for Literature-based Mathematics
Curriculum & Support



Research Advances
The current traditional math curriculum -- The Pipeline Model

Curriculum analysts have characterized the current school math system as a pipeline design and have attributed much of the school´s failure in the math arena to the limitations of the pipeline mentality.

In this design, the student population is presented with a single path to follow, must pass each milestone in prescribed sequence, and if they are diverted from this route, have no re-entry point to the process and are lost to the system.

The casualty count of this rigid design has a horrendous cost to society as well as to the individual and has led to the development of a defensive layer of propaganda and abusive tactics to re-inforce the entrenched position of the system against criticism for its failures. Most telling evidence in this fray beyond the surveys of wasted skills is the starkly contrasting view of mathematics in the eyes of those most knowledgeable in the subject.

 Escape from the Pipeline -- Mathland and Reality

The reality of what math is and how it historically has developed totally undermine the legitimacy of the pipeline and offer a whole new approach to math curricula. Professional mathematicians view mathematics as a world with many diverse territories as in Ivars Peterson´s Mathematical Tourist. Mathland is far from linear with many contiguous boundaries between conceptually diverse cultures, with many bridges and amazing commerce of ideas.

The origins of these vast knowledge areas with barely outlines yet explored are equally varied, from agriculture and astronomy to warfare and mortality. The common theme is the skill of pattern discovery, which is the theme of Keith Devlin´s Mathematics: The Science of Patterns, which gives a taste of the art of many mathematicians and the history of their work´s origin.

Although the pipeline´s flaws, the need, and the replacement paradigm are acknowledged by education researchers and they promote change from the pipeline to mathland, the machinery and entrenched personnel will take a wholly unacceptable time to turn around, leaving the public to depend on individual initiative to open new doors for their family members and selves.



The Mathland Curriculum Direction -- Diversity & Empowerment

The goals to be sought in all Mathland curricula are pattern appreciation, numeracy, logic and quantitative skills for personal life empowerment. Historically math develops to solve the developer´s dilemmas hence personal life empowerment is the direction curriculum development should move in.

Personal empowerment comes from a high probability of use in living so selecting topics by encountering their need in the individual´s chosen lifestyle, or in lifestyles that sound intriguing, is a reliable method of developing a Mathland curriculum.

In order to expand the content or a skill´s development beyond the discovery of its use in a chosen lifestyle, the ideal is to derive examples from individual personal experience. The structuring of the math pattern from a real life situation is a vital skill absent from the current artificiality of pre-abstracted problems characteristic of programs limited to standard textbooks. This important element of traction where the "rubber meets the road" is most likely to receive attention from the learner who is pursuing personal empowerment.

Developing a Mathland Compass -- The Power of The Story and Context

To facilitate learning the range of math concepts and results discovered and deemed major (theoretically) in a field, the techniques of encapsulating the high points for an executive overview or the techniques of tracking the history of some significant struggle in the concept´s acceptance have been effective.

For example, the main concepts of trigonometry can be developed in an afternoon for a learner stalled in understanding elements of physics. Similarly, an analysis of the sole winning strategy in longterm, zerosum games is an impressive highpoint in game theory, that opens the door for gamers, strategists and activists while displaying a daunting amount of unexplored territory in mathland.

Alternatively, the history of the struggle for acceptance of Maxwell´s equations can inspire on many levels. Canvassing mentors and professionals in the learner´s field of interest for those books and stories that infused their personal experience with meaning will uncover a wealth of motivation.

This recognition of the power of story-telling and mentoring has led to a new/old form of transmission of mathematical insight and skill, the literature-based curriculum. Not only is this approach to the accumulation of skills and insight natural, and part of our species´ own original mode of education, but the depth is automatically adjusted by the learner who skims the deeper areas of the subject until they get their bearings without losing the threads of the big picture or the joy of identification. These gifts of the story, the grand overview and the image internalization, make the delight of understanding more likely and the repetition and penetrating study required for skill acquisition seem like play.



