Two levels of workshops
will be
scheduled as needed.
Purpose: The workshops support
students wishing to deepen personal
understanding of the foundations to set theory, arithmetic and
algebra; or to understand first semester algebra at a deeper
level.
Length: The length of the workshops
are four weeks. Each
student will make a commitment to attend as if a regular class.
These workshops are open to any student enrolled in any freshman
mathematics class.
Basic workshop: In
this year's basic workshop uses topic mapping and demand side theory to
teach the minimal foundations of set theory,
arithmetic and
algebra.
Advanced workshop: The
advanced workshop uses topic mapping and demand
side theory to teach theory and notation related to the study of
polynomial expressions, equations and functions.
Opening Access:
A path may be opened to a secure foundation
and an extension from this firm foundation into all areas of academic
excellence. The
comprehension of the foundations transfers to other classes, and will
assist the student in the study of college algebra.
Topic Listing
Listing topic descriptors and then specifying the
relationships between topics will increase the certainty about our
studies and fulfill our desire for understanding. This technique is
taught in the workshop.
Topic mapping is a process that starts when a student lists
the topics of the curriculum.
For examples, let
C = { topics in
the standard curriculum in Chapter }
Topic mapping is part of a comprehensive philosophy about
educational processes. This philosophy has a demand side and a
supply side, as will be explained in the workshop. The demand
side
philosophy suggests many changes in how the classroom is managed, how
students study and how testing is conducted.
An example of a year long Lifting Strategy
curriculum :
Draft
Textbook
Uniqueness and group
study
The list developed by each individual will be unique,
reflecting the fact that we are each unique human beings. We are
'particular' persons and our experiences are rooted in personal
experience. The creation of a list is a process in which the
individual identifies sharable concepts. This identification
process may be formalized so that the process may be done exceedingly
well. Once formed, individual lists may be compared. This
comparison is a social activity that increases time-on-task. [1]
Group study, as well as a demand side philosophy, directly
increases the quality and amount of time-on-task. The knowledge
that one has is celebrated and refined through several processes, some
social and some individual. The supply side is also important,
and is honored by reading the textbook and working exercises from the
textbook. The instructor supplies clear knowledge of and motivation to
the topics. Each student is encouraged to spend time outside of class
on his or her study of college mathematics. The use of this time
may be structured by the methodology discussed in class.
Let us be clear. Study occurs best when there is an
absorption by the individual in the topic. Complete absorption
likely occurs only when there is a non-awareness of the passage of
time. It is like prayer. We know that prayer may
spontaneously start to occur as one walks between classes, or when one
is alone. Prayer may also occur in small groups or in
assembly. The lists of topics will contain some common topics.
Sharing will focus on positive experiences where clarity exists.
The respect given to our academic studies may be seen as a type of
devotion.
Only topics understood by at least one student are to be
discussed while in the study groups! This rule is very important
for reasons that will be discussed in class. So how does the
supply side work to bring new concepts? The answer comes from the
experience, but also from some analysis of the nature of formal
reasoning. As we share our lists between members of our small
study group, a social consensus will develop over what ‘should be’ in
this list.
A subset of the elements in the union of all of the lists, in
a study group, is composed of topics that may not be well understood at
first. These are set aside and may form the basis for questions
to be asked in class. The demand side is experienced directly,
enriching each student’s understanding of mathematics as well as
enhance the objectives that study groups may attempt. The supply
of new topics is provided by textbooks, by Internet searches, and by in
class lectures.
The study group helps each student shift responsibility for
learning from the textbook and teacher to the peer group and then to
the individual student. This sets up the demand side philosophy,
since individuals learn to express choices about what is important or
not. As this occurs, the freshman finds personal capability and
success. The choices reflect individual backgrounds, and the
classroom and study group brings these backgrounds into harmony with
college level expectations. As we will see, establishing an expectation
that all tests will be actually created by the student reinforces this
responsibility.
By demanding of ourselves a comprehension of the curriculum
we find positive results from standardized tests. So how might
these mutual goals be supported? Standardized tests cover the
curriculum in the chapters of our textbook. Our final will be a
standardized test. We will, however, study for the sake of
comprehension not ‘just’ test outcomes. A firm foundation is
sought so that a high level of comfort may be realized with the
curriculum.
