Professor B. A. Wade
wade@uwm.edu
www.uwm.edu/~wade
414.229.5225
Current Teaching Activities
Mathematics Teaching & Learning
These two sides of teaching and learning are indeed
complementary, since the best teacher every time approaches the job as if he/she were learning it all anew,
while the most successful learners actually teach themselves. Both parties need to understand
the whole process to avoid that amazingly stupid version of education that today's system sometimes
encourages, whereby the information is efficiently transferred from the professor's notes to the
students' notes without anyone needing to actually think.
No one learns by watching others do it. A teacher must set
conditions of practice with solid variety and well chosen examples and
discuss the ideas in context with development of the intuition. A student must
realize that he/she is really teaching himself/herself, that the teacher is only a facilitator.
Thus, active and serious effort is always needed.
People these days have no patience to listen to long
explanations or read long
paragraphs. Therefore, writing in a handout or on the board should be
bullet-point style accompanied by sufficient repetition of the important items.
The teacher should give different views of the same thing if possible. Consider this common
experience, that one does not start a new thing by reading a technical manual completely through, but rather one
attempts to use it first to become familiar with it. Then when the need arises we look back
at the instruction manual. Why, then, do some think students attempt to learn mathematics by reading the whole theory first? Have students start immediately to work on problems before they are technically ready, and give the explanations as you go.
Related to the above, do not study "linearly," straight
through the book. Any such order is artificial and designed for convenience of the presenter, not the
learner. Learn "non-linearly," skip things here and there to come back to them later. Learn the easier
parts of later topics occasionally before the hard version of the earlier topics. Look at the
book before hearing the lecture, etc.
Students and teachers both: Anything important must be written down.
The handwriting and organization is important, so take time and effort to write neatly and
make a plan for a well-organized presentation and with good classroom management (including usage of time).
Students: re-copy lecture notes to perfection, or
re-write what is in the text book. Take the time! It will pay off later.
This creates an active connection to the material as
well as sets out a sequential aspect that is not part of ordinary reading. Try to recreate a sequential
appreciation for the item in from of you, do not make the mistake of trying to take in a snapshot of the
whole page of logical progression.
When
lecturing be sure to make the plan of the presentation very, very clear right
away by giving an overview at the beginning, and repeat yourself frequently.
Be sure to give short explanations with a lot of variety (color, tone, cadence, etc.),
using bullet-point style. Use color effectively, but not too many colors-
tell the students to carry the same number of colors to class for notes, three colors is a good number.
Keep it as simple as possible, but no simpler (Ref. A. Einstein),
leaving complications to the exercises or for later
lectures (or, better, never be excessively
complicated, since most of the time a course is a first time through the
material).
Examine
anything... at its heart is a simple idea. Communicate that idea (when teaching).
Identify that idea (when learning).
NEVER believe that there is not a simple idea underneath anything.
A teacher
should teach in small groups. Classes of
15 or more usually have bad dynamics, although frequently the budget doesn't
allow fewer students. If one does not have a small group to teach, just do
your best by working with the students after class and inviting them to your
office! Do not consider the lecture to be the beginning and end of instruction.
Invite students to
interact together in small groups away from your influence.
Use class email reflectors, etc., to facilitate this.
Be
enthusiastic and adapt yourself to the audience, or risk losing them. Be very
conscious to the time management plan you should have designed ahead of time.
This is one of the most difficult things, yet most important.
The
tortoise beats the hare.
Practice and practice, be patient with yourself (isn't this often the most difficult part?), organize your time well, do not
expect quick, easy results.
Master the art of working hard but staying relaxed. It is
very important to "try easy."
Be sure to take into account that some things are just plain
difficult. If you expect it to be hard, and it is, you will not be frustrated! If, on the other hand,
you make the mistake of trying to do something intrinsically difficult
while hoping it will be quick and easy, that will be a major source of frustration and trepidation.
Be sure to get organized for academic success. There are many
variations, so here is one idea. Have at least two loose-leaf notebooks for each course or research
endeavor (soft-sided preferrably, they fit in backpacks better). One serves to archive notes and
handouts, etc. The primary notebook should have many sections or divide it by paperclips for quick access. Keep every handout and take notes on administrative things in one section. Another will have notes from class. Each project or homework can be done in its own separate section of the primary notebook. Move things to the archiving folder as they become dated. The goal is to encourage you to write everything and have everything easily retreivable. You can review notes easily and a project or homework in progress will have developmental notes easily accessible.
Keep computer files always in a separate subdirectory, with as many subdirectories under that one for each project and administrative information. Back-ups should be done very often to avoid catastrophic loss of important work. Backing up eat least very week is a must, perhaps more often.
The human brain is not a computer, but rather an associative learning system, or
neural network. That is, everything cognitive
is really an impression, an image, an emotion, an intuition. Strict "computer style" logic is an illusion.
Information is stored
in a distributed fashion throughout the incredible number of neurons in the brain.
Thus, we should concentrate on building our intuition and trusting our intuition. Do not over-task yourself, either, as
that is fundamentally the cause of so-called dumb mistakes. Write everything down and be very
organized in order to place less burden on your memory process.
Summary: The principles of successful learning:
1) Study actively. Do it, don't just watch; write everything, re-copy notes and text material, reproduce
sequential aspect of new material; organize yourself (in space and time);
be neat, write down all ideas and steps in order to avoid multi-tasking.
2) Learn nonlinearly. Persevere, keep on moving with your own best order of topics.
3) Build and trust your intuition. Your brain is not a computer, use it to its full power the way it was designed.
Don't 'think' while 'hitting.'
4) Try easy, be relaxed.
Here is the 3 word version summarizing everything I've said:
Detail, perseverance, intuition.