Arthur Benjamin's formula for changing math education | Video on TED.com

[Owen Densmore]

<base href="data:">

This is kinda cool and less than 3 minutes long!

http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.html

The thesis is a different spin on my claim that modern Math Notation (MN) is the roman numerals of our times.  Arthur Benjamin clearly explains that statistics and probability should be the "pinnacle" of our basic math education, not calculus.  His reasoning includes the discrete vs continuous argument that resonates with my MN vs Algorithm (or MN vs script) concern, which I'd love to see resolved in a parsable reworking of MN.

    -- Owen

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[Owen Densmore]

Hi joe.

> However, I don't understand your comment that math notation is the  
> roman
> numerals of our times. Which branch of math do you have in mind?  
> Certainly
> not calculus, where, as you know, we use Leibniz's elegant notation.

The core problem is the clash between two cultures: not the humanities  
vs the sciences, but that between mathematics and computing.  Or more  
precisely, between mathematics and algorithms.

This is a large topic: it includes the lack of good mathematical  
languages (like APL of old, and J today)
   http://www.jsoftware.com/
   http://www.jsoftware.com/jwiki/Guides/Getting%20Started
..which bridge the gap between symbolic computing and MN. It also  
refers to the impossibility of parsing mathematics .. it is ill-
defined as a language.  I.e. AB may mean A * B or the single variable  
named AB.

It extends to the "Asymptotic Assumption" made by many mathematicians  
when a discrete problem is more easily solved by converting to  
continuous. (Reminds one of ABM vs Math modeling)  Knuth has a good  
discussion on this in his book Concrete Mathematics (CON=Continuous,  
Crete=Discrete).  Basically he makes the case that, although the leap  
is reasonable at some point, it generally is taken too quickly.

So Roman Numerals == notational roadblock.  MN is not only is  
impossible to parse (and apply semantics to), it does not include any  
notion of "scripting" .. i.e. pseudo-code.

> I also don't follow your comment about discrete versus continuous.
> Among theoretical computer scientists, people who want to understand  
> the power of the computer and questions such as P vs NP study discrete
> problems whereas people like me who want to solve problems
> coming from, say, physics or computational finance think about  
> solving continuous problems such as path integration.

See above on Asymptotic Assumption and MN vs scripting.  Certainly  
computing, intrinsically discrete, provides wonderful approximations  
to continuous problems.

Interestingly enough, the Sage system:
   http://www.sagemath.org/
.. was originated by mathematicians who *required* open source so that  
their theorems could be solved knowing the system on which they were  
built.  Sage is the first system I know of that has variable  
declarations of Ring, Field, and so on.  What would happen if Euclid  
were propriatorey and only the results, not proofs were public  
knowledge?

Computer use by mathematicians remind me of Statistics use by social  
scientists.  Often the techniques are used without understanding the  
domain within which they are valid.  If nothing else, the power law  
distribution made many of us run back to see if our assumptions were  
reasonable.  Economics has fallen prey to this, the Black–Scholes  
model apparently assumed a Gaussian where a fatter tail was needed.

This rant is a long one, but the summary is simple enough: Mathematics  
and Computing/Algorithms need to be reconciled.  Modern MN needs  
(minor) changes to be at least machine readable.  Computing languages  
for mathematics need to bridge the gap between pseudo-code and  
symbolics.  APL/J are close.

How about a beer or glass of wine over this fascinating topic!

     -- Owen


On Jun 29, 2009, at 8:19 PM, Joseph Traub wrote:

