general have to deal with them in order to extract something useful out of them.

Saying that "Mathematica is wrong" just because of the different ways they handle those expressions would be a little similar to saying the Arabic language is wrong for writing right-to-left, or that the Chinese language is wrong for writing top-down.

Also, keep in mind the fundamental power inequality implicit in one of the common uses of the word "wrong"

Teacher vs student:
The teacher gets to mark the student's answers as "right" or "wrong." Not the other way around.

Top-level math-software-package (corporate machine) vs global netizen (individual human)
The program gets to determine whether the user input is "right" or "wrong." Not the other way around.

Cheers!

Darius Clarke

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Jan 26, 2013, 11:27:21 PM1/26/13
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I remember studying Psychology in college in the 80's when they were challenging Piaget. The challenges came up as qualifying the course materials taught about Piaget's discoveries in my undergrad classes. However, the crux of the challenges was the supposition that certain stages occurred at certain ages. Apparently different youth proceed through the various stages at widely different rates, and stall at different stages for widely different lengths of time. My professors pointed out that the value of Piaget's discoveries was more that we found for the first time that youth go through qualitatively different stages of cognition and are not just mini-adults, and that everyone goes through the stages in sequence. We don't skip around the stages out of sequence, or skip back to another, or skip over stages. 

One professor, whom I studied under, compared Piaget's stages to how adults think about politics. His preliminary finding was that a majority of adults in the U.S. only think of the political structures of our society in a very simplistic, early childhood thinking kind of way (rather than a complex matrix or directed graph of influence relationships kind of thinking) ... which possibly might help explain why elections have become popularity contest rather than choosing a leader. 
___

Being a software programmer I value how certain precedence rules helps to accurately communicate with the deterministic nature of our computational hardware. But, I've always valued the organic, creative, intuition of our species over that. Though, I know several technologists who bemoan the fact that humanity isn't as deterministic and their hardware. 
“I can understand the motions of heavenly bodies but not the madness of men” - Isaac Newton

However, few technologists, except those who program in multiple programming languages, know that the precedence rules are not universal but can vary from language to language. They were indeed designed for the convenience of stack based (FIFO) parsers for efficiency. One outlier is Smalltalk which gives priority from right to left and throws beginners for a loop. (I can explain why Smalltalk is different if you're interested. Just in general, Smalltalk's origin is based on an organic, cellular message passing paradigm.)

- Darius

kirby urner

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Jan 26, 2013, 11:54:01 PM1/26/13
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On Sat, Jan 26, 2013 at 8:27 PM, Darius Clarke <soci...@gmail.com> wrote:



Darius writes:
 
However, few technologists, except those who program in multiple programming languages, know that the precedence rules are not universal but can vary from language to language.

Yes. E. Chelin and I often rave (positively) about the J language and I quoted from one of its teaching manuals to this effect a few posts back.  In J, you have lots and lots of operators and it'd be madness to try ranking them all and remembering the rules.  So there's a simple rule:

"There is in J no rule such as multiplication before addition. We can always write parentheses if we need to. However, there is in J a parenthesis-saving rule, as the example of 3*2+1 above shows. The rule, is that, in the absence of parentheses, the right argument of an arithmetic function is everything to the right. Thus in the case of 3*2+1, the right argument of * is 2+1."

Which is why 3 * 2 + 1 evaluates to 9, not 7. Not incorrect, just a language game with different rules.

Kirby

Ted Kosan

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Jan 27, 2013, 12:11:54 AM1/27/13
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Michel wrote:

>is 2x/2y = 2*x/2*y?

>
>According to what we teach, they should be, right?
>
>Well, check out these Wolfram Alpha pages:
>
>http://www.wolframalpha.com/input/?i=2x%2F2y
>
>http://www.wolframalpha.com/input/?i=2*x%2F2*y
>
>Is Mathematica wrong?

The Mathematica TreeForm function can be used to unambiguously determine what 2x/2y and 2*x/2*y “mean” to Mathematica:

http://reference.wolfram.com/mathematica/ref/TreeForm.html

Unfortunately, Wolfram Alpha does not appear to include the TreeForm function, and I don’t own a copy of Mathematica, so I can’t provide the Mathematica expression trees for these two expressions. However, here is the expression tree that MathPiper creates for 2*x/2*y :

Inline image 1


>I think this would fit very well in a discussion about what we mean by
>mathematical 'rigor'. Lots of kids have been marked 'wrong' in the past by
>teachers considering themselves 'rigorous'. But I'm starting to see that as
>a mistake. The world the kids will be entering requires the ability to
>think more flexibly. The picture that gets painted for them is that
>mathematics is a set of rigid rules, but in the real world, it's actually
>quite flexible. It is exact, yes, and precise. But it's not anal.

My wife and I homeschooled our two sons, and I was responsible for teaching them mathematics. Since I had almost complete freedom in the way I presented the materials, I decided to take what I had been learning about teaching computers how to do mathematics, and use it to teach my sons mathematics. I found that using extremely rigorous methods enables mathematical knowledge to be learned much more quickly than nonrigorous methods can achieve.

For example, I taught the order of operations of expressions using expression trees because this notation provides an unambiguous version of the information that is present in an infix-form mathematical expression. Whenever I suspected that one of my sons was having difficulties with the order of operations, I simply had him draw the expression tree for the expression. The expression tree made it easy to diagnose any conceptual problems that he had. It also made it easy to show exactly where the conceptual problem was, and how it could be fixed.

