Perspectives in Mathematics

A Portfolio of Mathematical Understandings

Introduction and General Instructions:

The following standards are essential components of your knowledge base as a future teacher of grades 5-12 mathematics. Yet, they may not appear explicitly on the syllabus for any of the St. Olaf mathematics/statistics courses required for students seeking licensure to teach grades 5-12 mathematics, because:

most students have achieved the standard in pre-college work; or
the standard calls for understandings across several courses; or
the standard calls for learning that is best done with individual study.

Before you apply for a St. Olaf recommendation for licensure to teach mathematics in grades 5-12, you must compile a portfolio that demonstrates your knowledge of the concepts and procedures listed below. You may demonstrate your knowledge in a variety of ways, including:

Submitting work from previous classes, with cover statements as to how they demonstrate the standards claimed;
Presenting complete work on a series of problems provided by a the mathematics department, with a cover statement as to how your work demonstrates attainment of the standards claimed;
Conducting an experiment or mathematical study and writing up the results;
Writing a research paper or reading and writing a response to articles that address mathematical history or perspectives.

You will complete this portfolio as part of your work for Education 350, but you may obtain advice earlier about how you may demonstrate each item. The portfolio will be assessed by a panel of Mathematics Department faculty convened by the Instructor of Education 350.

Required Items in Your Portfolio:

Standard C: NUMBER SENSE

Demonstrate your understanding of number sense, including:

(C1) an intuitive sense of numbers including a sense of magnitude, mental mathematics, place value, and a sense of reasonableness of results;
(C2) an understanding of number systems, their properties and relations including whole numbers, integers, rational numbers, real numbers, and complex numbers;
(C3) translation among equivalent forms of numbers to facilitate problem solving;
(C4) application of appropriate methods of estimation of quantities and evaluation of the reasonableness of estimates;
(C5) a knowledge of elementary operations, application of properties of operations, and the estimation of results;
(C6) geometric and polar representation of complex numbers and the interpretation of complex solutions to equations;
(C7) algebraic and transcendental numbers;

Standard D: SHAPE AND SPACE

Demonstrate your understanding of geometry and measurement from both abstract and concrete perspectives, including:

(D0) identify real world applications and to use geometric learning tools and models, including geoboards, compass and straight edge, rules and protractor, patty paper, reflection tools, spheres, and platonic solids;
(D1) shapes and the ways shapes can be derived and described in terms of dimension, direction, orientation, perspective, and relationships among these properties;
(D2) spatial sense and the ways shapes can be visualized, combined, subdivided, and changed to illustrate concepts, properties, and relationships;
(D3) spatial reasoning and the use of geometric models to represent, visualize, and solve problems;
(D4) motion and the ways in which rotation, reflection, and translation of shapes can illustrate concepts, properties, and relationships;
(D7) attributes of shapes and objects that can be measured, including length, area, volume, capacity, size of angles, weight, and mass;
(D8) the structure of systems of measurement, including the development and use of measurement systems and the relationships among different systems;
(D9) measuring, estimating, and using measurements to describe and compare geometric phenomena;
(D13) elementary topology, including topological properties and transformations;
(D15) unit circle trigonometry.

Standard G: MATHEMATICAL PROCESSES

Demonstrate your ability to reason mathematically, solve problems mathematically, and communicate in mathematics effectively at different levels of formality and your knowledge of the connections among mathematical concepts and procedures as well as their application to the real world. In particular, demonstrate your ability to:

(G1) solve problems in mathematics by:

formulating and posing problems;
solving problems using different strategies, verifying and interpreting results, and generalizing the solution;
using problem solving approaches to investigate and understand mathematics; and
applying mathematical modeling to real world situations;

(G2) reason in mathematics by:

examining patterns, abstracting and generalizing based on the examination, and making convincing mathematical arguments;
framing mathematical questions and conjectures, formulating counter-examples, and constructing and evaluating arguments; and
using intuitive, informal exploration, and formal proof.

G3) communicate in mathematics by:

expressing mathematical ideas orally, visually, and in writing;
using the power of mathematical language, notation, and symbolism; and
translating mathematical ideas into mathematical language, notations, and symbols; and

(G4) make mathematical connections by:

demonstrating the interconnectedness of the concepts and procedures of mathematics;
making connections between mathematics and other disciplines;
making connections between mathematics and daily living; and
making connections between equivalent representations of the same concept.

Standard H: MATHEMATICAL PERSPECTIVES

Provide evidence from a variety of experiences to demonstrate your understanding of the following perspectives:

(H1) understand the historical bases of mathematics, including the contributions made by individuals and cultures, and the problems societies faced that gave rise to mathematical systems (this standard will most likely be met by writing a paper on this topic);
(H2) recognize that there are multiple mathematical world views and how the teacher's own view is similar to or different from that of the students (this standard can be met by reading and writing a written response to the paper "The Centrality of Mathematics in the Western World);
(H3) understand the overall framework of mathematics including the:

processes and consequences of expanding mathematical systems;
examination of the effects of broad ideas, including operations or properties, as these ideas are applied to various systems;
examination of the same object from different perspectives; and
investigation of the logical reasoning that takes place within a system; and

(H4) understand the role of technology, manipulatives, and models in mathematics.

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