Theory and Pedagogy
It may be pointed out that there is a glue-like substance
that holds the memorized, but isolated, facts together, and that this
glue is theory. An understanding of theory is opened to the
student through well-structured methodology. [2]
In the same way as words in a natural language point to the
objects of the natural world, the elements of elementary mathematical
theory are pointed at by the elements of the set of topics. Once an
individual has looked over the chapter and highlighted what he or she
has identified as a phrase or word ‘pointing to’ a standard topic, this
word or phase may be written onto an index card. On the back of the
card may be placed information about the topic. This is topic
mapping, a powerful tool.
The Lifting Pedagogy provides additional parts to a
methodology for the study of mathematics. A positive experience
develops as the individual separates from all topics those topics that
he or she is most comfortable with. Positive experiences are a
fundamental principle to the methodology. Contrary to initial
expectations, the learning strategy is to study only those topics that
one is comfortable with. The other topics are ‘set aside’ but are
looked at now and then simply to remind him or her about the names of
these topics. Known topics are rehearsed as students prepare for the
blank paper test. Grades are based on the clarity of these
presentations.
Demand side philosophy
In-class testing is a realization of a demand side
educational philosophy. The student prepares a presentation of
those topics that he or she is comfortable with. This practice
empowers the individual to leave all feeling of uncertainty
behind. Power is centered at the door of the individual’s own
intentions. The methodology supports this empowerment in multiple
ways.
Paradoxically, those topics for which there is an absence of
clarity are set aside and are not to be studied outside of
class. The nature of theory and of mathematics itself is
revealed as topics that are not known reveal themselves to the
individual as the individual rehearses and refines grounding in what is
comfortably and clearly known. This learning from the inside is deeply
Socratic in nature and is the foundation to the demand side educational
philosophy.
The preparation of topic cards involves three perspectives on
the topic. This set of three will be used as we discuss the use
of a framework for the study of each topic.
P =
{ notation, theory, application }
Mathematics notation includes things like the set notation,
{ } , notation for membership, the notation for the set of
real numbers, the set of integers, some logical notation, and the
notation for functions and variables.
Theory includes the listing of properties like the
commutative law, theorems and definitions.
Application examples illustrate the topic by demonstrating
how the topic plays a role in some specific application. Illustrations
should be individualized and relevant to individual student interests.
Taking the cross product of C and P creates a topic
framework. This cross product is being linked to a web based
knowledge management system and the standards of the National Council
of Teachers of Mathematics. [3] In our classes, we will examine
the notion of cross product as the basis for the study of functions,
and illustrate other uses of this mathematical concept to national
educational standards.
Demand side educational philosophy may only be realized after
the student has a clear sense of the nature and meaning of a specific
set of topics. The topics cohere into a view about mathematics
and about the self that is learning mathematics. The
self-efficacy, image of the self, is a topic that education majors may
wish to study.
One of the topics we talk about in class is the supply side
and demand side economic and political philosophy. Excesses can occur
when there is an absence of balance between demand side and supply side
pressures. The nature of generalization is also discussed.
As part of the contextualization of supply and demand pressures, we
also talk about the nature of the particular and the universal, as
represented in natural philosophy. Students are encouraged to
develop topical representations of these sophisticated topics for
presentation onto blank paper. This pair of topics opens the
student’s awareness to the nature and origin of mathematics, and in
fact to the nature and origin of natural language and all of the
academic disciplines.
Topic mapping and blank paper tests
The lifting pedagogy has a methodology for the study of
elementary areas of mathematics. The methodology involves the following
elements:
1) Enumerative Description of Topics: Look over the chapter
and find all words or phrases that seem to be significant(e.g., create
a vocabulary).
2) Elaboration: For each word or phrase, create a glossary
type description, explanation and illustration.
3) Topic Mapping: Write the word or phrase on the front of an
index card and the glossary information on the back.
4) Build a Foundation: Sort the cards into two categories:
comfortable with and set the other topic cards to one side.
5) Deepening: Initially study only these topics that one
knows. Become clear in one’s knowledge so that a foundation may
be rehearsed with confidence. One may find that one comes to love
what one knows. In class you will see new topics. Take good
notes. Listen carefully. Convert your notes to topics and
illustrate these topics using the examples found in your textbook.
6) Illustration: Use the textbook to identify word problems
that illustrate each of the topics you are comfortable with.
Rehearse these topics and share your understanding with your study
mates.
7) Extension: Remember the words or phrases in the stack of
cards that are set aside, but realize that these may be studied when
the knowledge already known is reviewed and rehearsed.
8) Understanding how your mind works: Mentally observe
as topics move between the two stacks. This may occur at random times
like when you are walking or driving, or when you are in your study
group.
