# Mathematics Objectives Skills and Attitudes

Learning mathematics is a complex job, but it is not an
impossible one. We hope that laying out these course goals and

objectives will be helpful to you.

• You will learn some basic terminology.

• You will learn and practice reasoning and mathematical computational skills.
We will show you how to get started,

and we will show you examples, but most of your learning will come from
practice.

• You will adopt some attitudes. You will not be graded on these, but we put
them on the list for two reasons .

1. Because having the these attitudes will be to your advantage in meeting the
other objectives

2. Because having these attitudes will be to your advantage in the rest of your
education and in your

professional career.

**Introduction to Logic**

Students will be able to construct a truth table to
determine the truth values for a given compound statement containing any

combination of connectors (negation, conjunction, disjunction, conditional, and
bi-conditional).

Students will be able to determine whether a statement is a tautology,
contradiction, or neither.

Students will be able to construct the inverse, converse, and contra-positive of a statement.

Students will be able to translate an argument into symbolic form and determine its validity.

**Set Theory **

Student will be able to construct new sets by performing
set operations (complement, union, intersection and Cartesian

product) on given sets and determine the cardinality of the new set.

Students will be able to form a partition of a set, draw a
Venn diagram to represent the partition, and determine the number

of subsets that can be formed.

Students will be able to solve word problems using sets, Venn diagrams, addition principle , and DeMorgan's Laws.

**Combinatorics **

Student will be able to draw a tree diagram to represent
the outcomes of an experiment and determine the number of

outcomes by using an appropriate counting formula (Fundamental Counting
Principle, factorials , permutation, ordered

partition with indistinguishable objects - letter problems, circular
permutation, combination, unordered partition).

Students will be able to solve word problems involving counting.

**Probability **

Student will be able to construct an event (collection of
equally-likely outcomes), determine the probability of the event

(using the properties of probability, the addition rule of probability, and
Bayes’ formula), determine the odds for and against

the event, and determine if two events are disjoint or independent for a given
experiment.

Students will be able to solve word problems involving conditional and binomial probability .

**Introduction to Statistics**

Student will be able to compute the central tendency
(mode, median, and mean), values of a random variable , probability

density function, expected value, and standard deviation of a given set of data.

Students will be able to find the mean and standard
deviation of a binomial random variable, perform a Z-score conversion,

and use Table B (table of areas under the Standard Normal Curve ) to determine
the probability a randomly selected score is

below, between, or above certain values.

**Linear Equations and Matrix Algebra **

Student will be able to solve a system of linear equations
with the All-Integer Method, including systems that are inconsistent

or dependent.

Students will be able to perform matrix operations
(addition, multiplication, scalar multiplication, transpose, and inverse)

when possible.

Students will be able to write a system of linear
equations in matrix form (AX = B), find the inverse of matrix A, and use the

inverse to solve the system.

**Markov Chains **

Student will be able to construct the transition diagram,
transition matrix, and initial state vector for a Markov process

(chain).

Students will be able to find the state vector after n-repetitions of a Markov chain.

Students will be able to determine if a Markov chain is regular, and if so, find the steady state vector.

**Skills **

**Read and understand an English language description of
a problem.** You will learn to determine what physical quantities and

processes are involved. You will learn to determine which quantities you may
presume to be known , which may be neglected,

and which you are responsible for determining.

**Analyze the problem**. Most problems cannot be solved
in a single step. You will learn to break problems down into

subproblems that can each be solved independently. You will learn to recognize
which laws are relevant , and you will learn to

apply the laws to solve the subproblems. You will also learn to reassemble these
solutions into a solution to the whole

problem.

**Describe the problem.** In order to break down the
problem and reassemble the solution efficiently, you will learn to describe

the problem in several useful ways. You will learn to use appropriate diagrams,
graphs , mathematical formulas and

terminology to describe problems and solutions in a way that emphasizes the
physics and suppresses the complexities and

ambiguities of everyday language.

**Explain your results.** The last link in this chain
is to take your solution, which is usually in a mathematical form, and restate
it

in English. Taken together, these first four skills are what we call "problem
solving."

**Organize your knowledge. **Like most college courses,
there is a lot of material covered in this class. You will learn to

recognize how this knowledge fits together to make a whole subject.

**Picturing situations**. In physics, we often deal
with complex situations in which several objects interact. You will learn to

picture these situations as a step towards analyzing them.

**Modeling.** This is a complex skill, which we will
only begin to teach you. The idea is to take complicated real-world situations

and create models of those situations that are useful. To be useful a model must
be simpler than the real thing: enough so

that it can be analyzed. But the model cannot be too simple. It must
reflect the aspects of the situation that make it worth

studying.

**Connecting your knowledge.** You will learn how
physics is connected to other courses you have taken, to other areas of

knowledge, and to knowledge you have gained informally (" common sense ").

**Attitudes **

**Remember that knowledge is cumulative. **

Many technical subjects are structured like a pyramid. Each idea is built on the
foundation of the last. Therefore, you cannot

forget a formula or idea just because the test is over. That idea may be vital
to your understanding of something new next

week, next month, or next semester. Learn this subject for life.

**Be bold. **

We all know this is a difficult subject, and sometimes you will just be clue
less. That is ok, it happens to all of us, faculty

included. If you do not understand something, ask a question in a big room in a
loud voice. The lecture hall is ideal.

Remember the saying "the only stupid question is the one not asked." In a
similar vein, turn in every homework assignment,

even if it is all wrong. Turning in incorrect work looks bold. Not turning it in
looks lazy.

**Persevere. **

At times , you may feel like we are asking too much, or that the material is just
too hard. At these times, you need to suck it up

and work extra hard. Put in an extra hour working problems, take an extra hour
to talk to one of the faculty, whatever it

takes. We try to make sure everyone succeeds in this course. However, we also
want you to reach as far as you can.

Everyone's ability should be stretched by this course. It isn't supposed to be
easy.

**Get serious. **

When you are doing something difficult, you need to use all your skills, and you
need to devote plenty of time. Organize you

work so you don't lose things. Do all the assignments. Use every resource
available; that includes faculty office time, the

MAC, the library, the internet, and other students. Do every assignment as if
you were going to frame it. If it is going to be

difficult, you want to be able to brag about it later.

**Be curious. **

Ask yourself why we are studying each idea, concept, formula, etc. Mathematics
is one of the most finely tuned subjects in

the undergraduate curriculum (This isn't ego, it is age. Mathematics is simply
an older subject). Everything in the course is

there for a reason. If you can figure the reason out, you will learn something
about the way the subject is structured, and that

will help you succeed. If you can't figure it out, ask.

**Be skeptical.**

When you finish a problem, be prepared to back it up. Ask yourself "how can I
prove this is correct"? There are ways to do

this (checking units, testing limits, etc.); we will demonstrate in class. When
you use these methods , you help yourself by

looking at the problem in new ways, and by examining how it fits with other
ideas. Also, backing up your ideas is an extremely

useful skill in the workplace. Practice now.

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