results of math dept 'quiz'

[michel paul]

At a math dept meeting I gave my colleagues a 'pop quiz'. I've forwarded the results below. I was curious to see what would happen. I've noticed over the years how our traditional curriculum produces boxed-in mathematical thinking, not just in students but also in teachers. I noticed it in myself when my understanding of mathematics broadened under the influence of computational thinking.  

For example, a couple of years ago I went around asking my colleagues whether or not a translation was a function. Interestingly, most of them said it was not! I kind of suspected that, as I figured their reasoning would be based on the limited notion our curriculum presents of function where a function relates single numbers to single numbers, and their graphs have to pass the 'vertical line test'. So in that way of thinking, translating a point vertically doesn't at all register as a 'function'. But in very simple programming, it makes total sense to think of a function that will consume an ordered pair and return a new ordered pair. Even younger kids can understand that. But our traditional curriculum screws this all up, presenting functions in a very limited sort of way, just one topic among many. For the kids who get to polar coordinates, they finally get to see functions whose graphs don't pass the vertical line test, but even then, I would say, the understanding persists that functions relate single numbers to single numbers rather than mathematical structures to mathematical structures.

Anyway - here's the report I sent them:   

---------- Forwarded message ----------

Here are the 'quiz' results from the 13 who responded.  Note - very interesting - no one got either of the first two correct!  Although s

ome people were intellectually honest and chose C, and some were close with B or D:

Question #1 - correct answer is E.

I got 6 As, 2 Bs, 3 Cs, 2 Ds, and 0 Es!

Question #2 - correct answer is A.

I got 0 As!, 2 Bs, 2 Cs, 1 D, and 8 Es

Question #3 - correct answer is E

I got 7 As, 5 Bs, 0 Cs, 0 Ds, and 1 E

Question #4 - see Peano Axioms

And here again are the questions:

Answer questions 1 -  3 as follows:

A - definitely true

B - perhaps true

C - I really do not know

D - perhaps false

E - definitely false

1.  A B C D E:  Mathematics is in principle a formal logical system.  This was conclusively demonstrated during the 20th century and became the foundation for computer science.

2.  A B C D E:  Mathematics in principle will never be reduced to a formal logical system.  This was conclusively proven in the 20th century in both mathematics and computer science.

3.  A B C D E:  Mathematics is sequential in nature.

4.  What is the definition of the natural numbers? 

(Note -> a set of examples is not a definition.)

Yeah, believe it or not, in the 1930's Kurt Godel proved conclusively that it is impossible to reduce mathematics to a formal logical system.  His work is considered to be just as important in the foundations of mathematics as Einstein's was in physics.  (He and Einstein were actually good friends.)  But for some reason this news has still not reached secondary mathematics education!  Isn't that bizarre?  Imagine if secondary science education wasn't aware of Einstein?

Regarding math being 'sequential in nature' - being 

'sequential in nature' means that there is always a 'next' term, right?  Well, that is not the case for either the reals or the complex numbers!  What is the 'next' number after pi?

Mathematics also did not evolve 'sequentially' in history.  It was more like various centers of mathematical activity gradually reaching out and communicating with each other.  And that's also how it happens in our brains.

However, the defining characteristic of the natural numbers IS that they are sequential in nature!  : )  That's the point of the last question.  The Peano axioms provide a recursive definition of the naturals.

So this is a good way to illustrate what computational thinking means - in ordinary math the naturals are kind of a boring subset of the reals, but in computational reasoning, EVERYTHING is ultimately defined, constructed, in terms of the naturals 0 and 1. 

So, though mathematics as a whole is not sequential in nature, computational mathematics is!  

- Michel

 

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

- Richard Feynman

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

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

[kirby urner]

On Fri, Nov 4, 2011 at 9:54 AM, michel paul <python...@gmail.com> wrote:

<< snip >>

> So, though mathematics as a whole is not sequential in nature, computational
> mathematics is!
> - Michel
>

I enjoy reading your insights.