Expression trees also provide a very simple and unambiguous method for solving simple equations:



I was so impressed with how effective expression trees can be for teaching mathematics, I asked my university students what they thought about them. I was shocked to discover that none of them had ever seen expression trees before!

Ted
exp3.png

kirby urner

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Jan 27, 2013, 12:46:34 AM1/27/13
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On Sat, Jan 26, 2013 at 9:11 PM, Ted Kosan <ted....@gmail.com> wrote:

<< SNIP >>
 
I was so impressed with how effective expression trees can be for teaching mathematics, I asked my university students what they thought about them. I was shocked to discover that none of them had ever seen expression trees before!

Ted

--

But you'd get 'em (parse trees) if we had more computer science / discrete / computational thinking in the mix, including pre college.

Before over-specialization occurs, cover Unicode.  Small world after all.  Could be up beat.  We get darker when doing SQL.**

Fun links to "parenthesis free" reverse polish, and prefix notation in LISP don'tcha know.

More fun for the kids.  So how do we get their teachers motivated?  Should Big Bird take this on?  Bert & Ernie?

Kirby


** I'm alluding to my way of weaving lore / history in with technical skills, with tabulation machines heralding abuses by fascism

michel paul

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Jan 27, 2013, 11:10:33 AM1/27/13
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Yes, very relevant! Thanks for this.

The first thing that occurred to me when reading through this was how binary expression trees naturally fit in here. And then that's exactly what Ted brought up. 

The need to specify order of operation only occurs with infix notation. In a math culture using pre- or post-fix notation, no rules of order would be necessary. And then picturing a set of operations as a binary tree unifies all of this beautifully.

- Michel


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michel paul

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Jan 27, 2013, 11:11:02 AM1/27/13
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On Sat, Jan 26, 2013 at 5:37 PM, Phillip Kent <philli...@gmail.com> wrote:
I was taught BOMDAS, MDSB, SOH CAH TOA and others in middle grades in
the late 1970s in England.

Yeah, I think that's when this stuff started to become popular. Though acronyms have been used all through history, there was a time in our culture when their use suddenly exploded and continued through today's LOLs.
 
I don't know for sure but I think these have
been a staple of English maths education for a long time - I learnt
'Silly Old Hitler Caught A Headache Through Our Army' (and still say
this to myself!) which must have a fairly well-defined date of
introduction!

Well, not necessarily. Even that could have been coined in the 70s.
 
My take on Michel's first question is like this. I don't see that
knowing precedence rules helps your initial (and most likely incorrect)
interpretation of what Wolfram Alpha is doing. What happens for me is
that the tight/loose heuristic (as Michel calls it) overwhelms any use
of precedence; it dropped me into a conflicting situation which I was
able to recover from only by getting the heuristic out of the way.

I think tight/loose can still apply to what Wolfram is doing. In '2x/2y' vs. '2*x/2*y' implied multiplication is a bit tighter than explicit multiplication. And I think that does correspond to how we actually think of monomials like '2x' and '2y'. We perceive the '2' and the 'x' as more tightly bound than we do in an expression like '2*x'.

The key issue I think about is the essential ambiguity of conventional
written maths notation.

Right. And the interesting thing is, this ambiguity occurs only with infix notation. With prefix or postfix notation there is no ambiguity at all. And this relates directly to Ted's mention of expression trees.
 
In the 80s-90s when maths education started to
embrace computer algebra/symbolic maths systems (Mathematica, Maple,
DERIVE etc) one of the huge pedagogical benefits was that things like
precedence rules became more than apparently authoritarian impositions,

Right. And as Kirby points out, in J there's a quite unique way to implement order of operations.
 
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michel paul

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Jan 27, 2013, 11:11:19 AM1/27/13
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On Sat, Jan 26, 2013 at 6:53 PM, Juan <here...@gmail.com> wrote:
 
I fail to see a context where the word "wrong" in your question "Is Mathematica wrong?" would mean something.

The context is our standard schoolish mathematics where PEMDAS is considered gospel. According to what is typically taught, Mathematica is wrong.

However, of course Mathematica is not wrong. It is the perspective of schoolish math that is wrong.

The question was meant with a sense of humor.

-- Michel

michel paul

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Jan 27, 2013, 11:22:57 AM1/27/13
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I love this. Yeah, I have felt for a long time that we should teach expression trees when discussing order of operations with students, and I think it's really great that MathPiper shows this.

An expression tree is a mathematical structure that can be represented in horizontal notation in various ways, and it is only with the operator infix form that any ambiguity might arise.

What I love about an expression tree is that it so nicely turns into functional expression. The expression tree for '2*x/2*y' can be functionally represented as 'product(product(2, quotient(x, 2)), y)'.

I think when we teach order of operations we should dump PEMDAS and create expression trees. And I think we should even discuss the various ways of horizontally representing these expression trees using in-fix, pre-fix, and post-fix notations.

If we want to integrate technological and mathematical literacy, expression trees are a beautiful way to do that.

- Michel


On Sat, Jan 26, 2013 at 9:11 PM, Ted Kosan <ted....@gmail.com> wrote:

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exp3.png

michel paul

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Jan 27, 2013, 11:40:47 AM1/27/13
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Hi Darius! You were helping run the Camtasia workshop at Pepperdine a couple of summers ago, right?

On Sat, Jan 26, 2013 at 8:27 PM, Darius Clarke <soci...@gmail.com> wrote:

One professor, whom I studied under, compared Piaget's stages to how adults think about politics. His preliminary finding was that a majority of adults in the U.S. only think of the political structures of our society in a very simplistic, early childhood thinking kind of way (rather than a complex matrix or directed graph of influence relationships kind of thinking) ... which possibly might help explain why elections have become popularity contest rather than choosing a leader. 

