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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Towards Effective Remediation

Do we have a vision of effective remediation … a model which minimizes the pre-college level work for students, in total, while providing an opportunity for all adults to be included in the process of completing credentials leading to better employment and quality of life?  Based on some 39 years in developmental education, what would I suggest?

I have been thinking, as hard as I can, lately on the problems caused by policy makers looking for a simple solution.  Often, the policy makers’ interest in remediation has been prompted by reports issued by groups like Complete College America (CCA); the CCA “Bridge to Nowhere” report is excellent use of rhetorical tools, but is not a good foundation for building policies in support of effective remediation.  The simple solutions involved are usually crafted by groups that do not include people with expertise in developmental education.  Somehow, the viewpoint that we present, as experts, is difficult to understand by non-experts; perhaps some policy makers are worried that experts will only want to preserve the current system, or that we will suggest that even more courses be provided in our field.

Effective remediation involves providing the appropriate learning opportunities for each learner so that the learner reaches college courses with adequate preparation.  Traditionally, we establish remediation in discrete content areas (reading, writing, math), with an independent decision in each area based on a placement test.  Some promising practices have evolved recently with efforts to link developmental content courses, and efforts to include learning skills.   Especially within mathematics, considerable effort has been invested in creating a modularized approach; modularization is a topic of its own.  However, two observations might help us:

1. Each student is considered for 0 to 4 developmental courses in each of the 3 content areas, usually based on one placement test in each area.
2. The content is the developmental courses is often severely constrained by the historical roots of the system; especially in mathematics (though still true in reading and writing), the focus is on mechanics and procedure, with less emphasis on reasoning and analysis.

For us to develop a vision of effective remediation, we need to understand the deeper problems with the existing system such as those suggested by these observations.  In order to provide appropriate learning opportunities (whether courses, workshops, or other experiences), we need a more advanced conceptualization of remediation itself.  We need to more beyond a simple binary choice independently made in discrete content areas based on one test in each.

I suggest that we consider the following framework:

• Students roughly within a standard error of placing in to college work in a content area be provided just-in-time remediation and register for the college course.
• Students over one standard error away are placed into one of two populations based on other measures (such as high school GPA).  Some might be placed into the ‘just-in-time’ remediation group.
• The low-intensity developmental students are placed into a one-semester ‘get ready for college work’ course in one or more content areas.
• The high-intensity developmental students are provided a year of connected course work which blends reading, writing, math, and learning skills designed to address content and thinking needs.

The first two categories involve a significant portion of our developmental students, who have less intense needs; their remediation can be quicker than we often provide now.  For those who ‘almost place’ into college work, the ‘discontinuity’ research on placement tests suggests that we might be able to avoid any developmental enrollments in that content area.   The low-intensity developmental students are those who are not predicted to succeed in college but have limited needs; within mathematics, this group would include those who can review areas of algebra and quantitative reasoning in one course with minimal support outside of the class.

The high-intensity developmental group would include students with broad needs across multiple content areas.  These are the students who now struggle to complete developmental courses.  However, their educational needs are not limited to the content area skills; reasoning skills and study skills are a problem for many students in this group.  I am envisioning a two-semester package of courses (three or four each semester) with intentional overlap of cognitive skills being addressed … the math course, the reading course, and the writing course would all address issues of inference and concise language use.  This high-intensity group would also have a student-success type course to prepare them for the academic demands of the college course that lie outside of a content area.

Here is an image of this model:

A goal of this model is to make a better match between student needs and the remediation that they receive.  Our traditional system is designed for the ‘low-intensity’ type of student, and I believe that these students are well served on average.  The just-in-time remediation group is the source of our current problems from the policy makers; because these people exist in our current developmental program when the evidence raises questions about this practice, policy makers generalize the conclusion to all developmental students. 

The biggest change, and our largest opportunity, comes from the high-intensity group of students.  In our society, these are often called ‘low-skilled’ adults; they might be functionally literate (or perhaps not), and generally have few options in the economy.  Our traditional developmental program tends to be either limited in helpfulness or a problem for these students.  In a mathematics class, the high-intensity students have difficulty with both the mathematical ideas and the language factors in the work.  We tend to expect some magical cognitive growth in these students, as if working on discrete content areas will generate spontaneous global changes in the brain; I have no doubt that this does, in fact, happen to some students … I have seen it.  However, we do not create conditions for the larger cognitive changes.