> Owen,
>
> I find nothing to argue with in Benjamin's talk. He says that students
> studying economics, science, engineering, or math should learn  
> calculus
> but that it may not be needed by other students who should study
> probability and statistics.
>
> However, I don't understand your comment that math notation is the  
> roman
> numerals of our times. Which branch of math do you have in mind?  
> Certainly
> not calculus, where, as you know, we use Leibniz's elegant notation.
>
> I also don't follow your comment about discrete versus continuous.
> Among theoretical computer scientists, people who want to understand  
> the power of the computer and questions such as P vs NP study discrete
> problems whereas people like me who want to solve problems
> coming from, say, physics or computational finance think about  
> solving continuous problems such as path integration.
>
> Best, Joe
> <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
>
> Joseph F. Traub,   Edwin Howard Armstrong Professor of Computer  
> Science
>                   and External Professor, Santa Fe Institute
>
> [hidden email]          http://www.cs.columbia.edu/~traub
>
> Phone: [phone]    Messages: [phone]    Fax: (212)  
> 666-0140
>
> Columbia University
> Computer Science Department
> 1214 Amsterdam Avenue, MC0401
> New York, NY 10027
> USA
>
> Administrative Assistant: Sophie Majewski
> [hidden email] [phone]
>
>
> **************************************************************
>
> From: Owen Densmore <[hidden email]>
> Date: June 29, 2009 12:07:14 PM MDT
> To: The Friday Morning Applied Complexity Coffee Group <[hidden email]
> >,
> General topics & issues <[hidden email]>
> Subject: [FRIAM] Arthur Benjamin's formula for changing math  
> education |
> Video on TED.com
> Reply-To: The Friday Morning Applied Complexity Coffee Group
> <[hidden email]>
>
> This is kinda cool and less than 3 minutes long!
> http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.h
> tml
>
> The thesis is a different spin on my claim that modern Math Notation  
> (MN) is
> the roman numerals of our times.  Arthur Benjamin clearly explains
> that  statistics and probability should be the "pinnacle" of our  
> basic math
> education, not calculus.  His reasoning includes the discrete vs  
> continuous
> argument that resonates with my MN vs Algorithm (or MN vs script)  
> concern,
> which I'd love to see resolved in a parsable reworking of MN.
>
>    -- Owen
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[Eric Charles]

While much of the conversation below is steeped in issues I only peripherally understand, from a pedagogical perspective I am in complete agreement with Benjamin. A basic understanding of probability and statistics is more likely to be achieved by students, and would be more useful in most of their lives than a basic understanding of calculus. Calculus is a big stumbling block even for many students who enjoyed the math before that. I'm not sure how the high school curriculum would change to accommodate the new agenda, but I'd be really interested in finding out.

Eric

On Mon, Jun 29, 2009 11:19 PM, Owen Densmore <[hidden email]> wrote:

Hi joe.

> However, I don't understand your comment that math notation is the  
> roman
> numerals of our times. Which branch of math do you have in mind?  
> Certainly
> not calculus, where, as you know, we use Leibniz's elegant notation.

The core problem is the clash between two cultures: not the humanities  
vs the sciences, but that between mathematics and computing.  Or more  
precisely, between mathematics and algorithms.

This is a large topic: it includes the lack of good mathematical  
languages (like APL of old, and J today)
   http://www.jsoftware.com/
   http://www.jsoftware.com/jwiki/Guides/Getting%20Started
..which bridge the gap between symbolic computing and MN. It also  
refers to the impossibility of parsing mathematics .. it is ill- 
defined as a language.  I.e. AB may mean A * B or the single variable  
named AB.

It extends to the "Asymptotic Assumption" made by many mathematicians
 
when a discrete problem is more easily solved by converting to  
continuous. (Reminds one of ABM vs Math modeling)  Knuth has a good  
discussion on this in his book Concrete Mathematics (CON=Continuous,  
Crete=Discrete).  Basically he makes the case that, although the leap  
is reasonable at some point, it generally is taken too quickly.

So Roman Numerals == notational roadblock.  MN is not only is  
impossible to parse (and apply semantics to), it does not include any
 
notion of "scripting" .. i.e. pseudo-code.

> I also don't follow your comment about discrete versus continuous.
> Among theoretical computer scientists, people who want to understand  
> the power of the computer and questions such as P vs NP study discrete
> problems whereas people like me who want to solve problems
> coming from, say, physics or computational finance think about  
> solving continuous problems such as path integration.

See above on Asymptotic Assumption and MN vs scripting.  Certainly  
computing, intrinsically discrete, provides wonderful approximations  
to continuous problems.

Interestingly enough, the Sage system:
   http://www.sagemath.org/
.. was originated by mathematicians who *required* open source so that  
their theorems could be solved knowing the system on which they were  
built.  Sage is the first system I know of that has variable  
declarations of Ring, Field, and so on.  What would happen if Euclid  
were propriatorey and only the results, not proofs were public  
knowledge?