Yep. And I think that was part of the initial criticism of democracy. It's why some thinkers didn't take it seriously or saw it as fundamentally flawed.

However, few technologists, except those who program in multiple programming languages, know that the precedence rules are not universal but can vary from language to language.

And then an interesting thing is the mathematical structure of an expression tree. It is an object that can be represented in various ways.
 
-- Michel

Linda Fahlberg-Stojanovska

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Jan 27, 2013, 11:43:59 AM1/27/13
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I really have no idea what an expression tree is so my question might be stupid.

Do you actually write: 'product(product(2, quotient(x, 2)), y)' ?

I ask because my kids hate it when I write (sin(x))²+(cos(x))²=1 for sin²x+cos²x=1 (which is so much shorter, but which of course they don’t understand).

 

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of michel paul


Sent: Sunday, January 27, 2013 5:23 PM
To: mathf...@googlegroups.com

image001.png

michel paul

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Jan 27, 2013, 11:47:09 AM1/27/13
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On Sun, Jan 27, 2013 at 8:43 AM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:

I really have no idea what an expression tree is so my question might be stupid.


Take a look at Ted's post. He has a graphic of a binary expression tree there.
 

Do you actually write: 'product(product(2, quotient(x, 2)), y)' ?


Well, that would just be a functional representation of a tree. And this functional representation can easily be thought of as operator-prefix notation. In contrast, you can have operator post-fix notation, and that gets you into the world of RPN and David Chandler's love of HP calculators. : )
image001.png

David Chandler

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Jan 27, 2013, 12:07:06 PM1/27/13
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Prefix and Postfix are essentially parenthesis-free notation systems, so the function notation with parentheses everywhere looks a little strange.  The programming language Forth also uses RPN.  2*x/2 * y would be written like this on a stack:

2, x, *, 2, /, y, *

The operators *, and / are binary operators, so they operate on the two immediately prior numbers on the stack.  If x is 3 and y is 5 this would be processed:

2, 3, *, 2, /, 5, *   -->  6, 2, /, 5, *   -->   3, 5, *   -->   15

(2*3)/(2*5) would be written:

2, 3, *, 2, 5, *, /   -->   6, 2, 5, *, /   -->   6, 10, /   -->   0.6

The temporary result 6 just floats on the stack until it is needed.

--David Chandler



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michel paul

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Jan 27, 2013, 1:35:52 PM1/27/13
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On Sun, Jan 27, 2013 at 9:07 AM, David Chandler <david...@gmail.com> wrote:
Prefix and Postfix are essentially parenthesis-free notation systems, so the function notation with parentheses everywhere looks a little strange. 

Right. My background was originally with Scheme, and there you would express it as '(* (* 2 (/ x 2)) y)'. That tends to be clearer than the English terms. Still uses parentheses, but no commas, and that provides for more clarity than when there's a hodge-podge of parentheses and commas.

michel paul

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Jan 27, 2013, 4:08:25 PM1/27/13
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OK, fasten your seat belts. Here is what Mathematica as opposed to WolframAlpha says:

In[22]:= 2*x / 2*y

Out[22]= x y

In[23]:= 2 x / 2 y

Out[23]= x y

I decided to check out what Ted had to say about TreeForm in Mathematica, and I kept getting identical tree forms for either expression. I found that puzzling.

So, this is really bizarre. Is Mathematica wrong? Well, it seems to contradict WolframAlpha!  : )


On Fri, Jan 25, 2013 at 9:12 PM, michel paul <python...@gmail.com> wrote:
According to what we teach, they should be, right?

Well, check out these Wolfram Alpha pages:

I think this would fit very well in a discussion about what we mean by mathematical 'rigor'. Lots of kids have been marked 'wrong' in the past by teachers considering themselves 'rigorous'. But I'm starting to see that as a mistake. The world the kids will be entering requires the ability to think more flexibly. The picture that gets painted for them is that mathematics is a set of rigid rules, but in the real world, it's actually quite flexible. It is exact, yes, and precise. But it's not anal. 

-- Michel

===================================
"What I cannot create, I do not understand."

- Richard Feynman
===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================



--
===================================
"What I cannot create, I do not understand."

- Richard Feynman
===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

Phillip Kent

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Jan 27, 2013, 5:32:49 PM1/27/13
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I don't have access to Mathematica, but I was not surprised these behave
identically, as space is interpreted as implicit multiplication. I would
think if you type 2x into Mathematica in conventional input mode you
will get a syntax error ? And if you used the 'natural math' input mode
for 2x I guess you will get same behaviour as WolframAlpha ?

(I remember telling students using Mathematica in the mid-90s: don't use
the implicit multiply as it will lead you into trouble)

Juan

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Jan 28, 2013, 2:17:38 AM1/28/13
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Michel,

Well, as far as how one of those two expressions is evaluated, yes, I would say WolframAlpha contradicts both Mathematica and the TI-89.
It still would not occur to me to say that either one of them is "wrong."

The inconsistency between them may be puzzling but it probably stems from tweaking / fine-tuning each product with the goal of targeting its perceived "average user."

Who buys Mathematica? Seriously. Universities, scientists, engineers, architects, companies doing research or technological design, mathematicians, university professors, and that would be about it, I say. Who else has the money, and wants to sink it into such an abstract, theoretical, mathematical investment?
You may not like PEMDAS, or teaching it, or having to teach it but not because you don't understand how to apply it. My point is that most Mathematica users tend to have some expectation that the program will process the expression 2x/2y as equivalent to xy.

