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Below are lists of the most important math
proficiencies needed in first-year Calculus...
General
Proficiencies
Post-secondary instructors insist that the general
proficiencies in the table below are extremely important for anyone wishing to
succeed in a Calculus course. In many ways they are more valuable
proficiencies for the student entering a post-secondary Calculus course than are even a
large number of individual, specific mathematics skills. Although general
proficiencies are difficult to measure, the effects of these qualities and attitudes
towards success in any Calculus course is difficult to over-emphasise.
General
Proficiencies Essential For Calculus |
General
Proficiency |
Related Proficiencies |
Comments |
| Thinking Skills |
Critical, Logical Thinking |
Orderly, analytical, inquiring thought processes. |
| Solving Word Problems |
|
Experience and comfort in setting up, solving and writing up word
problems. |
| Multi-step Problems |
|
Multi-step problems involve two or more stages to arrive at a
solution. They are not necessarily word problems. |
| Modelling |
|
Finding appropriate constructs or methods to solve a problem.
Example: writing a function that captures the components of a problem. |
| Integration of Topics |
Combining concepts
Synthesising |
Using two or more concepts in combination. Example: the
concept of an inverse of a function with the exponential function. |
Abstracting/
Generalising |
Concept development |
Seeing common characteristics in separate settings.
Abstractions lead to the creation of a system of symbols. |
| Symbolism |
Symbol Manipulation |
Familiarity and comfort with mathematical symbols. |
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Specific Proficiency Categories
The survey completed by college/university instructors listed broad
proficiency categories containing more-specific sub-proficiencies. These
main proficiency categories of specific mathematics topics that students typically
encounter in senior secondary courses are ranked in the table below. Each category
has also been given an importance rating of: Essential, Very
Important, Important, Nice to Know or
Not Important, in accordance with the survey results.
Code |
Mathematics Proficiency Categories Rated for Calculus |
| L |
Understand and use the Function Concept. |
Essential |
| A |
Understand and use Polynomial Expressions. |
Essential |
| C |
Understand and use Exponential Expressions. |
Essential |
| M |
Understand and use Straight Line and Linear
Functions. |
Essential |
| E |
Solve Equations and Inequalities. |
Essential |
| S |
Understand and use Circular Trigonometric
Functions. |
Essential |
| B |
Understand and use Rational Expressions. |
Essential |
| H |
Understand and use Triangle Trigonometry. |
Essential |
| N |
Understand and use the Quadratic Function. |
Essential |
| R |
Understand and use the Logarithmic Function. |
Essential |
| D |
Understand and use Radical Expressions. |
Essential |
| F |
Understand and use the Geometry of Lines and
Points. |
Essential |
| Q |
Understand and use Polynomial Functions. |
Very Important |
| P |
Understand and use Quadratic Relations. |
Very Important |
| K |
Understand and use Sequences and Series. |
Very Important |
| G |
Understand and use the Geometry of Circles. |
Very Important |
| T |
Understand and use some Concepts of the
Calculus. |
Nice to Know |
| J |
Understand and use Data Analysis. |
Nice to Know |
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Specific Sub-Proficiencies - THE TOP 20
Each Specific Proficiency Category contains several, individual
sub-proficiencies. The twenty individual proficiencies rated highest
for entry to university/college Calculus are given below:
THE
TOP 20 MATH PROFICIENCIES FOR CALCULUS - IN ORDER OF IMPORTANCE |
| Code |
Mathematics Proficiency |
Rating |
| L1 |
Understand and use function notation. |
Essential |
| M1 |
Graph and write the equations for linear
functions. |
Essential |
| M2 |
Understand and use the slope of a line. |
Essential |
| A1 |
Simplify polynomial expressions. |
Essential |
| A2 |
Add and subtract polynomial expressions. |
Essential |
| A3 |
Multiply and simplify polynomial expressions. |
Essential |
| C1 |
Use the Laws of Exponents to simplify
expressions. |
Essential |
| E1 |
Solve linear equations. |
Essential |
| C2 |
Write radical expressions in exponential form. |
Essential |
| F3 |
Understand and use the Pythagorean Theorem. |
Essential |
| C3 |
Apply the Laws of Exponents to expressions
with rational exponents. |
Essential |
| E2 |
Solve linear inequalities. |
Essential |
| A4 |
Factor binomials and trinomials. |
Essential |
| B2 |
Multiply and divide simple rational
expressions. |
Essential |
| B3 |
Add and subtract rational expressions. |
Essential |
| H1 |
Understand and apply the primary trig ratios
as related to triangles. |
Essential |
| L3 |
Graph basic R R functions like linear, absolute value, etc.on the Cartesian
plane. |
Essential |
| E5 |
Solve quadratic equations using the quadratic
formula. |
Essential |
| B1 |
Simplify rational expressions with polynomial
numerators and denominators. |
Essential |
| R1 |
Write exponential expressions in logarithmic
form. |
Essential |
Each of the above specific proficiencies is essential for success in a Calculus course.
The next 21 proficiencies are also rated as Essential. To view the specific
ratings and the entire list of topics, including ones not specifically mentioned in the
survey, you may use a pdf reader to download the entire Mathematics Proficiencies
Report from HERE.
The Mathematics Proficiencies for Post-Secondary Mathematics/Statistics Courses
Project Report may be freely quoted when the source is credited.
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