M4O8C AND D: CALCULUS I AND II: RELEVANCE TO FUTURE SECONDARY TEACHERS
What is calculus?
Calculus is a branch of mathematics that deals with various interconnected topics involving the concept of limit. These topics include rate of change (which in turn includes the concepts of velocity and acceleration), areas of curved regions, and infinite sums. Here are some more detailed answers to this question:
How do these courses address the Texas State Board for Educator Certification (SBEC) Standards for Secondary Math Teachers?
Standards II (Patterns and Algebra), III (Geometry and Measurement), and V (Mathematical Processes) of the SBEC Secondary Math Standards include the following standards regarding the content of M408C and D:
"The beginning teacher of mathematics is able to:
- (2.20s) analyze the properties of sequences and series and use them to solve problems involving finite and infinite processes;…
- (2.23s) analyze attributes of functions and, relations (e.g., domain, range, one-to-one functions, composite functions, inverse functions, odd and even functions, continuous functions) and their graphs; ...
- (2.24s) describe linear, quadratic, and other polynomial functions, analyze their algebraic and graphical properties, and use these to model and solve problems ...
- (2.25s) describe exponential [and] logarithmic ... functions algebraically and graphically, analyze their algebraic and graphical properties, and use these to model and solve problems ...
- (2.26s) describe trigonometric and circular functions algebraically and graphically, analyze their algebraic and graphical properties, and use these to model and solve problems using a variety of methods ...
- (2.27)s) describe rational , radical, absolute value, and piecewise functions algebraically and graphically,, analyze their algebraic and graphical properties, and use these to model and solve problems using a variety of methods ...
- (2.28s) investigate and solve problems using techniques of differential and integral calculus …
- (2.30s) apply the properties of vectors and vector algebra to solve pure and applied problems
- (2.32s) demonstrate an understanding of analysis (e.g., analytic geometry and calculus) and its relationship to secondary mathematics ... (pp. 8-9)
- (3.14s) relate geometry to algebra and trigonometry by using the Cartesian coordinate system and use this relationship to solve problems; ...
- (3.15s) use calculus concepts to answer questions about rates of change, areas, volumes, and properties of functions and their graphs...
- (3.20s) show how differential calculus is used to answer questions about rates of change and optimization;
- (3.21s) use integral calculus to compute various measurements associated with curves and regions in the plane, and measurements associated with curves, surfaces, and regions in three-space;
- (3.22s) illustrate geometry from several perspectives, including the use of coordinate systems, transformations, and vectors;... " (pp. 11 - 12)
- (5.15s) explore problems using verbal, graphical, numerical, physical, and algebraic representations;
- 5.16s) recognize and use multiple representations of a mathematical concept (e.g., a point and its coordinates, the area of a circle as a quadratic function in r, ...)...
- (5.17s) apply mathematical methods to analyze practical situations; and
- (5.18s) use mathematics to model and solve problems in other disciplines, such as art, music, science, social science, and business." (p. 19)
How do these courses relate to the Texas Essential Elements for secondary mathematics?
The TEKS web page Mathematics Course Selections shows a flow diagram of mathematics secondary math courses. One of these is AP Calculus. It is taught in many Texas high schools. You can link from the TEKS page to more information about AP Calculus. M 408C and D cover the material of the AP Calculus courses, so future teachers who may be teaching calculus definitely need to take these courses.
Moreover, the flow diagram shows that most of the high school math courses are prerequisites for calculus. Thus even a teacher who will never teach calculus needs to be familiar with the subject in order to be able to prepare her or his students who will be taking calculus.
Also, the following items in the TEKS for Precalculus are topics in M 408C and D:
- (1 E) investigate continuity, end behavior, vertical and horizontal asymptotes, and limits and connect these characteristics to the graph of a function.
- (2 B) perform operations including composition on functions, find inverses and describe these procedures and results verbally, numerically, symbolically, and graphically;.
- (3 A) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data.
- (3 C) use properties of functions to analyze and solve problems and make prediction
- (4 A) represent patterns using arithmetic and geometric sequences and series
- (4 B) use arithmetic, geometric, and other sequences and series to solve real-life problems;
- (4 C) describe limits of sequences and apply their properties to investigate convergent and divergent series; and
- (4 D) apply sequences and series to solve problems
- (5 A) use conic sections to model motion.