On the other hand, who uses WolframAlpha? On average, a whole lot more students from around the world looking for free help with their math homework. A whole bunch of kids, and in general, many more people who are not as mathematically literate as the average Mathematica user.

I am not trying to stereotype anyone, and I have no evidence but my guess / gut feeling says there are differences regarding the size of those two markets (Mathematica vs WolframAlpha), as well as the average mathematical skill of their consumers / users.
For a student not wanting to become detail-obsessed (as likely their math teacher is), it is a lot more natural to want to read 2x/2y as a "horizontal version" of the "vertical fraction" (2x)/(2y), without giving it much thought. It is a naive impulse, a perfectly comprehensible, natural, inexperienced, naive impulse. So, probably WolframAlpha doesn't see anything wrong with letting its users use it that way.

- Juan
Message has been deleted

Linda Fahlberg-Stojanovska

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Jan 28, 2013, 2:53:42 AM1/28/13
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Hiya Juan,

On my way to work. I couldn’t get blog to open but will try to see what the problem is later. Thank-you.

 

The kiddies don’t like all the extra writing in (sin(x))²+(cos(x))²=1, but can understand how to work it. If you ask them to calculate with the shorter-for-writing version (or in this case check the identity), they cannot do it because they don’t understand what it says.

For example:  I say: x=45degrees. Check the formula (with or without calculator).

 

(This is separate from the problem that they think anything with x must be a function and do not understand that this is an identity.)

Warm regards, Linda

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Juan
Sent: Monday, January 28, 2013 8:45 AM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Re: is 2x/2y = 2*x/2*y?

 

Hi Linda,

For what it's worth, I have one example of an expression tree for the quadratic formula here in this old blog post:
https://www.blogger.com/blogger.g?blogID=35228231#editor/target=post;postID=4425377811696517051

just scroll down to a little below the middle of the page and you will see it.
Hopefully looking at the tree and the formula at the same time will make it self-explanatory.

I also have a question for you, as I try to understand what you wrote about your kids:

First you write sin²x+cos²x=1 and they don't understand it, then you re-write

it as (sin(x))²+(cos(x))²=1 and then they hate it?

Is that it?
If so, what do you do? Do you keep going back and forth between the two forms until they finally settle for the short one?

- Juan




On Sunday, January 27, 2013 8:43:59 AM UTC-8, LFS wrote:

I really have no idea what an expression tree is so my question might be stupid.

Do you actually write: 'product(product(2, quotient(x, 2)), y)' ?

I ask because my kids hate it when I write (sin(x))²+(cos(x))²=1 for sin²x+cos²x=1 (which is so much shorter, but which of course they don’t understand).

 

--

Juan

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Jan 28, 2013, 4:01:47 PM1/28/13
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Hi Linda,

You are welcome. Have you tried to see how they react to x=cos(t) ;  y=sin(t) ;  and  x²+y²=1 ?

- Juan

Juan

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Jan 28, 2013, 4:17:38 PM1/28/13
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Ooops! I gave you the wrong link to that blog post. I'm sorry. This is the right one:

http://sdmath.blogspot.com/2009/07/is-there-connection-between.html

You are welcome.

- Juan


On Sunday, January 27, 2013 11:53:42 PM UTC-8, LFS wrote:

Darius Clarke

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Jan 28, 2013, 4:25:21 PM1/28/13
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Hi Michel,

Yes, that's me. :)


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Linda Fahlberg-Stojanovska

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Jan 28, 2013, 4:29:03 PM1/28/13
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Hiya Juan

I really liked the first example. I thought for sure it was 1/(a^2)

The handwritten expression tree was also very clear and I grasped the idea immediately (and I know what a bore it is to get images on a blog).  That is why I love my cheapo graphics tablet when I teach. Handwritten is just so much easier for me to understand – although I didn’t get the 1/(a^2) … I am always repeating to my kiddies. “Watch the level of your fraction lines and your operators, e.g. what is the derivative of sqrt(3x)?  I am always getting 1/sqrt(3x)*3 , but the question isn’t the one asked in this thread, but “Where does the * go? ”.

Great fun this thread.

 

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Juan
Sent: Monday, January 28, 2013 10:18 PM
To: mathf...@googlegroups.com
Subject: Re: [Math 2.0] Re: is 2x/2y = 2*x/2*y?

 

Ooops! I gave you the wrong link to that blog post. I'm sorry. This is the right one:

--

Ted Kosan

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Jan 28, 2013, 6:12:32 PM1/28/13
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Juan wrote:

>For what it's worth, I have one example of an expression tree for the quadratic
>formula here in this old blog post:
>
>http://sdmath.blogspot.com/2009/07/is-there-connection-between.html

This is an excellent explanation for the relationship between
mathematical expressions in traditional form and expression trees! I
have bookmarked it for future reference.

Here is another example which shows how an expression tree can be used
to compute the value of an expression in an unambiguous way:

http://math.hws.edu/javanotes/c9/expressionTree.png

This method of computation is so clear, I think that even young
children should be able to use it with little difficulty.