Computer use by mathematicians remind me of Statistics use by social  
scientists.  Often the techniques are used without understanding the  
domain within which they are valid.  If nothing else, the power law  
distribution made many of us run back to see if our assumptions were  
reasonable.  Economics has fallen prey to this, the Black–Scholes  
model apparently assumed a Gaussian where a fatter tail was needed.

This rant is a long one, but the summary is simple enough: Mathematics  
and Computing/Algorithms need to be reconciled.  Modern MN needs  
(minor) changes to be at least machine readable.  Computing languages
 
for mathematics need to bridge the gap between pseudo-code and  
symbolics.  APL/J are close.

How about a beer or glass of wine over this fascinating topic!

     -- Owen


On Jun 29, 2009, at 8:19 PM, Joseph Traub wrote:

> Owen,
>
> I find nothing to argue with in Benjamin's talk. He says that students
> studying economics, science, engineering, or math should learn  
> calculus
> but that it may not be needed by other students who should study
> probability and statistics.
>
> However, I don't understand your comment that math notation is the  
> roman
> numerals of our times. Which branch of math do you have in mind?  
> Certainly
> not calculus, where, as you know, we use Leibniz's elegant notation.
>
> I also don't follow your comment about discrete versus continuous.
> Among theoretical computer scientists, people who want to understand  
> the power of the computer and questions such as P vs NP study discrete
> problems whereas people like me who want to solve problems
> coming from, say, physics or computational finance think about  
> solving continuous problems such as path integration.
>
> Best, Joe
> <>
>
> Joseph F. Traub,   Edwin Howard Armstrong Professor of Computer  
> Science
>                   and External Professor, Santa Fe Institute
>
> [hidden email]          http://www.cs.columbia.edu/~traub
>
> Phone: [phone]    Messages: [phone]   
Fax: (212)  
> 666-0140
>
> Columbia University
> Computer Science Department
> 1214 Amsterdam Avenue, MC0401
> New York, NY 10027
> USA
>
> Administrative Assistant: Sophie Majewski
> [hidden email] [phone]
>
>
> **************************************************************
>
> From: Owen Densmore <[hidden email]>
> Date: June 29, 2009 12:07:14 PM MDT
> To: The Friday Morning Applied Complexity Coffee Group
<[hidden email] 
> >,
> General topics & issues <[hidden email]>
> Subject: [FRIAM] Arthur Benjamin's formula for changing math  
> education |
> Video on TED.com
> Reply-To: The Friday Morning Applied Complexity Coffee Group
> <[hidden email]>
>
> This is kinda cool and less than 3 minutes long!
>
http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.h
> tml
>
> The thesis is a different spin on my claim that modern Math Notation  
> (MN) is
> the roman numerals of our times.  Arthur Benjamin clearly explains
> that  statistics and probability should be the "pinnacle" of our  
> basic math
> education, not calculus.  His reasoning includes the discrete vs  
> continuous
> argument that resonates with my MN vs Algorithm (or MN vs script)  
> concern,
> which I'd love to see resolved in a parsable reworking of MN.
>
>    -- Owen
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org


============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

Eric Charles

Professional Student and
Assistant Professor of Psychology
Penn State University
Altoona, PA 16601

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[Owen Densmore]

Agreed.

There is an interesting shared concept that I hope does not get lost,  
a sort of "join" between the continuous and discrete.  That is the  
limit, and the epsilon/delta argument.

