MAT246 LEC5101 Vision and Learning Objectives
Winter 2023
The ultimate objective of the course, is summarized in its title:
to present a rich collection of mathematical concepts, to enrich our language, and to
sharpen our problem solving intuition. With this enriched language one can understand
and express complex ideas.
Such process, of learning and working with mathematical concepts, opens the door to new
cognitive frontiers. This tool kit of abstract mathematical concepts brings our thinking
ability to a finer, higher level, more powerful capacity.
The course is organized around educational goals as well. The following goals serve to
train certain personality that is a part of the characteristic of university education. We
set these goals as our learning objectives. Please
- reflect on the goals, and be justified about them; know why these goals are necessary
for a complete experience. Nothing short of a true justification can build enough
energy and motivation for a fulfilling journey in the course.
- take advantage of the course activities that facilitate the journey towards achieving
the goals. This is done by continuously being involved and being reflective about the course
activities, and measuring one’s success and the success of the activity against the goals and
objectives of the course.
Course Objectives:
The course is designed to help developing awareness in the following fronts.
Note: To enforce a continuous awareness of these objectives, students are asked to write
a reflection essay on their journey in the course with clear and meaningful reflections
on the objectives.
Types of Knowledge: to distinguish between three different ways one can ‘know’,
and to have a good understanding of the purpose of each way of ‘knowing’. Major
misunderstandings resolve when such distinction is made.
Please see the slides on ‘Types of Knowledge’, ‘Intuitive Knowledge’, ‘Passive Knowledge’
and ‘Formal Knowledge’. Reflection on these types needs to continue throughout the course.
Course Content: to understand abstract concepts; to know why they are intro-
duced, and to integrate them into one’s thinking process as tools for thinking.
This can be tied to the ‘intuitive knowledge’. Please refer to the set of puzzles presented in
the course. Puzzles help with this integration process: they justify the need for mathematical
concepts, and formulate our thinking process. So please don’t shy away from the Puzzles.
Presentation/Communication: to build a solid foundation for formulating and
presenting a formal argument. This goal can be achieved by consistent practice and
awareness of the format of Problem sets and Quizzes. (Note, this is tied to the ‘Active
Knowledge’.)
Theory Building: to understand how mathematical theories are born and de-
veloped, and how they are used in problem solving. (Note, this goal is relevant to the
‘Passive Knowledge’.)
See the systematic fashion in which the course slides present and establish the facts.
Attitude of Critical Reading: (of a mathematical text) this is the ability to read
a text with a full awareness of the theory, and “reading between the lines” with an
eye on the background and the future development of the theory. (Note, this item
is very much relevant to the ‘Passive Knowledge’. Also see notes on “Explicit vs Implicit”.
Reading material are designed to guide the reader to discovering what is implicit in a text.)
Further, personalized goals: Students are to be aware of the following ideas while
they are engaged in the course activities. These are not goals directly enforced by the
course, but they are to be practiced at a personal level, via a conscious engagement
in the course activities. So please reflect on the following qualities when engaging in
the course activities:
Being Present in the Activity: this quality is not specific to University education;
it is needed in all other life activities. This is about having full Attention and
Focus during an activity, be it ‘Reading a text’ (see critical reading goal), lecture
attendance, solving puzzles, writing a proof, etc. Try to develop longer and longer
attention span. Multi-tasking goes against this quality ...
To remain focused during the lectures the pop quizzes will test simple consequences
of the ideas presented in the lecture, be it a minute ago, or the synthesis of various
ideas in the same lecture. However, students are encouraged to:
- first allow the lecture to legitimately own and occupy the allocated time period;
don’t share this time with other thoughts! This is facilitated by the regular
questions posed and students are encouraged to respond to these questions,
and to take them for a kind of discussion.
- Second, have some idea about the most important ideas being discussed, and
remain anchored and focused on these pivotal ideas. Know which ideas are
central, and which ones are details; try to connects the main ideas together
and don’t let the details confuse and distract you. This is an important task,
which is crucial in general for attending an important, informative meeting in
the future of every professional person. The course tries to facilitate this we
give preparatory reading material and questions earlier, either as part of the
online quizzes or some question sheet in preparation for lectures.
Student Centered Learning: Traditionally, lectures present a simplified, believ-
able map of the subject. This can lead to an illusion of understanding. While such
an illusion is the beginning of a journey, far too many of our students stop at this
illusion. This “illusion” alone does not lead to a true, deep understanding of the sub-
ject. To help with achieving the above goals we practice the alternative paradigm of
Student Centered Learning. If executed properly, the benefits of this paradigm
include deep, individualized learning, and ‘learning to learn’. The term “student
centered learning” is not enforced but facilatated via workbook model, and provid-
ing students chance to discover the idea by answering questions. This does not mean
changing the classroom into a seminar in which participants discover the major ideas
and directions in the course.
Workbook model: Lectures and the textbook are accompanied and complemented
by slides, which contain a guided tour of the subject. These slides are organized into
workbook, leading the reader in a ‘Reflective Process’, which facilitates a ‘Critical
Reading’, with added features of exposure to theory building as well as revealing the
implicit content of the course/textbook. The lectures will touch upon these slides
but the slides will be available days before each lecture, and they provide students
a chance to deepen on what is ahead.
Implicit vs Explicit: A textbook in mathematics explicitly presents concepts
and techniques, and implicitly presents process of mathematical thinking as well as
the culture of the subject. Please note that the primary purpose of mathematics
education, is not in general, the explicitly presented tools and technique, but it is
the implicit thinking process. The course slides try to explicitly discuss the ideas
that are implicitly mentioned in the textbook. The discrepancy between the volume
of material in the textbook and the slides indicates the volume of implicit ideas
needed in this course. While the slides outline and stress on some of the implicit
ideas, the list of implicit idea might be different for each person. Therefore it is
extremely important that students have read and mastered as much of the slides
before attending the lectures and tutorials to benefit most from the activities.