Ted

Ted Kosan

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Jan 28, 2013, 6:28:08 PM1/28/13
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Michel wrote:

>I love this. Yeah, I have felt for a long time that we should teach
>expression trees when discussing order of operations with students,
>and I think it's really great that MathPiper shows this.
>
>An expression tree is a mathematical structure that can be represented in
>horizontal notation in various ways, and it is only with the operator infix form
>that any ambiguity might arise.
>
>What I love about an expression tree is that it so nicely turns into functional
>expression. The expression tree for '2*x/2*y' can be functionally represented
>as 'product(product(2, quotient(x, 2)), y)'.
>
>I think when we teach order of operations we should dump PEMDAS and
>create expression trees. And I think we should even discuss the various ways
>of horizontally representing these expression trees using in-fix, pre-fix, and
>post-fix notations.
>
>we want to integrate technological and mathematical literacy, expression trees
>are a beautiful way to do that.

I did a search with Google Scholar for research on using expression trees in mathematics education, and one of the few documents I found on the subject was a short paper titled “Computer Presentations of Structure in Algebra”:

http://pat-thompson.net/PDFversions/1987StrucInAlg.pdf

The paper describes research that was conducted with a program called EXPRESSIONS that enabled students to manipulate expressions using expression trees. Here are a couple of paragraphs from this paper:

“Typical errors found in previous studies of students' errors in algebra suggest that students studying algebra frequently fail to realize that formulas in mathematical symbol systems have an intrinsic structure. In algebra, expressions are structured explicitly by the use of parentheses, and implicitly by assuming conventions for the order in which we perform arithmetic operations. It is hypothesized that many of students' errors in manipulating an algebraic expression are due to their inattention to the expression's structure.”

“Finally, it should be noted that in eight days of instruction these leaving-seventh grade students went from essentially no working knowledge of order of operations to deriving algebraic identities, and did so with some depth of understanding. Even with the limitations stated earlier in this discussion, the fact that such coverage is possible makes us question assumptions that are built into traditional junior high school pre-algebra and algebra curricula about what one can expect of junior high school students in the United States.”

Since using expression trees to teach various parts of mathematics seems to be a good idea, I wonder why they are seldom used for this purpose? This email discussion has caused me to become very interested in the idea of conducting some “research” on using expression trees in mathematics education.

I don’t have access to mathematics students, but I think I can create an application that is similar to EXPRESSIONS that teachers on this email list can use with their students. The program can be designed to automatically collect assessment data, and I am confident that we can come up with ways to manage this data that are in compliance with FERPA (and similar) regulations.

Is anyone interested in pursuing this idea with me?

Ted

michel paul

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Jan 28, 2013, 8:06:58 PM1/28/13
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On Sun, Jan 27, 2013 at 2:32 PM, Phillip Kent <philli...@gmail.com> wrote:

I don't have access to Mathematica, but I was not surprised these behave
identically, as space is interpreted as implicit multiplication. I would
think if you type 2x into Mathematica in conventional input mode you
will get a syntax error ?
 

Interestingly, no, at least not in the current version. Entering either '2x' or '2 x' in Mathematica 9 produces the same result.

 
And if you used the 'natural math' input mode
for 2x I guess you will get same behaviour as WolframAlpha ?
 

Right. Using 'natural math' input, '2x/2y' yields x/y.

For school math fundamentalists, the world is now a scary place.   : )

(I remember telling students using Mathematica in the mid-90s: don't use
the implicit multiply as it will lead you into trouble)

On Sun, 2013-01-27 at 13:08 -0800, michel paul wrote:
> OK, fasten your seat belts. Here is what Mathematica as opposed to
> WolframAlpha says:
>
>         In[22]:= 2*x / 2*y
>
>         Out[22]= x y
>
>         In[23]:= 2 x / 2 y
>
>         Out[23]= x y
>
> I decided to check out what Ted had to say about TreeForm in
> Mathematica, and I kept getting identical tree forms for either
> expression. I found that puzzling.
>
>
> So, this is really bizarre. Is Mathematica wrong? Well, it seems to
> contradict WolframAlpha!  : )



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michel paul

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Jan 28, 2013, 8:23:00 PM1/28/13
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On Sun, Jan 27, 2013 at 11:17 PM, Juan <here...@gmail.com> wrote:

For a student not wanting to become detail-obsessed (as likely their math teacher is), it is a lot more natural to want to read 2x/2y as a "horizontal version" of the "vertical fraction" (2x)/(2y), without giving it much thought. It is a naive impulse, a perfectly comprehensible, natural, inexperienced, naive impulse. So, probably WolframAlpha doesn't see anything wrong with letting its users use it that way.

- Juan

Well, here's a very interesting statement: "the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[5] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[6] Additionally, Wolfram Alpha considers that implied multiplication without parentheses precedes division, unlike explicit multiplication or implied multiplication with parentheses. 2*x/2*x and 2(x)/2(x) both yield x2, but 2x/2x yields 1.[7] The TI 89 calculator yields x2 in all three cases."

This is from http://en.wikipedia.org/wiki/Order_of_operations. It covers all the bases we've mentioned, plus more. So it seems that Feynman and the Physical Review also think like Alpha.

michel paul

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Jan 28, 2013, 8:32:03 PM1/28/13
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On Mon, Jan 28, 2013 at 3:28 PM, Ted Kosan <ted....@gmail.com> wrote:

Since using expression trees to teach various parts of mathematics seems to be a good idea, I wonder why they are seldom used for this purpose? This email discussion has caused me to become very interested in the idea of conducting some “research” on using expression trees in mathematics education.

I don’t have access to mathematics students, but I think I can create an application that is similar to EXPRESSIONS that teachers on this email list can use with their students. The program can be designed to automatically collect assessment data, and I am confident that we can come up with ways to manage this data that are in compliance with FERPA (and similar) regulations.

