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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Mile Wide … Mile Deep!

Actually, I wanted to say “kilometer wide … kilometer deep!” — but then some people would not get the reference. 

At the recent AMATYC conference, I attended a session by Xiaoyi Ji titled Investigation of Math Teaching in the U.S. and China which I found inspiring.  One of her main points to explain the large gap in ‘performance’ between Chinese and US students is the Chinese committment to depth AND breadth.   You can see her presentation at http://www.amatyc.org/Events/conferences/2011Austin/proceedings/xiaoyiS75.pdf , and you can see the entire list of proceedings at http://www.amatyc.org/Events/conferences/2011Austin/proceedings.html.

Our recent drive to avoid a ‘mile wide & inch deep’ is a false dichotomy.  The implication is that we can not have both depth and breadth.  This is one that I think the Chinese system has right — we truly need a kilometer wide and a kilometer deep; depth without breadth results in students who know a fair amount about isolated pockets of mathematics … and I suggest that this is a self-defeating goal.  We create more problems than we solve.

Breadth refers to two dimensions — one is the domains or categories of mathematics, the other the major areas in each domain.  Within polynomial algebra, for example, we have some areas which receive most of our attention (simplifying, solving) while other areas are neglected (conic sections come to mind).  We often see ‘functions’ and ‘modeling’ as alternatives, when both have a purpose.  We often omit other basic forms (exponential, trigonometric).  As a result, we create pockets of knowledge and chasms of ignorance … and wonder why our students have such fragile knowledge.

Depth refers to levels of knowledge, and we actually do not share a good understanding of what this means.  Too often, we look at surface features of the questions we ask (skill, application) rather than a more sophisticated analysis.  When better work is done, it is sometimes framed within Bloom’s Taxonomy which is not particularly well suited.  A better framework for the depth of knowledge is the ‘five strands of mathematical proficiency’; you can see an excellent presentation (in fact, the original) in an online book at http://www.nap.edu/openbook.php?isbn=0309069955.  This material was originally written for a school mathematics audience; however, I think you will find the concepts transfer to our level quite nicely.

Of course, we can not achieve ‘depth and breadth’ in one or two college mathematics classes.  On the other hand, we can ensure either an inch deep or an inch wide in one course  by the choices we make.  Let us all contribute to both depth and breadth at every opportunity.

 
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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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Memorize This!

Slope formula?  Area of triangle? Quadratic formula? Basic number facts?

What is the proper role of ‘memorization’ in learning?  Specifically, is memorization needed in a math class?

Let me start with a short anecdote from today’s class.  A student needed to divide 16 by 8; he pulled out his graphing calculator.  Another student needed to know what the common factor was between 16 and 40; he also got out his calculator, and tried dividing each by 4 …

We have been stuck on a rejection of ‘rote’ learning, with a poor association of ‘memorizing’ with ‘rote’.  Now, there are actually times when rote learning is fine — though most of us (myself included) do not use this very often, in favor of more active learning.

This has gone so far as to result in students being told to NOT memorize; one of my colleagues tells students that they can always look it up (in the ‘real’ world).  On balance, this has harmed far more students than it has helped.  Let me explain why this might be true.

First of all, the human brain ‘wants’ to remember things (including formulas and facts) — you dial an arbitrary sequence of 10 digits more than once, and your brain is likely to work on remembering that phone number.  Telling somebody to ‘not memorize’ comes very close to telling them ‘turn your brain off!’.  We can’t condemn memorizing and condemn lack of learning; they go together.

Secondly, the progression from novice to more expert states involves a process called ‘chunking’.  We, as mathematicians, have a very large chunk size in domains where we have practiced and thought; while a student sees 15 steps in a series, we see 2 collections of steps.  When faced with a novel problem, we bring these chunks and our understanding of their connections to the problem.  Experts in any field have a large chunk size, often numbering 10 to 15 specifics in a chunk.  Telling a student to ‘not memorize’ is telling them “it is okay to have a chunk size of 1 (one)” — which means that they are likely to appear as a novice in that field, no matter how much they work.  [Memory, especially clusters of connected memories, seem to be a critical building material for our ‘chunks’; some writers call these ‘schema’ instead of ‘chunks’.]

Thirdly, and fortunately, it does not work to tell somebody to not memorize (see the first point).  The bad part is that some students actually listen to us, and they remember less because of it.  Basically, this is saying that our advice has damage that is limited by accident, not design.

In all my reading of learning theory, over a period of decades, I have yet to find a cognitive scientist say that ‘memorizing is bad’.  From a learning point of view, it is all just learning.  If a person memorizes a formula, without having practiced in varying contexts and without connecting it to other information, then they will be limited in how they can apply this formula; if a person does not memorize a formula, they have to organize their learning around other information — not connected to a formula.  We see students who have a vague notion that area is length times width, and connect all ‘area’ information to this; this incomplete learning creates unnecessary barriers.  If students know multiple formulas for area, as an example, they connect all of these to their understanding of ‘area’; they become better problem solvers … and transfer of learning is much more likely with this more complicated mental map of ‘area’.  The best situation is one where students have several area formulae available from memory, all connected to a concept of ‘product of two measurements, and perhaps a constant.

Mathematics is not the only domain with an interest in (not-)memorizing.  Language learning has also dealt with this, as well as others (see http://scottthornbury.wordpress.com/tag/cognitivist-learning-theory/, and you might also enjoy http://thankyoubrain.com/Files/What%20Good%20Is%20Learning.pdf).

The part that actually bothers me the most, however, is the attitude resulting from students not remembering basic information.  As long as he has to get a calculator for ’16 divided by 8′, he is going to feel dumb about math.  A sense of proficiency and competence goes a long way towards persevering.  Our students do not need a barrier added to their challenges, a barrier constructed out of our good intentions when we say DO NOT MEMORIZE!

Memorizing does not need to be ‘rote’, as we all know from personal experience.  Memorizing happens due to time on task, with a little reflection on the learning involved.  Memorizing is a natural process for a human brain; we need to take advantage of this capacity.

Memorizing alone will not be enough, and never has been.  However, without memorizing we limit the long term mathematical development of our students; we reinforce negative attitudes, and we create learners who have trouble transferring their learning.  Let’s keep a healthy balance — some memorizing, a lot of understanding, building connections, and enough practice to build competence. 

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