Should mathematics course requirements be satisfied by computer science courses?

[michel paul]

​I find this disappointing:​

​ ​

Should mathematics course requirements for high school graduation be satisfied by computer science courses?

I do understand the intent, but something seems wrong with it. They make the statement "it is widely agreed, especially by computer scientists and mathematicians, that computer science is not a subfield of mathematics, or vice versa". 

I can understand that they might not be regarded as subfields of each other, but the relationship between them is quite deep, especially given the nature of functional programming and Voevodsky's work in type theory. If he is successful, then mathematics actually would become a kind of computer science. 

The spirit of the NCTM position seems much different than Dijkstra's in Programming as a discipline of mathematical nature. He describes in there his "impression of "the standard mathematical curriculum" (whatever that may be), I come to the following differences in stress: 

1. In the standard mathematical curriculum the student becomes familiar (sometimes even very familiar!) with a standard collection of mathematical concepts, he is less trained in introducing new concepts himself.
2. In the standard mathematical curriculum the student becomes familiar (sometimes even very familiar!) with a standard set of notational techniques, he is less trained in inventing his own notation when the need arises.
3. In the standard mathematical curriculum the student often only sees problems so "small" that they are dealt with a single semantic level. As a result many students see mathematics rather as the art of organizing their symbols on their piece of paper than as the art of organizing their thoughts."

I love the spirit of that. He wrote that back in 1973. I was still in high school.

Again, though I do understand NCTM's intent, I would like to see them emphasize more the deep relationship that does exist between math and CS. "One and one and one is three ..." only if each 'one' is of the same type. We can only count those things to which we can give the same name. That's kind of important, both in mathematics and CS.

--

​ Michel

​

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

- Richard Feynman

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

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

[kirby urner]

On Thu, Mar 31, 2016 at 1:21 PM, michel paul <python...@gmail.com> wrote:

​I find this disappointing:​

​ ​

Should mathematics course requirements for high school graduation be satisfied by computer science courses?

I find the thinking somewhat quaintly retro, but understand the need for compromises in a national institution that doubles as a safekeeper of career-related credentials.  What if computer science people take it upon themselves to teach our math?  These are tribes with turf.  It's anthropology.

Allowing a computer science course to substitute for a mathematics course in states with a graduation requirement of only two mathematics courses (beginning with Algebra 1 or the equivalent) would surely undermine students’ mathematics preparation, while doing so in states that require four mathematics courses would be unlikely to have an adverse impact on college and career readiness in mathematics.

It's this language of "substitution" I find somewhat gross, but it naturally follows from the either / or mentality i.e. either it's math, or it's computer science, black or white, red or blue.

What's long overdue, is what the NCTM has been very shy in producing:  a new kind of math course wherein technology is used with growing mathematical sophistication.

The change is so overdue that now there's a need to concede ground:  if it's a three or four year state, i.e. it requires three to four years of math credits, then it's probably OK to experiment with this newfangled content. That's what I see them saying.  That's actually quite a concession.

Rather than focusing on NCTM, I think the question is more "what's the hold up?"  Why has "technology in the classroom" been so vapid, such that little to no coding has been introduced, and scientific calculators rule the roost?   That's only true on average in that significant change *has* occurred in some areas.

I see the NCTM basically giving a green light to most states to barrel ahead with stronger technology.  To me, this is all about finding a right mix within STEM. 

The T has been weak, except maybe in places like Singapore where the E might be stronger too.

Kirby

[Peter Farrell]

On Thu, Mar 31, 2016 at 1:21 PM, michel paul <python...@gmail.com> wrote:

​I find this disappointing:​

​ ​

Should mathematics course requirements for high school graduation be satisfied by computer science courses?

I find the thinking somewhat quaintly retro, but understand the need for compromises in a national institution that doubles as a safekeeper of career-related credentials.  What if computer science people take it upon themselves to teach our math?  These are tribes with turf.  It's anthropology.

I was going to weigh in but Kirby summed up my opinion. The NCTM is all about protecting the M in STEM. Don't be disappointed. The weavers' guilds weren't so hot on the automated Jacquard loom, either.

Peter 

[Ted Kosan]

Michel wrote:

> The spirit of the NCTM position seems much different than Dijkstra's in
> Programming as a discipline of mathematical nature.

Dijkstra also wrote that the main difference between mathematical
formula and computer programs is their length:

"It really helps to view a program as a formula. Firstly, it puts the
programmer's task in the proper perspective: he has to derive that
formula. Secondly, it explains why the world of mathematics all but
ignored the programming challenge: programs were so much longer
formulae than it was used to that it did not even recognize them as
such. Now back to the programmer's job: he has to derive that formula,
he has to derive that program. We know of only one reliable way of
doing that, viz. by means of symbol manipulation.” (“On the Cruelty of
Really Teaching Computer Science”, Edsger Dijkstra,
https://www.cs.utexas.edu/~EWD/transcriptions/EWD10xx/EWD1036.html)

In the future mathematics, computer science, and symbolic logic will
be taught together by teachers who are knowledgeable in all three
subjects.

> Again, though I do understand NCTM's intent, I would like to see them
> emphasize more the deep relationship that does exist between math and CS.

I think MathFuture is the new NCTM.

Ted

[Joseph Austin]

On Mar 31, 2016, at 4:21 PM, michel paul <python...@gmail.com> wrote:

​I find this disappointing:​

​ ​

Should mathematics course requirements for high school graduation be satisfied by computer science courses?

...

Again, though I do understand NCTM's intent, I would like to see them emphasize more the deep relationship that does exist between math and CS.

Hear! Hear!  

[Full Disclosure: I'm a Computer Scientist.]

I would tell my programming students that programming is a "life skill."  It's the skill of following and giving instructions.  So no matter what career they ended up in, even if they never touched a computer again, what they learned here would be worthwhile.

If "career readiness" is the criterion, I think we could make the case that Computer Programming *should* replace Algebra for non-science majors.

[My wife in fond of the T-shirt that proclaims: "Yet another day has passed I didn't use algebra once!']

Back in the day, I believed programming was a better "exercise for the mind" than proving theorems in Plane Geometry, which is what my HS geometry course was.

When I was in grad school, FORTRAN could even substitute for the Foreign Language requirement (but not for CS majors!)

As for "qualified to teach": in our school district a HS teacher was fired because she was not "qualified" to teach the Pascal Programming Class.

The teacher only had an MS in Computer Science.  She would need a Business degree, because the district classified Computing as a Business course!

And the state SPI was too busy fighting political wars to do anything about it.

My main concern with the math curriculum is that it doesn't prepare average citizens to judge science or economic policy.

Citizens need to make informed decisions about things like Global Warming and Ecological Degradation and Government Budgets and National Debt 

and Threat Assessments of various kinds.  (Which is the biggest danger to our way of life: terrorism? drugs? global warming? the national debt?)

These involve quantitative reasoning for which the average college graduate, much less high school graduate, doesn't have a clue.

Instead of just "solving" formulas, we ought to be teaching "creating" formulas and building and validating quantitative models.

Call it math or call it computing or call it science, if we can't "predict", we can't "prepare" or "prevent".

Joe Austin

[Murray]

I agree with much of what you say, Ted!

@Kirby: I just had to weigh in on the Singapore comment. 30 years ago the best and brightest in Singapore chose engineering - these days they head for medicine, law, business and finance. See http://www.stjobs.sg/career-resources/personalities/engineering-losing-its-shine-in-singapore/a/199055

Computer science has also taken a dive and most institutions struggle to fill available places.

Considering Singapore is one of the most wired places in the World, math here is largely still taught very conventionally, and very much to meet exam requirements. 

I give a lot of talks on the issue of technology use in mathematics education here. I get one of 2 responses usually:

(a) Those who "get it" sigh and say it will never change because of the exam system; and

(b) The mathematics teachers say "What about the steps? How can we grade them if we can't see the steps?"

The latter case is really the same as Peter's comment - it's all too threatening for them.

Regards

Murray

[kirby urner]

On Fri, Apr 1, 2016 at 8:06 PM, Murray <murray...@gmail.com> wrote:

I agree with much of what you say, Ted!

@Kirby: I just had to weigh in on the Singapore comment. 30 years ago the best and brightest in Singapore chose engineering - these days they head for medicine, law, business and finance. See http://www.stjobs.sg/career-resources/personalities/engineering-losing-its-shine-in-singapore/a/199055

Thanks for you thoughts.

I have to agree with you, sure.  Singapore Math is actually something rather conventional, though sometimes lip service is paid to the importance of Information Technology. 

At least the rhetoric sounds more promising sometimes. 

I tend to use "Singapore" almost as a figure of speech for "technological Shangri-La".  I use similar science fiction language in my role of IT Clerk for a regional religious group (I write about "Singapore Friends" as really having their act together).

The way we map disciplines / professions / walks of life tends to get in the way sometimes.  Is this person "a farmer" or "an agricultural engineer"? 

A business may be so dependent on code as to require its people to think like software engineers (or computer scientists) even if that's not what they signed up for exactly.

 

Computer science has also taken a dive and most institutions struggle to fill available places.

In some countries the perception is that "computer science" as a degree program is not turning out the class of web developer most wanted in industry. 

There's a cultural disconnect or maybe its more a feedback loop.

Sometimes a degree or certificate program might partner with a private sector player, bringing in fresh ideas and rejuvenating the curriculum.
 

Considering Singapore is one of the most wired places in the World, math here is largely still taught very conventionally, and very much to meet exam requirements. 

I give a lot of talks on the issue of technology use in mathematics education here. I get one of 2 responses usually:

(a) Those who "get it" sigh and say it will never change because of the exam system; and

If you're OK with a certificate instead of a degree, or if your potential employer is, then we have more options to discuss. 

As I often mention, the code school I taught in for the last three to four years had no grades and no exams, but did require turning in completed projects (perhaps multiple times) and passing tests (eventually).

 

(b) The mathematics teachers say "What about the steps? How can we grade them if we can't see the steps?"

We get to watch them write and run code, then get to run it ourselves.  A line of code is a "step" in this other world.

In project-based learning, we still have lots of measurables.

How much emphasis do we place on students helping each other, versus mostly rewarding those who "only do their own work"?  That might vary by challenge.

 

The latter case is really the same as Peter's comment - it's all too threatening for them.

Regards

Murray

Newer institutions have a way of growing up around those that don't or won't change.

Thanks again, good discussion.

Kirby

[Maria Droujkova]

What can we take home from this discussion for the projects we can influence directly? 

Take the ideals we develop via (theoretical) discussions of high-level policies. Turn them into know-how for direct action within our own projects.

What will you do with these ideas - next month?

Cheers,
Dr. Maria Droujkova

NaturalMath.com

-- .- - ....

[Joseph Austin]

On Apr 2, 2016, at 9:33 AM, Maria Droujkova <drou...@gmail.com> wrote:

What can we take home from this discussion for the projects we can influence directly? 

Maria,

I think Kirby said it:

"Newer institutions have a way of growing up around those that don't or won't change."

Keep doing what you're doing, and eventually the rest of us will catch up!

Joe 

[Ted Kosan]

Maria wrote:

> What can we take home from this discussion for the projects we can
> influence directly?
>
> Take the ideals we develop via (theoretical) discussions of high-level
> policies. Turn them into know-how for direct action within our own projects.
>
> What will you do with these ideas - next month?

