Fractions Lament

I am a fraction, and woe is me.  Nobody seems to understand my talents.  People have a prejudice against fractions, you know.  [I hesitate to say that there is ratial profiling, but it’s not pretty.]

People on the street say they hate me, without ever bothering to really look at me.  Is it my fault that they only remember their own difficulties with mechanics and procedures to generate answers involving things that resemble me?  If only they would look at the beauty and usefulness of fractions like me … they would find that fractions can be their friend.  [Look for me on Facebook!]   Fractions are much better than those elitist snobs, the integers; integers think that they are the greatest thing since sliced bread … but let me tell you that there are an infinite number of fractions for every integer.  The world could live without integers, but take away fractions and people would be back in their caves — without their iPads and Blackberry.

Then there are the math teachers!  You would think that math teachers would be enthusiastic about showing people how great fractions can be.  What do they do instead?  They tell students that you have to learn about greatest common factors and least common multiples, before you can ‘work’ with fractions.  Don’t they know that the GCF and the LCM are part of the integer conspiracy?  They make a big deal of mixed numbers; hey, if I want to hang around with an integer I will let you know — until then, I am happy being a fraction, thank you very much.  And ‘reducing’ fractions?  You (teachers) should talk; have you looked in the mirror lately?  I don’t think I am the one that needs reducing.

I am a fraction.  I can show much more than how many cups of flour you need to make 3/4 of a recipe; that’s boring stuff.  The good stuff is when a fraction lets you compare the rate of different groups of students to make sure that they are all getting the benefit of passing that math class.  A fraction lets you communicate about a rate.  [Did I tell you that my first cousin Marcel is a second derivative? He is one beautiful fraction!]  That reminds me … fractions make it pretty easy to convert one measurement to different units; you’ve just got to line up the units to get rid of versus the ones you need in the answer.  A fraction can also tell you what the chances are for having 2 boys and 2 girls in that family you want; that’s a beautiful thing by itself, isn’t it?

I am a fraction.  Don’t show me that cute picture with 8 parts and 5 shaded; even if I was 5/8 I would not like that picture.  Sure, I can show how many parts are there, just like certain integers can show how many pieces of candy are in the bag.  How would you like it if somebody showed a stick figure and said “this is Jennifer”?  You are more than a stick figure; I am much more than parts shaded in a drawing.

I am a fraction.  It’s time you saw me, and understand me.  I can’t make you love me, or even respect me.  However, I promise that I will do my part if you give me a chance/

 
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Graphing and Models

One of the current trends in mathematics is ‘models’, often connected as ‘functions and models’.  What do students bring from their work on graphing in beginning algebra (often linear graphs) to this broader work?  Is this an easy transition?  Do we face challenges or hidden dangers in this work?
One thing I have noticed is that we often assume facility with basic graphing based on the linear function graphing included in a beginning algebra course.  A student can generate a table of values and use those to graph; a student can graph the y-intercept and use slope to find more points on graph to create the line.  I suggest that we face a significant gap in knowledge when we present a model to graph on their own.

This is the type of thing I am talking about:

A company finds that it costs $2.50 per glass, in addition to a basic set up cost of $80.  Write the linear function for the total cost based on the number of items (glasses).  Graph this function for a domain 0 to 100.

The typical beginning algebra class does not prepare students for this work.  Here are some of the gaps:

This is not a scientific analysis of the knowledge needed for this problem; there are details at a finer grain of analysis that would show more gaps.

Essentially, this is a problem caused by “Bumper Mathing” (see an earlier post on that).  We constrain the graphing environment to the extent that the resulting knowledge is not applicable in any realistic situation.  We can do better than this.

“Graphing”, as a collection of related concepts and procedures, is fairly complex yet very useful … and is worth doing well.  We can certainly make more room in the algebra course so that students leave with good mathematics and knowledge that transfers.