Colleges might create a one-semester option for the less intense of the high-intensity group — those who can accomplish the goals with a one-semester package.  Smaller colleges might have difficulty with the logistics of this, while larger colleges would probably benefit from having two categories of high-intensity students.  Part of the rationale for the design for high-intensity need students is that preparation for them, is a more complex challenge.  Some will have had special services in the K-12 schools, and some will have significant learning disabilities.  This is the group most at risk; if community colleges are to serve all adults, then our remediation design needs to provide an appropriate pathway through to college work.  The alternative is to have a significant group of adults who will always be economically and socially vulnerable.  This high-intensity group are the ones that we need to educate policy makers about, so that they can understand the needs better — both the student needs, and our needs if we are to truly help them.

If you would like to do some reading on research related to this model, much of what I am thinking of resides in reports from the Community College Research Center (CCRC) at Columbia (http://ccrc.tc.columbia.edu/). Two specific articles: placement tests in general (http://ccrc.tc.columbia.edu/Publication.asp?uid=1033) and skipping developmental based on discontinuity analysis (http://ccrc.tc.columbia.edu/Publication.asp?uid=1035).    An article of interest by Tom Bailey and others on state policy appears at http://articles.courant.com/2012-05-18/news/hc-op-bailey-college-remedial-education-bill-too-r-20120518_1_remedial-classes-community-college-research-center-remedial-education.

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Neat Knowledge? Messy Landscape?

We all spend quite a bit of time talking with students, and we also look at massive amounts of student work.  Sometimes, we get in to “homework system mode” where we only provide feedback on the answer.  The answer, by itself, is very weak as a communicator of the knowledge a student possesses. 

I have been thinking about how messy student knowledge is (about mathematics); perhaps this is true for all domains.  In my classes, I try to create an atmosphere where people feel safe asking questions and participating; of course, sometimes the questions asked indicate significant misunderstandings … which can temporarily cause some damage to the overall learning in the class.  In this post, I want to focus more on the implications of errors in student work.

Okay, in our intermediate algebra class we just had a test on ‘quadratics’.  The material is a mixture of procedural and conceptual, with a few ‘applications’ included.  One of the applications involves compound interest (annual) for 2 years — so it creates a quadratic equation like this:

 Most students managed to write this (based on the verbal description and the provided formula).  The most common error?  Subtracting 4000 from each side, a disturbing error.  Since this problem was on a test, I did not have an oppportunity to discuss the error with the students to identify the source.  My primary suspect:  An early rule (early in education) to the effect that “large and small numbers mean divide, similar size numbers mean subtract”.  Every  one of these students can solve linear equations requiring division (like -4x=30), and most solved a quadratic involving a common factor (like 2x²=100).  What triggers  the “we must subtract” response?

In another class (the quantitative reasoning course), we have been doing geometry this week.  As for other topics, the formulas are provided — we are much more interested in the reasoning involved.  One of the problems dealt with finding the perimeter of this shape:

Two consistent problems came up.  First, students thought that ALL perimeter problems dealt with “P = 2L + 2W” … causing them to count the widths of the rectangle (which are interior, therefore not part of ‘perimeter’).  Second, even when convinced that some perimeter problems dealt with a different relationship, they did not see  why we should omit the perimeter (they still wanted to include the interior dimension).  Since I was able to discuss these issues, I have some idea of what is behind them. 

My point here is not to complain about what students do or don’t do, nor to suggest that they have had bad teaching  [I mean, beyond having ME as a teacher :)].  Rather, perhaps we need to think more about the root cause for many student difficulties: 

The basic problem is an oversimplification of a connected set of ideas down to a single ‘if-then’ clause.

Students are sometimes desperate to learn math, and we want to help them.  Too often, this results in only dealing with one thing (or one thing at a time) — which simplifies the learning, but which avoids building connections (contrast, similar).    The geometry instance of this is easily described:  by only dealing with one shape for ‘perimeter’ we encourage students to connect just one formula with that work — and to disconnect this from the concept involved (‘distance around’).   Our procedural work in algebra is even more prone to the oversimplification, because the correct learning is abstract about a combination of ‘properties’, ‘undoing’, and ‘maintaining value OR balance’. 

I’ve got to anticipate what some readers are thinking (due to that last sentence) … are manipulatives the solution for this problem in algebra?  Not really.  The research I have read on ‘kinetic learning’ suggests that this might provide some of the scaffolding to abstract ideas — but is counter-productive if the manipulative remains a focus.  This is a very difficult task, to shift from manipulative to concept; the stages vary with each learner.  If you can do manipulatives on an individual basis with continual feedback (conversation) with a highly skilled faculty, then I believe this can support better learning.   The skills involved are complex, and most faculty will not have them — including me; this is combining the skills of a cognitive researcher, qualitative researcher, and math faculty.  The problem with most manipulatives in learning is that the entire process is over-simplified to be practical in a group setting; a group of 3 students is too large, and possible 2 students is too large.

