Date: Nov 29, 2012 12:13 PM
Author: kirby urner
Subject: Re: Some important demonstrations on negative numbers

On Wed, Nov 28, 2012 at 5:14 PM, Robert Hansen <> wrote:

> On Nov 28, 2012, at 11:45 AM, Joe Niederberger <>
> wrote:
> "computation" absolutely did arise from noticing how the merely formal
> could faithfully adhere and correspond to the concrete. So it is "grounded"
> in that sense. There continue to be arguments of how best to ground the
> formalities.
> See, that is the gist. 1 + 1 = 2 is formal. Not merely formal, actually
> formal. It requires the student to understand and own a formal system. The
> set of integers and the additive operation. So what we are really talking
> about here is what actually occurs when a student takes ownership of a
> formal system. What difference does it make if one apple and one apple
> always makes two apples in the real world if 1 + 1 isn't always 2 in the
> student's head, all by itself? Over time, that formal system grows and
> becomes more involved and more self sustaining.

However, as we learned in New Math, 1 + 1 = 2 is premised on concepts of
base and we need to know what set we're operating in. You say integers and
it's good that you say that, as in a modulo arithmetic game, 1 + 1 + 1
might equal 1, as we're adding modulo 2. Or 1 + 1 might equal 10, because
we're using base 2.

I do not regret learning the above at a young age. I learned that
"addition" is not just one operation, nor is multiplication, but there's a
family resemblance across language games. Similarities in usage, allow us
to use the "addition" as the in-common name for these ops. That's how we
develop our formalisms.

Learning about bases is still highly practical, as computers work in base 2
and their set of integers is sometimes closed in the sense of modulo. In
some integer type implementations, adding 1 to a maximum flips you to a
maximum negative, so that you go in a circle.

In some math classes, playing with robotic devices such as the Arduino is
de rigeur (a required / expected activity), so learning about a closed int
type makes plenty of sense. Fat paper-wasting math textbooks that don't
cover that should not be where the money goes.

School boards that buy truck loads of heavy textbooks instead of Arduinos
and Raspberry Pis are likely made up of civilian know-nothings who waste
money out of ignorance.

These kinds of extra-curricular centers, where more practical / relevant
math is taught, are spreading, staffed by geeks who may or may not be
unionized teachers (more likely not these days): (Portland) (San Francisco -- one of our OCN faculty was
just visiting)

Students with this kind of background are more likely to know about
projects like this one: (has an MIT

> When you say "how to best ground the formalities" what you really mean is
> how to reach students unaware of the formalities, but the formalities
> themselves are grounded in one thing only, formal thinking (is that better
> than "reasoned thought"?). If you fail with the formalities then you have
> failed period and you fail with the formalities when you have failed to
> invoke formal thinking in the student.

I think formalities often come through with contrast. Compare formalities
with one another.

In Euclidean geometry, there's the formality of the infinitely tiny point,
so small that any point you make is too big to really be a point.

In non-Euclidean geometry, that concept may not survive intact. Perhaps
the formality is you have "lumps" and these are defined to always have an
inside and outside, so one may speak of a point's "interior" (Karl Menger,
mathematician, describes such a geometry).

There may be a concept of a zero point or empty point but it's only
fleetingly achieved i.e. there's a sense of passing through and never
pausing at zero. That sounds like a different metaphysics, and it is.

The important thing is to realize that a line going to infinity in both
directions, that is infinitely thin, and made of points of zero dimension
-- that's all metaphysics too, i.e. philosophy. These concepts are not
"proved" but are axiomatic. They're inherited from ancient Greek

Younger students may not remember "board games" but older adults do. When
you buy a new board game, or take one off the shelf at your assisted living
facility, you read the back of the box or under the lid for directions.
The directions spell out the rules of play.

Maths are a lot like that: a huge stack of board games that interconnect
with one another in various logical ways (or don't -- it's more of a
disconnected graph with islanded connected parts, if we want to bring graph
theory into it).

> You said that the "mapping" between formal thinking and the real world is
> important. How so? I spend hours, days, and even weeks pondering some
> problems with nary a single thought of the real world. But I agree, when we
> apply mathematics to the real world, that mapping is very important. But
> this is irrelevant to

Maybe you should give an example.

> my claim. If you have not developed formal thinking then any notion of
> mapping is moot, is it not? The real issue that I am concerned with here is
> what role does common sense, namely concrete examples, play in the
> development of formal thinking. After searching deep, the only role I can
> come up with is as an

The usual answer is special case examples have things in common that
remain, when you abstract away the differences, and these commonalities
are what we call the formal aspects.

For example, we look at one container after another, starting with simple
balls and/or polyhedrons.

When inside, the surface tends to curve toward us and focus reflected light
towards a center. There's a concentrating aspect.

When outside, the surface tends to bend ways from us and light is
dispersed. We look at this over and over in different special case
circumstances and from these experiences we develop the concepts of
"concave" and "convex" respectively.

What we notice about concavity and convexity is they go together i.e. are
two aspects of the same thing. This is fairly common in formal
conceptualization: two concepts will co-define one another.

For example, the United States is understandable as having an inside and an
outside. Inside and outside go together. The interface between inside and
outside is the world of borders, customs inspections, screening,
filtering. It's also the space of treaties, agreements, relationships with
other sovereignties. Drive across the border, into the Warm Springs
reservation, and you're in another nation with its own laws and practices.

> aid in communicating the basic properties of the formal system you are
> trying to teach and as a tangible check against which you test the formal
> reasoning you are developing. I gave examples of things (coordinate
> transformations etc.) that are entirely of formal origin and I could fill
> page after page of similar examples. This tells me that formal reasoning,
> once developed, is entirely grounded in its own domain, separate of the
> real world.
> Bob Hansen

You may make the mistake of thinking there's only one logic and only one
reasoning process that might be considered formal. We don't want students
to end up in a mental cul de sac of that nature, thinking "my way or the
high way" i.e. we don't want exclusionary (closed minded) thinkers to
result from our training and/or curriculum.

For this reason, I introduce many segments and topics designed to remind
that "all math is ethno-math".

This is especially important for North Americans, who tend to be
mono-lingual in many cases, and are often not well traveled.

Their perspective is narrow and parochial "out of the box" or rather is
"cosmic" at the outset (the natural human condition) but tends to be
narrowed and/or specialized really quickly by whatever state the student
lives in. This "narrowing by the state" is to be resisted and countered in
my book.

If they turn into grownups who say that 2 x 2 *must* be called "squaring" and 2
x 2 x 2 *must* be called "cubing" (instead of *may* be called), then we may
have failed them as teachers / trainers.