It is generally placed in the Continuous realm and thus most students  
do not see how it is used in the Discrete.  But the limit of a  
sequence is just one example of its use in the discrete.
   http://en.wikipedia.org/wiki/Limit_of_a_sequence
   http://en.wikipedia.org/wiki/Cauchy_sequence

     -- Owen


On Jun 30, 2009, at 7:19 AM, ERIC P. CHARLES wrote:

> While much of the conversation below is steeped in issues I only  
> peripherally understand, from a pedagogical perspective I am in  
> complete agreement with Benjamin. A basic understanding of  
> probability and statistics is more likely to be achieved by  
> students, and would be more useful in most of their lives than a  
> basic understanding of calculus. Calculus is a big stumbling block  
> even for many students who enjoyed the math before that. I'm not  
> sure how the high school curriculum would change to accommodate the  
> new agenda, but I'd be really interested in finding out.
>
> Eric
>
> On Mon, Jun 29, 2009 11:19 PM, Owen Densmore <[hidden email]>  
> wrote:
> Hi joe.
>
> > However, I don't understand your comment that math notation is the
> > roman
> > numerals of our times. Which branch of math do you have in mind?
> > Certainly
> > not calculus, where, as you know, we use Leibniz's elegant notation.
>
> The core problem is the clash between two cultures: not the humanities
> vs the sciences, but that between mathematics and computing.  Or more
> precisely, between mathematics and algorithms.
>
> This is a large topic: it includes the lack of good mathematical
> languages (like APL of old, and J today)
>    http://www.jsoftware.com/
>    http://www.jsoftware.com/jwiki/Guides/Getting%20Started
> ..which bridge the gap between symbolic computing and MN. It also
> refers to the impossibility of parsing mathematics .. it is ill-
> defined as a language.  I.e. AB may mean A * B or the single variable
> named AB.
>
> It extends to the "Asymptotic Assumption" made by many mathematicians
>
> when a discrete problem is more easily solved by converting to
> continuous. (Reminds one of ABM vs Math modeling)  Knuth has a good
> discussion on this in his book Concrete Mathematics (CON=Continuous,
> Crete=Discrete).  Basically he makes the case that, although the leap
> is reasonable at some point, it generally is taken too quickly.
>
> So Roman Numerals == notational roadblock.  MN is not only is
> impossible to parse (and apply semantics to), it does not include any
>
> notion of "scripting" .. i.e. pseudo-code.
>
> > I also don't follow your comment about discrete versus continuous.
> > Among theoretical computer scientists, people who want to understand
> > the power of the computer and questions such as P vs NP study  
> discrete
> > problems whereas people like me who want to solve problems
> > coming from, say, physics or computational finance think about
> > solving continuous problems such as path integration.
>
> See above on Asymptotic Assumption and MN vs scripting.  Certainly
> computing, intrinsically discrete, provides wonderful approximations
> to continuous problems.
>
> Interestingly enough, the Sage system:
>    http://www.sagemath.org/
> .. was originated by mathematicians who *required* open source so that
> their theorems could be solved knowing the system on which they were
> built.  Sage is the first system I know of that has variable
> declarations of Ring, Field, and so on.  What would happen if Euclid
> were propriatorey and only the results, not proofs were public
> knowledge?
>
> Computer use by mathematicians remind me of Statistics use by social
> scientists.  Often the techniques are used without understanding the
> domain within which they are valid.  If nothing else, the power law
> distribution made many of us run back to see if our assumptions were
> reasonable.  Economics has fallen prey to this, the Black–Scholes
> model apparently assumed a Gaussian where a fatter tail was needed.
>
> This rant is a long one, but the summary is simple enough: Mathematics
> and Computing/Algorithms need to be reconciled.  Modern MN needs
> (minor) changes to be at least machine readable.  Computing languages
>
> for mathematics need to bridge the gap between pseudo-code and
> symbolics.  APL/J are close.
>
> How about a beer or glass of wine over this fascinating topic!
>
>      -- Owen
>
>
> On Jun 29, 2009, at 8:19 PM, Joseph Traub wrote:
>
> > Owen,
> >
> > I find nothing to argue with in Benjamin's talk. He says that  
> students
> > studying economics, science, engineering, or math should learn
> > calculus
> > but that it may not be needed by other students who should study
> > probability and statistics.
> >
> > However, I don't understand your comment that math notation is the
> > roman
> > numerals of our times. Which branch of math do you have in mind?
> > Certainly
> > not calculus, where, as you know, we use Leibniz's elegant notation.
> >
> > I also don't follow your comment about discrete versus continuous.
> > Among theoretical computer scientists, people who want to understand
> > the power of the computer and questions such as P vs NP study  
> discrete
> > problems whereas people like me who want to solve problems
> > coming from, say, physics or computational finance think about
> > solving continuous problems such as path integration.
> >
> > Best, Joe
> > <>
> >
> > Joseph F. Traub,   Edwin Howard Armstrong Professor of Computer
> > Science
> >                   and External Professor, Santa Fe Institute
> >
> > [hidden email]          http://www.cs.columbia.edu/~traub
> >
> > Phone: [phone]    Messages: [phone]
> Fax: (212)
> > 666-0140
> >
> > Columbia University
> > Computer Science Department
> > 1214 Amsterdam Avenue, MC0401
> > New York, NY 10027
> > USA
> >
> > Administrative Assistant: Sophie Majewski
> > [hidden email] [phone]
> >
> >
> > **************************************************************
> >
> > From: Owen Densmore <[hidden email]>
> > Date: June 29, 2009 12:07:14 PM MDT
> > To: The Friday Morning Applied Complexity Coffee Group
> <[hidden email]
> > >,
> > General topics & issues <[hidden email]>
> > Subject: [FRIAM] Arthur Benjamin's formula for changing math
> > education |
> > Video on TED.com
> > Reply-To: The Friday Morning Applied Complexity Coffee Group
> > <[hidden email]>
> >
> > This is kinda cool and less than 3 minutes long!
> >
> http://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education.h
> > tml
> >
> > The thesis is a different spin on my claim that modern Math Notation
> > (MN) is
> > the roman numerals of our times.  Arthur Benjamin clearly explains
> > that  statistics and probability should be the "pinnacle" of our
> > basic math
> > education, not calculus.  His reasoning includes the discrete vs
> > continuous
> > argument that resonates with my MN vs Algorithm (or MN vs script)
> > concern,
> > which I'd love to see resolved in a parsable reworking of MN.
> >
> >    -- Owen
> >
> >
> > ============================================================
> > FRIAM Applied Complexity Group listserv
> > Meets Fridays 9a-11:30 at cafe at St. John's College
> > lectures, archives, unsubscribe, maps at http://www.friam.org
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org
>
>
> Eric Charles
>
> Professional Student and
> Assistant Professor of Psychology
> Penn State University
> Altoona, PA 16601
>
>
> ============================================================
> FRIAM Applied Complexity Group listserv
> Meets Fridays 9a-11:30 at cafe at St. John's College
> lectures, archives, unsubscribe, maps at http://www.friam.org