Becoming a Knowledgeable Tourist -- Numeracy

Numeracy, like literacy, has many facets and develops with exposure. Getting a feel for a part of mathland can be as pleasant as subscribing to a special interest book club. Experiencing the delight, as a native from mathland describes the scenery and relates the tidbits of personality and history, in a conversational popular book makes more in-depth research intelligible and potentially establishes links to many concepts already in the learner´s knowledge base. In this way, hyperspace concepts can become part of the learner´s manipulable math vernacular by cruising Michio Kaku´s book club recommended exposition of the history of the idea´s development.

Another productive exercise is the exploration of table books and reference works in math. The learner gets a feel for the diversity of resources and how information is organized, gaining both knowledge of the applied territories as well as pattern skills and reference work skills.

The real Mathland is a developing, growing world. As it´s always been. And those at the frontiers are where the excitement is. The distance to the frontiers is not always far. Access to mentors is a natural treasure in finding where the opportunities wait.

Whether the frontier is near because of your experience, interest or viewpoint, the way to develop the skill to create math is for learners to frequently do it themselves when they find something that excites them "to find out". Even when it´s been done before. A pleasant struggle is instructive, much more so than having the answers served up pre-digested when the actual experience is attainable.

In the balance, are time and desire. Life is too precious to rebuild every math bridge even if the learner´s goal is to become a recognized native instead of a knowledgeable traveler. Arbitrary goals for some supposedly typical learner to build a prescribed set of bridges, when what the individual learner really needs is a passport to use the bridge, is offensive to the respect owed the learner and to the learner´s chosen lifework.

There is an inherent lack of respect in the pipeline curriculum, that it is imperative to replace. Even though knowledge of the geography of mathland, however acquired (assuming the learner survives), is useful whether the learner is a native or a tourist, treating the learner as a free-person is more productive, qualitatively (in the broader picture of mathland learned) and quantitatively (in the number of successful learners) -- rather than a system in which the learner is an unwilling slave on a bridge-building gang making a prescribed sequence of temporary duplicates of existing bridges along the traditional pipeline route. The benefits of Mathland are more sure and more fine tuned. No more lost learners, no more social costs because the tourist is an accomplished math user in his own field and an educated consumer of other native´s math skills.

This is not an untested relationship, as many professions considered mathematical, like engineers and actuaries, don´t learn to build math theory. It is already accepted that gaining the quantification skills for a chosen profession does not require learning to build math bridges.



Recognizing the Myriad Faces of Known Math

Even within the traditional mathematics programs, the myriad faces of mathematics begin to emerge for the student, provided you include the doors that open for the student in graduate school and research reading. There´s real analysis, complex analysis, differential equations and measure theory, functional analysis and approximation theory among the analysis family. In the algebraic family are algebraic theories of all sorts, lattices and groups, rings and more. There´s topology, math logic, set theory, number theory and crossbred beasts like algebraic topology.

So much more, and on the perimeters of the theory areas are the more applied areas like actuarial science, financial planning, econometrics, game theory, knot theory, tiling, cybernetics, cryptography, theory of ruin, formulas for fairness, quantitative business, operations research, decision analysis, statistics, probability, ballistics, engineering, stochastic processes, chaos theory, complexity and fractals.

And each family has its own *culture*, so many interesting places to visit. Some with long histories, some fledgling. The vaunted, feared calculus is just the applied grandsire of the more esoteric theoretic real analysis. Gambling´s offspring, probability, belongs to the measure theory clan. That miserable pipeline just makes both those limited to its confines as well as those off the route feel isolated and alienated.



Sensing the Omnipresence of Math

Once you´ve begun touring, you begin to sense the omnipresence of mathematics. It has emerged from so many diverse disciplines. The math in weaving is an ancestor of computing. Mathematical patterns appear in physiology, clinical psychology, construction, biology, demographics, operations management, warehousing, graphic design, purchasing... even in creative writing and story development.