Is anyone interested in pursuing this idea with me?

Yes! This is definitely cool. 
 
-- Michel

Juan

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Jan 30, 2013, 3:44:32 AM1/30/13
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Hi Linda,


On Monday, January 28, 2013 1:29:03 PM UTC-8, LFS wrote:

Hiya Juan

I really liked the first example. I thought for sure it was 1/(a^2)

It looks like that, right? Some expressions can get really confusing when you vary the size or position of some symbols.
 

The handwritten expression tree was also very clear and I grasped the idea immediately (and I know what a bore it is to get images on a blog). 

Thank you!

… I am always repeating to my kiddies. “Watch the level of your fraction lines and your operators, e.g. what is the derivative of sqrt(3x)?  I am always getting 1/sqrt(3x)*3 , but the question isn’t the one asked in this thread, but “Where does the * go? ”.

Above or below the fraction bar? That is their question, right? Or, do they write the answer in a single, horizontal line?
Do they also usually miss the 1/2 factor that comes from the power rule?

- Juan

Juan

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Jan 30, 2013, 4:06:48 AM1/30/13
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On Monday, January 28, 2013 3:12:32 PM UTC-8, tkosan wrote:
Juan wrote:

>For what it's worth, I have one example of an expression tree for the quadratic
>formula here in this old blog post:
>
>http://sdmath.blogspot.com/2009/07/is-there-connection-between.html

This is an excellent explanation for the relationship between
mathematical expressions in traditional form and expression trees!

Thank you, Ted! I'm glad you liked it.
 
Here is another example which shows how an expression tree can be used
to compute the value of an expression in an unambiguous way:

http://math.hws.edu/javanotes/c9/expressionTree.png
 
Thank you for that other link. Yes, the expression trees take the ambiguity out, and clearly show the order of operations.
 
This method of computation is so clear, I think that even young
children should be able to use it with little difficulty.
 
Well, most of them yes but it always depends on the individual student. I show expression trees to students when they are not getting it any other way. Sometimes, when I show them the expression tree to illustrate the order of operations, some students react defensively, asking: "Do I have to do that? Is that what the teacher is expecting me to do? Will they mark me down if I don't include the graphics?" Many students assume they will be tested on everything their instructors show to them, and some worry exponentially to the amount of material covered in class. Any new topic, or any additional method for solving problems, tends to increase their math-test-anxiety by a multiplicative factor greater than one.

- Juan

Juan

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Jan 30, 2013, 4:30:59 AM1/30/13
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On Monday, January 28, 2013 5:23:00 PM UTC-8, michel paul wrote:

Well, here's a very interesting statement: "the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[5] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[6] Additionally,
 
This is from http://en.wikipedia.org/wiki/Order_of_operations. It covers all the bases we've mentioned, plus more. So it seems that Feynman and the Physical Review also think like Alpha.
 

Oh, physicists!

OK then, it seems so far regarding the precedence of implicit multiplication in the order of operations, the opposing teams line up as follows:

{PEMDAS, TI-89, conventional-input-mode-Mathematica, school math fundamentalists, ... }

versus

{natural-math-input-mode-Mathematica, Wolfram Alpha, physicists, a very great many students, ... }

Interesting, indeed.

- Juan

michel paul

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Feb 7, 2013, 12:26:38 AM2/7/13
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OK, fasten your seat belts again.

As of now, 2x/2y == 2*x/2*y in Alpha.

What's more, the natural input result from Mathematica has changed! As of Jan 28 it was the case that natural math input returned 'x/y' for '2x/2y'. However, as of today natural math input returns 'x y' for '2x/2y'.

This is fascinating. Also a little creepy, but mostly fascinating. Something like that could make a person think they were crazy.

It sure is a good thing that I know that this did in fact change.

Apparently someone at Alpha reads mathfuture.  : )

- Michel

Linda Fahlberg-Stojanovska

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Feb 7, 2013, 5:15:14 AM2/7/13
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I know Maria organized a mathfuture event /talk with a wonderfully interesting colleague who was someone important at wolframalpha. I cannot remember her name. Linda
Sent from my HTC
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Juan

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Feb 9, 2013, 4:05:50 AM2/9/13
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So now Mathematica / Wolpham Alpha are consistent with themselves, with PEMDAS, and with the TI-89, at least in regards to the evaluation of 2x/2y

Interesting change, indeed, plus the timing of it. This kind of temporal coincidence reminds me of movies like "Ruby Sparks," "Stranger than Fiction," "The Matrix," and "The Truman Show."

- Juan

Linda Fahlberg-Stojanovska

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Feb 9, 2013, 5:44:18 AM2/9/13
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Thanks Michel for this thread and all the responses.

I was added this to a webinar I am preparing. One of the things I really like about wolframalpha is that the first line Is: input interpretation.

So it tells you how it is interpreting 2x/2y as 2 *(x/2)*y

 

 

 

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Juan
Sent: Saturday, February 09, 2013 10:06 AM
To: mathf...@googlegroups.com
Subject: [Math 2.0] Re: is 2x/2y = 2*x/2*y?

 

So now Mathematica / Wolpham Alpha are consistent with themselves, with PEMDAS, and with the TI-89, at least in regards to the evaluation of 2x/2y

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Maria Droujkova

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Feb 10, 2013, 8:06:29 AM2/10/13
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On Thu, Feb 7, 2013 at 12:26 AM, michel paul <python...@gmail.com> wrote:

Apparently someone at Alpha reads mathfuture.  : )

 I do not know if the good Wolfram people still read Math Future email group, but we've had this event lead by Crystal Fantry: http://mathfuture.wikispaces.com/WolframAlpha and this with Kelvin Mischo: http://mathfuture.wikispaces.com/WolframDemonstrations

Also, Math Future archives are linkable and searchable, so if anyone monitors what people are saying online about them, our messages will come up...


