Second School

Work on creating four week
Lifting Strategy Workshops

Students often enter college without a comprehension of the foundations of set theory, arithmetic and algebra. My work suggests that a single reason may be found for this social reality. The single reason is that educational theory has become firmly grounded in a supply side theory. In spite of well meaning efforts by teachers and school systems, the individual is more often than not disenfranchised by educational practice. The mismatch between the mission of school systems and recent outcomes point to a thesis on why student preparation for college is poor. This thesis has been grounded in biological science in my 1988 PhD thesis and in published research and research posted on the web.

Scholarship suggests that student under-preparation may be associated with specific cultural impacts as well as failures in educational philosophy. The demand side theory takes the position that in both cases the structural fault is an absence of balance between supply side causation and demand side causation. The nature of this imbalance, and the mechanisms involved in human learning, is the subject of our research supporting a new demand side theory of learning.

My work is in an advanced stage. However, the results have been hard to publish into the mainstream literature. Part of the difficulty has been the wide scope of the work, work that involves cognitive neuroscience, immunological research, systems theory, physics, foundations of mathematics, philosophy, and computing theory. Classroom practice has also reached a mature stage.

This year, I have suggested that four week workshops, on the missing foundations, should be offered several times during each semester. I have recommended that enrollment eligibility be campus wide, and efforts made to motivate students to enroll in these workshops. The workshops will be tuition free, would have an enrollment process and require a work book. I also am working on a summer program that would assist colleges in recruitment and enhancement of student preparation, and students in the selection of colleges.

Initially, reasonable opposition to this workshop has come from three quarters;

* from some individual students who are under prepared and frustrated,

* from some faculty who may not understand the full theory, and

* from some college administrations who may have lost sight of what a college is for.

The concerns may be positively addressed.

* Students may be shown that their experience in high school and in their young lives has mislead them and now unnecessarily limits their potential.

* Faculty may be shown that new methodology may be put into place that alters the purposes of mathematics instruction. These purposes may be aligned with current cultural needs. A realignment of the curriculum and pedagogy may be seen as essential if the nation is to recover from a failed system in mathematics education.

* Administrators may be shown a positive path forward on what now appears to be intractable concerns. The results of a campus wide switch to demand side practice and a revision of curriculum may be seen immediately using outcome metrics.

Given the difficulty of problems from which reasonable frustration arises, we must prepare for several approaches if the full impact of demand side theory is to be felt. A national program designed to create web based infrastructure for a bridge between high school and college level instruction is being considered.

The origins of opposition to the demand side theory is one of the reasons why a National Bridge between high school and college may be developed as an independent not-for profit entity. The two primary approaches are

(1) an independent action using a corporate structure or

(2) working within the community of colleges.

Ideally, the demand side theory movement will be successful in both approaches.

Campus based work

We hope the lifting strategy workshops will be seen as a means to transform the nature of mathematics training on one campus (2008-2009). If this may be accomplished, the program here will serve as a proto-type for the national program. The history of the demand side theory included exploratory development of the principles of the pedagogy at three colleges; 1988-1990, 1993-94 and 2007-2008. The pedagogy has become mature and well defined.

Work this year continues based on this history and on work on human knowledge representation and management technology addressed within the government agency communities (1990-1993 and 1994-2006). Significant work remains on the design and implementation of the knowledge management technology being designed to support the demand side theory. This work is suggested in the next section, and may be discussed in detail in faculty meetings and seminars.

Seminars and talks at many campuses will be scheduled towards the end of this semester (Fall 2008) and during the next semester.

National Infrastructure

Looking forward, we see the evolution of new types of support tools.

* In the web based version, the enrollment process as well as supporting tools and resources would be managed via the second school web site. This web site is being developed to support a not for profit corporation and individual campus administration of freshman mathematics programs.

* Technical aspects of the web site is discussed in a tutorial on ontological modeling. <link> [1] This technology is itself innovative. <link> <link> One key element of this technology will allow students to easily scan, and upload to a private and password protected web site, presentations and classroom notes written in pencil on blank typing paper. As this occurs, students working within a workshop, or summer based bridge program, will co-create a single electronic textbook enumerating and presenting a modern curriculum for college level mathematics.

* A more advanced technology will use an electronic white board where portions of the board may be selected and automatically uploaded to the web, and then placed within a formal ontology. <link>

* The form of the student presentations follows a specific standard, examples of which will be posted soon. <link>

The topic mapping component

The ideal web based system will allow a student to enter any phrase or word, indicating one of the topics, and either get a statement asking for clarification or a specific introductory view of that topic. The topics will often focus on theory and notation rather than examples. One topic will lead to one or more additional topics and may in fact to linked eventually to every other topic in the set

C = { topics in the standard curriculum in Chapter } = { t_i | i = 1, 2, . . . n }.

The notation used above indicates that there are n topics listed and indexed by the counting numbers. This type of notation is easily learned by freshman students, once they begin to understand the new rules governing the demand side classroom.

Each workshop is a discovery process where students themselves create the topic descriptions, writing on blank typing paper, and then posting these to a common but secure and private web space. Using topic descriptions members of the workshop may share presentations with other members. Thus the mastery of a small set of notational tools is the first objective of the workshop.