At some level, the very meanings of the words "formal" and "system"
are up for grabs, i.e. it's possible to get ahead of ourselves in
imagining the limitations we think we might be up against.

Apropos of these remarks:

http://mathforum.org/kb/message.jspa?messageID=7601612&tstart=0

Here we're taking up, once again, the meaning of "=" as in "equals"
and "=" as in "bind name to object" (assignment).

When you get down to basic use cases, you're also getting down to core
meanings (as in "core Python").

"Not every math notation is working from the same center" -- that may
sound like heresy in some schools.

Again, I liked what you said about functions, as I'm seeing a lot of
convergent thinking.

Kirby

[David Chandler]

I have heard of the Peano axioms but had not worked with them formally before.  Please clarify.  It seems that "successor" is an undefined term, not tied to n+1, so the axioms could equally apply to the non-negative numbers (successor of n = n+1), the non positive numbers (successor of n = n-1), the non-negative even numbers (successor of n = n+2), etc.: any set isomorphic to the natural numbers.  Is this correct?  Do the Peano axioms, then, distinguish the natural numbers from these other sets?
--David Chandler

On Fri, Nov 4, 2011 at 9:54 AM, michel paul <python...@gmail.com> wrote:

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[michel paul]

On Fri, Nov 4, 2011 at 12:20 PM, David Chandler <david...@gmail.com> wrote:

I have heard of the Peano axioms but had not worked with them formally before.  Please clarify.  It seems that "successor" is an undefined term, not tied to n+1, so the axioms could equally apply to the non-negative numbers (successor of n = n+1), the non positive numbers (successor of n = n-1), the non-negative even numbers (successor of n = n+2), etc.: any set isomorphic to the natural numbers.  Is this correct?  Do the Peano axioms, then, distinguish the natural numbers from these other sets?
--David Chandler

Right, 'successor' doesn't have to be specifically 'n+1', 'successor' is basically 'next'. I haven't formally studied the Peano axioms either, except for an introduction in a Philosophy of Math course in college. You just need a place to start and a way to designate 'next'. Peano himself started at 1, and more recent versions start at 0, but it ends up not mattering. So you have a starting point, and then a 'next' term, and then a 'next' term, indefinitely. In a way, it is a computational description of a list. In fact, read the Peano axioms replacing 'number' with 'position' or 'index'. That's kind of interesting.

 The spirit of the Peano axioms inspired a computational exercise: 

• Consider two black box functions previous(n) and next(n), or predecessor(n) and successor(n).
• 'Black box' means you don't have to care how they're implemented. 
  
• Using only 
• these two functions, 
  
• an origin, 'zero'
• conditional reasoning, 
  
• and a test for equality, 
  
• define sum(a,b).   

A possible solution would be: if a is zero, then b is the sum. Otherwise, the sum will be the sum of the predecessor of a and the successor of b.

Here's a possible implementation in Python:

>>> def sum(a,b):
           if a == zero: return b
           else: return sum(pred(a),succ(b))

We can also define things like difference(a,b), greater_than(a,b), etc. Basically we can define arithmetic, although division gets pretty gnarly and raises interesting questions.

All we need is a way to define 'zero', 'pred()', and 'succ()'. Once we have them we can do something like this:

>>> one = succ(zero)
>>> two = succ(one)
>>> four = succ(succ(two))
>>> three = pred(four)
>>> one,two,three,four
('/', '//', '///', '////')

>>> sum(three,four)
'///////'

We are creating arithmetic from scratch! We are not using any built in arithmetic operators. Now here are the particular definitions for zero, succ(), and pred() used in this particular case:

>>> zero = ''
>>> def succ(n): return n+'/'

>>> def pred(n):
           if n == '': return None
           else: return n[:-1]  # simply means everything in n except for the last item

But the point of the exercise is not these definitions but to think of zero, succ(), and pred() as abstract, so that if they were to be implemented differently, our arithmetic functions would still work. For example, if they got redefined as:

