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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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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Math is Non-Linear

Within our current curricular structure, there is premise of linearity — that the topics need to be studied in a particular order (in general), AND that students will not be able to understand a later topic that depends on some earlier topic.

This is not true.

At the global level, we have “pre-algebra” → “beginning algebra” → “intermediate algebra” → “college algebra” → “calculus”.  We have an existence proof that this is not true … students (in larger numbers than we’d like to admit) can perform as well in the college math class without taking the prior course ‘required’ according to placement tests. 

At the micro level, each course is constructed (normally) by a list of topics and then outcomes within each topic.  If the topics were basically linear, however, we would always teach these topics as connected to other topics (prior and future); we do not do so, making the non-linear nature evident. 

You might consider both of these points ‘logic chopping’, and classify this as a worthless post.  So … let’s move on to a different analysis.

One of the imbedded linear conditions is ‘fractions’.  Let us assume, for the moment, that we have sufficient rationale to justify the inclusion of rational expressions in the algebra curriculum.  Back in pre-algebra, we cover various operations; we suggest to our students that they need to master simplifying arithmetic fractions prior to simplifying rational expressions (‘algebraic’ fractions).  In between these two topics, we covering factoring … and feel good about the parallels between prime factoring in arithmetic and polynomial factoring in algebra.

I see two reasons why the sequence of these topics is not linear.  First, arithmetic fractions deal with place-value numbers; students need to transform these additive forms into multiplicative forms to simplify by factoring … and this is a more advanced topic than algebraic fractions (which are often multiplicative in the first place).  [Just show a group of students the fraction “54/24” and see what fun they have with these hidden binomials… compared to ‘8x²/4x’ (obvious monomials) or ‘(6x+24)/(8x+32)’ (obvious binomials).]

Second, the fraction topics are not linear because of the extra rules often imposed for arithmetic — improper fractions and mixed numbers.  These rules do not exist for algebra.  Because arithmetic is a relatively advanced topic, we often cover these topics with a series of guidelines or procedures for each type of problem; few of these items transfer to algebraic fractions.

Personally, I would rather help my students understand a mathematical topic like ‘algebraic fractions’ without having to cope with layers of bad learning relative to arithmetic fractions.  In this area, I can not expect this to occur.

I suspect that you have noticed some of what I am talking about.  We cover linear forms before quadratic before exponential — and yet some students ‘get’ the more advanced topic while they still struggle with the earlier one.  You might have noticed, in particular, that some students just don’t understand these properties of real numbers (associative, commutative, etc) … and then they start simplifying expressions with terms and parentheses, and they get an insight into what the properties were saying.

Math is not linear.  We are not building machines that have a clear dependency in design; we are dealing with human beings working with ‘mathematics’ — a collection of scientific domains dealing with different types of objects.  Our job is to identify the most important mathematical concepts appropriate for each course, and allow the course to be non-linear; we can revisit concepts, bring in a new perspective, look at a different context.

Math is not linear.  Our curriculum tends to consist of a series of two to five courses, presumed to be linear … and we create many opportunities for students to leave math.  We tend to put a lot of “preparing for the future” procedures in each course, which tells students that they will not study the good stuff of mathematics until much later (if at all).  Our job is to show good mathematics to students in all courses; rather than seeing mathematics as a difficult set of hundreds of procedures, they might just see mathematics as interesting.

We can not expect to inspire students by using a linear curricular model which ensures that the early courses cover very little of interest … and even less of intellectual value.
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