Cheers,
Dr. Maria Droujkova
919-388-1721

michel paul

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Feb 12, 2014, 2:23:27 PM2/12/14
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Deja Vu!

As of right now Wolfram Alpha presents the result of a / ax as 1/x and the result of 2 / 2x as x.

About a year ago this thread began with a similar observation, and in the end Alpha removed the inconsistency. Let's watch and see if the same thing happens this year.

- Michel


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Donald Cohen

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Feb 12, 2014, 4:24:12 PM2/12/14
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Wolfram Alpha Pro also, when a 7th grader was asked to factor x^2-5x+6 and they gave x(x-5+6/x), she was told this answer was incorrect. They are still trying to fix that, a month later. They said WAP was expecting (x-3)(x-2), as if I didn't know that. Having worked with PLATO, I understand their problem. We were told that an hour of student work would take 40 hours of preparation; we found this was about 400 hours, if you want to do it right. They are not ready for the real world.

Don



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Christian Baune

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Mar 8, 2014, 3:44:41 AM3/8/14
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The first thing to teach is that rules are not engraved into stone and can be broken if the intention is clear and the break meaningful.

Kirby gave an example with APL which does indeed only use functions. Those functions are variadic and they can be also unary and binary. Unary argument can be on the left or on the right.
So the language designer did deliberately broke all the Maths rules concerning writing. Look at floor function, only the left "bracket" is used.

Or you can even find things like this:
(/+ X)÷#X

This expression is the mean of list X.
It does violate all Mathematics rules regarding writing but it is short to type and with a bit of habits, you read it as easily as the summation symbol.

The fantastic part is that you can select the minimum value of a list like
/$ X

("$" is not the right symbol for min function but my phone lack such symbols)

It works because 4$5 means take the min from values surrounding "%".

If
X<- 2 7 9 1 5 3
Then
/$ X
Becomes
2$7$9$1$4$3
And in APL we "reduce" from right.
2$7$9$1$(4$3)
2$7$9$1$3
2$7$9$(1$3)
...
1

Of course, if you do a bit of APL, you will find all of these breaks very useful.

APL is listed on the esoteric languages page (esolang) but esoteric does not mean "difficult" but "unusual".

This is called "codification" and this is inherent to thinking. Anyone willing to solve a problem must have a good codification system which is personal.

We can't teach codification but we can make students aware of it and how important it is.

Perhaps introducing esoteric languages and asking students to write in some simple expressions could enlighten them and avoid the "computer said ... thus you are wrong" or "computer did output the wrong value". It should lead to student being aware that the system they use may not work as they expect and that they should adapt their input to it!

Kind regards,
Christian

kirby urner

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Mar 8, 2014, 10:27:45 AM3/8/14
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On Sat, Mar 8, 2014 at 3:44 AM, Christian Baune <progr...@gmail.com> wrote:

The first thing to teach is that rules are not engraved into stone and can be broken if the intention is clear and the break meaningful.

Kirby gave an example with APL which does indeed only use functions. Those functions are variadic and they can be also unary and binary. Unary argument can be on the left or on the right.
So the language designer did deliberately broke all the Maths rules concerning writing. Look at floor function, only the left "bracket" is used.


APL was my favorite language for awhile.  APL terminals were scattered about campus and I felt I'd walked into a science fiction novel.  Iverson, his son, Roger Hui at al later invented the J language (jsoftware.com).

J gives you a right-to-left pipeline of functions with no "precedence rules" as with so many language primitives it would be crazy-making to try to remember how to rank them all.  APL is like that too:  no precedence rules.

In "ordinary" mathematics notation we say 2 + 2 * 4 is 10 because we go 2 + (2 * 4), not (2 + 2) * 4.

In J:

4 * 2 + 2

16


i.e. since it's right to left, it's 2 + 2 giving 4, then * 4, giving 16.


 

Of course, if you do a bit of APL, you will find all of these breaks very useful.

APL is listed on the esoteric languages page (esolang) but esoteric does not mean "difficult" but "unusual".

Are you sure?  Didn't see it.  I think it probably should be simply on the basis of using non-standard characters (but these symbols are now part of Unicode, unlike in the early days).

Kirby


Message has been deleted

Donald Cohen

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Mar 13, 2014, 4:38:43 PM3/13/14
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Have you seen I Practice math , solving equations:

x + 3 = 9 (original equation )
x = 9 – 3 (take the constant term to the other side so that the variable x is isolated.) 
x = 6

I don't think the statement take the constant term to the other side so that the variable x is isolated is mathematics.

Don Cohen


On Wed, Mar 12, 2014 at 9:13 AM, Alice Lewis <alice...@gmail.com> wrote:
I think the whole setup of making kids misunderstand the task by refusing to discuss what they meant, how they understood what you meant, etc.
seems so nasty! Then in the eighties all those studies started to come out "debunking" this or that claim of children's developmental limitations.
People still publish such studies now.
___________________________________
Math Worksheets For Kids | 7th Grade Math Practice


On Sunday, January 27, 2013 2:11:55 AM UTC+5, LFS wrote:

It’s odd you say tight and loose. I don’t remember being taught that, but I do remember the “idea”.

