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'.
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.To post to this group, send email to mathf...@googlegroups.com.To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
--
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
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
we mathematicians must stop being righteous about things like "order of operations" and shortcuts and just make ourselves clear.
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…
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
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…
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.
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.
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.
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
--
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
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,
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
I fail to see a context where the word "wrong" in your question "Is Mathematica wrong?" would mean something.
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
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.
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.
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
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)' ?
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Prefix and Postfix are essentially parenthesis-free notation systems, so the function notation with parentheses everywhere looks a little strange.
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
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.Out[23]= x y
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
===================================
"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
===================================
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).
--
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
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:
--
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)
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! : )
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
To unsubscribe from this group, send email to mathfuture+...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
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
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?
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).
… 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? ”.
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!
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.
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.
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
--
Apparently someone at Alpha reads mathfuture. : )
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
Visit this group at http://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/groups/opt_out.
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
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.
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".
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
===================================
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture?hl=en.
For more options, visit https://groups.google.com/groups/opt_out.
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.
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.
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.
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
--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at http://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.