Variables Less Understood

In a traditional beginning algebra course (like one I am teaching), we spend much of the time working with variables in expressions and equations … and functions.  The course first shows variables within simple expressions with a progression towards complexity and equations.  One problem (and the student errors on it) really caught my attention this semester; it’s actually a simple problem, not requiring any procedural steps.

Here it is:

Solve   -6k + 3 = 3 – 6k

I have to admit that I did not emphasize ‘reading the equation to see what general statement it is making’, though we did actually talk about equations where the variable term was equal on both sides.  One of the common errors is shown below:

Every student making this error could some an equation like  ‘2y – 5 = 4 – y’.  What was causing the error?

Sadly, the problem was that many students are learning the ‘algebra dance’: Duplicate these steps, record the result.  Part of the dance is to write the opposite of the variable ‘thing’ on the other side to get one variable in the problem.  Students used this dance to solve a number of equations to produce quite a few correct answers.  For this problem, part of the dance was the ‘get one variable’ — the student knew that -6 + 6 was zero, so we just have the letter.  The variable was less understood than we thought, based on the consistent correct answers to other problems.

It’s very likely that you can list some mistakes that are similar in showing a less understood variable concept.  One of the errors I am seeing is “5 + 2x” becoming “7x”; the numbers and letters become the whole story … the operations are not even being read.

If you are curious, there is a wide body of research on learning variable concepts; for one summary, see http://www.nctm.org/news/content.aspx?id=12332 (an NCTM item).  Some particular research items:  http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol3KoiralaEtAl.pdf and http://www.merga.net.au/documents/Steinle_RP09.pdf  and http://elib.mi.sanu.ac.rs/files/journals/tm/16/tm915.pdf. 

What I am focused on, however, is not the research nor the particular misunderstandings — rather, I am thinking about WHY this happens.  It seems the problem is most likely when students have a higher motivation for ‘correct answers’ compared to their valuing of understanding (which is a combination of desire to understand and confidence in being able to).  In my classes, I often say that I am not that interested in the answer they get; I am more interested in the knowledge you have about that type of problem (the understanding).  Obviously, this statement from me does not change the drivers of student motivation (answers or understanding); I need to create instructional spaces where the understanding is the result being assessed directly.  I suspect that I will be using some type of writing for this purpose; this will be a challenge, given the range of writing abilities in the class.  For one reference on writing in math, see http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm. 

However, I can count on a basic human trait:  We (meaning our brains) naturally prefer to understand the world around us.  Knowledge organized by understandings is easier to maintain and use, compared to knowledge that is random memories.  The problem with ‘variables less understood’ is that this natural desire to understand has been subverted, perhaps caused by messages about ability … perhaps reinforced by social messages that math is about formulas and answers.

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Modules in Developemtnal Mathematics — pro and con

I am hearing about colleges either adopting or considering modules in their developmental mathematics program.  Sometimes, this is done as part of an ’emporium model’; however, other designs make use of modules.  Perhaps it would help to have a brief exploration of the pros and cons of modules.

The word ‘modules’ does not have a uniform meaning for us.  In general, a ‘module’ could be another name for a ‘chapter’ — each being a sub-unit within a larger organization of material.  However, most uses of the word ‘module’ refer to one of two approaches to content — uniform sequence of modules or customized sequence.

The difference between the two uses can be subtle, such as a case where the customized exit point is the end of a ‘course’ — some modular programs designate ‘modules 5 to 8’ as a course, and that is where the exit point is.  Customizing is done by either changing the ending module within a course or changing the entry point (starting module) within the course.  Conceptually the contrast for the two designs is important due to the fact that a customized program prevents a summative assessment common for all students.

Over the past several years, I have had discussions with faculty involved in a type of modular program.  Via this obviously non-scientific method, I have developed some pros and cons for modularization.  Most of these apply to either type (uniform or customized).

A modularized approach is usually based on an assumption that the delivery mode is a major source of problems, sometimes stated “we can’t teach this to them the same way they saw it the first time”.   I have not seen any evidence of this being true; it’s not that I want to teach them “the same way” (whatever that means) … it’s that this assumption about the delivery mode often precludes examination of larger issues about the curriculum.  Modularized tends to reinforce notions that ‘mathematics’ is about knowing the procedures to obtain correct answers to problems (often contrived and overly complex).  Our professional standards (such as the AMATYC Beyond Crossroads  … see http://beyondcrossroads.amatyc.org/) begin the discussion about mathematics by describing quantitative literacy.  This aspect — of modularization tending to limit the mathematics considered — is the largest factor in seeing this approach as being weak and temporary.

The other major area of concern, suggested somewhat in the pros and cons, is the professional status of faculty in developmental mathematics.  Administrators and policy makers often do not understand the professional demands of being developmental mathematics faculty; in the modularized approaches, faculty tend to look a lot like tutors.  This similarity then suggests to some that faculty are not necessary, and we can provide a larger pool of tutors.  Our professional standards call for us to see the work in math classrooms as being rigorous in both mathematics and education.  This aspect — the professionalism of faculty — is the most common concern reported by faculty engaged in a modularized program.

Summary:
The attractiveness of modular approaches is easy to understand.  However, the typical implementation of modular approaches will reinforce a traditional content with a weaker assessment system combined with a generally lower faculty professionalism.  When implemented, modular programs will tend to be temporary solutions.  The emerging models — New Life, Carnegie Pathways, Dana Center Mathways — provide a clear alternative to address the problems based on professional standards to create long-term solutions.

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PEMDAS and other lies :)

We use ‘correct answers’ as a visible indication of knowledge.  If the learning environment focuses on homework systems, correct answers may be the only measure used.  However, even when we ‘look at the work’, we may confuse following a procedure with knowing what to do.

