[ a first draft published to MathFuture, a Google Group ]

My talks at PyCons (Python conventions) and OSCONs (Open Source conventions) going back show me mounting the stage to decry "calculus mountain", by which I meant the obstacle course and sometimes source of disappointment occasioned by the "forced march" through "delta calculus" if wanting to get into a four year college, and by extension into a STEM field (or "work for NASA" as some put it).

My contention, in those years, was those like me who'd made it over calculus mountain -- I got into honors calc at Princeton, skipping the regular, had Thurston the topologist for a prof; we used Spivak's,

An example of such a "cool pool", no delta calc required, is Group Theory. Permutations at play. More generally, what we today call "discrete mathematics" was screamingly relevant, in terms of the doors it opens, compared with "integration by parts", or so I believed. I found other voices sang with mine. I was part of a choir. I also joined forces with those wanting to teach "how to code" (computer program) for core credit.

Fast forward and I'm in Oregon, joining a small lobby group to say to the legislature: "You know those three years of math you require for a high school diploma? We want to bring in discrete math topics and open up more space in STEM than just pre-calculus / calculus, whaddya say?" As far as I can tell, pretty much all the states said "OK" in unison i.e. they had no concrete objections to math-credit computer science. So ruled without objection, right?

When it comes to marketing the possibility in practice however, that's another matter and big publishing gets involved. I like to float mnemonics, easy to use -- but we hope not misleading -- memes. Calling the conventional pre-calc / calc track "delta calculus", as I do above, is not how most people write. The letter "delta", though used in differential / integral calculus, has not been made to "brand" it in that way. High school calculus, as typically taught, does not go by any Greek letter.

My innovation, then, was to cast "delta calc" versus "lambda calc" as two flavors of "calc" (computation), like chocolate and vanilla. No one said a given student can't do "swirl" (both flavors mixed together).

Lets be clear: lambda calc already exists, one might point to the Princeton Institute for Advanced Study, a couple generations back: Alonso Church and company. So where do we want to take it next?

Lets remember that so-called "delta calculus" was at one time esoteric and not introduced pre-college. In formalizing the work of Newton and Leibniz, as filtered through generations of refinement, thank the French, we got it into the form of pre-college deliverables, a spiral, a ladder, featuring Riemann sums and Leibniz notation for derivatives, first, second, third and so on. Different school systems have different histories. Many narratives criss-cross (inter-weave).

In canning calculus ("canning" as in "canning tuna" not as in "firing from a job") for the pre-college crowd, we sliced out most of the history. We take for granted that mathematics, being a "universal language" is somehow too eternal to be subjected to a merely tempo-real -- as in historical -- treatment. That's considered aftermarket trade book reading. Textbooks must peek at history only in sidebar, or in footnotes maybe.

Whatever the rationale for so sanitizing the subject, we neglect a most important bridge to the humanities i.e. here is where C.P. Snow's chasm, betwixt the humanities and sciences begins to yawn. History provides at least a rope bridge across, whereas many find themselves trapped, on one side or the other (reading maths or reading history, but never the twain together thanks to illiteracy barriers).

Were we to restore more of the history, we would discover more of the all too human drama of contention. But is debate a bad thing? Isn't moving forward a matter of dialog, or "dialectic" as the educated say?

Mathematics has not evolved quietly, without argument. Newton and Leibniz argued with each other (a priority dispute) while Bishop Berkeley attacked the whole idea of infinitesimal quantities and proofs based on them (a conceptual integrity dispute). Cantor and Kronecker took different sides in some arguments. There's ferment, disagreement, or at least alternative ways to go.

But that's precisely what gets white washed in going over this material: schools find it expedient to agree there's at least one subject on which everyone agrees. Or, if they don't, at least none of the boiling-over arguments should touch the kids. That math is contentious gets "dirty secret" or "in the closet" treatment. But Thomas Kuhn in his talk of "paradigms" at least made it OK / safe to question the caricature of a "steady advance" did he not? So why be so timid? Why all the shielding?

Anyway, in rolling up "lambda calculus" into a more popular form, like was done already with the Newton - Leibniz stuff, I've focused on "composition of functions" as the primitive notion, with the multiplication operator very soon introduced as a "composer operator", so that we further develop that sense of polymorphism around operators, the ground of Abstract Algebra.

