Functions, Models, and Dev Math

First of all, if you are not ‘doing exponentials’ in developmental mathematics then you are missing many great opportunities.  From finances to environmental studies to biology, repeated multiplying is a very common process — and is at least as practical as repeated adding (linear).

This post is about two issues … first, what functions are relevant for which developmental math course, and second, how to present the distinction between functions and models.

A pre-algebra level course should include practical experience with linear and exponential situations.  Linear relationships can be used as part of working on proportional reasoning, where the rate of change (like ’12 in/1 ft’) can be written in two forms depending on which value is the input.  Various representations are accessible to students, so an understanding of graphs of data can be included — even without dealing with concepts like slope and intercepts.  Exponential relationships can be used starting from a practical context such as compound interest or indices such as the “CPI”, where the multiplier can be written as a percent added to 1 … including negative changes.  Representations can be included.

A first algebra course should formalize the practical work with these functions to include the symbolic forms normally seen, and concepts related to graphing — slope, intercepts, base, initial value.    The first algebra course can introduce quadratic relationships based on geometry, but it is more important that students understand function terminology and some notation. 

A second algebra course should ease away from practical contexts to deal with topics from a more scientific point of view; half-life and doubling-time would be appropriate.  The second algebra course could include work on conic sections, especially if the course serves to prepare students for pre-calculus.

The distinction between functions and models should be included in both algebra courses.  Even a pre-algebra course should have measurement concepts such as precision and accuracy.  The first algebra course can use this to describe the distinction — functions represent data where the only variation is due to measurement error and show a known relationship between inputs and outputs, while models represent data where other sources of error cause variation and reflect an educated guess about the possible relationship between inputs and outputs. In the second algebra course, students should have experience in judging the distinction between functions and models for themselves.

Functions and models should form a significant part of any algebra course, with more attention than symbolic manipulations.  This emphasis should be especially strong in the first algebra course; the second algebra course can reasonably incorporate relatively more symbolic work.  A pre-algebra course should be mostly about developing a quantitative sense concerning numbers in relationships.

The “New Life” model courses have learning outcomes that reflect this point of view, at least partially.  Whether you can do New Life courses, or can just make changes to existing courses, I encourage you to strengthen the work done in your developmental math courses on linear and exponential relationships.

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