9) Express your knowledge on blank paper: Select some subset
of the cards from the comfortable with stack and express on ‘blank
paper’ where each of these topics is completely described and
illustrated. Tell a story. Rehearse this story and extend it.
10) Getting the grade: Prepare for an
in-class blank paper test. At the beginning of the test, list the
topics to be covered. Then present these topics including
examples that you have memorized from the text. Demonstrate
the depth of your understanding.
11) Tell your story: Each student may keep a
journal about his or her total experience. This journal may be used as
part of the grading criterion. The journal may include
individual views about the class, about one’s learning of mathematics
and about the success that is found by following the Lifting Strategy.
12) Apply to other disciplines: The demand
side philosophy will work in all disciplines. Apply the
principles discovered in your study of freshman mathematics to all
areas of your academic career.
Classroom Procedures
The ideal class session starts by taking attendance. After
the doors are closed attendance is taken. Late students knock before
entering, and when admitted, will quietly take a seat so as to not
disturb the class. At the end of the session, these students will
report to the teacher’s desk to be included in attendance
information. Under no circumstances will a student talk unless
recognized by the instructor. This rule is so that every minute of time
may focus on the goals of the classroom.
Students are asked if anyone would like to present parts of a
blank paper test on the board. Individual students will request and
will be given a specific amount of time. During the allocated time, the
student has the floor. After student presentations, the
instructor asks the students if there are any topics that they would
like to see presented by the instructor. If a student is well
qualified, the instructor may ask a student to present the new
topic. An atmosphere of seriousness is enforced. Students obtain
new grades by turning in a presentation on paper, written in
pencil. Consistent with the demand side philosophy, the number of
grades depends entirely on the demand of the individual student as does
the nature and depth of the topics. Grading is based on clarity,
precision and completeness.
Study Group Procedures
All students will wish to form a study group having 3 to 6
students in each group. Each group will wish register the names
of the group with the instructor, and self report attendance at the end
of each six weeks. Groups may reorganize each six weeks.
Attendance issues are to be reported to the instructor by the
group. Students will meet a minimum of two hours per week, but
are encouraged to meet the number of hours required so that each member
of the group makes a grade of A or B for that six weeks.
Web based resources for the support of this type of study
group is being designed during the academic year 2008-2009. The
web support system will then be deployed also in support of the Second
School High School to College Bridge program in mathematics,
humanities and English speaking and writing skills. [4]
Ideal classroom and web based
support
The constitution of small study groups indeed supports
increased time-on-task, and increased success. However, the learning
process may be seen as a seven days a week,twenty-four hour a day
experience. The primary reason is that the human brain is not
necessarily recognizing the very concept of classroom and study
time. If learning is occurring, the brain will remember
experiences related to what is being learned. In this case,
topics known about but not clear in the past may all of a sudden
crystallize as if from nowhere. This is a natural process, which
if not occurring generally means that learning is blocked.
Rehearsing only topics well understood creates a ‘field of remembrance’
which may spontaneously arise.
Thus communication with someone in the class or the
instructor is a means to optimize the learning experience.
Supporting this communication is to be done using cell phones, mobile
devices, and computers. Ideally the communication should focus on a
specific topic, not a word problem, unless the word problem is seen as
an exemplar of a specific topic theory. The descriptor set
P =
{ notation, theory, application }
has an order to it. The notation required to communication
about that topic is needed, then the general theory, and then the
exemplars.
The ideal web based system will allow a student to enter any
phrase or word and either get a statement asking for clarification or a
specific introductory topic. This introductory 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 }.
This ideal web based system will also have a linkage or
mapping between
C X P
and the National Council of Teachers of Mathematics K-12
focus elements.
The classroom and faculty offices have ideal resources. The
ideal condition of a classroom includes large chalkboards or dry eraser
boards, comfortable environment and conditions related to sound and
line of sight. Students and instructors will be able to hear well
and see the board presentations. The ideal faculty office could be a
classroom, as is done in the early grades of K-12, and in many ways
this is the most desirable. The reasons include clarity regarding
where students may meet out side of class times and availability of
dedicated wall space to use. In the ideal case, each instructor
should have a highly functional printer, scanner, Internet connection
and reference library.
The reasons why each of these resources is needed are
discussed in various papers on pedagogy. However, the reasoning behind
the requirements follows from an understanding of topic mapping, blank
paper tests, and demand side learning theory.