What is the answer?  We need to clearly articulate the few critically important learning outcomes in a math class (perhaps a dozen for a course), and provide diverse learning experiences around those outcomes.  “Simple” is not the solution; simple is part of the problem.  The solution is a planned sequence of activities to deeply explore the ‘good stuff’ of mathematics.

 
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Rotman’s Race to the Top

A few year’s ago, Paul Lockhart wrote an essay entitled “A Mathematician’s Lament”, sometimes known as “Lockhart’s Lament”.  I find this essay to be incredibly inspiring.

If you have not read it, here is a copy: lockhartslament

As far as I know, Lockhart never published this essay. I actually know very little about Paul Lockhart, besides the information that he went to teaching in a K-12 school after being a university professor for a time.  Some brief bio stuff is at http://www.maa.org/devlin/devlin_03_08.html

Okay, back to the essay.  Paul Lockhart is presenting a vision of mathematics as an art, not as a science, where learning is very personal and where inspiration is a basic component.  A fundamental process in Lockhart’s vision is the personal need to understand ideas, often ideas separate from reality.  The ‘lament’ aspect of the title for the article refers to the view that school mathematics has made mathematics into ‘paint by numbers’ instead of an exciting personal adventure.  Where we might focus on insight and creativity, we instead deliver an assembly line where students receive upgrade modules to their skill sets; Lockhart presents the view that these skill sets have nothing to do with mathematics.

I have two reasons for highly appreciating this essay.  First of all, people often assume that I believe that mathematics should be presented in contexts that students can relate to (probably because I advocate for “appropriate mathematics”).  Contexts are fun, and we can develop mathematics from context.  However, contexts tend to hide mathematics because they are so complicated; mathematics is an attempt to simplify measurement of reality, so complications are ‘the wrong direction’.  When I speak about appropriate mathematics, I am describing basic mathematical concepts and relationships which reflect an uncluttered view of mathematics; applicability might be now, might be later … or may never come for a given student.  Education is not about situations that students can understand at the time; education is about building the capacities to understand more situations in the future.

The second reason for appreciating Lockhart’s Lament is the immediate usefulness in teaching.  If we believe that students want to understand, then we can engage them with us in exploring ideas and reaching a deeper understanding.  Here is an example … in my intermediate algebra class this week, we introduced complex numbers starting with the ‘imaginary unit’.  Often, we would start this by pointing out that ‘the square root of a negative number is not a real number’ and then saying “the imaginary unit i resolves this problem.”

Think about this way of developing the idea instead (and it worked beautifully in class). 

1. Squaring any real number results in a positive … double check student’s understanding (because quite a few believe that this is not true).
2. The real numbers constitute the real number line, where squaring a number results in a positive.
3. Think of (‘imagine”) a different number line where squaring a number results in a negative.  Perhaps this line is perpendicular to the real number line.
4. We need to tell which number line a given number is on; how about ‘i‘ to label the number?
5. Let’s square an imaginary number and it’s opposite (like 2i and -2i).
6. Squaring an imaginary number produces i²; since the imaginary numbers are those we square to get a negative, i² must be -1

Any context involving imaginary or complex numbers is well beyond anything my students would understand.  I would always want to explore imaginary numbers; first, it provides a situation to further understand real numbers … second, the reasoning behind imaginary numbers is accessible to pretty much all students.

You might be wondering why I gave this post the title “Rotman’s Race to the Top”.  One reason is just not very important … I wanted a alliteration comparable to “Lockhart’s Lament”; well, I tried :).  The important reason is this:  The “Race to the Top” process is based on  some questionable components that do not connect directly with the learning process (common standards [a redundant phrase], standardized testing, and outcomes.  The world we live in, as teachers of mathematics, is a personal one: We interact with our students, we create situations that encourage learning, and we want our students to value learning for its own sake.  We also want our students to understand some mathematics.  Inspiration and reasoning have a lot more to do with this environment than any law or governmental policy.

Rotman’s Race to the Top is the process of teachers showing our enthusiasm, modeling the excitement of understanding ideas, and helping each student reach a higher level of functioning.  Building capacity and a maintaining a love for learning are our goals.  Rotman’s Race to the Top is about our students and their future.

Will you join this Race to the Top?

 
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