============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[Russell Standish]

On Tue, Jun 30, 2009 at 09:19:54AM -0400, ERIC P. CHARLES wrote:

> While much of the conversation below is steeped in issues I only peripherally
> understand, from a pedagogical perspective I am in complete agreement with
> Benjamin. A basic understanding of probability and statistics is more likely to
> be achieved by students, and would be more useful in most of their lives than a
> basic understanding of calculus. Calculus is a big stumbling block even for
> many students who enjoyed the math before that. I'm not sure how the high
> school curriculum would change to accommodate the new agenda, but I'd be really
> interested in finding out.
>
> Eric
>

I don't see why its an either/or thing. From what I recall of my years
11/12, we had two maths subjects, one which covered things like
algebra, trig and logarithms, and the other covering probability,
statistics, differential and integral calculus.

The next level down (1 unit maths as opposed to 2 unit) left out the
calculus part, but did do statistics and probability, along with
reduced algebra etc.

Of course the whole thing was a complete snap for me. I learnt the
entire calculus syllabus during a two week period in year 11 whilst on
a trip to France, sitting in cafes drinking beer. My teacher was a
gifted year 12 student!

Where I agree with the original poster is that more use should be made
of writing programs that illustrate the maths concepts from a discrete
point of view. There's nothing like watching a series converge on a
computer screen to see how the discrete maps to the continuous. In
fact I purchased a TI58 calculator in year 12, and writing programs to
do these sorts of things formed part of my "extension". These days,
one would probably do it in Matlab (or Octave).

 --

----------------------------------------------------------------------------
Prof Russell Standish                  Phone [phone] (mobile)
Mathematics                        
UNSW SYDNEY 2052                 [hidden email]
Australia                                http://www.hpcoders.com.au
----------------------------------------------------------------------------

============================================================
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Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org