Since the applications are math´s origins and are ubiquitous, so are the opportunities for math development. You will find it lurking in art and music and craft work. Especially games. Working with children and gaming leads to probability, statistical conjecture, confidence intervals, random walks so totally naturally. Children´s fascination with nature will carry you into discussions of chaos and fractals and complexity, to activities like building models and orreries. The awesome spectacle of natural disasters challenge us with resource management models, sustainability calculations, risk analysis, family survival unit studies. There is no legitimate fear that mathematics will not develop if learners are allowed to explore the world of their experience. And the delight of discovery and adventure in mathland is a treasure to share with our learners.


Appreciation of the Beauty

As tourists in mathland, the learner develops an appreciation of the beauty of mathland and the world it comes from. Math is an art and there are styles characteristic of each area because of the media the mathematicians there are working with. Mathematical induction, proof by contradiction, essential singularities, limits. But even within an area, there are diverse approaches, which is what it takes to unlock new areas. Learners become conversant in styles that appeal to them.

Even the natives have distinct preferences, some analytic, some synthetic. Meeting the personalities, collecting the stories of discoveries, the histories of the *big* conflicts, the secret societies like the Bourbaki, the contests for fame, the powers who drove the excitement, the powers who stifled, the loners who struggled, the ones who made it, the costs, the adventurers who developed weaponry for military powers. These fascinating stories are accessible to readers with book club memberships if natives are not as close as neighbors. Libraries are golden with resources.


Gender & Environment Clues

Is there a gender issue in mathematics? Yes, but not an intrinsic one. Categorically, without reservation, women are as capable mathematically as men.

So why do the data sometimes look so skewed? Why do the studies of IQ and aptitude show a preponderance of males at the high end? Why do the history books and math conventions seem almost exclusively male?

In spite of all the skewing forces in our society and our ancestry, the tests actually show no difference in average ability; the apparent preponderance of male high-end performance is balanced by a similar preponderance of male low-end apparent inability. Assuming the tests measure what they´re purported to measure.

Nor is demonstrated performance more reliable than predictions of aptitude which research such as Valian´s Why So Slow amply demonstrates. The book documents the mind viruses from unfiltered environmental and historical sources that vitiate good intentions, limiting responses, and permit the greed or power-motivated in select decisionmaking positions to perpetuate the distortions in our social systems.

Hugely significant are the mathematical simulations that show that multi-tier corporations with a mere 1% bias in promotions would likely deteriorate from an equitable initial position to a severely skewed organization within one generation. Considering that our initial workforce distribution is far from proportionately equitable and the biases greater than a percent or two, the dynamics indicate neither reliability nor promise for measuring anything labeled ´demonstrated performance´.

The most profound source of measurement skewing though is the definition of mathematics. The establishment legitimized the pipeline and draws its power from areas designated as "male". Conversely, any area that becomes a source of power is confiscated or annexed as "male". Computers are an example where the earliest theorists and programmer/operators would be today considered "surprisingly" female, especially since the pool of women working outside the home was variously restricted. Until the field gained acceptance, then the "detail oriented" requirement became a need for "engineering aptitudes". If anything, today´s over-civilized hardware requires no more machine-fathoming aptitudes than the early program designers demonstrated. These historical patterns of rationalized gender shifting have been seen in many fields, from telegraphers, to teaching, to librarians.

Since the establishment sanctioned math curriculum pipeline is built through "male" work areas, it is no surprise that it´s faces are male. What´s being measured is pipeline usefulness, not individual ability. The realization that the fundamental essence of math is pattern recognition and manipulation frees the learning process from this quagmire of prejudice of all stripes. From the gender viewpoint, there´s math in all work areas, in various states of application and theory development.



But the gender-education research has even more to offer. Besides ability questions, there´s a whole raft of environmental baggage to deal with, from the dynamics of organizationally overlooked harassment and sociological hazards of imbalanced groups, to the misplaced testosterone of individual behavior in mixed group endeavors and the prevalence of conflicting expectations.