A permutation, a mapping of objects to themselves in another or same order, is a primitive function, a set of ordered pairs. It's one-to-one, bijective. And permutations may be "chained" (composed). As a topic, they thicken the soup of whatever computer language, giving a gym to work out in.

"Why use the multiplication operator at all?" That's where the "cool pool" of Group Theory comes in. This is material we currently try to get into around Algebra 1 and 2, just a little, but we're in a hurry to dive into delta calc.

We have no time for passing functions as arguments to other functions, as we do in Python and other languages (that's a good introduction to delta calc too, through the gate of functions with function-arguments).

But now we

I'm under no illusions that with the snap of some fingers, even more than just mine, these "reforms" might be injected in short order. Rather I'm providing a road map for like-minded to reference, when explaining to the general public or intelligent layman what the strategy is. "All the computer stuff we don't currently manage to squeeze in? -- we've got a way now, and I can explain it in terms of two Greek letters, lambda and delta".

I've "rehabilitated" obscure disciplines before, too early to assess with what success. General Systems Theory (GST) is all over the place (somewhat like Tensegrity), as a management philosophy, as a kind of ecology, who knows? It has a high caliber pedigree but where does it go from here?

I noticed how Economists and Economics tend to have monopoly status when it comes to advising the financial markets regarding guns vs. butter, and suggested GST muscle in under the banner that "monopoly breeds inefficiency" owing to lack of serious competition. GST was about giving Economics a run for its money. Still is. That's easy to understand, no?

Courses in GST could just as well provide rungs of that "climbing some business world ladder" as more science-oriented than Econ in some dimensions, including around issues of climate change we might hope.

Finally, another axis or spectrum I've contemplated, as have many, is what oft goes by "left brain" versus "right brain" as a dichotomy. My track record is riddled with slides talking about "lexical versus graphical" by which I somewhat mean the same thing as the brain hemisphere people do. We're talking about bridges again.

In STEM a goal is to have noodling-with-symbols (call it "algebra" or "being lexical") match up with visualizations and other experiential presentations or summaries. We want to understand what we're looking at when interpreting all that data.

Control panels, dashboards, instruments, sensors... we have a kind of model, view, controller architecture to consider, where what we reason about and codify using semi-numerical algorithms is the model, the business logic, and what we view and measure is feedback regarding our direction, as a company or enterprise or whatever.

We hope for some kind of decision-making or steering capacity, where choosing a more promising direction, over a less promising one, remains a possibility. We're hoping to be more like pilots, not just witnesses to the inevitable, spectator-fatalists with no active role. "Activism" is not a negative, but informed and effective activism is better yet. "Passivism" is not an English word, but needs to be, as many are militantly passivist in their anti-activism. I think "reactionary" is getting tired and needs a rest.

One of the best left-right i.e. lexical-graphical connectors I've found is using string substitution in lexical computer code to build a script that, when rendered, provides a ray tracing and / or perhaps a 3D printable object. VRML and POV-Ray scene description language were often my two top choices, with similar choices in Elastic Interval Geometry land (a branch I was following).

The 2D fractal, the Mandelbrot Set in particular, coupled with some historical timeline information, is a perfect topic, a sweet spot. A strong lambda calculus course could set its sights in that direction: doing the vector math lexically, with overloaded operators (like + and *), yet driving graphics on the screen. Gerald de Jong's "creatures" provide a great example, of "math puppets" turning logic into animations.

Another approach to bridging model-lexical with viewable-graphical is to simply build up the skills to create a dynamic web page, where things happening graphically are driven by things happening lexically in the code. In my Digital Mathematics outline, I cover that in "Supermarket Math" which would cover "e-commerce" (but regular commerce as well, as brick and mortar stores use SQL just as much). Mine is but one more sandcastle on this pretty big beach -- just take a few ideas, flatter me by copying.

The goal is to forge these left-right connections, even as we bridge the C.P. Snow chasm by remembering to share more history.

My talks at PyCons (Python conventions) and OSCONs (Open Source conventions) going back show me mounting the stage to decry "calculus mountain", by which I meant the obstacle course and sometimes source of disappointment occasioned by the "forced march" through "delta calculus" if wanting to get into a four year college, and by extension into a STEM field (or "work for NASA" as some put it).

