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On this page you will
find information to help you successfully complete not just MATH
1046/2046
but all of your university courses. This section is quite long so
a table of contents is provided for quick access to the concept you
wish to read about.
TABLE OF CONTENTS
It is recommended that before lecture you read
through the sections that will be covered that day. The purpose
of this is to familiarize yourself with the concepts that your
professor will be lecturing on, thus allowing you to better understand
the lecture. If you are not exactly sure which sections will be covered in
a lecture, at the end of class, ask your professor which topics he/she
hopes to cover in the next lecture. This will give you a fairly good idea
of how much you should read.
Remember that you are just reading to
familiarize
yourself with the topic, not fully comprehend every aspect of it.
Keeping this in mind will make it seem less of a pre-lecture
chore. More information on how to read the textbook can be found
below in the Reading the Textbook
section. The reading should be
done either on the day of the
lecture or the night before so that it is still fresh in your mind when
you arrive for lecture.
Right before a lecture it is a good idea to review
your notes from the last 10 or 20 minutes of the previous
lecture. By doing this you will be ready to learn the next
concept right away or if the last lecture ended in the middle of an
example, you will know what the example is about. A great time to
do this reading is in the few minutes before class while you are
waiting for the professor to start the lecture.
There are many strategies for reading a textbook.
Two of the most popular are listed here.
When reading to prepare for a
lecture (see Lecture
Preparation above) an easy-going approach is used. Since the
focus of this reading is on familiarizing yourself with concepts, some
areas of each section should be read and some can simply glanced over.
In
order to better explain this method and what should be read or glanced
over,
here is a walkthrough of how to read a section:
READ the introduction
Reading the introduction is beneficial
because
it introduces the concept of that section in the most basic terms. This
allows you to easily understand the basics behind the concepts covered.
GLANCE over theorems/formulae/proofs:
These items
often involve detailed and
sometimes complex parts which may be confusing and in the case of
proofs, might not be fully relevant to the most basic comprehension of
the
concept. This does not mean that proofs are unimportant and don't
apply to a concept. In fact they are just the opposite!
Understanding a proof is one of the best ways to understand a
concept. However, this reading is not for understanding. It is
meant so you can better follow a lecture. Therefore you should
just skim over the theorems, formulae, and proofs so you will have an
idea what is going on when your professor goes over them in detail.
TRY examples before glancing over them:
Before reading the walkthrough
solution of
examples provided throughout sections try to complete the example by
yourself. If you get stuck, look at the next step in the textbook
and try to finish the example from there. By doing this exercise,
you reinforce your knowledge on the subject and if that example is done
in class, you will be able to follow along easier.
READ the conclusion:
The conclusion restates the concepts in a
simple form to enhance basic understanding of the concept. If
your textbook does not have a conclusion at the end of the section,
look after theorems or formulae are stated or proved. Complex
theorems and formulae are often restated there in simpler terms.
TRY a few problems:
Look through the problems at the end of the
section to see general question types and what questions based on the
concepts covered in the section look like. Then choose a few
problems to try. Don't attempt questions that look too
difficult here. Remember, you just want to know what the section
is about, not fully understand it. So choose questions that can be
easily answered.
Above
all, remember that this reading is only meant to
give you some background knowledge on the subjects that will be covered
in the lecture. If you don't understand something, make a note of it,
pay extra attention at that part of the lecture, and ask your professor
during or after lecture if you still need clarification. Don't worry
about completely understanding the concepts while reading before the
lecture.
The other reading strategy to
be
mentioned is how
to read the textbook in preparation for tests or exams. This is
more intense reading than the lecture preparation reading. The
introduction and conclusion should be read in the same manner as before
but the other parts of the chapter have significant changes in how they
are read. Here is a list of the changes from the lecture
preparation reading.
UNDERSTAND theorems/formulae:
Anything which is in a box such as a theorem,
formula, fact, axiom,
definition, etc. should be understood to
the best of your
ability. These are often the items needed to solve problems, so
understanding them and how to apply them should be the focus of this
reading.
READ proofs:
One of the best ways to understand a concept
is
understanding where it comes from and why it works. If you are
able to understand the proof of a concept you will be better able to
apply that concept in problem solving. Hence you should try and
understand the proofs provided in the textbook.
COMPLETE examples before glancing over them:
When preparing for lecture you were
encouraged
to attempt examples once with the aid of the textbook. This time
you should again try the examples but without the aid of the
text. If you need the aid of the textbook to complete examples, then
use it. After you have read the rest of the chapter, attempt these
examples again without the textbook. This should be repeated until you
can complete all the relevant examples without referring to the
textbook or your lecture notes.
TRY a few problems:
When choosing problems to try do not choose
the
easier ones that you know you can solve. Instead, focus on moderate
questions that give you some difficulty. When you can do those
easily without referring to the textbook, try some harder
questions. The harder questions often require you to have a
deeper understanding of the concepts, so in attempting to complete them
you are reinforcing your knowledge of the concepts of the section.
