*[*

Course approved by Faculty Senate.

University Studies Course Approval Proposal

Unity and Diversity – Critical Analysis

The Department of Mathematics and Statistics proposes the following course for inclusion in University Studies, Unity and Diversity, Critical Analysis at Winona State University. This was approved by the full department on Thursday, January 4, 2001.

Course: Technology Based Geometry and Other Essential Mathematics for Elementary Teachers

MTED 201 4 s.h.

Catalog Description:

Study of additional mathematical topics essential to mathematics in the elementary and middle school grades including Euclidean geometry. This course is designed to fit the requirements of the University Studies category, Critical Analysis. Prerequisite: MTED 125. Offered each semester.

This is an existing course, previously approved by A2C2.

Department Contact Person for this course:

George Gross; email wngros144@winona.edu

Critical Analysis

Critical Analysis courses in the University Studies program are devoted to teaching critical thinking or analytic problem-solving skills. These skills include the ability to identify sound arguments and distinguish them from fallacious ones. The objective of these courses is to develop students’ abilities to effectively use the process of critical analysis. Disciplinary examples should be selected to support the development of critical analysis skills.

** **

This course includes requirements and learning activities that promote students’ abilities to

;*evaluate the validity and reliability of information*- analyze modes of thought, expressive works, arguments, explanations, or theories;
- recognize possible inadequacies or biases in the evidence given to support arguments or conclusions; and
**advance and support claims**.

Generally, inherent in working with mathematical theories, problems, and concepts is a necessity to question the validity and reliability of given information as well as the calculated conclusions. If the conclusions do not make sense when the given information is used in making these conclusions, then the student must make decisions as to whether the information or the context in which it is given is invalid and/or unreliable, or if there is some other reason for the inconsistency(s).

For instance when using the Geometer’s Sketchpad to investigate geometric concepts to, say, make measurements, there are times that these measurements given by the software are not exact, and this presents opportunities to discuss why they are not given exactly correct. When calculating empirical versus theoretical probability the students need to judge the validity and reliability of the information given or found experimentally and why it may be different than the theoretical probability. When working any kind of an applied problem in the geometry area, the given data must be consistent with the context of the problem, and be realistic and consistent with concepts involved. The students need to be continuously on the alert for this.

Generally, the course deals with geometric and other mathematical arguments and explanations that lead to theories (usually called theorems), and then these theories and other conclusions reached are applied to solving problems in these mathematical areas. These arguments are sometimes present by the instructor, sometimes read by the student, and sometimes constructed and/or discovered by the student. To understand mathematics it is necessary, that as arguments are made each student has the opportunity to analyze these arguments for accuracy and validity.

Students in this class are frequently required to use this process in situations involving congruency of triangles, for validating the accuracy of geometric constructions, in discovering the Pythagorean relationship, etc.

In general terms, the students are asked to be on the lookout for such inadequacies and/or biases in the mathematical arguments and conclusions as they build and reflect on their own mathematical understanding. Typically they will evaluate classroom discussion, readings, and/or student discovery exercises.

For example, when working on a geometric investigation on computer software,

it is important for the student to be aware of the methods used, the accuracy and appropriateness of the information that is input, the order of which the sketches are done, etc. too avoid any biases or inadequacies in these inputs(evidence) in making

their conclusions. In the process of producing geometric arguments the students need to recognize any inadequacies in the information that is assumed.

In this course conjecturing (advancing and supporting claims) permeates much of the course in order to enhance mathematical understanding leading to conceptual knowledge of mathematical content.

Sketchpad investigations consist predominately of advancing conjectures and attempting to support them via making measurements on the diagrams formed from the dynamic software and or arguing their results synthetically. Paper and pencil investigations and arguments are also done throughout the course.

WINONA STATE UNIVERSITY

COLLEGE OF SCIENCE AND ENGINEERING

DEPARTMENT OF MATHEMATICS AND STATISTICS

Course Outline-MTED 201

** **

**Course Title**: Technology Based Geometry and Other Essential
Mathematics for Elementary Teachers.

**Frequency of Offering**: Each Semester **Prerequisite(s)**: MTED
125

**Grading:** For elementary education majors and minors, offered on a
"Grade Only" basis.

P/NC is available to others.

**Course Applicable**: Elementary Education specialty, required
graduate credit available for

in-service teachers.

**Catalog Description**:

Study of additional mathematical topics essential to mathematics in the elementary and middle school grades including Euclidean geometry. This course is designed to fit the requirements of the University Studies category, Critical Analysis. Prerequisite: MTED 125. Offered each semester.

**Number of Credits**: 4 Semester Hours

**Required Textbook(s): **

Billstein, et al, 2001, __A Problem Solving Approach to Mathematics
for Elementary School Teachers, Addison Wesley Longman __

**Additional Readings**: None

**Standards Included**: Minnesota law requires that Elementary
Education programs

include topics appropriate to issues encountered in grades K-8.

This course is designed to help students develop competencies

outlined in the Minnesota Standards of Effective Practice for

Beginning Teachers. In particular4 this course will address the

following Standards:

Standard 1-Subject Matter

Subpart 3C (1, 2, 3ab, 4, 6a, 7, 8)

Subpart 4B (1f, 2b, 4, 6bcd)

**Additional Requirements**: None

**Course Description**:

1. The major focus of this course is to provide students with not covered in MTED 125,

2. a foundation in probability and geometry for elementary education,

3. expertise in the use of technology to explore, discover, and learn geometry.

