Design Teams for Mathematics Focus Area Courses
Design teams of mathematicians, mathematics educators, and teachersinresidence worked collaboratively to improve the mathematical preparation of teachers. Listed below are the math focus area courses for elementary education majors that were developed and taught by the design teams at the request of the Department of Curriculum and Instruction at UWM. The Department of Curriculum and Instruction requires elementary education majors to choose a focus area and to take additional courses in that area. These courses would strengthen the Mathematical content knowledge of students choosing a mathematics focus area.
Mathematical Problem Solving and Critical Thinking
This course seeks to build a strong foundation for the teaching and communication of mathematical concepts by the handson development of critical thinking skills via problem solving experiences. It provides a guided opportunity for the implementation of problemsolving instruction in a classroom. Students are encouraged to become better problemsolvers by experimenting with various problems. Students are required to maintain a journal with entries for each class and problem project.
Course Objective: In Principles and Standards for School Mathematics, published by the National Council of Teachers of Mathematics, five Process Standards are identified that highlight ways of acquiring and using mathematical content knowledge. These Process Standards are Problem Solving, Reasoning and Proof, Communication, Connections, and Representation. In this course, we seek to address all five standards and:
 Become better problemsolvers by experimenting with and solving various problems
 Develop confidence in our mathematical understanding of basic concepts by problem solving and discussion.
 Emphasize the process of doing mathematics as well as the product
 Encourage multiple methods of solution and divergent thinking
 Build experience and skills in problem solving strategies
 Explore problems indepth
 Foster collaboration and other forms of constructivist learning
 Learn to use various technology tools to aid understanding
 Express mathematical ideas in a clear concise language to classmates and our students.
Sample Problems from this Course
Geometry for Elementary Education Majors
The goal of this course is to develop the students’ familiarity and facility with various aspects of geometry. Students will be exposed to the basic facts and techniques of geometry by having them solve practical and tangible problems involving spatial objects and patterns. Also, students will confront the logical web of geometry based on simple axioms by having them formulate definitions and conjectures, complete proofs and search for counterexamples.
Course Objective: Geometry is a vast subject with a several thousand year history and mention of geometric methods can be found in some of the earliest written human records. Geometry is, on the one hand, a very applicable subject comprising a collection of methods for studying properties of figures in space such as length, angle, area, volume, visual patterns and symmetry. On the other hand, geometry is a highly interconnected logical subject in which a few simple basic principles give rise to huge variety of geometric facts and techniques. The goal of this course is to develop the students’ familiarity and facility with both of these aspects of geometry. Students will be exposed to the basic facts and techniques of geometry by having them solve practical and tangible problems involving spatial objects and patterns. Also, students will confront the logical web of geometry based on simple axioms by having them formulate definitions and conjectures, complete proofs and search for counterexamples.
Sample Problems from the Course
Discrete Probability and Statistics
This course provides an examination of the principles of Probability and Statistics. The goal of the course is to enable students to determine the likelihood of various events occurring and to make inferences based on such likelihoods. This course aims to provide content knowledge in probability and statistics to elementary educations majors who choose Mathematics as one of their focus areas.
Course Objective: Dramatic advances in technology have launched the world into the Information Age, when data are used to describe past events or predict future events. Whether in the business place or in the home, as producers or consumers of information, citizens need to be well versed in the concepts and procedures of data analysis in order to make informed decisions. This course provides an examination of the principles of Probability and Statistics. The goal of the course is to enable you to determine the likelihood of various events occurring and to make inferences based on such likelihoods. Additionally, this course aims (1) to provide mathematical content knowledge in the area of probability and statistics to preservice teachers so that they in turn can facilitate their students’ achievement of the Wisconsin performance standards, (2)to provide preservice teachers a “big picture” of how knowing frequency of random occurrences enables one to make meaningful inferences, and (3)to provide realworld examples of problems in which one needs to understand the patterns of randomness in order to fully develop a complete understanding of the solution to the question.
Sample Problems from the Course
Algebraic Structures Course
Through this course, prospective teachers develop a deep understanding of the structure of algebra—not just what algebraic steps to take when solving a problem, but why it makes sense to take those steps, what is the underlying algebraic reasoning, and where in mathematics do the same structures appear. Course topics included language and mathematical logic, set theory, functions, operations, structures (such as groups, rings, and fields) and number theory.
Capstone Course
This course for prospective high school teachers connects major strands in courses for math majors with high school mathematics. It is an opportunity for prospective high school teachers to look deeply at fundamental ideas of mathematics, to connect topics that students often see as unrelated, and to develop important mathematical habits of mind. The course connects logically to the course work in mathematics education as well as making
a strong positive contribution to the mathematical understanding and pedagogical
skills needed by all successful teachers of high school mathematics. The course topics include: a history of mathematics, Number systems, Dynamics and Fractals, and Descartes’ “Big Idea"