>>> zero = []

>>> def succ(n): return [n]

>>> def pred(n):

if n == zero: return None

else: return n[0]

>> zero

[]

>>> one = succ(zero)

>>> one

[[]]

>>> two = succ(one)

>>> two

[[[]]] 

Leaving our definition of sum(a,b) intact, without changing anything, it still works with the new definitions of zero, succ(), and pred():

>>> three = sum(one,two)

>>> three

[[[[]]]]

So we can define natural arithmetic operators in terms of the notions of 'previous' and 'next'. And then, whenever we're navigating web pages, we will be reminded of this. : )

The cool thing is - an exercise like this has us thinking about both foundations in mathematics and also functional abstraction in CS.

- Michel

 

[tkosan]

Michel Paul wrote:

> I've noticed over the years how our traditional curriculum produces boxed-in
> mathematical thinking, not just in students but also in teachers.

I have been working on finding ways to increase the use of computer
algebra systems (CASs) in schools since 2008. It took me a long time
to realize that one of the main reasons that most schools don’t use
CASs is that the traditional mathematics curriculum teaches “boxed-in”
mathematics (perhaps it can be called boxematics?) while most CASs use
unlimited mathematics. Since most schools don’t teach unlimited
mathematics, these schools don’t have much use for CASs.

I see that Maria will be attending (and speaking at) the “Computer-
Based Math Education Summit” on November 10th and 11th:

http://computerbasedmath.org/events/londonsummit2011/speakers.html

I am looking forward to hearing her thoughts on the Summit after it is
over to see what potential solutions they come up with for the
“boxematics” problem.



> The spirit of the Peano axioms inspired a computational exercise: <snip>

Your Peano axioms example looked interesting, so I created a version
of it in MathPiper just for fun:

%mathpiper,title="Peano arithmetic."

//Define rulebases for a function called s and two new infix operators
// named @+ and @*.
RulebaseHoldArguments("s",{x});

RulebaseHoldArguments("@+",{X,Y});
Infix("@+",1);

RulebaseHoldArguments("@*",{X,Y});
Infix("@*",1);


//Define some rules for the new @+ and @* operators.
10 # _X @+ 0 <-- X;

20 # _X @+ s(_Y) <-- s(X @+ Y);

10 # _X @* 0 <-- 0;

20 # _X @* s(_Y) <-- X @* Y @+ X;


//Examples which use the new rules.
Echo("2 + 2 = ", s(s(0)) @+ s(s(0)) );

Echo("2 * 3 = ", s(s(0)) @* s(s(s(0))) );

Echo("3 * 3 * 3 = ", s(s(s(0))) @* s(s(s(0))) @* s(s(s(0))) );

%/mathpiper

%output,preserve="false"
Result: True

Side Effects:
2 + 2 = s(s(s(s(0))))
2 * 3 = s(s(s(s(s(s(0))))))
3 * 3 * 3 =
s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(s(0)))))))))))))))))))))))))))

. %/output



Here are some of the rules that define the + operator in MathPiper for
comparison:

