“But I’ve always been good at
math!”
How High School and University Math
are different and what you need to do
to succeed in University Math
High School Math: Imitation and
Memorization.
What you may have been good at in high school
is imitation and memorization. The teacher
shows you a problem, and then you are given
the same problem on the test with different
numbers. You are expected to remember the
pattern and change the numbers. Very little
thinking is required.
High School Math : University Math
= Microwaving a frozen meal : Cooking.
University Math: Conceptual
Understanding and Creative
Application.
At the university level, you are expected to understand
the underlying concepts and be able to apply them
independently to fresh problems that you may not yet
have not yet encountered in that exact form. The only
review you may get is a list of concepts. If you get a
review test, the review test will cover the same concepts,
but it will not be the actual test with just a few numbers
changed.
University math emphasizes critical thinking over
memorization and imitation.
Examples of High School Math vs.
University Math
High School Math: find the derivative of the
function 𝑓 𝑥 = 𝑥 2 .
University Math: Use the limit definition to
determine whether the function 𝑓 𝑥 = 𝑥 is
differentiable at x=0. Explain your reasoning.
High School Math: evaluate sin 90° .
University Math: explain why, for all angles 𝛼,
sin 𝛼 = cos(90° − 𝛼).
Common Reactions to Misunderstood
University Math
“The tests were unfair!”
Some students think it is unfair to be expected to master the concepts and to have to
demonstrate independent, critical thinking. Like it or not, that is the nature of higher
education. If universities are the wrong place for cultivating understanding and critical
thinking, then where is the right place?
“The test asked trick questions!”
A trick question is a question that is intended to make you fail. Most alleged trick questions
in university math are actually just concept questions – they ask you to demonstrate that
you understand the underlying theory, and are perfectly answerable if you acquired the
necessary understanding through persistent and proper study.
“We never covered this in class!”
In a way, that’s the point. The exact question was not supposed to have been covered in
class. What was covered in class, though, were the principles that enable you to solve the
problem. If the exact question, or a nearly identical question with just a few numbers
changed, had been rehearsed in class, the question would no longer test your
understanding, just your ability to memorize and imitate.
You are responsible, not the teacher
• In high school, the teacher is responsible for your learning,
and most of that learning happens in the classroom. At the
university, you are responsible for your own learning, and
much of that learning happens outside of the classroom.
There are many resources to help you learn, but you need
to take the initiative and take advantage of them.
• High school math proceeds slowly and offers redundant
review and repetition. University math is fast paced and
offers minimum repetition. If you don’t understand
material even after it has been covered in class, and fail to
make up for it, the class will move on without you. Each
lecture proceeds from the assumption that you have
understood the contents of all previous lectures.
Math is a “Cumulative” Subject.
Concepts build on one another.
If you fail to understand one concept, everything on top of
that will be incomprehensible. It is therefore necessary that
you review and thoroughly understand the concepts that were
taught in each lecture before you come to the next lecture.
It’s like climbing a mountain – each step is a prerequisite for
the next step. To get to the top, you need to get up early and
then do the hard work of climbing, step by step. If you sleep in
and relax for most of the day, thinking that a quick sprint in
the late afternoon will get you there, you will suffer a rude
awakening when you realize that this mountain is much taller
and steeper than it looks, and has much more difficult terrain
than the similarly named hill you climbed in high school.
Proper Study Habits
In University math, doing homework alone will usually not prepare you for
the exams. You must study the textbook and your notes with the goal of
understanding the theory. In particular, you must commit definitions and
theorems to memory; but not in the mindless fashion of just memorizing
empty words that mean nothing. The definitions and theorems and their
interplay must come alive in your mind.
To create these kinds of mental associations, you must actively engage with
the theory. Each time you study, you should make or expand a list of
important concepts (definitions, theorems and algorithms), and learn them
like you would learn vocabulary in a foreign language.
This type of mastery is not achieved overnight. It can only be achieved
through persistent, regular study, starting on the first day of class. Lastminute cramming will get you nowhere and practically guarantees failure.
Do not fall into the trap of thinking that because the material seems
superficially similar to material you may have learned in high school, the
university course is just a review.