I think the way forward is implied by Alan Kay’s “The best way to
predict the future is to invent it.” quote which has been on the
MathFuture website from day one. Inventing something, even if it is
vastly superior to what currently exists, is usually not enough to
have the invention adopted. This truth is relayed by the following
quote by another one of the pioneers of computing:

“If your ideas are any good, you'll have to ram them down people's
throats.” Howard Aiken

The mathematics education establishment will never change unless it is
forced to. From the first day MathFuture was started it has been on an
inevitable course of eventually needing to bring about a change in the
way mathematics/computer science/symbolic logic is taught by “ramming
[the new educational ideas it came up with] down people’s throats.”
That day has now arrived.

The step-by-step equation solver I have been working on provides us
with a ram that is powerful enough to do the job. I have some
strategies in mind for how to use this ram effectively. The question I
have is are the people on this list willing to go up against the
mathematics education establishment? I definitely am.

Ted

[Maria Droujkova]

For my part, I am strongly dedicated to peace. I am not going to go against entities, or force things down people's throats without their prior consent. 

I think math ed, in particular, can do with more love and care, because it has plenty of harshness already. 

Having said that, inventing, building, and growing something of our own is very appealing. 

Ted, what can I do next month to care and nurture your equation solver in the context of mathematics?

Cheers,
Dr. Maria Droujkova

NaturalMath.com

-- .- - ....

[kirby urner]

On Sat, Apr 2, 2016 at 12:31 PM, Ted Kosan <ted....@gmail.com> wrote:

“If your ideas are any good, you'll have to ram them down people's
throats.” Howard Aiken

Echoing Maria's sentiments, "ramming" usually leads to even more entrenched resistance. 

In the Cold War Era, the fact of Sputnik and the prospect of "falling behind" was used as leverage by SMSG ("New Math"), or at least the atmosphere of "needing to get catch up" was strong. 

But then came the backlash.  Teachers resented the top-down imposition of so much alien material without proper training.  New Math would have to go.

Parents were frustrated too: not knowing how to help with junior's homework can be upsetting.  All this newfangled stuff about the unions and intersections of sets was not in their own background.

One force that keeps mathematics relatively unchanging is an unwillingness to accept future shock. 

Not just mathematics is kept straitjacketed by this fear.

When it comes to HTML / CSS and the Document Object Model (DOM), I think to myself "why aren't English teachers tackling this?" topic.  We're talking about documents after all.

A poet I knew, Gene Fowler, considered HTML just more punctuation, like semi-colons and quote marks.  Why is this content absent from language class?

Every gap or shortcoming we perceive suggests a possible competitive advantage, in a world where schools and curricula actually get to compete.

In circling SQL and web stuff, I'm trying to compensate for a lack of literacy across the board.  SQL is record-keeping, whereas web stuff is communicating.  Isn't school about both?

 

The mathematics education establishment will never change unless it is
forced to. From the first day MathFuture was started it has been on an
inevitable course of eventually needing to bring about a change in the
way mathematics/computer science/symbolic logic is taught by “ramming
[the new educational ideas it came up with] down people’s throats.”
That day has now arrived.

The step-by-step equation solver I have been working on provides us
with a ram that is powerful enough to do the job. I have some
strategies in mind for how to use this ram effectively. The question I
have is are the people on this list willing to go up against the
mathematics education establishment? I definitely am.

Ted

I'm definitely willing to toss my hat in the ring as a competitor and am always on the lookout for co-conspirators.

More about my oft stated agenda (fresh writing the BestThinking blog):

https://www.bestthinking.com/thinkers/kirbyurner?tab=blog&blogpostid=23723

Kirby

[Ted Kosan]

Maria wrote:

> For my part, I am strongly dedicated to peace. I am not going to go against
> entities, or force things down people's throats without their prior consent.

In primary school, secondary school, and post secondary school I hated
math because it did not make sense to me at all. My dream was to
become a computer scientist, but I was forced to drop out of the CS
degree I began in college because I could not do the math that was
required. I felt awful about myself for years because I thought I was
too stupid to understand math. Years later while I was in the midst of
figuring out how computer algebra systems worked, I discovered that
almost all math teachers have absolutely no clue how mathematics
actually works. I then realized that the reason I never grasped the
"math" I had been taught was because my teachers didn't grasp it. I
was not very happy about this discovery.

Over the past century the lives of millions of student's have been
ruined by so-called math teachers who were no better than witch
doctors. The sad thing is there were some mathematicians during this
time that understood quite well the brutal harm that math teachers
were doing to students, but these individuals stood by and let this
form of child abuse go on.

It turns out I am also a peaceful person, and I feel bad even when I
step on a bug accidentally. However, if I saw someone abusing a child,
I could not stand by and let it continue, and I would do anything in
my power to stop them.

I recently wrote the following short fairy tale to capture part of the
purpose I have in mind for MathPiper:

---------------------------- "Humpty Dumpty sat on a child on a wall"

(Humpty Dumpty is wearing a square academic cap and a gown which has
mathematical symbols on it. The child is wearing a dunce cap with “I
hate math!” written on it in red crayon. The child is moaning.
MathPiper looks similar to the Pied Piper of Hamelin.)

MathPiper: Why are you sitting on that child?

Humpty Dumpty: Because I am being paid to.

MathPiper: But, you are hurting the child. Can’t you hear them moaning?

Humpty Dumpty: (Looks down at the child and smiles) Moaning and
complaining are easily silenced with a little punishment. (Humpty
smacks the child’s hand with a thin rod, and the child stops moaning).

MathPiper: (Unsheathing a sword and pointing it at Humpty’s chest)
Free the child at once or I will smite you with this symbol sword.

Humpty Dumpty: (Laughs) Symbol shmimbol. I have been sitting on this
child for decades. Many have tried to topple me from my perch, but
none have succeeded. I doubt you will succeed in unseating me with
mere symbols.

----------------------------

Thankfully, the first step I have in mind does not involve ramming and
throats. This first step consists of simply explaining to computer
programmers who feel awful because they think they are bad a math,
that its not their fault.

> Having said that, inventing, building, and growing something of our own is
> very appealing.
>
> Ted, what can I do next month to care and nurture your equation solver in
> the context of mathematics?

You can start learning the MathPiper language which I will begin
teaching sometime next week.

Ted

[Ted Kosan]

Kirby wrote:

> Echoing Maria's sentiments, "ramming" usually leads to even more entrenched
> resistance.
>
> In the Cold War Era, the fact of Sputnik and the prospect of "falling
> behind" was used as leverage by SMSG ("New Math"), or at least the
> atmosphere of "needing to get catch up" was strong.
>
> But then came the backlash. Teachers resented the top-down imposition of so
> much alien material without proper training. New Math would have to go.

Intelligent Tutoring Systems and experiments like Khan Academy have
shown that technology now makes it possible to bypass conventional
teachers if needed. However, I am hopeful that a significant number of
math teachers will learn and then go on to teach the "New New Math" we
are creating because of its beauty, simplicity, and effectiveness.

> I'm definitely willing to toss my hat in the ring as a competitor and am
> always on the lookout for co-conspirators.

An interesting thing about this conspiracy is it will all be done in
the open with open source software.

Ted

[Maria Droujkova]

I'd like to make two points.

First, individual collaborators, even in positions of some authority (such as teachers or parents), are not causes of problems. Problems are inherent in systems. The systems are too complex to even have cause-effect scenarios. These grown-ups, if anything, are fellow victims of children, and also need help.

Second, it's a special challenge trying to stop a form of abuse that is perfectly legal under the current law of the land, and widespread. How do you go about that? 

Because of these two points, I like peaceful direct action that involves adults and children, and shows them doable, practical ways to live better. Not actions against, but actions for. 

Toward that end, learning MathPiper seems awesome. 

Will children be involved too?

-- 

Cheers,
Dr. Maria Droujkova

NaturalMath.com
-- .- - ....

[Peter Farrell]

On Saturday, April 2, 2016 at 6:45:33 PM UTC-7, tkosan wrote:

My dream was to
become a computer scientist, but I was forced to drop out of the CS
degree I began in college because I could not do the math that was
required. I felt awful about myself for years because I thought I was
too stupid to understand math.

I've met too many people with similar stories, and they were not trying to be computer scientists but businesspeople and forest rangers! Regardless, they were sent to the math department to study calculus as if they were going to be pure mathematicians and they failed. 

No other department could get away with this, could they? If a businessperson were to fail a requirement because they couldn't solder a circuit or remember what the Treaty of Ghent said, they'd sue the school. 

My experience trying to get a grown woman through a calculus course to get her MBA 6 years ago first inspired me to create an alternative. My course would be a corrective to the current 12-year program of unnecessarily abstract symbol-wrangling with no application in the real world. It wouldn't teach you everything you need to be a computer scientist, but it would get you past all the Greek stuff by using real-world tools and occasionally getting creative.

Peter

[Ted Kosan]

Maria wrote:

> [...] it's a special challenge trying to stop a form of abuse that is

> perfectly legal under the current law of the land, and widespread. How do
> you go about that?

My plan is to use a "emperor has no cloths" strategy to solve this
problem. It involves proving that the high-level authorities who are
in charge of forming mathematics education policy are incompetent.



> [...] learning MathPiper seems awesome.

>
> Will children be involved too?

It turns out children have been involved in the development of
MathPiper since its inception. I have had high school teachers teach
MathPiper classes on a regular basis so I could progressively modify
the MathPiper language to meet the needs of children, and in 2013
Michel Paul (from this Math Future group) had one of his classes
experiment with an early version of the step-by-step equation solver.
This is the feedback they had on it:

http://patternmatics.org/research/student_feedback/student_feedback_spring_2013.html

The conventional parts of MathPiper are designed to be similar in
complexity to Logo or BASIC, so in theory whatever the youngest age is
at which children have been able to learn Logo or BASIC would also be
the youngest age they could start learning MathPiper. The simpler
aspects of the expression tree parts of MathPiper, on the other hand,
should be able to be taught to even very young children.

Since I do not have direct access to children, it is difficult for me
to teach MathPiper to children directly. However, over the next month
or two if the people in this group who decide to learn MathPiper want
to experiment with teaching some of it to children, that would be
great. I would be especially interested to receive feedback on how
MathPiper's automatic program grading capabilities work with children.

Ted

[Ted Kosan]

Peter wrote:

>> My dream was to
>> become a computer scientist, but I was forced to drop out of the CS
>> degree I began in college because I could not do the math that was
>> required. I felt awful about myself for years because I thought I was
>> too stupid to understand math.
>
> I've met too many people with similar stories, and they were not trying to
> be computer scientists but businesspeople and forest rangers! Regardless,
> they were sent to the math department to study calculus as if they were
> going to be pure mathematicians and they failed.
>
> No other department could get away with this, could they? If a
> businessperson were to fail a requirement because they couldn't solder a
> circuit or remember what the Treaty of Ghent said, they'd sue the school.

It would be very useful if the Math Future group could start
collecting and publishing stories from people who had their careers or
self-image damaged by the current approach to teaching mathematics.
Maybe if more people were aware of the amount and seriousness of the
damage that is being caused by the current approach, utilizing the
more extreme "ramming" measures I have in mind might be unnecessary.

Ted

[Maria Droujkova]

On Sun, Apr 3, 2016 at 1:53 PM, Ted Kosan <ted....@gmail.com> wrote:

Since I do not have direct access to children, it is difficult for me
to teach MathPiper to children directly. However, over the next month
or two if the people in this group who decide to learn MathPiper want
to experiment with teaching some of it to children, that would be
great. I would be especially interested to receive feedback on how
MathPiper's automatic program grading capabilities work with children.

Awesome!

Would you like me to invite Natural Math families and math circles to join the fun?

Would you like a webinar room? We have one you can use, just need to coordinate schedules. This way you can watch people at it, talk, see faces.