A more in depth look at which problems should be
chosen, as well as more test preparation hints, are provided in the Test/Exam Preparation section.
Assignments are usually longer and have harder
questions than tests. Because of the increased length and
difficulty many students have problems with assignments. Here are
a few suggestions to help in completing them.
Work in Groups
Working in groups is a great way to help
each other. The questions that you can easily solve may be different
from the questions that one of the others can solve. By working
together, you reinforce your knowledge of the questions you know and
you get an explanation to help you understand the questions you
don't. It should be noted that working in groups does not mean
copying each other's work. If someone has the answer to a problem
that you don't, do not just copy their answer. Ask the person to
explain how they got their answer and why or how they did the steps
that you
don't understand. Then try the question on your own, asking for
their help only where you need it. Doing this not only protects
you from plagiarism (which is taken seriously at this university) but
also increases your knowledge and understanding, which will help you
on tests and exams.
Take a Break
When working on assignments it is not recommended
that you work for too long in one sitting. Take breaks whenever
you are completely stuck, feel youself getting tired of math, or are
starting to get unproductive. These breaks should last between 15
and 30 minutes and should be spent thinking about something other than
math. Go have something to eat, watch a television show, play a
video game, or whatever relaxes you and takes your mind off math.
You should not work on an assignment for more
than a few hours at
a time (meaning the entire session including breaks and all time spent
working on the assignment). You should find that by taking breaks
and not working for extended periods of time, you will be able to solve
more questions. By leaving and coming back to a question, you have a
clearer perspective on it. Questions will also be quicker to solve
because you will not be as mentally tired.
When You're Hopelessly Stumped, Move On
If you have tried your best on a question and are
simply stumped, move onto another question. You can attmept the blank
questions again at a later time. You may find that in completing other
questions, you find techniques to help solve the one you are having
problems with.
Visit the Math Drop-In
If you still cannot answer some questions, visit
the Math Drop-In Center. The Drop-In is run by math professors and
upper year math students. They are all there to help you with your
problems. If your professor is not at the drop-in, ask another
professor or a math student for help. The upper year students have
taken your courses and have shown considerable understanding of the
concepts. The Drop-In is a place that almost every math student visits
at some point throughout their studies. Everyone needs help at times,
so don't be afraid to 'drop-in'.
Visit the Algebra Website
This website can also be an invaluable tool to
help with assignments. The tutorials, tests, sample problems and other
sections of this website were created to help students that are having
problems with algebra. The examples, sample problems and test
questions were designed around probable questions that you may see on
tests or assignments. Most of the examples have full, detailed
solutions that may help you solve some of your problems.
Ask Your Prof
If your professor wasn't available or was busy
helping other students in the Math Drop-In Center, go see him/her
during office hours and ask for help. Each professor has certain office
hours, during which they are more than happy to help students with
problems that they are having.
Never Leave a Question Blank
If you are still stuck after attempting all of the
suggestions above, you should remember one last thing: never leave a
question blank when handing in an assignment. By this point you have
probably put hours of work into the solution, so show your professor
this by writing down what you know. Otherwise, the professor has no way
of knowing that you even attempted the problem. Even if all you know is
a formula that might possibly apply to something at some point in the
solution, write it down. You never know what you'll get part marks for.
These part marks could be the difference between a passing or failing
grade, so never leave a question blank.
When preparing for tests and exams, it is
suggested
that you follow the Test Preparation
Reading Strategy explained above. The key components
mentioned in that reading strategy are understanding of the concepts
and being able to complete the proper questions without external aid,
such as asking for help or referencing the text.
A common complaint of students is so called
"math
amnesia", where the pressures of testing, such as time
constraints
and
the need to score high to keep grades up, cause students to forget
formulae or make mistakes in solutions that they normally would not
make. One step in overcoming this is a mental exercise to try and
relieve the stresses of test writing. Convince yourself that your score
on
this test will not harm your overall grade in a large way.
Another step is understanding the concepts that will be tested.
By understanding where a formula or process comes from (I.E. the proof
of the formula or explanation of it in your textbook) you can always
use that information to complete a question even if you forget the
formula that should be used. Another bonus of understanding where
the formula comes from is that you are less likely to forget it in the
first place since you have a deeper understanding of it.
Here is an example of how understanding where a
formula or process came
from allows you to complete a question, even though you may forget the
actual formula or process.
Question: 3x4=? (Now say you forget
how to multiply but you know the concept behind multiplication)
Answer: 3x4 'means' 3 groups of 4. So I can add 4
three times to get the answer. So 3x4=4+4+4=12. Therefore
3x4=12.
Although this is a simple example, it still shows
that understanding a concept can allow you to answer questions even if
you forget how to answer them.
