CALCULUS
by Florian Colceag
Educating students is a two stages process. The first one is depending by teacher
skills and consists in creating a good bias about the subject, introducing new concepts,
and showing examples. Students who need to practice until they master these new
concepts and methodologies make the second one. They need to learn.
How can we help our students to learn? This question is frequently asked, and many
solutions were developed in time. The solution proposed here considers the stages of
learning and understanding described by the modern psychology (see Edward de Bono,
Six thinking hats).
These stages are connected with several cerebral activities:
-1) Accumulation of data. These data are selected considering an initial selection
filter. For example if somebody is interested in a subject, the selected data will consider
the field of interest. So data related directly or indirectly with this field of interest will be
selected with priority and accumulated.
-2) Processing new information. Data accumulated in the previous step will be
processed conforming to processing algorithms already existing, using trial and error
methodology. If some data will not correspond with any algorithmic ability, will be
probably rejected, or stocked in a different folder of unknown methods.
-3) Using and applying processed data. Data that are already processed will be
transferred in a new database containing possible applications. These possible
applications can be extended for new processed data, extending this application field.
-4) Expressing new things. New applications need to be expressed in an accurate way
such that to be logically clear their history including selection filters, processing
methodologies and application fields.
-5) Assessing what you expressed. These formulations might be incomplete,
inaccurate, or local. This is why a new critical stage regarding these characteristics is
required.
-6) Developing new methodologies or understandings. This stage permits to extend
your understanding to new domains, connections, or to develop phenomenological
understanding, which can give methods applicable in various contexts. As a result of this
last stage a new horizon of understanding will permit another kind of selection filter for
information. Information considered as irrelevant before will be relevant in the new
context.
The cycle will repeat now starting from a new selection filter. Data that were stocked
as unknown might be processed in the second cycle considering new methodologies
developed in the 6th step of the previous stage.
How this philosophy will be transferred in computerized education is not a difficult
task. Without computer De Bono tested this a methodology related to this theory with
very good results. Students develop curiosity and desire for understanding, increase their
level of intelligence, and try to find out new thinks without being forced by their teachers
to do this.
A computer application program in mathematics will follow this methodology also in
six steps. Let’s consider an example and a potential situation.
Problem
Find the value of the derivative of f (x) =…. At the point 5
Connected links for each problem:
-1) what definitions do you need (filter of selection)
-2) what algorithms, rules and theorems do you need (processing new information)
-3) how will you apply these algorithms, examples (using and applying)
-4) which is the standard form to express your solutions (expressing new things)
-5) how can you be sure that you solved in a correct way (assessing procedures)
-6) what is behind this problem (mathematical philosophy)
Each student may have the possibility to solve alone, to ask specific help, or to go
forward. The first three steps will help students who don’t understand very well, to
recuperate their knowledge. The last three steps will help student to go forward faster and
to follow their own understanding. For example from assessing methodologies they will
have also six steps to select. They will be able to understand assessing definitions,
assessing rules, assessing applications, assessing standard expressions, assessing
algorithmic procedures, and assessing philosophy.
Many links will remain opened as a potential new problem. Introducing information
in such a new field will be rewarded with points, because this will mean personal
research and interest in understanding. A new kind of manual on the web can be
developed starting with a mathematical traditional manual like calculus. This manual will
be collective, contributing to its development each teacher or student. Teachers will make
evaluation for the quality of data introduced. Problems will interfere to each other, each
problem being processed as a result of the development of different other problems.
Pushing the first button of filter of selection each student will go to the previous problem
that developed new definitions, pushing on the second button will go to results obtained
considering these new definitions, etc. The manual will automatically self develop. This
manual will need few hundred problems to be previously introduced on a web page, and
connected to each other, to different knowledge required by the algorithmic system
explained behind, and letting a lot of uncovered links for the future. These links will be
covered by teachers who will solve problems if they will connect these problems with
different other knowledge, by students who will be glad to develop a mathematical work
(each new result will have the name of the discoverer), and by outsiders. This kind of
manual will have later very specific obligatory demands for students, but will reward
creativity publishing new results and noting progress.
From the technical perspective this manual can be written as a web pages net and will
need a specific computer program. This computer program will permit development of
the manual, grading new results, grading progress, and grading accumulation of old data.
Teachers’ work can also be assessed by this computer program, considering the accuracy
of information introduced and not criticized by others, and recognized by teacher as
valuable. The program needs also to be able to identify old problems using specific
words.
Teachers’ role will be to teach concepts, to create a good scientific bias, to teach
students how to work with this manual, to introduce new data and to assess data
introduced by students in the manual.
The program can be developed in several steps. The first step will be the arrangement
of a web page in the format required. Assessing progress for students will be the second
stage, this stage considering an official perspective about how to grade students’
progress. The third stage will consider assessing teachers. This stage needs to be
negotiated with teachers’ organizations. Several other steps will be required by the
system later. The first step, which is the most important, is not restrictive and doesn’t
follow a specific educational policy. All the other steps are optional, but the first step is
necessary.
The program will permit education for everybody encouraging creativity, flexible
understanding, complex understanding and development of specific giftedness. The
program will be useful not only in gifted education, but also in regular education.