The most interesting antidote for these impediments to have turned up in recent research is the startling performance of women whose teen and college experience was in all-girl schools and classes. The results show that the substantial majority of women in top leadership positions in any field had come from all-girls schools. Nor was there evidence of a "Radcliffe" effect. Confirming this dynamic, girls have been shown to do substantially better even in traditional math classes that were all-girls, even when the school itself was coed.

Comparing this to the research data behind SchoolGirls and Reviving Ophelia which underscore the extensive problems in classroom and school dynamics, leads to the question of what´s different about all-girl classrooms. Information is accumulating to suggest that older adult classroom dynamics are more similar to all-girl classrooms, possibly because of the earlier maturity of young women. Among the analyses of successful adult learning environments is Peak Learning, a book on strategies for thriving, which is what a replacement curriculum for the pipeline should be seeking... peak learning.



Peak Learning, Sac´Caiders and Unschoolers

The fundamental tactic of Peak Learning is to acquire the skills of an autodidact in order to enter the classroom or workshop knowing the material in advance. And contrary to the author´s expectation, this is demonstrably doable by children, suggesting a complete role reversal in the classroom. This stunning realization leads to a curious, but parallel, reversal that happens in the academic world in graduate school, the academy´s pinnacle.

The typical master´s program requires the demonstration of self-acquired learning in your specialty in the form of a suitable thesis. The PhD, or other doctoral programs, require the demonstration of the ability to create a significant addition to the knowledge base in your specialty in the form of a successfully defended dissertation. No amount of classroom work prepares you for these tasks, a fact that academia tacitly acknowledges in their promotion of post-doctoral work. "Once is not enough." Explicitly, the learner has done it only once and that skill requires more practice, which the learner´s classroom work is apparently poor preparation because the doctoral skills are the reverse of all those years of classroom experience with few exceptions.

If self-acquired competence and creativity in information development (SAC´CAID) are the ultimate goal of academic life, why is it not the theme of the classroom? Is it not doable at an early age? Must the classroom be limited to indoctrination because of student incompetence? Far from it.

The most momentous message from that portion of the homeschooling population, currently dubbed "unschoolers", is the demonstration of this doability. The fact that major universities, examining unschoolers´ portfolios, has led to active recruitment of such a bizarre minority establishes that unschoolers have succeeded, not only in becoming sac´caiders in their areas of special interests -- some academic or some not -- but have done so from the beginning ages, K-up with comfort and ease in nearly every imaginable variation of circumstances. A reading of their experience sharing in the various homeschooling journals, illuminates clearly the progress made in understanding and dealing with issues, from management of "downtime" to progress assessment, from finding their direction to organizing a wealth of resources. An impressive demonstration compatible with mathland´s philosophy and peak learning.



Crucial Life Skills
Crucial Decisions, Projections, and Planning

Of all the alternative math areas that could be recommended as a life skill, decision analysis, mathematical simulations, and operations management would rank very high on empowerment.

Choosing the elements of a lifestyle is full of complexities, from imperfect information to merely probable outcomes, from intricate financial maneuvers to non-quantifiables, yet decision analysis was developed for just such situations. With mathematical simulations to explore the financial maneuvers, with decision trees to evaluate the significance of probable outcomes, with sensitivity testing to focus attention on pivotal variables, and with tactics for calibrating intangibles, it´s difficult to imagine a more valuable preparation for life´s turning points.

Developed in the 1970´s, decision analysis is part of recent explorations of the math in social issues, economics, communications and organizing. Offering insights in negotiations, strategy, and risk assessment, the techniques liberate the user from those scenarios plagued with fear and complicating emotional issues. Comfortable in the world of high-tech expert assessments, it´s equally at home in the riddles of motherhood.

Similarly, project management techniques were developed in the world of warfare but have applications in businesses, organizations and household management. Developing the skills to see problems in terms suitable for powerful tools is play when life´s thrills come from reality experimenting.

Best of all, the key to today´s implementation of these concepts is the development of the art of spreadsheet design to accommodate the complexities. The spreadsheet is an amazingly powerful tool that welcomes the novice as well, and is generally part of the suite of software initially loaded on every computer.




