Developing School Algebra Through A Focus on

Functions and Applications

Results from focusing on the

“dance of symbols”

• Fragile mastery of limited technical skills;

• Training in special procedures for solving inauthentic problems;

• Strong distaste for the subject

Rethinking the roles of variables, expressions, and equations.

• Variables are quantities that change over time and in response to change in other quantities;

• Equations represent relationships among quantities;

• Expressions represent algorithms for calculating values of dependent variables.

Algebra in Context

What average ticket price will maximize

operating profit of the MLL all-star game?

Variables and Relationships to Consider

Ticket price, tickets sold, income, expenses, profit, …

Demand: n(x) = 5000 – 65xIncome: I(x) = 5000x – 65x2

Expenses: E(n) = 4n + 25,000 E(x) = 45,000 – 260x

Profit: P(x) = – 65x2 + 5260x – 45,000

Alternative Solution Options

Alternative Solution Options

Alternative Solution Insights

What ideas give coherence to algebrain secondary school mathematics?

The most important goals of algebraic reasoning are understanding and predicting patterns

of change in variables.

What ideas give coherence to algebrain secondary school mathematics?

The letters, symbolic expressions, and equations or inequalities of algebra are tools for representing what we

know or what we want to figure out about a relationship between variables.

What ideas give coherence to algebrain secondary school mathematics?

Algebraic procedures for manipulating symbolic expressions into equivalent forms are also means to the

goal of insight into relationships between variables.

Calculating tools offer powerful alternative methods to gain insight and solve problems.

Situations Outside of MathematicsPatterns in Various Branches of MathematicsAlgebraic ReasoningSymbolic ReasoningReasoning with Tables and GraphsCreate tables, graphs, and equationsSolve equations and InequalitiesEstimate by tracing graphs and scanning tablesExplore rates of changeOptimize, interpret, predictRepresentRelationshipsInterpret MathematicalResults

Sources of Patterns, Relationships, and Questions

Essential dispositions, understanding, and skills:

Disposition to look for key quantitative variables in problem situations and

relationships among variables that reflect cause-and-effect, change-over-time, or

pure number patterns.

Essential dispositions, understanding, and skills:

A repertoire of significant and common patterns to look for—direct and inverse variation, linearity, exponential change,

quadratic patterns, etc.

Essential dispositions, understanding, and skills:

Ability to represent relationships between variables in words, graphs, data tables and

plots, and symbolic expressions.

Essential dispositions, understanding, and skills:

Ability to draw inferences from represented relationships by estimation

from tables and graphs, by exact reasoning using symbolic manipulations,

and by insightful interpretation of symbolic forms.

Essential dispositions, understanding, and skills:

Judgment to translate deductions back to the original problem situations with

reasoned sensitivity to limitations of the original modeling process.

Reasonable Concerns

• Is this “algebra”?

• Does the function-oriented development serve well the variety of topics in which algebraic manipulation is useful?

• Don’t users of CAS need some personal skill to understand how to utilize the tool?

• Are CAS tools flexible enough and adaptable enough to serve all problem solving needs?

• Is “just in time” skill development feasible pedagogically?