Another common complaint is that while doing
practice problems, students either pick the wrong problems to do or
picked the right problems but forgot how they solved them. The
solution is simple but also quite laborious. When choosing
questions to do, use the "Suggested Exercises" sheets handed out in
class. Professors will often take questions directly from the suggested
problems, or will use slightly modified questions on tests. Of course,
you need
not do all of the suggested questions. If you have no problem
completing a certain problem type, choose only the hardest suggested
problem of that type. If you had difficulty with any of the
questions, redo them after you're done the rest of the questions until
you can do all the problems with minimal difficulty and no help from
any source such as a friend, classmate, or textbook.
One suggestion is to start a seperate notebook
reserved just for solving practice problems. This way you have
fast access to previous solutions and can see at a glance what problem
types you had the most trouble with. This is especially important
for final exam preparation as it allows you to minimize your time
searching for appropriate examples or solutions, giving you have more
time
to spend practicing examples.
As a final note on practice problems, do not be
timid about having to do an excessive amount of examples. The more
examples you
do, the better you will be at solving them. There is nothing wrong with
having to do 60 questions 3 times each if that's
what it takes to understand that topic.
As can be seen, test preparation can be a lot of
work. Don't save it for the last minute. You should start
preparing for the next test as soon as you know when it is. At
the latest start preparing a week in advance of the test.
Taking tests can often be a stressful situation
causing you to forget what you know and make 'silly mistakes' such as
-1-1=2, instead of -2. The most important thing to remember is to
RELAX.
At least a week before the test you should start
following the strategy laid out in the Test/Exam
Preparation section.
The last 30 minutes before you come to class
should be spent doing something fun and interactive. Studying
right up to the test/exam can often leave students
confused about concepts and stressed by what they don't understand,
leading to "math amnesia".
So play a video game, play a card game, do a crossword. Do anything
that makes you have to think and perform a task, as well as takes your
mind off the
test. Interactive activities such as these will help stop
concepts from getting confused as well as alleviate some stress.
Arrive at class at least 15 minutes before the
test starts so you can spend that time reviewing the concepts you need
to review the most and have time to ask your classmates for last minute
help.
If you are stuck on a question, skip it and come
back if there is time when you're done the other questions. Try
not to leave a question blank. Sometimes time constraints can
force you to leave questions unanswered, but if you have the time, fill
in what you know about questions you cannot answer. You may get
part marks for your partial solution and part marks add up.
Writing down partial solutions for a few questions you cannot answer
could mean the difference between a 40% and a 60%.
If you finish as much as you can do before time is
up, check over your solutions. Make sure you solved the right
questions and that you didn't make any 'silly mistakes'. Check if you
remember anything else that could help you solve those questions that
you didn't completely answer.
Above all else RELAX, STAY CALM, and DON'T STRESS
before or during the test/exam.
This section is devoted to tips which either don't
fit into any other category or were provided by past students who took
this course.
Don't Skip Sections
If you ever come across a concept or topic you
do not understand don't skip it and never return to it. Many concepts
in this course build on previous concepts so by skipping one concept
you may not have the basic understanding needed to understand future
concepts. Since this new concept you do not fully understand can also
be the basics behind another concept then you may not understand the
next one either. It is quite easy for this to continue perpetually
throughout the course and can make even the simplest topics covered at
the end of the term be beyond your understanding.
This is one of the hardest errors to correct as
well because the longer
you leave trying to understand the first concept you skipped, the more
you have to learn and the less likely you are to bother trying.
Therefore skipping topics should be avoided at all costs. Use the
resources on this website, ask your professor for help during office
hours, visit the Math Drop-In Center, ask a classmate, try sample
problems, or hire a tutor. Do everything in your power to understant
all the topics you can.
Do Lots of Examples
One of the best ways to learn is by doing. Take
advantage of this and tons of problems especially problems on topics
you are having difficulty with. For more information on which
problems to do see the Practice
Problems section.
Keep a Notebook of Problems
As suggested in the Practice Problems section
keep a seperate notebook for problems. Every problem you solve or
attempt to solve should be in this notebook. This lets you
quickly see where you went wrong, what you need to work on, and quickly
reference solutions to maximize studying time.
Don't Make Math a Chore
Try to make math as fun and exciting as
possible. If you make completing a math assignment or studying for a
test into a chore then you may not learn as much and the purpose of the
assignment or test (which is to help you learn and understand) may be
defeated. A great way to make math more exciting is having math
parties. Get together with a group of classmates and study together or
help eachother through the assignment. For more information on working
in groups, see the Completing
Assignments section.
Ask for Help
When all else fails, ask somebody for help. Go
see your professor during office hours, ask a classmate, go to the Math
Drop-In Center, hire a tutor, or ask anyone else who may be able to
help you. Sometimes all you need is to hear a concept explained in
different words to allow you to understand the concept. Asking for help
can be a valuable source of different explanations and can often help
with your understanding of a topic.
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