For Whom
...for Teens -- because their life is filled with turning points now!

Using the leverage of literature-based introductions, teens have a wealth of decisions to apply the concepts to, as well as an urgent need to communicate and negotiate solutions. Being able to examine the logic and values behind their choices facilitates rational communication with adults; and conversely, adult arguments can be similarly examined.

...for Mothers -- because operations management is their real expertise!

It is imperative that recognition of the high level of skill inherent in home and family management be accorded to those most talented in this form of applied mathematics. Though comprehension of this reality is beginning to surface in South America, it has yet to emerge on the scene in the US.

Like underwriters´ intuitive mathematical skills in risk assessment and valuation of intangibles, expert home management status is acquired with extensive practice. Once acquired, it translates well into business scenarios, whether home-based or large scale. The ease with which the home-based business trends developed bear testimony to the natural connection.

Access to the formalisms not only would promote this growth of independence but would offer opportunities for women to openly contribute to math concept development applied to those issues mothers care about. And last but not least, with the content knowledge in "women´s" work areas, those mothers who unschool will have in hand the formal concepts for those mysterious "teachable moments".

...for Artists, Activists and Counselors -- because their work impacts our futures!





Math Series
Penelope Press presents a comprehensive but flexible approach

Beginning with a literature-based text in a variety of formats, to suit different learning styles and content presentation requirements, the learner takes a tour of an area of mathland where the crucial life skills of decision analysis, mathematical simulations, or project management are practiced. Surrounded by an extensive development of context, the math concepts retain their contact with reality, from which the learner observes math intuition in action.

After a period of incubation, the learner is ready to explore the techniques applied in everyday and many life scenarios. To foster the development of these explorations Penelope Press holds occasional workshops online and offers our readers membership in an e-studygroup operated by the author of the series, J.H.Raichyk, PhD to discuss case histories.

Dr. Raichyk is a professional mathematician and decision analyst with over 25 years of applied experience in both the insurance and retail industries, as well as over 15 years of extensive reading and experience in education trends as a homeschooling mother. Active in alternative energy, natural landscaping and women´s tech, health and shelter issues, she lives with her children, now grown, in SW Ohio. She is also the author of the web-zine, LSR, and occasional speaker at conferences, meetings and workshops.

The titles in the Math Series reflect exciting forays into future trends and, as such, explore a world relevant to future citizen activists: We at Penelope Press have been observing the language and math skills of sac´caiders, autodidacts and unschoolers for an extended period and are convinced that the reading habits of these learners is not well served even by the language arts curriculum standards, much less the math programs. As a result, we have opted to serve this audience in the full glory of its learning development.

Our texts are mathematics for the verbally fluent, the linguistically talented. For those seeking the enjoyment of wordplay, phrasing finesse, role-playing imagery, and theater arts, while absorbing the powerful but elegant simplicity of the mathematics designed to implement the dreams of this generation of cultural creatives, our future rainmakers of the world.

Lastly we should emphasize that replacing the pipeline is not going to be uncomplicated by doubt but it is important to your success in a changing world. We must keep in mind that we have been indoctrinated, especially the current generation of parents and elders.

The odds are high that those who were lost from the pipeline, and that´s a large majority, will lack the confident intuition to seize the moment when math emerges in their child´s life -- the innocent comment. The butterfly will emerge and be gone unnoticed. How will children see it though they sensed its import?

It will take persistent reminders and the more images of such unconventional math experiences the better. What better support for a parent than learning to devine the shapes and use the concepts of an area of math that permeates everyday turning points, a desireable life skill in itself for any adult as well as their children.

Even those who successfully navigated the pipeline are not guaranteed that their children will follow easily in their steps through the pipeline. Without broadening their math skills to include a repertoire relevant to their children´s chosen path, transference of parental past experience is unpleasant. Why not learn together? Absorb the experience of those who have "been there" via literature-based math; then share the discovery of power and pattern in your own life.

Come explore the math that navigates the turning points in life, validates intuition and powers the dreams of planners. Take us home tonight.