[Ted Kosan]

Maria wrote:

> Would you like me to invite Natural Math families and math circles to join
> the fun?
>
> Would you like a webinar room? We have one you can use, just need to
> coordinate schedules. This way you can watch people at it, talk, see faces.

The MathPiper development environment is designed to collect data on
students as they create programs. For example, every time a student
runs a program, a copy of the program is saved into a log file so an
analysis can be done on the steps a student took to create the
program. This data is used to identify student misconceptions,
deficiencies in MathPiper's syntax, differences in the ways students
solve programming problems, deficiencies in teaching methods, etc.

I would like to hold off on teaching MathPiper to children in a formal
setting until some people on this list who have more experience
teaching children than I do have learned at least the conventional
parts of the MathPiper language. These people will then be in a better
position to determine ways to teach MathPiper to children that are
most likely to be effective pedagogically, along with being likely to
result in the collection of high-quality data.

Is there anyone in the group who has experience doing educational
research that involves children?

Ted

[Joseph Austin]

Ted,
What little I know about "experimenting" with kids is that you probably need a lawyer.
You will definitely need consent of the parents.
It's not something I would undertake lightly.
Just one more reason the status quo is the status quo.
Joe

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[Ted Kosan]

Joe wrote:

> What little I know about "experimenting" with kids is that you probably need a lawyer.
> You will definitely need consent of the parents.
> It's not something I would undertake lightly.

I agree. I read a book on FERPA regulations in order to prepare for
the time when MathPiper would be taught to children in a formal
setting. My understanding is not just statistics on students writing
programs is covered under FERPA, but the programs themselves are also
considered to be educational records:

"But what are considered "education records"? According to FERPA,
education records are "directly related to a student and maintained by
an educational agency or institution or by a party acting for the
agency or institution" and include "any information recorded in any
way, including, but not limited to, handwriting, print, computer
media, video or audio tape, film, microfilm, and microfiche." [5] This
does not include private notes intended for personal use, student
disciplinary, law enforcement or medical records, and certain student
employee records."
(http://www.dlib.org/dlib/january10/ramirez/01ramirez.html)

From the reading I did I determined that I would probably not be
directly teaching MathPiper to children in a formal setting. However,
MathPiper is being designed so that teachers can teach MathPiper to
students in a way which is FERPA compliant.

By the way, complying with FERPA regulations is one reason MathPiper
is not a server-based web application like SageMath and Wolfram Alpha
are. MathPiper worksheets are just plain text files that are edited on
a school's computer or on the student's computer. The data that is
collected while a student is working on a worksheet is placed into a
text-based log file which is stored in the same directory as the
worksheet is. These two files are then normally submitted to a teacher
using the school's course management system (like Blackboard or
Moodle) for grading. At no time are these files stored on a third
party server.

Ted

[Maria Droujkova]

This is the latest study we conducted at Natural Math: http://naturalmath.com/citizen-science-station/

Informed consent is important. Parents' in case of children.

Cheers,
Dr. Maria Droujkova

NaturalMath.com
-- .- - ....

[Julia Brodsky]

It would be very useful if the Math Future group could start
collecting and publishing stories from people who had their careers or
self-image damaged by the current approach to teaching mathematics.
Maybe if more people were aware of the amount and seriousness of the
damage that is being caused by the current approach, utilizing the
more extreme "ramming" measures I have in mind might be unnecessary.

Ted,

this could be quite an interesting anthology, if written well. I'd be happy to submit a piece.

You may want to reach out to people of different backgrounds, ethnic groups, social statuses, etc. to present a whole spectrum ( may be not only through Math Future, but Quora, etc). 

Best,

--

Julia Brodsky
www.artofinquiry.net

[Sue]

I'd contribute.

---

Date: Mon, 4 Apr 2016 09:25:02 -0400
Subject: Re: [Math Future] Should mathematics course requirements be satisfied by computer science courses?
From: in...@artofinquiry.net
To: mathf...@googlegroups.com

[michel paul]

The class in which I had students explore Ted's equation solver was a computational math class I was fortunate to be able to create and have going for a few years. We used Python, Sage, GeoGebra, and WolframAlpha as our primary tools for modeling mathematical ideas. It was a wonderful experience. Discussions on this list were immensely helpful.

For a long time I've promoted having students create expression trees for analyzing algebraic expressions. I saw that it was good to do for both mathematical and computer scientific reasons. Many times in Algebra classes I had students create expression trees and read them top down to translate 'infix' notation (standard algebraic expression) to 'prefix' notation (functional programming). Some students actually got intrigued, but others would complain that this wasn't what we were 'supposed' to be doing, so it was kind of a delicate balance. I was delighted when I finally could create a computational math class where this WAS what we were 'supposed' to be doing. I was delighted when I found that Ted Kosan had implemented this as an interactive equation solver in Math Piper. I was so glad to see this kind of stuff happening. We discussed it, and I used it in my computational class. Since the students had already had some experience of coding on their own so had first hand knowledge of the conceptual challenges it offers, and since early in the course we had worked with expression trees, they really were impressed with Ted's equation solver.

Unfortunately politics can be ugly and ruthless, and my course no longer exists. Skipping details, a little over a year ago I experienced what it means to hit the wall, and I am now in early retirement. Reflecting over the last year, I can see that I have been going through a grieving process.

The entrenched stupidity that is able to comprehend programming only as a 'tool' that mathematicians might 'use', that it is not an intrinsic mathematical activity in itself, that sure, we can 'use' it, but we shouldn't confuse it with 'math' makes me so deeply furious that I'm brought to tears whenever I now try to write about this. However, that's an improvement. Earlier, I couldn't even write. For years I kept weathering the stupidity thinking that it would change, but it only got worse.

That is why I felt motivated to post the disappointing NCTM link that began this thread. I'm encouraged by the discussions happening here. Though I no longer am in a classroom, at least for now, I do not know how the future will unfold, I'm still deeply interested in these things.

Sincerely,

Michel

Ted

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[kirby urner]

Hey Michel, this is all news to me about how your flagship Computational Math came under attack and went down in flames with all hands.  Metaphorically speaking. 

I treasure memories of our crossing paths at the Chicago EduSummit at Pycon.  https://flic.kr/p/e3c1BX

I've not seen you on edu-sig in awhile.  That tends to be about sharing source code.  I'm one of the listowners these days.  We talk about functional programming and what is code school etc.  Feel free to join us.

You've been a leading avatar in this cultural debate over what constitutes passing the torch to our children, collectively, as a society.  Said debate is of course a large number of debates, many of them involving religion, however your views and innovative curriculum fall into that smaller circle where "mathematics" and "computer science" get used as core concepts, and advancing waves of... what to call it, the future?.... make us deal with the need for change.

I just want to throw it out there that we shouldn't give up on the adults in this picture, by which I mean, so much of the experimental / innovative / Wolframish computational stuff you were cramming into a high school framework (where it belongs) is of huge interest to men and women long out of high school in some cases, who want interesting work, even if employed currently -- and maybe even more interested if "retired" or whatever. 

People are living longer than ever, and sustaining any highly technological society requires a willingness to learn new skills and retrain, and so many many adults are in the mode (and the mood) to get on with it and buckle down, but where are the opportunities? 

You've been honing your skills, maybe you just need to shift your target demographic?

One of my stated objectives in working with PDX Code Guild here in Portland is to help PSU open up more opportunities for Oregon math teachers at all levels, to get "code school" to count as "professional development", in turn a required feature along many career paths. 

I'm not talking about a twelve week "boot camp" necessarily, so much as something custom-formatted. 

However nor am I necessarily talking about some leisure "summer camp" either, as the time pressure is enormous, so evening classes, even middle of the day workday classes, with teachers offered van trips from remote areas (to a more local code school lets imagine, not necessarily ours on Corbett) to these better equipped training facilities. 

Alternatively, might we send a caravan of "training vans" to the host school, with sponsor decals?  I've storyboarded such scenarios in my journals and know there's historical precedent, so I know it's not either/or.  Right now it's more like neither (the schools are under-served).

So... the question is how far have I've gotten, with either the above, or that property near Fossil, Oregon, a place to prototype outdoor GPS scouting / boot camp / GIS computer school technology for different groups -- picture a "retreat center" with lots of turnover around a more slowly turning over core staff? 

Hmmm.... that's a difficult computation, perhaps NP hard or harder.  http://mathworld.wolfram.com/NP-HardProblem.html

I mention these two projects (professional development for teachers, code school in the wilderness) in the same breath because I've mentioned the Fossil one here on mathfuture before, should be somewhere in the archives, if memory serves. 

I'm still committed to that vision, even if Brian Sharp (one of those "Wanderers" -- that group I meet with in the Linus Pauling House) is currently off my radar.  He's the owner of said property.  A Yorkshireman, like my friend Steve Holden (former Python Software Foundation chairman).

Brian was an avid soccer player in his youth.  I've known him in his later years, a "deep ecologist" (by training as well as temperament), always deeply concerned with the planet's health, an expert in birds in particular, and also a musician.

Anyway, I see our strategy as two-pronged: 

(A) reach out directly to students of all ages through podcasts and Youtubes, not involving or implicating administrators, combined with

(B) working directly with administrators on plans and programs to enhance career advancement opportunities for already-employed adult teachers, and not only of mathematics or STEM [I see a looming debate as to whether Language Arts and Communications should take over a lot of the Document Object Model stuff, as an extension of Rhetoric and Grammar]. 

In Portland, we have a fascination with Finland and how they pass the torch there.  Our teachers fly over there and study, even do internships.  I myself flew to London to advise a group planning South Africa's evolution (they had code schools branded "Tux Labs" -- not sure of the picture today). 

The consequences of future shock are global and are being coped with in different ways around the world, including in our focus area, where torch passing happens.

Not surprisingly, then, I've been looking at code school experiments outside the US as well, most recently advising a network in Vietnam regarding plans that mirror the two prongs described, in terms of building up freely accessible libraries of computational communications materials (JavaScript, HTML, SQL), but then also mediating through classroom instructors by negotiating with academic institutions to share space and cross-fertilize in creative ways. 

Code schools need not fight the existing ecosystem so much as find the right API (interface, control panel), which has to be designed (crafted).  Communications channels don't just happen by accident. The emerging matrix seems to be a "business accelerator" program that runs in conjunction with a code school, because we associate technology cultures with "startups" and "ventures". 

When governments want to "stimulate" a local economy and/or find new work for people, this matrix gets a lot of focus, with good reason I think.

In sum, I hope we'll be able to enlist your skills and support as we move towards focusing on adult learners who in turn may already serve as faculty in a school setting. 

I'm trying to get my head around our direct outreach assets in a more coherent way.  I'm not sure "outreach" is the best word either.  In some respects it's plain old advertising in the sense of playing up the advantages of learning to code. 

You've probably seen some of the genre:  https://youtu.be/STRPsW6IY8k

Ted's about lame teaching preventing his getting on with the subjects that really interested him reminds me a lot of Stephen Wolfram's testimony in that address to the audience in Mumbai that I cited recently.  Stephen confided he wasn't all that efficient at the symbolic noodling and that was slowing him down, yet so much of the computational process was purely mechanical, begging the question why not let a machine do those parts that one could -- leaving plenty of room for "the non-computational" aspects as Roger Penrose readers might call it.

Getting a plus or minus sign wrong or something else way back in the steps could derail the whole thing yet take forever to debug (I'm now paraphrasing Wolfram, plus speaking from my own experience). 

A script or computer program is like a perfect studio recording of a piece of music, where you've got no mistakes at the end of the process, and where you've deliberately left the door open to different inputs to a deterministic piece -- one gets to piggy-back on (capitalize on) previous efforts to get it right, and moreover frozen in that form for others to enjoy and benefit from, as something recorded and reusable. 

