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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Friendly Placement Testing

Like most community colleges, my college requires students to take a standard math placement test to determine their math level.  Like many, my college enforces the resulting score — students can not enroll for any course higher than what their placement test score qualifies them for.  How can this standard approach be done so that is fair to students and allows them to begin at the highest reasonable level of mathematics (for their knowledge)?

Let’s agree right at the start … the content of the standard placement tests is not aligned with the best mathematics in those areas of knowledge; the items tend to be basic procedures and basic concepts in a fairly narrow range of topics.  However, changing a placement test is a long term (and commercial) process.  As much as I would like to see (drastic) changes in the tests, that is not under our control and any changes to those tests will not be seen by students for a while (like 2 or 3 years).

Here are some observations about typical math placement testing systems that affect how friendly it is to students:

• Upon admission, community college students usually do not know what will be on the test.
• Community college students often do not understand how important the results will be.
• Our students usually take a math test without any review.
• Options for re-testing (challenging) are often limited, and we tend to not provide information on ‘what to do before retesting’.

You might not agree with all of these observations.  I hope you see enough truth in them to agree with this statement: “The advising for students prior to taking a math placement test is not currently adequate in most community colleges”.  In fact, many colleges are like mine … the first advising a new student receives is done at an orientation; students are required to complete placement testing BEFORE orientation (and advising).  There is a logical reason for this — advising tends to deal with specific questions about enrollment, and this means the results of testing are needed.  However, I would suggest that this approach is not student friendly.

If I could do so, I would make the initial advising a two-step process:

1. An orientation & advising (done in groups) which would cover information on math placement testing, followed by taking the placement test (different day)
2. Individual advising after placement testing, where possible re-testing is discussed (based on how the student sees their initial results aligns with their background).

Alas, I am not in charge of advising … as I suspect math faculty would not be in charge of advising in general.

In the meantime, here is one specific thing we could consider doing to make the process a little more friendly for our students: Make use of an online homework system for the review prior to retesting. 

At my college, some students who want to re-test for math are referred to the math department where they speak with an administrator (usually).  For many students, this results in them receiving access to a “MyMathTest” program that provides specific preparation for the placement test they want to retake.  We are able to do this because of the cooperation of the publisher, so there is no cost for the student; we are somewhat limited to a total number of users for this program, but the limit is high enough to accommodate the students talking to us about retesting.

We do not have specific results to share about how this is working (gains in math level or not).  I hope there are gains there.  However, I think this is a good thing to do just because it is student friendly.

If you are interested in using an online homework system as part of the review process for a placement test, start by talking to your book company representatives.  We have found the representatives to be helpful, and willing to look at doing something extra that would help students.

 
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Finding Statway(tm) materials (and Quantway!)

Are you looking for the Statway™ materials … so you can become familiar with them, and consider that kind of approach?  Are you intrigued by the Quantway™ ideas, and want to check it out?

You can email pathways@carnegiefoundation.org with the request; the Carnegie Foundation will send you a collection of lessons.  [Currently, a sample of Statway™ lessons are available; a set of Quantway™ lessons will be available soon.] 

Next year, all lessons will be available under a Creative Commons license.