I certainly don’t remember “pembas” or “rise before you run”, but I was so lucky. I had the “old” math in the US. The next year (in 1960) the New Math started…

 

From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of michel paul


Sent: Saturday, January 26, 2013 9:15 PM
To: mathf...@googlegroups.com

Subject: Re: [Math 2.0] is 2x/2y = 2*x/2*y?

 

On Sat, Jan 26, 2013 at 6:02 AM, Linda Fahlberg-Stojanovska <lfah...@gmail.com> wrote:

 

we mathematicians must stop being righteous about things like "order of operations" and shortcuts and just make ourselves clear.

 

Yes!


I'm old enough to remember that I was never taught PEMDAS. (Or maybe I'm old enough to be losing my memory.) Order of operations, sure, but the understanding I somehow developed as a kid was that products were 'tight' and sums were 'loose', similar to a chemical bond vs. a physical collection. So when I simplified an expression, I just perceived sums and products as different types of things. I didn't consciously follow a set of rules.

 

Just yesterday I had this very conversation with another teacher, which is what prompted the post, and she said that not only was she taught PEMDAS, she likes it and still uses it to this day and isn't gonna quit. I thought, "Really? Does she really scan an expression from left to right consciously applying PEMDAS? Really?" Funny thing, she considers herself a 'traditional' teacher, and the irony is, these acronyms like PEMDAS, SOH CAH TOA, FOIL, etc.are NOT traditional! I'm curious about when we started using them. I didn't see them until I began teaching, and at the time I thought, "How stupid." There was a long break between the time I graduated high school and when I started teaching. I went to college and then wandered around India and stuff like that. So when I again saw secondary math, the culture had shifted.

 

-- Michel

 

===================================
"What I cannot create, I do not understand."

 

- Richard Feynman

===================================
"Computer science is the new mathematics."

 

- Dr. Christos Papadimitriou
===================================

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David Chandler

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Mar 13, 2014, 9:10:46 PM3/13/14
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On Thu, Mar 13, 2014 at 1:38 PM, Donald Cohen <doncohe...@gmail.com> wrote:
Have you seen I Practice math , solving equations:

x + 3 = 9 (original equation )
x = 9 – 3 (take the constant term to the other side so that the variable x is isolated.) 
x = 6

I don't think the statement take the constant term to the other side so that the variable x is isolated is mathematics.


I don't entirely agree.  If that method is taught without justification, I agree it is not mathematics.  However, once you recognize that transforming an equation by moving a term from one side to the other while changing the sign is a valid operation, justified by the idea that you can add something to both sides, I see nothing wrong with using it as an efficient method.  You don't have to prove a theorem every time you see it.  Once a method is justified, the method has the status of a proved theorem and enters into the toolkit as something valid to do.

I think most mathematical practitioners in the real world do in fact do something like the old fashioned "transposing" method in their heads to solve equations, and it can be a much more fluent way to operate.  (Confession time:  I would be interested to know if most of YOU use this construct when doing equations on your own.)

--David Chandler

Donald Cohen

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Mar 14, 2014, 1:12:53 AM3/14/14
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David, 
How about asking 7th grade teachers, and say 7th graders, what they are doing, then when it gets slightly more complicated. Why not begin by asking kids to make up identities, they can shorten the list and these identities can be in their "toolkit". I have the kids make up identities for exponents, and logarithms, and moving circles left and right, and on and on…most textbooks tell the kids the identities first and they don't know where they came from and are just more things to memorize, without seeing the patterns, and without making mistakes. And I put their name on the identity they make up. 

On graphs, a 3rd grader the other day graphed x-y=2, saw the pattern in the points plotted and realized a point on this line was (1, neg1) and said 1- neg1=2. I can't raise the negative sign here. I thought that was cool, and I showed this to his Dad when he came to pick him up.

I seem to always be in the position of trying to justify my teaching methods; but then I realize most people I talk to haven't had my experiences, and most don't enjoy their teaching, except those who know everything- like Newton and Leibnitz and those guys who made lots of mistakes using infinite series. See Morris Kline's "Mathematics The Loss of Certainty", p142. 

And one of my 7th graders, in WolframAlpha, when asked to factor x^2-5x+6 put x(x-5+6/x) and was told this was incorrect (she made up 14 others). After I sent a message to the "team", it took a couple of weeks for the "team" to say it was a correct answer. They tried to tell me the program was expecting , you know, (x-2)*(x-3).

I'm just glad I work with kids of all ages and abilities and I learn from them. And can they do things 4 different ways?

Don


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kirby urner

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Mar 14, 2014, 1:19:47 AM3/14/14
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On Thu, Mar 13, 2014 at 1:38 PM, Donald Cohen <doncohe...@gmail.com> wrote:
Have you seen I Practice math , solving equations:

x + 3 = 9 (original equation )
x = 9 – 3 (take the constant term to the other side so that the variable x is isolated.) 
x = 6

I don't think the statement take the constant term to the other side so that the variable x is isolated is mathematics.

Don Cohen


The next question would be, what is it then, if not mathematics?  Forensics maybe?  

Wasn't it Einstein who didn't get algebra until some uncle made it a detective game?  Unmask that x!  

Police work.

So instead of calling it math, should we call it "math in a mix with solving murder mysteries"?  

Sounds promising.  Sherlock Holmes and all that ("It's mere Deduction, Watson!").

Kirby



Donald Cohen

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Mar 14, 2014, 9:14:40 AM3/14/14
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That sounds good! That's how I start equation solving. Try a number, test it to see if it makes the sentence true.

Don


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