Important Mathematics for Developmental Math Students

It’s all about rates of change. Some change is a constant amount, some change is a percent change. A few changes involve rates that are a non-linear function of the input.

What is some important mathematics for our students? We could accomplish a lot by looking at linear (additive) rates of change and exponential (multiplicative) rates of change. All of our algebra courses include linear functions and slope, and this is good. However, for exponential functions we often omit them or cover the less important manipulations.

Here is a basic conceptualization of content that would help our students have a deep understanding of the two most common relationships in their world (linear and exponential).

A linear context — data from a situation familiar to students (hourly pay, or distance-rate-time). Predict the next value … find the ‘adding value’.
Rate of change is slope — formalize the adding value; know that ‘$ per hr’ is a slope.
The y-intercept, and meaning in context
Graphing linear functions (exact relationship between input & output) … reasonable domain & range, reasonable axis values.
Graphing linear models (involves measurement errors) … reasonable domain & range, reasonable axis values.
Applications — identifying whether a situation is a function or a model, slope, y-intercept, graphing, predictions
An exponential context — data from a situation familiar to students (interest on a loan, or drug-blood levels). Predict the next value … find the ‘multiplying value’
Rate of change is a base for an exponent — formalizing the multiplying value; know that ‘% per year’ is an implied multiplying factor
The initial value, and meaning in context.
Graphing exponential functions (exact relationship between input & output) … reasonable domain and range, reasonable axis values.
Graphing exponential models (involves measurement errors) … reasonable domain & range, reasonable axis values. Include half-life, as one particular.
Applications — identifying whether a situation is a function or a model, base, initial value, graphing, predictions
Discriminating between linear and exponential rates of change, including a variety of contexts
Capstone experience — the surge function (often used to model drug levels in pharmacology)

This list might imply that there is a ‘fixed order’ — that is not needed. In one of my courses, we tend to cover corresponding steps for each function at the same time (1 & 7, 2 & 8, etc). Very little symbolic work is needed, outside of creating a function or model; the work tends to be numeric and graphical. [Too often, we connect ‘exponential functions AND logarithms’ due to a fixation on symbolic methods for exponential functions.]

Important mathematics satisfies two conditions: First, the mathematics involved is general and powerful; second, the mathematics involved will be useful in a variety of situations within and outside of academia.

Much of the ‘content’ described in the list above is embedded within “Mathematical Literacy for College Students” (MLCS), which is both New Life model course and part of Quantway™. I encourage you to look at those materials. Beyond that, I encourage you to think about why these concepts are important for our students.
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