Figuring out How to Solve It is an important step, but then Remembering How to Solve It may be just as critical, and "getting it done in time" may not wait.

Figure out a generic way to pipe the output of one such algorithm to the input of another and you have the basis for something like MathPiper or in Stephen's case Mathematica, a tool for taking the drudgery out of complicated routine processes that are easy to get wrong (what automation is good for).

I know in my own research I've depended a lot on convex hull finding algorithms (input a dispersion of points, get back a convex graph internalizing as many of those points as possible while using a set to make the system itself) -- just to underline how much we rely on shared tools.

I look forward to further co-venturing.  I hope we get to meet in person again.

Kirby

[Ted Kosan]

Michel wrote:

> Unfortunately politics can be ugly and ruthless, and my course no longer
> exists. Skipping details, a little over a year ago I experienced what it
> means to hit the wall, and I am now in early retirement. Reflecting over the
> last year, I can see that I have been going through a grieving process.
>
> The entrenched stupidity that is able to comprehend programming only as a
> 'tool' that mathematicians might 'use', that it is not an intrinsic
> mathematical activity in itself, that sure, we can 'use' it, but we
> shouldn't confuse it with 'math' makes me so deeply furious that I'm brought
> to tears whenever I now try to write about this. However, that's an
> improvement. Earlier, I couldn't even write. For years I kept weathering the
> stupidity thinking that it would change, but it only got worse.

After I read this earlier today, I felt like someone had just punched
me in the gut.

What this story brings to mind is the following painting by
Loutherbourg which some people describe as "aspiration outstripping
convention":

http://www.tate.org.uk/art/images/work/T/T01/T01138_10.jpg

It appears that the conventional thinkers at your school pulled you
off your horse.

> Though I no longer am in a classroom, at least for now, I do not
> know how the future will unfold, I'm still deeply interested in these things.

When I read this, the following quote came to mind:

"When one door closes another door opens; but we so often look so long
and so regretfully upon the closed door, that we do not see the ones
which open for us." Alexander Graham Bell

One of the main reasons that I have selected computer programmers who
think they are bad a math as the first large group to teach the "new
new math" we have been creating on this list to is because these
programmers will then want their children to be taught this
information. Since you are one of the pioneers of this new "teach math
and programming together" approach, I think it will be easy to
convince these programmers to have you teach their children math and
programming.

Ted

[Maria Droujkova]

On Mon, Apr 4, 2016 at 11:30 AM, michel paul <python...@gmail.com> wrote:

Unfortunately politics can be ugly and ruthless, and my course no longer exists. Skipping details, a little over a year ago I experienced what it means to hit the wall, and I am now in early retirement. Reflecting over the last year, I can see that I have been going through a grieving process.

The entrenched stupidity that is able to comprehend programming only as a 'tool' that mathematicians might 'use', that it is not an intrinsic mathematical activity in itself, that sure, we can 'use' it, but we shouldn't confuse it with 'math' makes me so deeply furious that I'm brought to tears whenever I now try to write about this. However, that's an improvement. Earlier, I couldn't even write. For years I kept weathering the stupidity thinking that it would change, but it only got worse.

Michel Paul, hugs to you!

How harsh. At least you can talk about it now, and there are people who'd listen and understand. 

One more story that demonstrates why interfacing with other people's systems is unreliable by itself, and we need to build our own. 

Toward that end, what are your thoughts about live independent courses, online?

[Joseph Austin]

On Apr 4, 2016, at 11:30 AM, michel paul <python...@gmail.com> wrote:

The entrenched stupidity that is able to comprehend programming only as a 'tool' that mathematicians might 'use', that it is not an intrinsic mathematical activity in itself, that sure, we can 'use' it, but we shouldn't confuse it with 'math' 

Perhaps we need a change in perspective.  Instead of focusing on "what" is math, or CS, shouldn't we instead consider "why" is math?

Once upon a time math was a "quantitative model" for commerce: 

instead of counting oxen or plots of land, we could count pebbles, then marks on a tablet.

In time, our models have become more far-reaching, and complex.  We model the courses of heavenly bodies, natural and artificial; the transformation of power, first in heat engines, now in thermonuclear reactions; the reflection and refraction of light, and now lasers.

Today we attempt to model global economies, the weather, the evolution of the universe, the biochemistry of disease and of the biosphere, the shifting sentiments of the body politic.

I think any fair-minded person would concede that the complexity of our models has far outstripped the advances of classical, formulaic mathematics.

And even the sophistication of that math has far outstripped our ability to teach it to a sufficient fraction of our population in a reasonable time.

Meanwhile, we have computing machines and "mathematical" methods of numerical approximation that enable accurate computation of immensely more complex and realistic models. And the ability to "do the math", that is, do the programming, for such models is demonstrably achievable by even elementary-age students.  The modeling itself, of course, requires skill and training in "science"; but on the other hand we can't do modern science without modern models, and in our age, that means computers.

So suppose we adopt a new vocabulary, a vocabulary focused on the objective instead of the method or the tool,

and call our discipline something like "Quantitative Modeling", leaving open both what we model and what we model with,

but definitely including not only the techniques of model-building but also the art of understanding and imitating the reality we undertake to model.

The classical liberal arts had as their foundation Grammar, Logic, and Rhetoric. It seems to me we have been losing the "rhetoric" and even "logic" analogues in math education, focusing almost exclusively on the mechanics and "rules" and ignoring the practical applications to human affairs.  "Math" must evolve to serve the needs of society, not vice versa.

Joe

[Peter Farrell]

On Tuesday, April 5, 2016 at 6:35:56 AM UTC-7, Joseph Austin wrote:

<snip>

 

And the ability to "do the math", that is, do the programming, for such models is demonstrably achievable by even elementary-age students.  The modeling itself, of course, requires skill and training in "science"; but on the other hand we can't do modern science without modern models, and in our age, that means computers.

Brilliantly put, Joe. 

[michel paul]

> what are your thoughts about live independent courses, online?

At the moment my thoughts are, yeah, that is definitely the world we are in now, and there are all kinds of ways to communicate. That's encouraging. It's just a matter of finding the right channels.

One thing I've learned so far - retirement will NOT be boring!

I've been thinking that a useful project might be to catalog the ways in which learning how to program in languages like Scheme and Python showed me how my own mathematical understanding was boxed in. This might help others. I knew that there were all kinds of things in higher levels of math that I had not studied, but learning a little bit about contemporary programming forced me to reconsider aspects of ordinary schoolish math that I had long taken for granted.

Not only that, it showed me how our whole math curriculum is conceptually unclear. Self proclaimed 'traditional' math teachers (whatever the hell that really means) like to pretend that they are intellectually rigorous, but they will run away as soon as you try to discuss things like the ambiguous ways we treat the '=' symbol. When you question the 'obvious', people think you're crazy. On this list I was able to have a wonderful discussion regarding that, and this is one of the most important points I want to make regarding the conceptual confusion of our math curriculum --> both students and teachers are at least a little bit confused about what '=' really means. Lots and lots of students think it means 'find the answer'. They even think that when they are in higher level math classes. They think the left hand side of an equation is the 'problem' and that the right hand side is the 'answer'. I suspected they retained a sense of that from elementary school, and yep, lots of them really do have that sense floating around in their background, even in higher level courses, and it causes all kinds of unnecessary confusion. When you program and have to understand the difference between '=', '==', and even '===', it really can raise some important mathematical ideas. It's not just shop talk about computers.

I'm truly grateful this list has existed. 

- Michel

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[Andrius Kulikauskas]

Joe,

I very much like your focus on "why" and your many beautiful examples.

They bring to mind a few more reasons for "why" we have math:
* As the basis for a caste system based on how much math you have
passed. A way of controlling who is in what profession, for example.
* As a way of treating people differently (through rates, credit scores,
incomes, etc.) without them having full knowledge or even understanding
what its all about.
* As a way of making our systems just incomprehensible enough to most
people so that they can't argue with them. For example, most people
think that banks loan out money based on the deposits they have. But
actually, the central banking system and participating banks are
chartered by the government to create loans in an amount ten times or
more than whatever assets a bank has; but nobody creates the money
needed to pay the interest on those loans, which grows exponentially;
which might be all right if the economy itself grew exponentially; but
we have thereby legislated the need to grow exponentially, naturally or
(when that fails) otherwise; thus the pressure to (artificially)
monetize everything in sight; and to prey on the most vulnerable (a
major reason why ghettos persist, I think). So that bubbles (based on
money for money's sake) are inevitable. Similarly, the recent housing
crisis was an application of math.
* Math also lets us model realities in ways that let us suspend thinking
about the underlying meaning. Which is essential for modern warfare.

Computers (and all systems) likewise allow us to ignore the underlying
meaning. Social software is in a large part a way to avoid human
contact by controlling it in very rigid channels.

Joe, my examples are negative, but I think the positive side would be
math for citizenship.

I suppose that a distinction can be made between what must be
taught-learned and what should be optional. I once thought that what
really need to be taught is ethics, what is right and wrong. For
example, language should be taught as a way of empathizing with others
and ourselves, of caring about them. Math should be taught as the study
of systems, especially the systems that we find ourselves in. It's
morally essential for citizens in our modern world to distinguish
between linear, exponential and periodic behavior and appreciate the
implications. Overall, I imagine having a required school of just maybe
two hours a day but that focused only on what is agreed to be absolutely
essential. Which I think would include drill of "math facts"
(multiplication tables, etc.) And most adults as well would be required
to regularly show competence. Then the rest of education would all be
optional.

About myself: I participated on this list about five years ago. I'm in
Lithuania now. I will reintroduce myself but I was moved by Joe's
letter and I could not keep from writing my own thoughts. I'm glad to
see this list so lively with participants from before and I think new
ones as well.

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

2016.04.05 16:35, Joseph Austin rašė:
>
>> On Apr 4, 2016, at 11:30 AM, michel paul <python...@gmail.com

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[kirby urner]

I agree with Joe (below) that mathematical modeling has gone beyond closed form differential equations based simulations, to modeling more like SimCity or Sims.  Jay Forrester and Club of Rome were turning points (1970s), leading to products like Stella and PowerSim.

http://www.iseesystems.com/softwares/Education/StellaSoftware.aspx
http://www.powersim.com/

We still have cause and effect, and bump ups here still result in bump downs or ups over there of some commensurate magnitude, i.e. we still have correlations and delta-x / delta-y ratios.  It's still the same can of soup in a lot of ways, that we learn about in calculus, wherein cycles drive other cycles, a wiring together of differences (deltas, delta calculus).

Picture a computer game board as "a planet" (hexagons + 12 pentagons for tiling, many millions of tiles, like pixels in some ways). 

http://vickijoel.org/hexplanet/

Each planet comes with "themed" optimization puzzles where you're trying to get electrical power and food to billions, non-humans as well as humans as an ecosystem is way more than just us.

At a young impressionable age is when we want healthy skepticism to stick (different from cynicism) in the sense that models may always be re-invented and need not be bowed down to as ultimate authorities, always a danger when people get lazy and forget how to make new models, finding it less work to just worship the old. 

Many older models work well and I'm not making the elementary mistake of thinking "just because it's newer, it has to be better". 

I just want STEM / Math to instill the confidence to "make it your own" as Maria puts it.  Or as my old boss Scott Gray would say, it's about learning to "make math" versus being just a passive consumer thereof.  Ditto with video:  if you watch it a lot, then also learn to make it. 

Ditto any media you rely on heavily?

Final thought for this post:  MVC i.e. Model View Controller. 