My contention, in those years, was those like me who'd made it over calculus mountain -- I got into honors calc at Princeton, skipping the regular, had Thurston the topologist for a prof; we used Spivak's,

*Calculus on Manifolds*-- found on the other side some "cool pools" i.e. an oasis of other mathy topics that*could have been*just as well taught in high school, providing alternatives to those not turned on by calculus.An example of such a "cool pool", no delta calc required, is Group Theory. Permutations at play. More generally, what we today call "discrete mathematics" was screamingly relevant, in terms of the doors it opens, compared with "integration by parts", or so I believed. I found other voices sang with mine. I was part of a choir. I also joined forces with those wanting to teach "how to code" (computer program) for core credit.

Fast forward and I'm in Oregon, joining a small lobby group to say to the legislature: "You know those three years of math you require for a high school diploma? We want to bring in discrete math topics and open up more space in STEM than just pre-calculus / calculus, whaddya say?" As far as I can tell, pretty much all the states said "OK" in unison i.e. they had no concrete objections to math-credit computer science. So ruled without objection, right?

When it comes to marketing the possibility in practice however, that's another matter and big publishing gets involved. I like to float mnemonics, easy to use -- but we hope not misleading -- memes. Calling the conventional pre-calc / calc track "delta calculus", as I do above, is not how most people write. The letter "delta", though used in differential / integral calculus, has not been made to "brand" it in that way. High school calculus, as typically taught, does not go by any Greek letter.

My innovation, then, was to cast "delta calc" versus "lambda calc" as two flavors of "calc" (computation), like chocolate and vanilla. No one said a given student can't do "swirl" (both flavors mixed together).

Lets be clear: lambda calc already exists, one might point to the Princeton Institute for Advanced Study, a couple generations back: Alonso Church and company. So where do we want to take it next?

Lets remember that so-called "delta calculus" was at one time esoteric and not introduced pre-college. In formalizing the work of Newton and Leibniz, as filtered through generations of refinement, thank the French, we got it into the form of pre-college deliverables, a spiral, a ladder, featuring Riemann sums and Leibniz notation for derivatives, first, second, third and so on. Different school systems have different histories. Many narratives criss-cross (inter-weave).

In canning calculus ("canning" as in "canning tuna" not as in "firing from a job") for the pre-college crowd, we sliced out most of the history. We take for granted that mathematics, being a "universal language" is somehow too eternal to be subjected to a merely tempo-real -- as in historical -- treatment. That's considered aftermarket trade book reading. Textbooks must peek at history only in sidebar, or in footnotes maybe.

Whatever the rationale for so sanitizing the subject, we neglect a most important bridge to the humanities i.e. here is where C.P. Snow's chasm, betwixt the humanities and sciences begins to yawn. History provides at least a rope bridge across, whereas many find themselves trapped, on one side or the other (reading maths or reading history, but never the twain together thanks to illiteracy barriers).

Were we to restore more of the history, we would discover more of the all too human drama of contention. But is debate a bad thing? Isn't moving forward a matter of dialog, or "dialectic" as the educated say?

Mathematics has not evolved quietly, without argument. Newton and Leibniz argued with each other (a priority dispute) while Bishop Berkeley attacked the whole idea of infinitesimal quantities and proofs based on them (a conceptual integrity dispute). Cantor and Kronecker took different sides in some arguments. There's ferment, disagreement, or at least alternative ways to go.

But that's precisely what gets white washed in going over this material: schools find it expedient to agree there's at least one subject on which everyone agrees. Or, if they don't, at least none of the boiling-over arguments should touch the kids. That math is contentious gets "dirty secret" or "in the closet" treatment. But Thomas Kuhn in his talk of "paradigms" at least made it OK / safe to question the caricature of a "steady advance" did he not? So why be so timid? Why all the shielding?

Anyway, in rolling up "lambda calculus" into a more popular form, like was done already with the Newton - Leibniz stuff, I've focused on "composition of functions" as the primitive notion, with the multiplication operator very soon introduced as a "composer operator", so that we further develop that sense of polymorphism around operators, the ground of Abstract Algebra.