- (5 B) use properties of conic sections to describe physical phenomena
- (5 C) convert between parametric and rectangular forms of functions and equations to graph them; and
- (5 D) use parametric functions to simulate problems involving motion.
- (6 A) use the concept of vectors to model situations defined by magnitude and direction; and
- (6 B) analyze and solve vector problems generated by real-life situations.
How do these courses relate to the National Council of Teachers of Mathematics' Principles and Standards for School Mathematics?
The Principles and Standards do not have a section specifically about calculus. However, many calculus topics are included in various content strands. Examples:
In what other ways will this course help prepare me to be a secondary mathematics teachers?
- Since calculus uses most of high school mathematics, these courses will help you see how high school topics are used as well as help you know them more thoroughly and understand them in greater depth.
- Many of the students you teach in high school will be taking calculus in college, so you need to have experience in calculus in order to be able to prepare these students to succeed in calculus.
- Calculus is important for having a deep understanding of probability and statistics, topics which are becoming increasingly important in the secondary curriculum.
- Calculus is an important part of a mathematical education and important in applications of mathematics to science, business, and some social sciences.
- In some schools, calculus concepts are being introduced into several courses in the secondary curriculum. See the SimCalc Project for some interesting examples of this.
How do these courses relate to other courses I will be taking?
- Calculus is used in M 362K, Probability I, which is a required math course. Improper integrals, series, and double integrals (topics from M 408D) are especially important in M 362K.
- Calculus is used M 427K and M378K. BS-Teaching Option students are required to take at least one of these courses as a depth course.
- Calculus is a prerequisite for most upper division math courses. The ones required for the Teaching Options that are not mentioned above do not always use calculus, but experience in calculus helps to develop the mathematical maturity needed for these courses and is a tool that instructors may draw on for examples or special topics in other courses.
- You are urged to take a calculus-based Physics course for your science requirement.
- Many of the options for the Breadth Course for the BS-Teaching Option use calculus
- The UTeach course Perspectives in Math and Science may discuss the development and impact of some of the ideas of calculus
How can I get the most out of these courses?
Most importantly, remember that calculus involves understanding concepts, reasoning, and setting up problems as well as following procedures.
Many students find calculus difficult. There are many factors contributing to this, including:
- Calculus involves new ideas
- Many students come to UT with inadequate preparation in prerequisite mathematics. In particular, many students are accustomed to thinking of math as just carrying out procedures, rather than understanding concepts, reasoning, and setting up problems.
- Because of high enrollments and budget constraints, we teach calculus in large sections
- Students usually take calculus in their first year, when they are adjusting to the change from high school to calculus.
Despite these difficulties, it is possible to succeed in calculus at UT (most students do), and gain a lot from the course. Here are some tips on how to get the most out of calculus.
- If possible, take M315C (Functions and Modeling) along with M408C. This will help you develop some important calculus and precalculus concepts in an activity-based lab setting.
- If possible, sign up for the Emerging Scholars Program. Students in this program work together to solve challenging problems that help them understand calculus concepts and practice calculus and precalculus skills.
- If you are not able to participate in the Emerging Scholars program, form a study group to help you learn by discussing your work.
- Be aware that college expectations on out of class work are different from high school expectations. The typical expectation in college is that a student will spend twice as much time on a class outside of the classroom as inside the classroom. This means that you should plan to spend about ten hours a week outside of class on calculus, in addition to attending class. Your homework assignments probably won't take all this time; the rest should be spent in studying the textbook and in reviewing, absorbing, and thinking about what went on in class.
- Consult KARL'S CALCULUS TUTOR for tips such as "Why Bother to Learn Calculus," "Stuff You Should Already Know, " and "Study Tips."
- As you take calculus, ask yourself where you could have used better preparation in your high school mathematics, and how you can a give your future students the preparation that will help them do their best in calculus.
- Take advantage of the opportunities (review sessions, tutoring, self-help programs) provided by the UT Learning Center