This is "code school speak" for having

(M) a database in the background, a rule-driving black box, that simulates a planet, an ecosystem, a business or whatever. 

(V)  the View consists of visualizations and sonifications (aural / audio track), other sensory feedback through instruments, about what's happening with the Model. 

(C) the controller is what gets in between, some glue language that allows the Viewer to manipulate / update / interact with, the Model. 

MVC architecture is about architecting these components as distinct / separate entities for the purposes of better organization and maintainability.

Kirby

[Peter Farrell]

More excellent points by Michel:

"Self proclaimed 'traditional' math teachers (whatever the hell that really means) like to pretend that they are intellectually rigorous..."

Whenever I hear the word "rigorous" I feel like I've been bitten by a vampire.

And Andrius:

" ...a caste system based on how much math you have 

passed."

John Taylor Gatto made it plain in The Underground History of American Education that compulsory schooling was designed and implemented to dumb us down and create a unified and easily governed populace.

I like Andrius' point about our institutions systematizing human contact to avoid its messiness. 

And Kirby points out an issue we techy types have to deal with: the active "math maker" vs. the passive consumer. Minecraft is a great motivator (I made something cool and mathy with it using Python today) but it's a challenge to keep the kids from just playing Minecraft. I showed a kid some things (a house, a city) I'd made as an example of what he can generate procedurally using code and he said, "That's not that great."

I don't know what it's like in Lithuania, but I have a feeling the students at the community school I taught at in Kenya years ago would die for some Raspberry Pi's and a little code coaching. 

Peter

[Ted Kosan]

Andrius wrote:

> They bring to mind a few more reasons for "why" we have math:
> * As the basis for a caste system based on how much math you have passed. A
> way of controlling who is in what profession, for example.
> * As a way of treating people differently (through rates, credit scores,
> incomes, etc.) without them having full knowledge or even understanding what
> its all about.
> * As a way of making our systems just incomprehensible enough to most people
> so that they can't argue with them. For example, most people think that
> banks loan out money based on the deposits they have. But actually, the
> central banking system and participating banks are chartered by the
> government to create loans in an amount ten times or more than whatever
> assets a bank has; but nobody creates the money needed to pay the interest
> on those loans, which grows exponentially; which might be all right if the
> economy itself grew exponentially; but we have thereby legislated the need
> to grow exponentially, naturally or (when that fails) otherwise; thus the
> pressure to (artificially) monetize everything in sight; and to prey on the
> most vulnerable (a major reason why ghettos persist, I think). So that
> bubbles (based on money for money's sake) are inevitable. Similarly, the
> recent housing crisis was an application of math.
> * Math also lets us model realities in ways that let us suspend thinking
> about the underlying meaning. Which is essential for modern warfare.

For years I have thought there must be reasons for why mathematics
education is (the seemingly crazy) way it is, but I was unable to
clearly identify them. The reasons you posted certainly get to the
core of the problem.

Ted

[Ted Kosan]

Kirby wrote:

> Or as my old boss Scott Gray would say, it's about learning
> to "make math" versus being just a passive consumer thereof.

Do you think Scott could be persuaded to join the Math Future group? I
think his input on the topics we are discussing would be very
valuable.

Ted

[kirby urner]

That's an excellent idea Ted.  I'd love to reconnect with Scott.  We've been out of touch.  I just Google up this bread crumb -- I have a trail to follow now.

http://www.foundersspace.com/mentors/scott-gray-founders-space-instructor/

Kirby

[Joseph Austin]

On Apr 5, 2016, at 11:50 AM, Andrius Kulikauskas <m...@ms.lt> wrote:

the central banking system and participating banks are chartered by the government to create loans in an amount ten times or more than whatever assets a bank has; but nobody creates the money needed to pay the interest on those loans, which grows exponentially; which might be all right if the economy itself grew exponentially;

I think examples such as this illustrate the value of models, and computer tools to solve them, vs. the traditional closed-form formulaic approach.

The formula for compound interest with regular payments, critical to all "savings and loan" applications, is messy to deal with, involving high-order powers and roots requiring exponentials and logs.  But compound interest can be "modeled" quite easily in  a spreadsheet with basic add subtract multiply divide and a simple recurrence relation.

And using the solver feature, it's easy to solve for time to pay off at a fixed payment or for payment needed for a fixed term, with total interest cost along with that.

From my experience, the standard curriculum spends maybe one day on compound interest compared with weeks on polynomials.  But which is more relevant to daily life? And an informed citizenry?  For example, a simple numerical model of "farmers and bankers"  for your example would show that, eventually, the bankers could end up with all the money without ever having plowed a single furrow.

Suppose instead of teaching "pre-Algebra" we taught "modeling with spreadsheets."  This would introduce "variables" (aka cells and cell names) and also  recurrence formulas, a topic essential to the foundations of mathematics (induction) but largely ignored in the traditional algebraic curriculum.

Further, one could discuss "binary search" methods of numerical solution by way of introducing "solver". And of course, emphasize the "morphism" between algebraic formulas and the physical relationships among the physical quantities represented by the formulas.

In this manner, by the time the students were introduced to axiomatic transformations of symbolic formulas, that is, conventional algebra, they would already understand what formulas represented and why solutions were useful, and also perhaps appreciate the convenience of expressing the formula in various forms.  They might also realize that the lack of an easy "closed form" solution is not necessarily an impediment to getting an arbitrarily accurate numerical solution.  Indeed, the penchant for closed-form solutions leads to 

making do with exact solutions to poor models of the real problems 

instead of close-enough solutions to good models of the real problems.

Joe

[Murray]

Then the rest of education would all be optional. 

In fact, that is largely the reality now, but not in the constructive way you mean.

Students now learn math (and other things) for examinations only to immediately forget it afterwards because they were not helped to see the relevance to anything real, and because much of it is tedious mechanical algebraic operations that should be done by computers.

What they really learn (for keeps) is the result of their activities outside of school (hobbies, sports, computer games, Facebook, whatever) since they feel some ownership for those activities.

One of the best assignments I ever designed required the students to investigate a math topic (something we hadn't done in class yet) by themselves, and then use a computer algebra system to solve a typical problem in that topic. (It was a kind of "flipped" learning before that had become a buzzword.)

Each student had to choose a unique topic (ensuring some ownership and individual "expertise"). Each time I conducted this assessment, I was impressed by the way students would begin on it immediately and I would get emails on the first evening from students keenly claiming their topic.

Students could submit as many drafts as they liked and there was really fruitful discussion with them as they came to grips with what their topic meant, and how to use the software to solve their chosen problem. They were also encouraged to be creative in their presentation.

Many students reported it was the best assignment they ever had to do. From my point of view, it was the most enjoyable to administer (I never heard any "why do we have to do this?" complaints), and even enjoyable to grade (since all were different, and I could see plenty of effort had gone into them).

So I like the idea where students conduct their own investigations (of problems within "systems", as you suggest) because of the possibilities for ownership, appropriate use of technology tools, and creativity - things which are lacking in too many math courses.

I agree with a lot of your sentiments, Andrius.

Regards

Murray

[Peter Farrell]

On Friday, April 8, 2016 at 7:06:37 PM UTC-7, Murray wrote:

So I like the idea where students conduct their own investigations (of problems within "systems", as you suggest) because of the possibilities for ownership, appropriate use of technology tools, and creativity - things which are lacking in too many math courses.

Terrific post! I agree wholeheartedly. Very inventive exploration, too.

Peter 

[Andrius Kulikauskas]

Joe,

I very much like your idea of using spreadsheets to model different
"savings and loan" arrangements.

I think that playing around with numbers is essential to building
intuition. And spreadsheets seem like an excellent way to do that,
especially for that kind of problem. And a very natural one that leads
directly into work skills.

I think a key thing to learn is the different kinds of families of
functions and their qualitative differences, such as linear vs.
exponential.

If math is a study of systems, then that includes especially realizing
when systems or models break down, when they apply and when they don't
apply.

Among my favorite exponential modeling problems are:
* Given the current population and current growth rate work backwards to
figure out when the first person appeared ("Adam"), the second person
("Eve") and so on.
* Native Americans in Manhattan (NY) are said to have sold the island
for $24 in 1624. Supposing they invested that wisely (model different
rates), what would their investment be worth today? Could they buy
Manhattan back?
* There is an old classic story about the inventors of checkers and
chess in India. The king offered them to name a gift they would like to
receive for each of 64 days, corresponding to the squares on the board.
The inventor of checkers wanted 1,000 gold coins per day. The inventor
of chess wanted one grain of rice on the first day, but the amount
doubled every following day. How much rice would that be on the last day?

Andrius

Andrius Kulikauskas
m...@ms.lt
+370 607 27 665

2016.04.07 17:13, Joseph Austin rašė:
>
>> On Apr 5, 2016, at 11:50 AM, Andrius Kulikauskas <m...@ms.lt

[kirby urner]

Big thumbs up for spreadsheets.

I would caution against "only spreadsheets" though.

Analogy:  spreadsheets are open plain, wheat fields forever, whereas databases are craggy mountains.  We wish to navigate in both terrain types.

Kirby

[Joseph Austin]

You can create basic relational databases in Excel.

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[kirby urner]

On Wed, Apr 20, 2016 at 7:01 PM, Joseph Austin <drtec...@gmail.com> wrote:

You can create basic relational databases in Excel.

You can....

But SQLite is free. 

People who don't think about license fees as much assume Microsoft Office is a given.  I never do.  We're lucky if we have it but not every student is so privileged.

I call it GNU Math (sometimes) for a reason.

Kirby

[Joseph Austin]

Good point.  Of course, computers themselves are not free.

I'd guess the $50 Amazon Fire tablet is one of the cheapest systems widely available. 

What can run on that?   Javascript, yes?  Java, no?  GeoGebra, no?

I know Minecraft does, but I'm not young enough to know whether that is a full-function programming language!

RPi is supposedly cheaper, but by the time you add a keyboard, screen, etc. I'm not so sure.

Joe

[kirby urner]

Don't forget the power of the browser to turn your tablet into a window to something on the server's side that has all these capabilities.

Even if I can't easily access the bash shell to the tablet's file system, if the device supports a browser then I may create such a shell account in the cloud.

To that end, a Raspberry Pi will work, but doesn't come with screen and keyboard.

What schools might provide are ubiquitous cubicles or pods insider of which you may reconnect to your cloud accounts and get on with your projects.

The Chromebook is what many schools are standardizing on here in the Pacific Northwest.  Smart choice.

Kirby

[Ted Kosan]

Kirby wrote,

> What schools might provide are ubiquitous cubicles or pods insider of which
> you may reconnect to your cloud accounts and get on with your projects.

The network is the school, so there is no need to gather large numbers
of students into large buildings anymore to access it.

Ted

[kirby urner]

Right, the needs are shifting.

We need to be honest about a core purpose schools serve:  both parents
(typically two, often one, I'm profiling i.e. stereotyping) need to work out of
the house and don't have a way to keep the juniors supervised.  Schools
are a way to outsource parental supervision, so just saying "the school
is the network" may not solve that problem.

The Nexus Academy model adopted by Michigan may be an aspect of

the solution.  Lets change the name to Node Academy just so I can speak

generally and weave some science fiction, without implying I'm actually

describing today's world. 

In Node Academy, the subscribers actually do get a lot of supervised
time, and parents are comfortable with their kids there, but to the extent
trusted home use is an option, the supervisory framework may extend,
with a home becoming a recognized node in Node Academy.

Just to do "extreme case manga" (storyboarding), you'll have street youth

in Portland living in tents and getting nutrition from Food Not Bombs,

which siphons excess and redirects it to those in need.  Such a youth

goes to school with no Internet at home.  The school is the only place

giving unfettered access. 

