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.

Join Dev Math Revival on Facebook:

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!

Join Dev Math Revival on Facebook:

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!!

The Capabilities of Developmental Students

What are our students capable of?

I think we end up taking a ‘bipolar’ position on this.  On the one hand, we believe that our students can achieve their goals; we encourage them, nudge them, motivate them, and suggest that they might be capable of higher goals.  Our greatest satisfactions come from watching our students — who needed developmental course work — graduate with a completed degree.  Gowns, in college colors, form a visible symbol of this hope for all of our students.

On the other hand, we seem to design courses which say “I get it … you can’t understand mathematics, really; so I will just expect you to recognize some patterns for which you have a solution in memory.”  We build instruction around the goal of maximizing correct answers for students.  We select textbooks which simplify the presentation and provide clear examples of the procedures, and avoid textbooks which discuss the ideas outside of examples.  We observe that our students do not remember much of what they had last semester, and conclude that this reinforces our design of ‘simplify’.

In fact, our ‘simplify design’ paradigm is part of the problem.  As long as learning focuses on remembering procedures, the powerful brain work that enables long-term changes and transfer of learning do not have a chance to occur (except by accident).  In some ways, most of our students leave our classrooms with the same condition that they arrived … summarized by the one word “unable”.

I can not accept the ‘simplify design’ of curriculum due to its message about the capabilities of our students.  Our students are capable of achieving much, and our society actually depends upon them achieving much.  We can not avoid building this capacity within our ‘developmental’ classrooms.  (It’s ironic that we call our courses ‘developmental’ but tend not to develop capacity.) 

Now, I am not under the influence of some ‘just be happy’ medication.  Obviously, students in developmental mathematics classes have some current limitations.  Our response must be to overcome limitations and build capabilities.  This response is not easy, certainly not just ‘pick the best homework system’.  Just like our students, we will achieve more than we thought possible when we face challenges directly.

And, just like our students, we will find the work is easier … and we understand more … when we work with each other.  You are not alone, and we are capable of designing courses which build capacity within our students.

Join Dev Math Revival on Facebook: