Free Online Materials from NROC?

The National Repository of Online Courses (NROC) has developed extensive materials for developmental mathematics, which are available to preview and pilot test and will be free for individuals to use this fall at HippoCampus.org. Currently, a good beta version of the material is being piloted around the country – with the full release of the material due later this year that will include assessment to individualize a learners’ path through the lessons. Right now, the beta materials for arithmetic and basic algebra are available for previewing – I spent some time looking, and am impressed. Even though the topics look old fashioned, the presentation is very different; the material is engaging and worth learning.

If you would like to get a preview, go NROCmath.org and look under the Higher Ed banner (right side); the materials are called NROC Developmental Math—An Open Program. On that NROCmath.org page, there are links to learn more and to view course now ‘learn more’ provides an overview of the course, and ‘view course now’ provides you access to preview (one unit) the materials. You can register for a password to preview all beta units. (Warning: This will also put you on the mail list to get news of the final release.)

To fully integrate the material into an institution’s curriculum, in-class, blended or online, as a supplement or full course, institutions, systems or agencies may become NROC Network Members and receive technical support, professional development resources and the rights to adapt and import all resources into their own learning management system to adapt and make available to all teachers and students.

Isn’t it great when people share good materials like this?
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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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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).

1. A linear context — data from a situation familiar to students (hourly pay, or distance-rate-time).  Predict the next value … find the ‘adding value’.
2. Rate of change is slope — formalize the adding value; know that ‘$ per hr’ is a slope.
3. The y-intercept, and meaning in context
4. Graphing linear functions (exact relationship between input & output) … reasonable domain & range, reasonable axis values.
5. Graphing linear models (involves measurement errors) … reasonable domain & range, reasonable axis values.
6. Applications — identifying whether a situation is a function or a model, slope, y-intercept, graphing, predictions
7. 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’
8. Rate of change is a base for an exponent — formalizing the multiplying value; know that ‘% per year’ is an implied multiplying factor
9. The initial value, and meaning in context.
10. Graphing exponential functions (exact relationship between input & output) … reasonable domain and range, reasonable axis values.
11. Graphing exponential models (involves measurement errors) … reasonable domain & range, reasonable axis values.  Include half-life, as one particular.
12. Applications — identifying whether a situation is a function or a model, base, initial value, graphing, predictions
13. Discriminating between linear and exponential rates of change, including a variety of contexts
14. 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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Stealth Percents

A student experiencing all of our developmental math courses will see quite a few percents, and probably not understand much of this — in spite of obtaining hundreds of correct answers.  We cover stealth percents, not real percents.

Percents are stealth when we use rules and do complicated conversions between percents and other forms.  The truth is that converting, by itself, is not that useful; we cover conversions in an attempt to build understanding.  However, the understanding is submerged — the rules become the content, not the percents.

Percents are stealth when we use “is over of” and solve dozens of problems based on the sentence structure “8 is 20% of what”.  Getting correct answers for these problems shows little knowledge of percents.

Percents are stealth when we cover ‘applications’ using specialized vocabulary such as ‘discount’, ‘mark-up’, and others.  We create stimulus-response connections, but little knowledge of percents.

Some of this emphasis on percents comes from a time when many of our students would be working in a retail environment without calculators — let alone computers.  Those days are gone, for the vast majority of our students.

Yet, percents are important.  We need to understand percents to deal with daily life and academics.  Understanding percents means that we know how to communicate accurately — that percents always have a base.  If two percents share a base, then it might make sense to combine them (like percent of income for different categories); if two percents have a different base, then it never makes sense to combine them directly — we need to compute a common measure (like dollars) before combining.

Understanding percents means that we know that mixing two strengths means a mixture that is intermediate, whether we are talking about chemicals — milk fat — or interest rates.

Understanding percents means that we can connect a percent change to a symbolic representation (algebraic term) and to a table of values.  We can deal with a sequence of percent change.

Understanding percents might also deal with the most common use in media — surveys and polls; these are different kinds of percents (though they always have a base!), and the primary use of these percents is to sway public opinion. 

If you think your students understand percents, give them this simple problem:  In 2010, all employees took a 10% pay cut … in 2011, all employees received a 10% raise.  How does their 2011 salary compare to their 2009 salary?

Let’s do percents right — drop almost all of the conversions, and the ‘percent sentence’ problems; focus on really understanding them!

 
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