PEMDAS may be the most commonly used tool in the teaching of mathematics experienced by our students.  I have seen PEMDAS written on work papers and notebooks; I have heard students say PEMDAS when explaining how to ‘do a problem’ … and I’ve heard instructors say that PEMDAS describes what to do with an expression.

The problem is that PEMDAS is a lie.  PEMDAS only provides a memory tool (a mnemonic) for steps that might apply to some expressions in some situations.  Previously, I have written about the issues with the “P” (parentheses) component of this tool (see https://www.devmathrevival.net/?p=301).  Today, I am thinking about some of the ways in which PEMDAS is false or incomplete.

Take a simple expression like -4².  PEMDAS does not give any interpretation of this expression.  The issue here is that the memory aid only deals with exponents and the 4 binary operations; the negation (opposite) involved here is outside of the rule.  If we established mathematical truth based on an agreement among students passing a course, the truth would be at risk on this expression — whether “16” or “-16” would win a majority would vary by semester.

PEMDAS is incomplete about operations in general, such as the negation above … or absolute value.  Given the visual similarity with parentheses, most students see that the ‘inside’ of an absolute value is simplified first.  However, what to do with an expression like  3|x – 2|?  Is there a choice to distribute?  As we know, and students are confused about, the order of operations provides one possible procedure … properties of numbers and expressions completes the story, and these properties are more important in mathematics.  Getting the correct answer to 8 + 5(2) in a pre-algebra course has nothing to do with being ready to succeed in algebra, or math in general.  Basic expressions like 8 + 5x are a challenge for many students, partially due to how strong the PEMDAS link is.

Another example:  what does PEMDAS tell us about mixed numbers?  This is a special case of the ‘parentheses problem’, where there is no symbol of grouping.  Fractions, in general, are an area of weakness.  We tell students that “you need a common denominator” or “cross multiply” — both of which appear to violate PEMDAS (we would divide left to right).  Properties are the important thing here as well; adding requires similar objects.  We focus so much on correct answers and perhaps ‘correct steps’ that we miss opportunities to address the mathematics behind the visible work.

The meaning of an expression with mixed operations is based on the priority of each operation; mathematically, the level of abstraction of an operation determines the priority.  Multiplying is abstracted from the concept of repeated adding, so multiplying carries a higher priority; exponentiation is abstracted from the concept of repeated multiplying, and has a higher priority.  Lowest abstractions are the basic concepts — add, subtract, negation.  For those of you involved with programming, this approach should sound familiar — computing environments are based on a detailed list of these levels of abstractions.  In mathematics, our world is defined by properties which provide necessary choices for types of expressions where equivalent forms can be created without using the prioritization.

The big lie in PEMDAS is that those 6 words say something important about mathematics.  Those 6 words do not say anything important about mathematics, only about an oversimplification that produces some correct answers to some expressions without understanding the mathematics.  Properties and relationships are the important building blocks of mathematics; a student starting from PEMDAS has to unlearn that material before understanding mathematics.   If our goal is to have students compute correct answers for any expression, then we would never use PEMDAS — it is woefully incomplete, and we would need the prioritization list like a computer program uses.  If our goal is to have students understand mathematics, we would deal with the concepts that determine the order along with the properties that provide choices; a focus would be on the correct reading and interpretation of expressions.

Do your students a favor; avoid using PEMDAS.  Use mathematics instead.

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Math Applications — Taking it Personally

Contextualized math is a current theme in our profession; some courses are taught strictly working from context — the story is the thing, and only the mathematics that relates to that story is developed.  Other courses emphasize context, while many of us take a moderated approach in which we blend context and abstraction.  Whatever the mix, these contexts are experienced as ‘applications’ or word problems by our students.  Do applications (or context) present issues of equity in our developmental math classes?

I would like to tell you about what two of my students (beginning algebra) are experiencing this semester.  First, a student to be called Mary.

Mary: (looking at a typical ‘distance’  problem about two cars)  I don’t know how to write the algebra for this, but I can figure out the answer.
Instructor: Okay, so tell me more about that.  How do you figure it out?
Mary: Well, the problem says that one car is going 10 miles per hour faster, so I put myself in that situation; I know that the speed limit is 70 miles per hour, so that must be the faster car.  The other car must have been going 60.
Instructor: I see.  What part of the problem told you that the cars were on a highway with a speed limit of 70?
Mary: The problem did not say that, but the only way I can understand the problem is to put myself in to it.

The second student will be called John (whose native language is Arabic).

John: (looking at a problem about a tree and a flag pole dealing with their heights)  This problem is really hard.
Instructor: What makes it hard?
John: Everything in the problem … I need to translate it into my language; it does not make sense to me.
Instructor: Are you talking about the individual words?
John: Yes, yes … they are confusing.

The prognosis for Mary is not as good as the prognosis for John.  They are both taking the applications personally; the difference is that Mary thinks that she has to see herself in the problem for it to make sense, while John thinks that he will understand the problem once he knows all of the words.

This experience made me think of some research I saw a few years ago dealing with how word problems in mathematics might raise issues of equity.  The research suggested that students from a ‘lower class’ (this was British research) get distracted by the details of the applications as they relate to their personal life.  My student, Mary, was doing exactly that.  Her learning skills, and her life experiences, provided a limited view of applications; some problems dealt with objects or situations with which she had no experience, and she did not know what to do … other problems activated related but not worthwhile information (like the car problem).   Clearly, we will need to work together (Mary, the class, and me) to help broaden the view and provide more resources.

Taking an application personally can create difficulties in forming a solution strategy; taking it personally highlights information (which might be trivial) and causes us to possibly ignore other information critical to a solution.  This situation deals with perception and motivation.  For those of us who are using high-context classrooms, I wonder if you are finding that the approach is equally accessible to all of your students.

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