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Teaching Portfolio
The keyword for me is ‘synthesis.’
When one looks in a dictionary for this term, one gets a number of
different meanings, but the one I mean here is ‘the combining of the
constituent elements of separate material or abstract entities into a
single or unified entity.’
Mathematics is a very analytical discipline, and analysis tends to break
things down rather than build things up.
Both levels of understanding are important, yet students often
struggle with rebuilding the unified whole on their own.
What I try to do is help that process along.
The study of mathematics is well known to depend on attention to detail in
computation. Even if a
student understands the ‘big picture’, they still need to be able to
execute a computation correctly, and not make careless errors like losing
negative signs or similar things that students know how to do.
Learning to catch those errors and work computations with care, and
check their own work, and so forth, are very important to the execution of
correct mathematical operations.
In concentrating on this aspect, however, students often forget the
reasons they are doing the computations in the first place.
They know that it is a requirement for their degree, but
particularly in developmental math courses, they often fail to see the
need for math in their careers or in their daily lives, or even how the
course connects to the other math requirements, like statistics, that they
may better understand the need for.
One of the great advantages of having taught nearly the entirety of the
community college math curriculum is that I know where students are going,
and where they have been, and can help them make some of those
connections. Having also
obtained degrees in social science, the humanities, and natural sciences
myself, I have a unique vantage point from which to see the whole playing
field. Understanding math in
context is a great advantage to students because they can see the thread
of logic that runs through math more readily and that makes them less
resistant to learning.
Students who know that they still have three more math classes to take,
and how learning what they are learning today will help them in those
classes are much more motivated to work harder on something they find
otherwise pointless to their daily lives because they know blowing it off
now will impact their grades in the future.
Students in their last math course want to understand why their
major requires this kind of math, and it motivates them to see how they
will use it in their lives or specifically their careers.
Students often don’t realize the level of math that will be
required in careers they think of as (relatively) non-mathematical.
Making the connections backwards in time to math they have done in
the past also aids in their recall process because they can remember maybe
other aspects of that course and it will help jog their memories.
Even making these connections within a single course helps drive
the narrative for them of what the course is ‘about’.
When I explain to my elementary algebra students how Chapter 2
(linear equations in one variable) connects to Chapter 3 (linear equations
in two variables), and how that connects to Chapter 4 (systems of linear
equations), they get a better sense of the structure of the course.
Without making these connections explicitly, students will get
bogged down in the details and are left with the impression that each
section is independent of all the other sections, and the rest of the
world. This is also true to
some extent even for students in programs with much higher mathematical
requirements like physics or engineering.
It’s easy to think that, sure, they will need to know the basic
stuff, but sometimes particular topics will come up again in their later
coursework, like using surface integrals in courses on electricity and
magnetism, that they had not expected to be so important.
This is traditionally one of the last topics that is covered in
multivariable calculus, but if students figure ‘oh, it’s the last topic;
it’s only one question on the final; I don’t have to worry about it too
much,’ they will find themselves in a course where they really need it and
have to learn it basically from scratch at that point (and from someone
who is not skilled at explaining math).
Helping them see that it’s coming back again later on helps them
focus and give that type of problem the attention it deserves while they
still have someone around to explain it properly.
The same kind of issue arises with topics within chapters.
Mathematics textbooks will lump together in the same chapter
various techniques used to address particular types of problems, and there
are certainly logical and organizational reasons for doing this.
For instance, in elementary algebra, we learn at least three
methods for solving systems of equations: graphing, substitution, and
elimination (in later classes, they will see still more methods, mostly
involving matrices). Students
who master the various methods are fine solving problems if you tell them
which method to use, but they may struggle to choose a good (and
efficient) method when given the choice.
They may fall back on whatever one it was they learned first, and
run into a variety of complications that might have been avoided in
another method (such as working with fractions).
If fact, they often don’t understand why the other methods exist at
all. While any system of
equations can, at least in principle, be solved by any of the methods we
learn (students lose only time working too hard), the same can’t be said
of integration techniques that are learned in calculus.
Most integration techniques are designed to work where other
methods fail, so it’s especially important to choose the right one.
This is, however, precisely where students often come up short.
Even if they can execute each method, they have a difficult time
knowing on their own, which one is likely to work in a given case.
This sort of thing comes up over and over again in math courses
throughout the curriculum.
I have, therefore, made a point of trying to address this.
Once all the methods are laid out, I spend some time, maybe an
entire hour depending on the course, looking at how to determine the best
method to be applied to a particular problem.
Students used to ask me, particularly in tutoring, about how they
could tell the difference between when they were supposed to use method A
and when method B or C. In
the beginning, I had a hard time answering them, because it was something
that I sometimes just recognized from seeing numerous examples.
However, being analytically inclined, I turned my analytical skills
on my own thought processes.
I had to ask myself, what was I seeing that made me recognize the correct
method that they could not see, and how could I explain it to them? Now,
when I try to knit the various techniques together into a kind of whole
through comparing and contrasting them, we spend a lot of time discussing
those features. How do you
know when using substitution is going to introduce fractions into the
problem and can you choose the variable to solve for that will help you
avoid them? How do you know
this is a u-substitution problem and not a by-parts problem?
How do you know that you are finding the tangent plane by this
method rather than some other method? Or is it a tangent vector and not a
tangent plane at all? For
many of my classes, some of the first handouts I’ve written for students
have been on these types of topics, so that I can lay out examples and
explain my reasoning for them to reference later on.
This is also a very useful skill to have when dealing with student
questions coming from a different perspective than I am.
Breaking a problem down and building it back up again in a new way
is essential in addressing differing learning styles.
Analysis and synthesis for me must really go hand-in-hand.
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rights reserved To contact the webmistress, email betsy@pewtergallery.com last updated: 2019 January 8 |
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