Fractions as Filler

In many of our developmental courses, we focus on building skills with fractions.  For some of us, ‘fractions is where we start’.  How have fractions earned such a place of honor?

To understand the place of fractions in our curriculum, think about what we mean when we say ‘fractions’ — identifying types (proper, improper), reducing, building-up, four operations, conversion to decimal form, and word problems.   My comments are based on what I see in the vast majority of courses & textbooks on ‘fractions’.

I see two fundamental problems with the role of fractions in our work.  First, the content lacks sufficient justification in the lives of our students.  Second, our curriculum focuses on the algorithms to an extent that precludes significant understanding.

Originally, fractions in our first courses were justified for occupational as well as mathematical reasons — many jobs involved working with fractions, and college mathematics depended on manual skills with fractions.  These occupational justifications have diminished to the point of being a specialty affecting a small proportion of students; various technology tools in occupations either perform the calculations or avoid fractions entirely.  The college mathematics justification was weak originally, and that target is valid for a small portion of our students; even if we could inspire most of our students to take college mathematics courses, the algorithms in developmental courses have little purpose … the understanding does.

Our curriculum with fractions is especially ‘procedurally bound’ within our codes of LCM, GCF, and rules.  My students get trained to respond correctly to problems with two fractions separated by operation symbols, but can not explain why they do those steps.  Addition of fractions is not connected to ‘like terms’ as a general concept; adding ‘3x’ and ‘5x’ is much simpler conceptually than adding ‘3/4’ and ‘1/8’, though we insist that students get right answers for the complex problem before we cover the concept behind it.  The other operations are also generally done on ‘auto-pilot’; no need to think too much here, just remember the steps that match the operation.  Every time we ‘cover’ fractions (say algebraic fractions) we pretty much teach the process over again … because it looks different, students do not connect the procedures. 

We use fractions as filler in the curriculum.  I conclude that we think we have more time than needed for any ‘good stuff’, so we use fraction work to fill in the open spots.  We certainly do not teach fractions in a way that transfers to other situations within mathematics or outside of mathematics.  I believe that we would not hurt anybody, and might help some, if we eliminated all work on fraction operations.

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Bumper Mathing

In bumper bowling, there is no ‘gutter ball’ — every attempt hits something.

I’ve been thinking about this concept related to mathematics; do our developmental courses create ‘bumper mathing’, where all students hit something most of the time?

My college offers a course that I totally love … it’s a mathematical literacy course, covering a collection of really good mathematics.  It has only a beginning algebra prerequisite.  A common theme in the course is ‘applying percents’.

All of the students in this course have ‘mastered’ percents.  They have converted percents to other forms, they have solved percent ‘problems’ (like 80% of what is 60), and applied percents to life situations; in our beginning algebra course, they ‘mastered’ mixture problems dealing with percents.

Here is a chronology of percents in my ‘math lit’ course:

10% increase from a known value, find new value … almost all are okay

Old and new values, find the relative change … almost all are okay

10% increase from original unknown amount, express new value … almost none are okay, almost all need remediation

10% increase each year, express as a function … almost none are okay, almost all need remediation

95% confidence interval dealing with survey results … half think the 95% has to be used in the computation

10% probability of A happening, probability of ‘not A’ … almost all are okay

10% increase from original amount (known or unknown), express as a function … still difficult

10% increase from a known original amount, graph the function … almost all need remediation

Notice that there are 3 times that we revisit the ‘10% increase, represent new amount’ concept.  Each time, the majority of students do not see why we get ‘1.10n’ … they’d like to see ‘0.10n’.  The problem is that they want to compute with the percent stated (10%), because that has worked almost all of time in the past.  Part of the process of ‘remediation’ is to work through concrete examples (like 6% sales tax leading to ‘1.06n’), but this is a slippery process: The prior learning keeps drawing them down to computing with the 10%.

In our pre-algebra course, we cover perecents in a very template driven way … convert % to decimal, ‘2 places left’; ‘is over of’, and others.  These templates increase the proportion of correct answers (bumper mathing), but disguise the lack of percent understanding.  Our course is not alone in this problem; our collective pre-algebra courses are supposed to prepare students for algebra, which is all about generalizing … but percents are template-taught.  There is no transfer of learning, because of bumper mathing.

To create mathemtically literate people, there needs to be a chance for ‘gutter balls’ (as in real bowling).  We see 100 correct percent answers, and conclude that there is a good understanding of percents; that is not the case most of the time.  I’ve had a lot of students over the years say “I used to be good at math, and now I am struggling”.  Perhaps we have enabled this disability by practicing bumper mathing!

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Online — “instant presentations” now include MLCS and ‘Transitions’

The Instant Presentation page now includes several videos (4 or 5 minutes each) on the New Life curriculum.  Today’s additions are the two videos on the Transitions course … the powerful second course in the New Life model.

Take a look, and enjoy!!

Basic Skills … what is important?

Okay, here is a quiz for you.  Which of these categories of skills is more important in today’s world?

    A) Basic skills (correct computation, solving routine problems, etc)

or

   B) Critical thinking (solving novel problems, connecting sets of learned material, metacognition, etc)

You see, I think we have a problem with “Goals”.  Philosophically, we believe that mathematics is a venue to improve critical thinking … operationally, we deliver a curriculum much more focused on basic skills.  To complicate matters, our college curriculum follows the school curriculum which is decidedly skill oriented (given the high stakes testing, understandably so).

Back in February, NPR ran a story about some reports dealing with how well students are prepared for college (see http://www.npr.org/2011/02/09/133310978/in-college-a-lack-of-rigor-leaves-students-adrift).  In their story, writing was connected more to critical thinking than math was … I would not disagree, but would hope that our math courses are part of the answer.

Our developmental mathematics textbooks are severely ‘skill bound’, and we sometimes choose solutions that exacerbate the situation — such as modularized programs that discourage integration and accumulation.

I am sure that we all share a goal of improving our students critical thinking, and the evidence indicates that the need for this work is greater than ever.    My own courses are, sadly, typical of what we are all doing.  Could we, if we wanted to, create something better?

Absolutely!  Development of critical thinking is a field with its own theories, research, and methodologies.  For starters, see the nice chapter by Diane Halpern at http://education.gsu.edu/ctl/FLC/Foundations/criticalthinking-Halpern.pdf .  Even a brief online search will provide you with more material.

We can do better!  Let’s work together so our students are better prepared for the problems they will face …inside academia and outside.

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