** **

**Course Objectives: **The objectives of this course include the
following.

The future teacher should be able to:

1. show an understanding of the properties and relationships of geometric figures,

2. formulate and solve problems from both mathematical and everyday situations,

3. write using everyday language and mathematical language, including symbols,

4. communicate mathematical ideas orally, using both everyday language and mathematical language,

5. use geometric learning tools such as geoboards, compass and straight edge, ruler, protractor, patty paper, reflections tools, spheres, and platonic solids,

6. show an understanding of geometry and measurement from both abstract and concrete perspectives and identify real world applications,

7. connect mathematics to other disciplines and real-world situations,

8. understand and apply the process of measurement,

9. use geometric concepts and relationships to describe and model mathematical ideas and real-world constructs,

10. collect, organize, represent, analyze and interpret data (in geometry),

11. use algebra to describe patterns, relations and functions, and o model and solve problems,

12. use estimation in working with quantities, measurement, computation, and problem solving,

13. use calculators in computational and problem-solving situations,

14.use computer software to explore, and discover geometric concepts, and solve geometric problems,

15. use concepts and techniques of discrete mathematics and how to use them to solve problems from areas of graph theory, combinatorics, and know how to help students calculate probabilities and trace paths in network graphs,

** **

Critical Analysis

Critical Analysis courses in the University Studies program are devoted to teaching critical thinking or analytic problem-solving skills. These skills include the ability to identify sound arguments and distinguish them from fallacious ones. The objective of these courses is to develop students’ abilities to effectively use the process of critical analysis. Disciplinary examples should be selected to support the development of critical analysis skills.

** **

This course includes requirements and learning activities that promote students’ abilities to

- evaluate the validity and reliability of information;
- analyze modes of thought, expressive works, arguments, explanations, or theories;
- recognize possible inadequacies or biases in the evidence given to support arguments or conclusions; and
- advance and support claims.

The lowercase, boldface letters a.) – d.) are used in the following outline of topics and subtopics to indicate course requirements and learning activities which address these.

** **

**Course Outline of the Major Topics and Subtopics**:

- Probability

A. Determining Probabilities

a), b), c), d)

- Multistage Experiments with Tree Diagrams and Geometric Probabilities.
**b), d)** - Using Simulations in Probability
**a), b), d)** - Odds and Expected Value
**b), d)** - Methods of Counting
**b), d)**

II. Rate of Change

d)III. The Geometer’s Sketchpad

- Guided Tours
- Investigations and Constructions
**a), b), c), d)**

IV. Introductory Geometry

- Basic Notions
- Introductory Logic
**b), d)** - Polygons
**a), b), c), d)** - More about Angles
**b), d)** - Geometry in Three Dimensions
**d)** - Networks
**d)**

V. Constructions, Congruence, and Similarity

- Congruence Through Constructions
**a), b), c), d** - Other Congruence Properties
**a), b), c), d)** - Other Constructions
**b), d)** - Similar Triangles and Similar Figures
**b), c), d)** - Lines in a Cartesian Coordinate System

VI. Concepts of Measurement

- Linear Measure
**a), d)** - Areas of Polygons and Circles
**a), b), d)** - The Pythagorean Theorem
**a), b), c), d)** - Surface Areas
**a), b), d)** - Volume, Mass, and Temperature
**a), b), d)**

VII. Motion Geometry and Tessellations

- Translations and Rotations
**b), c), d)** - Reflections and Glide Reflections
**b), c), d)** - Size Transformations
**b), c), d)** - Symmetries
**b), d)** - Tessellation of the Plane

**Method of Instruction**: Instruction will model a variety of methods
and strategies including

lecture, discovery, group activities, writing, reports, student presentations, teacher and/or student guided discussion, and seminar type sharing of information. The course involves discovery learning/exploration using the computer.

** **

**General Expectations**: All students are expected to attend class on
a regular basis, to be

active participants in class and to be readers of the texts and

related writings. Students will be expected to demonstrate their

knowledge of the subject through writing, problem solving, presentations,

peer teaching, projects, and at least three examinations.

**Method of Assessment: **Assessments will vary in style, including
teacher evaluation of written exams, written homework problems, written reaction papers,
in-class group and individual problems, computer investigations and/or projects.

Additional References: __Geometry in the Middle Grades__, from the
Addenda Series for Grades 5-8, NCTM.

Exploring Geometry(Key Curriculum Press).

Dealing with Data and Chance, from the Addenda Series for Grades 5-8, NCTM.

**Approval/Disapproval Recommendations**

Department Recommendation: Approved Disapproved Date

Chairperson Signature Date

Dean's Recommendation: Approved Disapproved Date

Dean's Signature Date

*In the case of a Dean's recommendation to
disapprove a proposal, a written rationale for the recommendation

to disapprove shall be provided to USS.

USS Recommendation: Approved Disapproved Date

University Studies Director's Signature Date

A2C2 Recommendation: Approved Disapproved Date

A2C2 Chairperson Signature Date

Faculty Senate Recommendation: Approved Disapproved Date

FA President's Signature Date

Academic VP's Recommendation: Approved Disapproved Date

VP's Signature Date

President's Decision: Approved Disapproved Date

President's Signature Date