100 # 0 + _x <-- x;
100 # _x + 0 <-- x;
100 # _x + _x <-- 2*x;
100 # _x + n_Constant?*(_x) <-- (n+1)*x;
100 # n_Constant?*(_x) + _x <-- (n+1)*x;
101 # _x + - _y <-- x-y;
101 # _x + (- _y)/(_z) <-- x-(y/z);
101 # (- _y)/(_z) + _x <-- x-(y/z);
101 # (- _x) + _y <-- y-x;
102 # _x + y_NegativeNumber? <-- x-(-y);
102 # _x + y_NegativeNumber? * _z <-- x-((-y)*z);
102 # _x + (y_NegativeNumber?)/(_z) <-- x-((-y)/z);
102 # (y_NegativeNumber?)/(_z) + _x <-- x-((-y)/z);
102 # (x_NegativeNumber?) + _y <-- y-(-x);
150 # _n1 / _d + _n2 / _d <-- (n1+n2)/d;
200 # (x_Number? + _y)_Not?(Number?(y)) <-- y+x;
200 # ((_y + x_Number?) + _z)_Not?(Number?(y) Or? Number?(z)) <-- (y+z)
+x;
200 # ((x_Number? + _y) + z_Number?)_Not?(Number?(y)) <-- y+(x+z);
200 # ((_x + y_Number?) + z_Number?)_Not?(Number?(x)) <-- x+(y+z);
210 # x_Number? + (y_Number? / z_Number?) <--(x*z+y)/z;
210 # (y_Number? / z_Number?) + x_Number? <--(x*z+y)/z;
210 # (x_Number? / v_Number?) + (y_Number? / z_Number?) <--(x*z+y*v)/
(v*z);
250 # z_Infinity? + Complex(_x,_y) <-- Complex(x+z,y);
250 # Complex(_x,_y) + z_Infinity? <-- Complex(x+z,y);
251 # z_Infinity? + _x <-- z;
251 # _x + z_Infinity? <-- z;
250 # Undefined + _y <-- Undefined;
250 # _x + Undefined <-- Undefined;

Ted

[kirby urner]

On Sun, Nov 6, 2011 at 12:16 AM, tkosan <ted....@gmail.com> wrote:
> Michel Paul wrote:
>
>> I've noticed over the years how our traditional curriculum produces boxed-in
>> mathematical thinking, not just in students but also in teachers.
>
> I have been working on finding ways to increase the use of computer
> algebra systems (CASs) in schools since 2008. It took me a long time
> to realize that one of the main reasons that most schools don’t use
> CASs is that the traditional mathematics curriculum teaches “boxed-in”
> mathematics (perhaps it can be called boxematics?) while most CASs use
> unlimited mathematics. Since most schools don’t teach unlimited
> mathematics, these schools don’t have much use for CASs.
>

Hi Ted --

I'd like to play devil's advocate and/or the skeptical shopper
for a school that's buying curriculum components.

We've bought into using computers in math class. Both
hardware and software are basically free in a charity-minded
open source culture, so it's mainly a matter of having a
coherent, well-designed, integrated set of experiences.

Many of our students are adults (mixed generation, place
based) but that shouldn't matter too much, given the
peer teaching and auto-didactic aspects of how we
operate.

Anyway: my algebra curriculum, so far, is abstract in
the sense that we look at math language games, as well
as music language games, as just that: language games.

So we start with notation and the need to follow notation,
to become fluent, to learn the vocabulary, the meaning
of each sign.

Music is a good beginning, even if we don't take the time
to develop true proficiency with any given instrument
(that's for other times in the day or night).

Where we branch away from a lot of the other schools
in the vicinity is we stay screen-based for most our maths,
and choose computer languages besides MathCad /
Mathematica for notation, which latter stay closer to
the older notations / typography.

I use Python for math / logic. Others in the school
use Mathematica. Scott Gray is on the Mathematica
side, I'm on the Python side of the fence (along
with Perl, HTML5 etc.).

So I guess you could say we're using CAS. It's just
that I'm not using it in Python much to speak of, yet
my algebra is pretty abstract, involving operator
overloading, group and number theory, with enough
depth to cover RSA (the cryptography algorithm)
with quite a bit of understanding.

I'm also the geometry teacher, like I was at SDA
(St. Dominic Academy, Jersey City).

'Mathematics for the Digital Age and Programming
in Python' could be one of our texts, though a lot
of our learning is outside the classroom.

Developing negotiating skills is important. You can't
earn high marks (higher rank) in our curriculum
without exerting yourself both physically and
mentally together -- "just sitting" is only allowed
if you're confined to a wheelchair for medical
reasons, like FDR was.

We might do geocaching for example, as Geography
and Geometry are so intimately conjoined in this
component.

Anyway, that turned into less of an anti-CAS manifesto
than I thought it would be, as I remembered Scott and
his tracks.