But then the school is a multi-story affair, between floor 9 and rooftop (20). 
Floors below are private offices but with some of the companies helping
sponsor the school floors above them.  Long story.

Every kid has a private cubicle, more than simply a locker, a place to chill
and choose which movies to watch, which readings to read, off a list of
homework assignments, but this is homework done in a school setting. 
This is Node Academy.

So back to my scenario, you have a houseless refugee youth living in
a tent, going to school to study, and having not much Internet back at
camp.  School also provides an atmosphere of conviviality, a peer group
consisting of people from many walks of life, and most important,

a supervised environment with an enforced code of conduct. 

One might lose one's cubicle privileges, or one's place in line for the

better ones.  One feels held back if not allowed to compete by being

a good collaborator in this school setting, this Node Academy.

That being said, maybe three days a week the average student stays
at home or simply walks to the home of a nearby neighbor (might be
rotating duties), guardians saying that's fine, maybe there's a trusted
guardian at home too.  Maybe other students come over and make a
peer group happen there.  Node Academy has ways to expand into
this home schooler market in an organic manner.  Word of mouth is

critical as people are more likely to adopt new behaviors in imitation

of people they already trust.

Where are the teachers in this picture? 

Back to Nexus Academy, they need not be in the building, in the nodes. 
They may be in teacher pods elsewhere, in different buildings.  Yet
their interactions with students may be in real time, not just asynchronous.

In-person classroom meetups still happen in this picture, but are not

the core activity associating with school.  There's a more one-on-one

flavor to the cubicle-based Ux (might be hexagon-shaped).

So you see where I'm going with this.  As long as the guardians are

satisfied there's security for their loved ones, they'll give their consent.

However security and working hard to get on with one's studies are

not mutually incompatible goals.  On the contrary, by having the

provision of a safe personal workspaces a goal, we're recommitting
to fostering a secure academic setting.  Everyone is free to study. 
Colleges and universities are seeking to foster this same atmosphere.

Kirby

[Ted Kosan]

Kirby wrote:

> We need to be honest about a core purpose schools serve: both parents
> (typically two, often one, I'm profiling i.e. stereotyping) need to work out
> of the house and don't have a way to keep the juniors supervised. Schools
> are a way to outsource parental supervision, so just saying "the school
> is the network" may not solve that problem.

The network is the supervisor. The network is able to supervise
children much more effectively than humans can, and this ability is
increasing exponentially. For example, here is a technology that
tracks everyone in a city using a single drone that has a very high
resolution camera:

http://www.radiolab.org/story/eye-sky/

> Where are the teachers in this picture?

Every student will be taught by a HAL-9000 like ITS which will be made
available them when they are born. These ITSs will be like the Greek
tutor slaves that wealthy Romans had.

I think this style of education is the one the world is quickly moving
towards. To me the real question is who is going to be in control the
ITSs?

Ted

[kirby urner]

On Mon, May 2, 2016 at 2:30 PM, Ted Kosan <ted....@gmail.com> wrote:

Kirby wrote:

> We need to be honest about a core purpose schools serve:  both parents
> (typically two, often one, I'm profiling i.e. stereotyping) need to work out
> of the house and don't have a way to keep the juniors supervised.  Schools
> are a way to outsource parental supervision, so just saying "the school
> is the network" may not solve that problem.

The network is the supervisor. The network is able to supervise
children much more effectively than humans can, and this ability is
increasing exponentially. For example, here is a technology that
tracks everyone in a city using a single drone that has a very high
resolution camera:

http://www.radiolab.org/story/eye-sky/

To me that sounds too much like reliance on AI breakthroughs that

might be just around the corner.  For me, a litmus test is "did we have

the technology ten years ago?"  The solutions I'm proposing fit that

criterion.

From a longer article I'm currently writing for CERM Academy:

"What would the ideal Code School look like?" -- that's a
question I frequently ask myself.  I start with what's true today
and think in terms of realistic deltas, developments that are
not out of reach.  I don't bank on some miracle deus ex
machina solutions that require some big breakthrough in
Deep Learning or AI.

 

> Where are the teachers in this picture?

Every student will be taught by a HAL-9000 like ITS which will be made
available them when they are born. These ITSs will be like the Greek
tutor slaves that wealthy Romans had.

You're writing what I call "far future science fiction" whereas "near future"

so so close as to be unrecognizable from today.  It's a spectrum, not

invalidating any particular frequency on this radio dial.  We also have the

Book of Revelation crowd.  There's plenty of room.

 

I think this style of education is the one the world is quickly moving
towards. To me the real question is who is going to be in control the
ITSs?

Ted

Who's to say anyone is going to control it?  By the sound of it, ITSs

will control us.  To some degree that's already true:  we're somewhat the

puppets of systems thought out for us by our ancestors, from freeways

to fast food, from television to billboards.  None of us really "control"

the environment, which as you say, is self-supervising, with or without

any HAL.

Kirby

[Ted Kosan]

Kirby,

> To me that sounds too much like reliance on AI breakthroughs that
> might be just around the corner. For me, a litmus test is "did we have
> the technology ten years ago?" The solutions I'm proposing fit that
> criterion.

AI step-by-step elementary algebra equation solvers that could solve
equations and show the steps taken better than any human who has ever
lived were created in the 1970s. The AI breakthrough has already
happened. The problem is not the AI. The problem is the ignorant
humans who teach in the complacent monopolist education system. I
don't see how the solution you are proposing will be able to break
this monopoly.

> You're writing what I call "far future science fiction" whereas "near
> future" so so close as to be unrecognizable from today.

Again, this AI future is already here:

http://www.cyc.com/mathcraft/

The people who can't see it yet are similar to the people in the early
20th century who could not see how automobiles would quickly replace
the horse and buggy.

> Who's to say anyone is going to control it? By the sound of it, ITSs
> will control us.

The programmers who create the ITSs will control the ITSs.

Ted

[kirby urner]

On May 2, 2016 15:02, "Ted Kosan" <ted....@gmail.com> wrote:
>
> Kirby,
>
> > To me that sounds too much like reliance on AI breakthroughs that
> > might be just around the corner.  For me, a litmus test is "did we have
> > the technology ten years ago?"  The solutions I'm proposing fit that
> > criterion.
>
> AI step-by-step elementary algebra equation solvers that could solve
> equations and show the steps taken better than any human who has ever
> lived were created in the 1970s. The AI breakthrough has already
> happened. The problem is not the AI. The problem is the ignorant
> humans who teach in the complacent monopolist education system. I
> don't see how the solution you are proposing will be able to break
> this monopoly.
>

If you're right about the current state of AI we'll know soon enough.

>
> > You're writing what I call "far future science fiction" whereas "near
> > future" so so close as to be unrecognizable from today.
>
> Again, this AI future is already here:
>
> http://www.cyc.com/mathcraft/
>

I looked at this...

> The people who can't see it yet are similar to the people in the early
> 20th century who could not see how automobiles would quickly replace
> the horse and buggy.
>
>
>
> > Who's to say anyone is going to control it?  By the sound of it, ITSs
> > will control us.
>
> The programmers who create the ITSs will control the ITSs.
>

Maybe... I've met my share of sycophantic programmers who just do the bidding of their managers. If we have programmers with enough independence to manage the ITSs without interference, then the world you predict may be possible.

I don't see it being a military project as there you have a culture into following orders. They'd surrender to the AI HAL, probably worship it with candles and churches.

Kirby

[kirby urner]

On Saturday, April 2, 2016 at 7:01:19 PM UTC-7, tkosan wrote:

Kirby wrote:

> Echoing Maria's sentiments, "ramming" usually leads to even more entrenched
> resistance.
>
> In the Cold War Era, the fact of Sputnik and the prospect of "falling
> behind" was used as leverage by SMSG ("New Math"), or at least the
> atmosphere of "needing to get catch up" was strong.
>
> But then came the backlash.  Teachers resented the top-down imposition of so
> much alien material without proper training.  New Math would have to go.

Intelligent Tutoring Systems and experiments like Khan Academy have
shown that technology now makes it possible to bypass conventional
teachers if needed. However, I am hopeful that a significant number of
math teachers will learn and then go on to teach the "New New Math" we
are creating because of its beauty, simplicity, and effectiveness.

In a lot of dimensions, Khan is a conventional teacher.  He lectures.  What's
unconventional is the delivery mechanism.  We see lots of math teachers
following Khan's example and moving their lectures to video, mixing in slides,
animations.  Everyone is making TV these days.  Students and teachers both.

You may need to disambiguate "New New Math" as that meme has already
been used quite a bit, and disparaged:

https://groups.google.com/forum/#!search/%22new$20new$20math%22

I think you're looking forward to something different, based more on AI.
 

> I'm definitely willing to toss my hat in the ring as a competitor and am
> always on the lookout for co-conspirators.

An interesting thing about this conspiracy is it will all be done in
the open with open source software.

Ted

Yes, and that's why I like the pun "gnu math" quite a bit.

Kirby

 

[Ted Kosan]

"gnu math" has a nice ring to it! I had planned on naming this new
approach to teaching math, computer science, and logic together
"Patternmatics" to make it clear that all three are based on patterns.
Maybe "Patternmatics, the gnu math" would work?

Ted

[kirby urner]

On Tue, May 3, 2016 at 8:56 AM, Ted Kosan <ted....@gmail.com> wrote:

 

>> An interesting thing about this conspiracy is it will all be done in
>> the open with open source software.
>>
>> Ted
>
>
> Yes, and that's why I like the pun "gnu math" quite a bit.

"gnu math" has a nice ring to it! I had planned on naming this new
approach to teaching math, computer science, and logic together
"Patternmatics" to make it clear that all three are based on patterns.
Maybe "Patternmatics, the gnu math" would work?

Ted

Patternmatics sounds interesting.  Makes for a pinpoint Google search.  I found your website right away.

http://patternmatics.org/examples/expressions/random/random_expressions_1.png

would make for some great wallpaper for a math-oriented website and/or Youtube channel.

Kirby

[Joseph Austin]

On May 3, 2016, at 11:56 AM, Ted Kosan <ted....@gmail.com> wrote:

"gnu math" has a nice ring to it! I had planned on naming this new
approach to teaching math, computer science, and logic together
"Patternmatics" to make it clear that all three are based on patterns.
Maybe "Patternmatics, the gnu math" would work?

Ted

I like "Patternmatics."  For one thing, it may liberate "math" from the plane of paper or screen.  I think "math" should be as comfortable explaining and transforming molecular structures  as prime numbers.  But put "science" into the mix as the "reality" behind the pattern. I'm lobbying for "math as model".

Joe Austin

[Ted Kosan]

Joe wrote:

> "gnu math" has a nice ring to it! I had planned on naming this new
> approach to teaching math, computer science, and logic together
> "Patternmatics" to make it clear that all three are based on patterns.
> Maybe "Patternmatics, the gnu math" would work?
>
> Ted
>
>
> I like "Patternmatics." For one thing, it may liberate "math" from the
> plane of paper or screen. I think "math" should be as comfortable
> explaining and transforming molecular structures as prime numbers. But put
> "science" into the mix as the "reality" behind the pattern. I'm lobbying for
> "math as model".

I think you are right that science should be put into the mix from the
beginning. Perhaps the ingredients of the Patternmatics mix are simply
Science, Technology, Engineering, and Mathematics taught together from
the very beginning, and logic is the glue that binds these areas
together [1].