A permutation, a mapping of objects to themselves in another or same order, is a primitive function, a set of ordered pairs. It's one-to-one, bijective. And permutations may be "chained" (composed). As a topic, they thicken the soup of whatever computer language, giving a gym to work out in.

"Why use the multiplication operator at all?" That's where the "cool pool" of Group Theory comes in. This is material we currently try to get into around Algebra 1 and 2, just a little, but we're in a hurry to dive into delta calc.

We have no time for passing functions as arguments to other functions, as we do in Python and other languages (that's a good introduction to delta calc too, through the gate of functions with function-arguments).

But now we

*do*have that time, because we have a fork in the road and the freedom to follow the lambda calc road instead, or in addition, to the delta calc road. The lambda calc road is certified legal and open to traffic, we just need more teachers to help out as guides.I'm under no illusions that with the snap of some fingers, even more than just mine, these "reforms" might be injected in short order. Rather I'm providing a road map for like-minded to reference, when explaining to the general public or intelligent layman what the strategy is. "All the computer stuff we don't currently manage to squeeze in? -- we've got a way now, and I can explain it in terms of two Greek letters, lambda and delta".

I've "rehabilitated" obscure disciplines before, too early to assess with what success. General Systems Theory (GST) is all over the place (somewhat like Tensegrity), as a management philosophy, as a kind of ecology, who knows? It has a high caliber pedigree but where does it go from here?

I noticed how Economists and Economics tend to have monopoly status when it comes to advising the financial markets regarding guns vs. butter, and suggested GST muscle in under the banner that "monopoly breeds inefficiency" owing to lack of serious competition. GST was about giving Economics a run for its money. Still is. That's easy to understand, no?

Courses in GST could just as well provide rungs of that "climbing some business world ladder" as more science-oriented than Econ in some dimensions, including around issues of climate change we might hope.

Finally, another axis or spectrum I've contemplated, as have many, is what oft goes by "left brain" versus "right brain" as a dichotomy. My track record is riddled with slides talking about "lexical versus graphical" by which I somewhat mean the same thing as the brain hemisphere people do. We're talking about bridges again.

In STEM a goal is to have noodling-with-symbols (call it "algebra" or "being lexical") match up with visualizations and other experiential presentations or summaries. We want to understand what we're looking at when interpreting all that data.

Control panels, dashboards, instruments, sensors... we have a kind of model, view, controller architecture to consider, where what we reason about and codify using semi-numerical algorithms is the model, the business logic, and what we view and measure is feedback regarding our direction, as a company or enterprise or whatever.

We hope for some kind of decision-making or steering capacity, where choosing a more promising direction, over a less promising one, remains a possibility. We're hoping to be more like pilots, not just witnesses to the inevitable, spectator-fatalists with no active role. "Activism" is not a negative, but informed and effective activism is better yet. "Passivism" is not an English word, but needs to be, as many are militantly passivist in their anti-activism. I think "reactionary" is getting tired and needs a rest.

One of the best left-right i.e. lexical-graphical connectors I've found is using string substitution in lexical computer code to build a script that, when rendered, provides a ray tracing and / or perhaps a 3D printable object. VRML and POV-Ray scene description language were often my two top choices, with similar choices in Elastic Interval Geometry land (a branch I was following).

The 2D fractal, the Mandelbrot Set in particular, coupled with some historical timeline information, is a perfect topic, a sweet spot. A strong lambda calculus course could set its sights in that direction: doing the vector math lexically, with overloaded operators (like + and *), yet driving graphics on the screen. Gerald de Jong's "creatures" provide a great example, of "math puppets" turning logic into animations.

Another approach to bridging model-lexical with viewable-graphical is to simply build up the skills to create a dynamic web page, where things happening graphically are driven by things happening lexically in the code. In my Digital Mathematics outline, I cover that in "Supermarket Math" which would cover "e-commerce" (but regular commerce as well, as brick and mortar stores use SQL just as much). Mine is but one more sandcastle on this pretty big beach -- just take a few ideas, flatter me by copying.

The goal is to forge these left-right connections, even as we bridge the C.P. Snow chasm by remembering to share more history.