Background

This is a content course, not a methods course. However, throughout the course we model and pay attention to pedagogical considerations

Objectives

To strengthen the background of those who teach or wish to teach middle school algebra (pre-algebra through algebra II) in algebraic thinking and in constructivist algebraic instructional design aligned with NCTM recommendations. We will analyze middle school algebra from a higher, teacher?s perspective (what comes in future courses in HS and college?)

Intended Audience

Pre-service and in-service elementary and middle school teachers admitted to SU's graduate program who wish to strengthen their mathematical background and confidence for teaching K-12 courses based on algebra (from pre "pre-algebra" courses, through Algebra, Algebra II, pre-calculus and beyond).

About the Course

This course provides a solid foundation in mathematical content (specifically, algebra) which is rooted in experiential learning, enabling its graduates to lead their students to experience algebraic thinking.

Emphasis is placed on mathematical meaning behind algebraic expression as well as on modeling real-world phenomena. Multiple paths towards solutions, and multiple perspectives from various contexts ground understandings of algebraic expressions at first, and ultimately of more general algebraic frameworks. These experiences, rather than memorized algorithms and rote manipulation of symbols, serves as the course's foundation.

Active communication by those enrolled in this graduate course play a central role and serve as a path towards deepened understanding as teachers rise to the challenge of expressing their ideas in a clear, comprehensible, and organized fashion. Participants complete written and oral-presentation assignments. Technology is used as a tool for student-centered investigations and explorations (e.g. the applets Creating, Describing, and Analyzing Patterns at http://standards.nctm.org/document/eexamples/chap4/4.1/). The internet also provides multi-disciplinary applications and dynamic representations of underlying concepts. Students experience algebraic concepts directly through the use of manipulatives, including Hands On Equations kits.

In alignment with NCTM's Standards 2000 (Algebra, 6-8, pp. 222-231) emphasis is placed on the use of tables, graphs, words, and symbolic expressions as well as the interconnections between them. These are supplemented with visual/geometric interpretations of algebraic properties ("picture proofs" of properties like FOIL or the Distributive Property).

These multiple representations are used to recognize, construct, analyze, and generalize a variety of patterns and rules, and to solve context-based problems. Linear, quadratic, and exponential functions are studied from these multiple perspectives. Such functions are not be studied as a collection of "black boxes" with names; rather, the conceptual and intuitive underpinnings of these functions are be explored. For example, when one variable increases at a constant rate, what happens to the other variable? Teachers enrolled in Math 541 explore the range of possibilities (e.g. constant-rate increase, increasing at a decreasing rate, etc, and asymptotic behavior versus increasing/decreasing without bound) in many (concrete) contexts. In this way students build a personal familiarity with, and intuition about algebra which supports an understanding of the rigorous mathematics algebraic models represent.

Special attention is given to linear relationships, slope, and the effect of scale, choice of variables and units on the look and shape of graphs and data plots. Connections will be drawn between algebra and geometry through patterns and other areas of overlap such as the area representation of multiplication and its ability to demystify algebraic properties such as the distributive law and commutative property.

These experiences (together with class discussions, written assignments, assigned Concept Maps/Webs, etc) allow us to construct rich mental landscapes of context and meaning which provide the critical support structure for mastery of algebraic techniques and general properties. As increased confidence emerges, more positive attitudes about mathematics naturally emerge as it is understood as based in the concrete world -- not based on memorization of "arbitrary" laws, but having rules which emerge from patterns and structures discovered in individual and collaborative learning explorations.