Model theory is part of mathematical logic
(https://en.wikipedia.org/wiki/Model_theory) , and I think its simpler
aspects can be taught very early. I have been collecting "new math"
mathematics books that were written in the 1960s, and some of them
have very clear explanations on how mathematical logic and
interpretations work. I am looking forward to putting these "new math"
explanations back into use again.

Ted

[1] "Logic is the glue that binds together methods of reasoning, in
all domains."
---David Gries and Fred B. Schneider

[Joseph Austin]

On May 4, 2016, at 6:21 PM, Ted Kosan <ted....@gmail.com> wrote:

Model theory is part of mathematical logic
(https://en.wikipedia.org/wiki/Model_theory) , and I think its simpler
aspects can be taught very early.

I suspect my notion of "model" is somewhat more informal than this.

I'm thinking of questions such as: What is the physical meaning of  "product of mass and velocity" (aka momentum)?  

For example, we say "momentum" is conserved in a collision, 

which means if the mass doubles, the velocity must be halved.

What I see in physics is that "everything" is conserved, just rearranged.

But "conservation" does not seem to be a sought-after property in mathematical systems.

But speaking of models, in CS I've encountered several "three operator" systems:

Logic: and, or, not

Set theory: union, intersection, complement

Regular Expressions:  selection, concatenation, repetition

Structured Programming: sequence, choice, repetition

Arithmetic:  add, multiply, exponentiate

But I'm not familiar with a standard mathematical system that treats such systems.

And if we note that the three logical "primitives" can be reduced to one,

either NAND or NOR, perhaps the same is true of the others.

Joe

[kirby urner]

On Wed, May 4, 2016 at 3:21 PM, Ted Kosan <ted....@gmail.com> wrote:

Joe wrote:

> "gnu math" has a nice ring to it! I had planned on naming this new
> approach to teaching math, computer science, and logic together
> "Patternmatics" to make it clear that all three are based on patterns.
> Maybe "Patternmatics, the gnu math" would work?
>
> Ted
>
>
> I like "Patternmatics."  For one thing, it may liberate "math" from the
> plane of paper or screen.  I think "math" should be as comfortable
> explaining and transforming molecular structures  as prime numbers.  But put
> "science" into the mix as the "reality" behind the pattern. I'm lobbying for
> "math as model".

I think you are right that science should be put into the mix from the
beginning. Perhaps the ingredients of the Patternmatics mix are simply
Science, Technology, Engineering, and Mathematics taught together from
the very beginning, and logic is the glue that binds these areas
together [1].

How about "intuition" in addition to "logic"?

We sometimes make leaps, then go back and show how to get there

with logic (if we can find a way).

The idea that, given a few axioms and a computer, we can thereby get

all theorems of interest is more science fiction than the documented

way in which science or mathematics has progressed.  Lets not base

our whole education system on a fantasy!

We've all read Thomas Kuhn right?  The Structure of Scientific

Revolutions?  The idea that the basic axioms were all set down by

the Greeks and everything since then is just working out the
consequences is clearly not the right idea. 

The idea that we have all the axioms we need, whether from the
Greeks or since, and just need to complete the theorem proving
machine that uses them is also not itself a provable model of

what's happening in STEM.

 

Model theory is part of mathematical logic
(https://en.wikipedia.org/wiki/Model_theory) , and I think its simpler
aspects can be taught very early. I have been collecting "new math"
mathematics books that were written in the 1960s, and some of them
have very clear explanations on how mathematical logic and
interpretations work. I am looking forward to putting these "new math"
explanations back into use again.

When I talk about models I'm sometimes picturing a computer game,

very literally. 

I'm suggesting we need a lot more games that start with a whole planet
(or "world"), not necessarily Earth (any template might do), but with
similar challenges:  how do you get power (as in electrical power) to
all these people?  How do you feed them? 

How do you keep them from engaging in suicidal behavior on a mass
scale (weapons of mass suicide are everywhere, on a hair trigger in
some cases)?  This last question suggests a rather mixed up world.

We might not start with Earth in the early grades, preferring to make

up unreal worlds that aren't as messed up or discouraging.

When we make a whole planet be the game board, then we're at last

thinking in the round. 

Our networks connect around in all circumferential directions, like shipping
routes, airline routes. 

This is something to focus on:  spherical networks, graphs on the planet
surface.  Circuits.  Or call it Motherboard Earth.

How do I think a computer game about a whole planet should look?

Lots of different ways, obviously, but one pattern I tout (speaking of

patternmatics) is called the hexapent, and you can see it displayed

here:

http://photos1.blogger.com/blogger/1134/545/1600/30freq.png

and here:

http://vignette1.wikia.nocookie.net/freeciv/images/f/f6/Screenshot.jpg/revision/latest?cb=20110823212257

and here:

http://www.combomash.com/wp-content/uploads/2014/02/2014_02_25_hextraction_banner.png

I favor the hexapent in part just because it's a logical extrapolation of

a 2D "flat" board game motif that's been used for ages.  Civilization [tm],
the game, uses hexagonal tiling.  Going sphereical with hexagons actually

requires using 12 pentagons.  The math behind this is something every

grade schooler should know.

I've called my campaign (HP4E) i.e. "hexapents for everyone", in part paying

homage to Guido's CP4E:  computer programming for everyone (accepted by

DARPA, just like Cycorp's proposal).  A closely related project, now complete,

was called the Hexagonal Awareness Project.  Here's the home page:

http://hexagonalawarenessproject.tumblr.com/

Kirby

[Ted Kosan]

Joe wrote:

> What is the physical meaning of "product
> of mass and velocity" (aka momentum)?

Before we can determine what the meaning of "product of mass and
velocity" is, we must first determine what the meaning of meaning is.
Since the formal definition of meaning is part of symbolic logic, this
is our starting point for this determination.

> Logic: and, or, not
> Set theory: union, intersection, complement
> Regular Expressions: selection, concatenation, repetition
> Structured Programming: sequence, choice, repetition
> Arithmetic: add, multiply, exponentiate
>
> But I'm not familiar with a standard mathematical system that treats such
> systems.
> And if we note that the three logical "primitives" can be reduced to one,
> either NAND or NOR, perhaps the same is true of the others.

According to Russell in "The Principles of Mathematics", all of pure
mathematics can be defined with implication. Of the operator systems
that are listed here, I am currently most interested in Structured
Programming's sequence, choice, and repetition. This is because I
suspect that these three exist at the meta-level of a programming
language instead of at its object level.

Ted

[Ted Kosan]

Kirby wrote:

> How about "intuition" in addition to "logic"?
>
> We sometimes make leaps, then go back and show how to get there
> with logic (if we can find a way).
>
> The idea that, given a few axioms and a computer, we can thereby get
> all theorems of interest is more science fiction than the documented
> way in which science or mathematics has progressed. Lets not base
> our whole education system on a fantasy!
>
> We've all read Thomas Kuhn right? The Structure of Scientific
> Revolutions? The idea that the basic axioms were all set down by
> the Greeks and everything since then is just working out the
> consequences is clearly not the right idea.
>
> The idea that we have all the axioms we need, whether from the
> Greeks or since, and just need to complete the theorem proving
> machine that uses them is also not itself a provable model of
> what's happening in STEM.

What the AI researchers who built the equation solver that I based my
solver on discovered was that people unconsciously use axioms at the
meta-level of algebra (along with meta-level inference) to solve
elementary algebra equations. The researchers suspected that many more
processes that are called intuition are actually unconscious use of
meta-level inference. The equation solving meta-level techniques they
identified are so simple that I think even young children could grasp
them. I am looking forward to explaining these meta-level techniques
on this list in the near future.

I have been thinking about this information and the PDX Code Guild.
After you have learned these meta-level equation solving techniques, I
am convinced you could offer a class on it for experienced
programmers, and interest in it would be so great that it would exceed
the capacity of the facility.

Ted

[kirby urner]

On Thu, May 5, 2016 at 12:05 AM, Ted Kosan <ted....@gmail.com> wrote:

Joe wrote:

> What is the physical meaning of  "product
> of mass and velocity" (aka momentum)?

Before we can determine what the meaning of "product of mass and
velocity" is, we must first determine what the meaning of meaning is.
Since the formal definition of meaning is part of symbolic logic, this
is our starting point for this determination.

This is where my background with Wittgenstein would come in.

Meaning is to be investigated and changes over time, has to do

mostly with usage patterns, how a symbol is used, or any tool

really. 

What's the meaning of a screwdriver:  it comes from how it gets
used, and not only to screw screws.

Russell never really understood Wittgenstein's stuff, according

to both of them.  I guess my training puts me in a different school

of thought from Bertie Russell then. 

I still appreciate formal logics though:  great language games.

Computer languages too (some of them).
 

Kirby

[kirby urner]

On Thu, May 5, 2016 at 12:40 AM, Ted Kosan <ted....@gmail.com> wrote:

<< edit >>
 

What the AI researchers who built the equation solver that I based my
solver on discovered was that people unconsciously use axioms at the
meta-level of algebra (along with meta-level inference) to solve
elementary algebra equations.

I'm not sure this is a falsifiable scientific hypothesis in the Popperian

sense.  More a proposal we use different terminology.  Anytime it's

"unconscious" we're typically introducing a "black box" full of

"mechanisms".  It's a grammatical move more than a scientific one.

 

I have been thinking about this information and the PDX Code Guild.
After you have learned these meta-level equation solving techniques, I
am convinced you could offer a class on it for experienced
programmers, and interest in it would be so great that it would exceed
the capacity of the facility.

Ted

I'm up to expressions as trees in the homework so far.

I don't see equation solving as a metaphor for thinking in general

though.  It's one thing people do -- using computers more recently.

I'm just as interested in RI (real intelligence) as in AI.

Kirby

[Joseph Austin]

> On Mar 31, 2016, at 8:57 PM, Ted Kosan <ted....@gmail.com> wrote:
>
> Michel wrote:
>
>> The spirit of the NCTM position seems much different than Dijkstra's in
>> Programming as a discipline of mathematical nature.
>
> Dijkstra also wrote that the main difference between mathematical
> formula and computer programs is their length:
>
> "It really helps to view a program as a formula. Firstly, it puts the
> programmer's task in the proper perspective: he has to derive that
> formula. Secondly, it explains why the world of mathematics all but
> ignored the programming challenge: programs were so much longer
> formulae than it was used to that it did not even recognize them as
> such. Now back to the programmer's job: he has to derive that formula,
> he has to derive that program. We know of only one reliable way of
> doing that, viz. by means of symbol manipulation.” (“On the Cruelty of
> Really Teaching Computer Science”, Edsger Dijkstra,
> https://www.cs.utexas.edu/~EWD/transcriptions/EWD10xx/EWD1036.html)
>
> In the future mathematics, computer science, and symbolic logic will
> be taught together by teachers who are knowledgeable in all three
> subjects.
>
> Ted
>
> --
Let me append Dijkstra's description (c. 1988) of such a Computer Science course:

QUOTE

... all by itself, a program is no more than half a conjecture. The other half of the conjecture is the functional specification the program is supposed to satisfy. The programmer's task is to present such complete conjectures as proven theorems.
...
Before we part, I would like to invite you to consider the following way of doing justice to computing's radical novelty
[i.e, (1) the size problem: 10^9 scale ratio; (2) the "chaotic" nature of binary systems, where 1 bit in a million can make a disastrous difference.]
in an introductory programming course.

On the one hand, we teach what looks like the predicate calculus, but we do it very differently from the philosophers. In order to train the novice programmer in the manipulation of uninterpreted formulae, we teach it more as boolean algebra, familiarizing the student with all algebraic properties of the logical connectives. To further sever the links to intuition, we rename the values {true, false} of the boolean domain as {black, white}.