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Good Algebra

A recent comment on this blog basically asked the blunt question:  Basic math seems fine, but WHY did I have to learn algebra?  Mathematicians know that the word ‘algebra’ has multiple meanings.  In developmental math courses, the ‘algebra’ is usually various procedures relating to polynomials with integer exponents, with a collection of procedures for rational exponents.  The traditional algebra course packages material that is either (A) thought to be important for pre-calculus and/or calculus, or (B) what students should have had in high school.

Given this, my honest answer to the question is “There is no good reason for you to learn that algebra.”  If you need calculus, we probably are not building your understanding deeply enough; we certainly are not developing your reasoning in the way you will need in calculus.  If you do not need calculus, what you experience in ‘algebra’ is unrelated to any mathematical need you might have (such as science classes, technical careers, or life in general).

A reasonable follow-up question would be: “If this algebra is sort-of okay for calculus bound students (and could be improved), and this algebra is not helpful to most students in the course, WHY does the profession maintain these courses built around an amazingly consistent content package?”

I believe that we, as a profession, are committed to helping students … that we want to provide the mathematics they need.  We seem to be ignoring a logical analysis of the situation; there must be a strong reason for us continuing the traditional ‘dev math’ package.  I believe that there are two processes which combine to create this reason (an illusion of a valid reason):

Myth 1: Algebraic manipulation is evidence of either understanding or mathematical reasoning; quick and correct execution are evidence of better understanding and/or mathematical reasoning.

Myth 2: Developmental students can not be expected to deal directly with abstractions (core mathematical ideas); the best we can do is provide basic skills.

For my college, we use a common departmental final exam for these courses … a practice which I support.  However, the final exam for our intermediate algebra course is a set of 40 problems to be completed in 2 hours; the 40 problems represent 40 learning ‘objectives’ in the course … no item on the final involves applying synthesis or learning based on multiple objectives.  Good algebra seems to be seen as quick algebra … good algebra seems to be seen as repetitive algebra.

Every day, people make mathematical claims.  Whether it is economics, environmental, or political … somebody says “this is growing exponentially”.  Do our algebra courses help students understand this phrase?  Would students have any idea what conditions allow truly exponential growth … could students tell when the phrase is being used as a rhetorical tactic?  Does the phrase “we expect 150000 jobs per month to be added to the economy” imply an equation for our students … could they estimate when we will have replaced the number of jobs lost in the recession?  Given a graphical representation of either an equation or data, can our students determine if the representation is accurate or if it is distorted (by inappropriate scales, for example)?

Yes, we have good algebra we can and should provide to our students.  Good algebra is not quick algebra (except for experts like us); good algebra involves abstractions and reasoning, and can be messy.  We need to have faith that our students are capable of doing good algebra; if we do not have this faith and act on that, we are enabling students to be ‘bad at math’ as a way of life. 

It’s time for us to step out of our constraints created by history and myths … step out of that cage, and build a new experience centered on good algebra for our students.

 
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Slope … Fast?

Well, I’ve been somewhat discouraged by the latest test in our ‘math – applications for living’ course.  This test is all about understanding linear (additive) and exponential (multiplicative) change.  In spite of approaching these models in different ways, followed up by repeated work, students had considerable difficulty translating verbal descriptions to anything quantitative or symbolic.

One basic problem seems to be that students did not start with much understanding of slope for linear functions.  Every student in the class had completed beginning algebra (usually recently), where we do the usual work with slope — calculating it, using slope to graph, and finding the equation of a line given information about slope.  When faced with non-standard problems with information on slope, the transition to quantitative or symbolic statements was not easy.

One part of this difficulty is the connection between input  & output units and units in slope.  Like most traditional algebra books, ours does not make an issue out of units for slope in ‘applications’; as you know, slope must involve two units.  Because of this difficulty, students would see a percent change as a linear change.

Mostly, this post is a “note to self”:  Learning slope is not really a fast thing.  Repeated use, alternate wordings, and non-standard problems need to be used to build a more complete understanding.  We too often spend a week or two on all linear function topics in a beginning algebra course; to get that ‘sufficiently good’ understanding, we need to allow more time so we can really dig in to what slope and the linear form are doing.

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