Plus I was helping my daughter with calculus last night
(I used to teach it full time for the Dominicans, as I
mentioned), which brought back more CAS-like
skills.

I sometimes almost forget about Analog Math (my
track is in Digital Math, mostly in teacher training).

I'm like one of the gym teachers in that I keep writing
to students: "the interactive console is the gym for
this course, where to build Python muscles".

I've written that hundreds of times (again, a lot of
these are adults, but we don't prevent minors from
going through this component, and they predominate
in other components).

Kirby Urner
Oregon Curriculum Network
PDX, Global U

For more background:
http://mathforum.org/kb/message.jspa?messageID=7577481&tstart=0
(Calculus & Mathematica)
http://mathforum.org/kb/message.jspa?messageID=7510611&tstart=0 (more
re our academy)

[michel paul]

Hi Ted!

It's fun seeing names pop up on this list from other lists. I remember your mention of MathPiper from the sage-edu.

On Sun, Nov 6, 2011 at 12:16 AM, tkosan <ted....@gmail.com> wrote:

I see that Maria will be attending (and speaking at) the “Computer-
Based Math Education Summit” on November 10th and 11th:

http://computerbasedmath.org/events/londonsummit2011/speakers.html

I am looking forward to hearing her thoughts on the Summit after it is
over to see what potential solutions they come up with for the
“boxematics” problem.

This is so cool. I'm so glad this conference is happening. It's about time!

I'm wondering how there maybe could be some sort of 'Occupy Math Education' movement? Not sure what the physical form would take, but it's interesting that the CBM summit will be occurring alongside all this other stuff. Coincidence? : )

> The spirit of the Peano axioms inspired a computational exercise: <snip>

Your Peano axioms example looked interesting, so I created a version
of it in MathPiper just for fun:

%mathpiper,title="Peano arithmetic."

Wow, this is cool. Yeah, defining simple arithmetic in a Peano-like way is something that we actually can implement these days at a high school level. It's really not that much more difficult than stuff we already want them to learn. Usually they wouldn't see stuff like this until college.

This is a perfect time to ask you - I've noticed before in MathPiper that you call attention to 'side effects'.  Is that specifically for things like display methods, or is there something more to how you use it?    

- Michel

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[Ted Kosan]

Kirby wrote:

>Where we branch away from a lot of the other schools
>in the vicinity is we stay screen-based for most our maths,
>and choose computer languages besides MathCad /
>Mathematica for notation, which latter stay closer to
>the older notations / typography.
>
>I use Python for math / logic. Others in the school
>use Mathematica. Scott Gray is on the Mathematica
>side, I'm on the Python side of the fence (along
>with Perl, HTML5 etc.).

If I were forced to make a choice, I would choose to be on your side
of the math-land/programming-land fence because I think that typical
programming languages provide a more powerful way to solve problems
than systems that mainly use traditional mathematical notation do.
However, general-purpose CASs do not force people to make this choice
because they are designed to straddle the fence.

One way they straddle the fence is that most general-purpose CASs are
able to do procedural programming, functional programming, and
rule-based programming. Many programming languages are able to do
procedural and functional programming, but not that many of them are
able to do rule-based programming. Here is what the “Mathematica” book
has to say about rule-based programming:

“One of the most powerful, and unique, features of Mathematica is the
ability to set up calculations by specifying collections of
transformation rules. The basic idea is to say that whenever
Mathematica sees an expression that matches a particular pattern, it
should transform it in a particular way. This approach allows you to
write out a sequence of rules that mimic quite closely the tables of
mathematical relations that you find in books.” (“Mathematica: A
System for Doing Mathematics by Computer”, Stephen Wolfram, 1988, p.
501).

The ability to hop over the math-land/programming-land fence at will
that a CAS provides is very helpful for showing students the full
range of capabilities that computers possess.

I could probably provide further advantages that general purpose CASs
have over a normal language like Python (for use in education), but I
do not have much of a heart for it because I really like Python :-)

Ted

[Ted Kosan]

Michel Paul wrote:

> This is a perfect time to ask you - I've noticed before in MathPiper that
> you call attention to 'side effects'.  Is that specifically for things like
> display methods, or is there something more to how you use it?

You are the first person since 2008 (when I forked MathPiper from
Yacas) who has asked me what the side effects are :-)

MathPiper is built on top of a custom Lisp interpreter and Lisp makes
heavy us of the functional programming style. Here is a good
description of what side effects are in the context of Lisp:

"The essence of functional programming is that programs are built
entirely of functions with no side effects that compute their results
based solely on the values of their arguments. The advantage of the
functional style is that it makes programs easier to understand.
Eliminating side effects eliminates almost all possibilities for
action at a distance. And since the result of a function is determined
only by the values of its arguments, its behavior is easier to
understand and test. For instance, when you see an expression such as
(+ 3 4), you know the result is uniquely determined by the definition
of the + function and the values 3 and 4. You don't have to worry
about what may have happened earlier in the execution of the program
since there's nothing that can change the result of evaluating that
expression." http://www.gigamonkeys.com/book/they-called-it-lisp-for-a-reason-list-processing.html.

When I first started working on MathPiper, the results it was
returning in the console were very confusing to me because some of
them could be assigned to variables and some of them couldn't. It took
me a while to figure out that MathPiper has a strictly functional
design and therefore all of its functions always return a value (even
if it is only True or False, which indicates the function succeeded or
not). Only some of the functions, such as Echo and Write, also return
side effect output.

Since my vision for MathPiper was for it to be specifically designed
for the needs of education, I decided to have it explicitly indicate
what output from a function is its result and what output is side
effect output.

Ted

[kirby urner]

On Sun, Nov 6, 2011 at 11:24 PM, Ted Kosan <ted....@gmail.com> wrote:
> One way they straddle the fence is that most general-purpose CASs are
> able to do procedural programming, functional programming, and
> rule-based programming. Many programming languages are able to do
> procedural and functional programming, but not that many of them are
> able to do rule-based programming. Here is what the “Mathematica” book
> has to say about rule-based programming:
>

Hi Ted --

Yes, I've steeped myself in the Mathematica literature quite a bit,
have done some time.

MathCad also, which fellow Wanderer and Applied Mathematician
David Feinstein prefers, more like a spreadsheet in conception,
but with fully tricked out 1800s notation (if we might call it that) --
same as Mathematica (a comfort, to see that familiar set of
symbols).

MathCad is Maple under the hood, as you probably know.

>
> I could probably provide further advantages that general purpose CASs
> have over a normal language like Python (for use in education), but I
> do not have much of a heart for it because I really like Python :-)
>

I do really simple CASsy stuff in my stash of source, like in
Polynomial objects you want to be able to enter just a simple
data structure, of coefficients and exponents.

Then what you want back should be (at least optionally) a
string looking something like "3 * x**2 + 2 * x - 3",
which is not especially beautiful or anything, but it
does execute / evaluate with values of x.

Just constructing a polynomial as a string is primitive CAS.
Then spit out the derivative while you're at it.

A core thing I'm into are integer like objects that add modulo N.

We also look at the guts of Rational Numbers, as if they
weren't native and had to be implemented, operation by
operation. Welcome to 8th grade (again).

Both types (Modulo Numbers and Rational Numbers) get
defined as Python classes with methods, and because of
operator overloading, we get to use + , -, *, / directly.

Then repeat: same thing with vectors, other math object
types.

They (the participants) see different meanings for the +
operator across types, but also commonalities, such as
additive identity, idea of an inverse.

We're in group theory *before* calculus, having lots of fun.

Kirby

[kirby urner]

On Mon, Nov 7, 2011 at 12:15 AM, Ted Kosan <ted....@gmail.com> wrote:

<< snip >>

> Since my vision for MathPiper was for it to be specifically designed
> for the needs of education, I decided to have it explicitly indicate
> what output from a function is its result and what output is side
> effect output.
>
> Ted
>

I was happy to see this post as several of our threads have
been poking at this functional versus imperative heuristic.
Some object to the word "paradigm". Talk of "side effects"
and "state" are intrinsic to the debate / discussion.