On the other hand, we teach a simple, clean, imperative programming language, with a skip and a multiple assignment as basic statements, with a block structure for local variables, the semicolon as operator for statement composition, a nice alternative construct, a nice repetition and, if so desired, a procedure call. To this we add a minimum of data types, say booleans, integers, characters and strings. The essential thing is that, for whatever we introduce, the corresponding semantics is defined by the proof rules that go with it.

Right from the beginning, and all through the course, we stress that the programmer's task is not just to write down a program, but that his main task is to give a formal proof that the program he proposes meets the equally formal functional specification. While designing proofs and programs hand in hand, the student gets ample opportunity to perfect his manipulative agility with the predicate calculus. Finally, in order to drive home the message that this introductory programming course is primarily a course in formal mathematics, we see to it that the programming language in question has not been implemented on campus so that students are protected from the temptation to test their programs.

And this concludes the sketch of my proposal for an introductory programming course for freshmen.
END-QUOTE

With due respect to Dijkstra, who after all gave us structured programming, I nevertheless don't accept that his solution addresses either of the "radical" features. (1) The types of problems typically encountered in beginning courses are not nearly the scale of "real" programs such as a Web Browser, much less MS Windows or (presumably) a global missile defense system. (2) The odds of a student completing a correct formal specification, derivation, and proof of even a toy problem with "zero defects" is questionable. As a teacher, I could not even guarantee that I could find all the errors in a student's solution.
And perhaps I could add a third "radical". (3) For contemporary distributed event-driven systems, there is a combinatorial explosion of possible sequences of events and interactions, even neglecting the possibility of "real" errors such as signal corruption and hardware failure, even deliberate sabotage, which of course we cannot neglect after all.

Of course, it does't solve the philosophical problem to bring in a mechanical proof-checker, because we can never have total confidence that the proof-checker is correct!

Nevertheless, Dijkstra's proposal provides a perspective for understanding Computer Science as a mathematical discipline
(or is it understanding math as a computational discipline?).

As a test case I usually propose to create a sort program.
What is the specification?
Given a set of elements over which a well-order relation (not necessarily strict) is defined,
produce a sequence of all and only the original set elements in non-decreasing order.

Now, how do you verify that the result meets the specification for every possible set?
You may do a proof by induction, verifying that every operation preserves the membership of the original set,
and that the algorithm terminates in a well-ordered set.
This, of course, differs markedly from the typical programmer "test case" of starting with a list of random numbers and displaying an ordered result.

I'd also consider an object-oriented "differential" solution: let each "element" exchange position with immediate neighbors to be "between" two neighbors in the correct order, and repeat whenever the identify of either neighbor changes.
But then we need to add a proof that the rearranging will eventually stop, and a method for determining that the rearranging has stopped.

The challenge, should we choose to accept it, is to produce an axiomatically-defined programming language in which such programs and proofs may be expressed. Tony Hoare once wrote a book purporting to offer such a language (similar to Pascal), but I was never convinced it was rigorous--not that my cursory inspection is a valid critique.

LISP might seem to be a good candidate for a provable language, but my experience with LISP suggests that writing correct programs with all those quotes and parens creates as many opportunities for error as it avoids, especially if done by hand, as Dijkstra suggests. Besides, the limitation to pushdown lists creates a severe constraint on algorithmic approaches.

Joe Austin

[kirby urner]

Dijkstra's views are interesting.  However he was helping establish computer science as a serious discipline (which it by now is) and in those days, harping on formal proofs was very prudent. 

There's a political dimension to everything.

Now that CS is well-established and we're looking at CS-friendly math, a more playful and open-ended approach recommends itself.

Maybe we can't "the program does what it's supposed to" because we're exploring, not writing to a spec.

Learning the tools to make 2D and 3D graphics, like the Turtle and Pi3D (both in Peter's book) sets the stage for undertaking investigations, trying stuff, like working with clay. 

Wolfram would be an influence in that the computer becomes a laboratory, a place of experiment.

Trigonometry comes in handy, they find out.  Vectors, matrices... because we're trying to construct something interesting to look at.  The "problem" we're looking to solve in software is.... "how to make interesting art" (not just ASCII art).

I'm not saying Dijkstra's views have been transcended or that formal, rigorous approaches are somehow wrong. 

I just don't consider them all the apropos given the changed political climate.

When it comes to exciting people about what technology might do for them, rather than what they might do for technology, a focus on individual creativity and expressiveness has its place.

Maybe that's why I talk about Art School and Art History in the same breath as Code School:  because strong design skills are a part of what we're looking for in polymath engineers. 

Making animations, visualizations, intelligible "control surfaces" (like for flying an airplane) is part of the territory.  Dijkstra did not live in quite this same world.

Kirby

[kirby urner]

Maybe we can't  [ prove ]  "the program does what it's supposed to" because we're exploring, not writing to a spec.

Missed my verb.

Kirby

[Joseph Austin]

Kirby,

I agree with what you are saying, as long as one understands whether one is doing "art" or "engineering".

Even engineers delve into the unknown, but they do so laboratories,

and have reasonable confidence their creations are at least safe, hopefully useful, before releasing them to the marketplace.

When teaching structured programming, I always emphasized that a well-structured program has "linear" complexity,

that is, the number of possible paths through the program (relating to the number or test cases) grows linearly with the length of the program,

not geometrically or worse, as "spaghetti code" did.  I think that was Dijkstra's greatest contribution.

As we enter the world of distributed, event-driven computing,  we are once again flirting with geometric complexity growth.

So we should continue to be concerned that we not build things to complicated to validate.

Joe

[kirby urner]

Good points Joe.

A lot depends on how we define "engineer", as well as "artist".

This word "artscience" -- all one word -- has some currency (I've seen at least one book by that title, and no, not by Bucky).

The kind of destructive engineering people engage in, undoing what that the constructive types do, is not necessarily controlled, nor predictable in its outcomes.

The atmospheric fission blasts conducted in the course of our planet's first nuclear war were done with full knowledge there were both known and unknown knowns in the picture, not to mention unknown unknowns.

The engineers went ahead anyway and yes, the ripple effects are still being calculated. Like a real number with chaotic digits, we have trouble second guessing the next twist.

As a species, we have centuries of work cut out for us, simply keeping ourselves safe from our new inventory of radio-toxins. Yes some of these occur in nature but some didn't in these quantities. Isotopes r us. I'm a strontium 90 baby.

Or were those tests the work of out-of-control artists? 

Semantics matter I guess.

Either way, I agree we need to ponder the consequences of our projects, look before we leap as they say.

Blindly tempting fate to "teach us physics the hard way" is pathological behavior no matter who this "we" is doing it.

So many gentler on-ramps, learning curves, thoughtful curricula, might serve. If we take fuller advantage of what's already in inventory, we might have an easier time making over this planet --something humans are clearly doing, with or without "climate change" (the biosphere has already transformed irrevocably in so many dimensions thanks to hominids, that's no longer debatable).

Kirby

[Joseph Austin]

Kirby,

I've come to appreciate the fact that "school" is a place where you can make mistakes that "don't count" 

(although few teachers seem to acknowledge that to the students.)

I think our schools could be better if we took ultimate outcomes (are you ready to graduate?) more seriously 

and intermediate outcomes (did you pass this week's quiz?) less so.

Once upon a time, so I understand, college lectures were optional, and you passed the course (or not) by taking the final exam.

Somewhat the way industry is now doing with "certifications".

Actually, it's the way "comprehensives" worked for my graduate degree.

I can remember "learning" entire courses I had never actually taken in order to pass the exams.

But that approach tended to make the economics difficult for ineffective teachers.

So now we grant diplomas based on how many hours you sat listening to the teacher.

Speaking of comprehensives, reviewing an entire course in one week tends to bring the basic principles into focus.

While taking a course, we spend much time looking at individual trees, and almost get to all of them by the end.

What seldom happens is taking a step back and having a look at the forest, 

and how the different species contribute to the synergy that makes the whole thing viable.  

The pity is, by the time we have the time for such (retirement?), it's too late to make practical use of the perspective! 

Joe

On Jul 15, 2016, at 11:36 PM, kirby urner <kirby...@gmail.com> wrote:

<snip>

So many gentler on-ramps, learning curves, thoughtful curricula, might serve. If we take fuller advantage of what's already in inventory, we might have an easier time making over this planet --something humans are clearly doing, with or without "climate change" (the biosphere has already transformed irrevocably in so many dimensions thanks to hominids, that's no longer debatable).

Kirby

[Bradford Hansen-Smith]

"So now we grant diplomas based on how many hours you sat listening to the teacher."

Good training for how we get paid later in the corporate world.  We sell our time and space; what we do is determined by ownership.

This is what the indebtedness of student loans are about.

Brad

--

Bradford Hansen-Smith
www.wholemovement.com

[kirby urner]

On Sat, Jul 16, 2016 at 6:36 AM, Joseph Austin <drtec...@gmail.com> wrote:

<< SNIP >>
 

The pity is, by the time we have the time for such (retirement?), it's too late to make practical use of the perspective! 

Joe

I dunno Joe, I think "retirement" is mostly cultural.  The physics meaning of "work" as distinct from simply "energy expenditure" is of interest.  Entropy fits in here somewhere right? [1]

They say Portland is where "young people go to retire" (that's a joke, started by the TV series Portlandia, which gently mocks Portland in a way Portland somewhat enjoys -- to where everyone in their uncle seems to be looking for digs).

However my experience is young people burn out and need downtown to retrain for something different.  The fact that high schools probably didn't teach them much HTML/CSS or SQL means basic record keeping and public sharing are more dependent on "web services".  Businesses want to be the ones providing, not just consuming.  So there's a lot to learn.

That's why the code school business is big here.  People are looking for work and discover these reading and writing skills are in demand, not just for developer jobs or "full stack engineers" but for office workers of all kinds.  Microsoft Access was just the beginning (the next step above Excel).

You'd think someone like me, who has a lot of the skills and am willing to share them, would have an easy time of it.  Not so.  Teaching is a strictly controlled, highly unionized business.  Rather than undercut teachers and compete for the same students, I'm hoping to help the most innovative schools innovate new curriculum, something beyond Pearson or anything else UK (not that I'm anti UK, just here in the Pacific Northwest we need to do our own thinking).

Anyway, I like the Global U model and cradle-to-grave work / study.  That's more realistic.  We need time to study all the way through life.  This idea we get the schooling part out of the way, then do a job, then retire, is not a workable idea in many lifetime scenarios, certainly not in mine.  But work / study, I can relate to.

Kirby

[1]  http://mybizmo.blogspot.com/2015/07/what-is-work.html

[kirby urner]

ADDENDUM:

They say Portland is where "young people go to retire" (that's a joke, started by the TV series Portlandia, which gently mocks Portland in a way Portland somewhat enjoys -- to where everyone in their uncle seems to be looking for digs).

"everyone and their uncle..."

 

However my experience is young people burn out and need downtown to retrain for something different. 

"... need down time..."  (time to study, catch up, practice, work with simulations -- like in school, mistakes OK by design)

That'll teach me to thumb lengthy postings on my HTC smartphone, just off campus from Whitworth University, home of the Pirates, in Spokane.  Smartphones have even more opinionated spellcheckers it seems to me, than say Gmail.

Just back in Portland, about a 6 hour drive with rest stops. 

I need to contact Peter Farrell about maybe making a guest appearance on my Python radio show in the remaining weeks.

Time to upload pictures from the trip.

I've read your later one Joe, about developing a discrete finite math that doesn't need extended precision in quite the same way real numbers might. 

Lots of thoughts, made the drive go faster (I read it before leaving Spokane).

Kirby