As the software system supporting second school is completed, we will post examples of student's work (without private information such as the name of the student) to illustrate and prove the second school assertions.

Some notation about the Lifting Strategy

Let P = { notation, theory, application } = { a_i | i = 1, 2, 3 }.

The ideal web based system will also have a linkage or mapping between the cross product between C and P,

C cross P = {( t_i , a_j ) | i = 1, 2, . . . n; j = 1, 2, 3 }

and the National Council of Teachers of Mathematics K-12 focus elements. This cross product is an example of a conceptual framework <link>. In this case, the framework may be uses to align Lifting Strategy workshops with national standards.

The student presentations on blank typing paper has the form of a thematic exposition that takes a series of identified topics and weaves these into a demonstration that the student comprehends the curriculum, can use notation to talk about these topics, knows some of the underlying theory and can create examples from scratch that illustrate the topics.

Four week workshop

The focus of the proposed four week workshop is on the nature of arithmetic, set theory and elementary abstract algebra. This material has been developed by me, as the core curriculum supporting [2] the lifting strategy and demand side learning theory <link>.

The first part of the workshop is exceptionally challenging without having any re-requests. The elementary notions of arithmetic is developed after positional notation is understood.

Counting, addition, multiplication, subtraction, division, factoring and solving linear equations are re-learned but this time in arbitrary bases, other than base ten. [3] This allows all of the classical properties of arithmetic to be developed in bases other than ten, and thus provides both intellectually challenging material, novelty and depth. Set theory is developed and then compared to arithmetic, while introducing elements of college junior level mathematics major abstract algebra and elementary number theory.

The rules of demand side learning <link> are developed to include shifting responsibility so that students ask informed questions, outline the curriculum as a set of topics, and prepare to exposit topics of the individual student's selection onto blank, standard copy or printer, paper. Examples of student's work (with names obscured) will be posted at the second school web site <link> soon.

Comparison to college programs that "study to the standard test"

As faculty of mathematics, we all know and must acknowledge that success FIRST AND FOREMOST depends on students' actual knowledge of curriculum material, and that this knowledge comes primarily from high school and college classes. If this knowledge and comprehension is not there, focused efforts on test passing becomes questionable. As educators, it may make us feel uncomfortable to help student study to the test when these same students do not comprehend the essentials of the standard curriculum.

Why is that?

When standardized tests are studied "for", there is a reduction in the validity of the measurement. As we all know, testing taking has become an art where comprehension is too often overlooked. The reason may be that comprehension is difficult to measure, or that the system is not measuring comprehension because the parts of the system are designed to avoid teaching for the purpose of gaining comprehension. Yes, by helping students studying for a specific type of test the college will make short term improvements in measured skill.

Yet, the underlying problems with student motivation and willingness to comprehend mathematics are being not addressed. A long term strategy is needed, one that does not methodologically avoid and ignore the causes of current outcomes. This strategy is designed to alter the educational practice.

It is that this long term strategy that the Lifting Pedagogy is directed. My effort is to create a national program based on this work.

Constructivist and Socratic practice

As in a Socratic teaching process, the student is asked to construct questions. These questions are properly constructed only when there is an internal synthesis of a deep inquiry.

So in class lectures are about how questions in text books are made up. An example of an in-class inquiry is about how quadratic expressions are to be factored by inspection. Given that only a very small percentage, actually 0%, of all quadratic expressions may be factored by inspection, how do the text book authors find these examples? How are quadratics that are to be factored by grouping found? How are these two methods related, and what is the purpose of the quadratic theorem in the task of factoring. My students have written essays about these topics, complete with notation, theory and examples that they construct while not referring to notes or text books.

The students in my classes learn that algebraic factoring is related to factoring in arithmetic. When this is realized, a light goes on in their mind. When this light goes on often enough then a complete change in motivation is observed to have happened. When a critical mass of the students in class change motivation, the class changes and when the class changes the word goes out that something unusual is occurring.

Attendance is going up in these classes. Why?

What is happening in class is that students all of a sudden are "seeing" mathematics properly for the first time, and finding motivation from this. Because I am not overly concerned with attendance, students not attending are encouraged by those attending to attend.

Attendance can be forgiven. In fact, in demand side theory, all of the rules are directed at opening the doors to real mathematical thinking. Even after only six weeks students are now starting to strongly respond to the hope that they might actually comprehend why mathematics is and how it is thought about.

I look forward to regular Friday meetings, and unless the Division has a meeting scheduled I would like to present and host a discussion on the demand side theory and the lifting pedagogy on Fridays at 11:00.

I have informal workshops in Room 208 Stone Hall even day at Noon and at 1 PM. Please invite your students to attend. These workshops sessions are each 35 mins in length.

Footnotes

[1] Founders of SecondSchool.net are seeking capitalization funding to complete this technology system. Business Plan notes: http://www.secondschool.net/beads/secondSchool/1.html
[2] For purposes of the proposed CORE National Bridge between high school and college.
[3] For examples see the sample curriculum posted at
http://www.ontologystream.com/beads/QuestionOfAccess/AQA.htm