I've seen it from the perspective of a lobbyist, one could
say. Quick story: joined a workshop of computer science
teachers from around the state ready to activate a for-credit
math offering that would be a lot like computer science,
-- but relabeled either digital or discrete math.

Roll the tape fast forward a couple years: I found funders
felt discouraged watching geeks fight, and there's some
bitterness in the functionalist vs imperativist debate that's
possibly holding up the show, scaring away recruits.

I don't feel we've gotten to the bottom of it on any of the
lists where I've explored the rift. math-thinking-l (public
archive) has probably gone deepest in my own experience,
on this particular topic. Until recently that is, when michel
paul started sharing his intuitions, which are quite sweeping
in scope and possibly moving us along again. At least
this thread is interesting.

Good to be tuning in,

Kirby

[Christian Baune]

Any maths boil down to patterns, isomorphism (pattern conversion) and logic (pattern discovery, deduction and abstraction).

The best thing that can be done is showing it, making it obvious.

There is a ceilling : IQ.
Either you are gifted and can deal with patterns or you have a profound memory and a fair ability to make links (pattern matching).

Maths are all that.

[kirby urner]

Lots of kinds of IQ. Some of our story problems require telepathic
abilities (let us say, for the sake of argument), and those with IQs
over 140 (as tested) have no special advantages. We find multiple
axes and talk about that openly, as that's another parameter. Lindsey
has a surplus of IQ, background with supercomputers, but she also
makes mistakes and has a criminal record, as do most of my faculty
(almost a requirement, if you're gonna teach RadMath and be admired as
a role model). Of course some of our "criminals" were doing civil
disobedience, or Robin Hoody kinds of things. One of our number set
fire to a logging truck and did time. He's not low IQ by any means
(on the contrary) but some story problems are closed to him, because
he's still not allowed to leave the state. And so on. That's why I
suggest a team-based a approach for a lot of these.

Kirby

[kirby urner]

Probably what distinguishes our curriculum is you don't just study
patterns, you insert yourself into them in an active way in order to
change them, an activist training (and deliberately so). We take a
critical approach to the historical pattern, which turned a lot of
math people into semi-paralyzed chair-sitters who talk a lot about
problem solving a lot, but when push comes to shove mostly just write
papers about it, that few if any have time to read. We've
deliberately altered the definition of mathematics such that we have
tautologies that allow us to say such chair-sitter passivists are
really posers, pretended to high level math skills they don't really
have (just read the resume). Our faculty tends to be dashing around
town in bike lanes, showing up at schools with lots of supplies,
cooking in churches, sleeping in tents. The gear is sometimes high
tech, sometimes imported, sometimes purchased on the web. I should
mention that Lindsey also has lots of experience with the military
industrial complex (jet airplanes for crime bosses 'n stuff).

Kirby

[Christian Baune]

Here it is :
Studying patterns is not about memorizing them.
It's about seeing how these patterns make a whole new one which fit boldly in another.

Numbers, geometry, algebra aren't always involved. I would even include Catell and Raven matrix in math !

Some people can do complex derivatived, others can workout them.
The later are slower bur are guaranteed to go further and deeper.

Most if not all curricula are not besed on ability to workout things but in stimulu/response scheme : I give you a problem, you match it with a pattern and apply another one. Sometimes, this has to be done more than once.

But the true part is finding the pattern itself. While the first suffice for 9/10 people, for the last one, there is a feeling like something lost.

I did teach in private course, using material not used/covered at school and always got successful. Even if the student told me many time : "it is no use".
Yes, it is. If something new arise, you'll be more equipped.

The overall math level to complete the last secondary school is very very weak ! We do it the hard way.

More people are able to memorize than "workout", so curricula is adapted yo them.