What does ‘sin(2x)’ mean? Or, “PEMDAS kills intelligence, course 1″

My department is starting a conversation about pre-calculus and intermediate algebra.  I’m very pleased we are taking this step, and it is great that I do not know where our work will take us.

In our discussion yesterday, one of the concerns expressed was that students tend to not understand functions … including function notation.  People referred to the classic error:

sin(2x) = 2*sin(x)

The problem runs deeper than functions and function notation.

√(a²+b²) = a + b              and it’s corollary

(a + b)² = a² + b²

We cause these problems ourselves by allowing and even encouraging students to learn procedures by categorizing symbols, without needing to apply the meanings of those symbols.  In arithmetic or pre-algebra, this occurs in both order of operations and basic variables.

“Use PEMDAS to determine the correct order of operations”

“To add like terms, combine the numbers in front”

In a basic algebra course, the comparable statements for exponents and radicals might be:

“Negative exponents mean you write the reciprocal”

“Take out the perfect squares”

Much of our work in classes deals with getting students to correctly process the symbols we place before them.  When they generate mostly correct answers, we conclude ‘they understand’.  Seldom is that the case … because we seldom take the time to focus on the meanings of these objects along with the various correct choices we have in working with them.

When I say “PEMDAS kills intelligence”, I am using the pneumonic as a place-holder for the prescriptive procedures that we focus on.  As mathematicians, we are all about choices.  When we see:

3x(x – 2) + x(x – 2)

We think of two choices (combine ‘like terms’ first, or distribute first).  The PEMDAS-mentality boxes students in to the ‘one correct [sic] way’ approach to mathematics; the PEMDAS approach also encourages students to perform procedures on symbols without much regard for what that particular expression or statement meant.  We use these approaches in all kinds of math classes, from elementary classrooms to university classrooms, and it has got to stop.

In recent years, some of the reform efforts have de-emphasized symbolic work … partially as a response to this problem.  I applaud that work, and have contributed to the efforts.  However, sometimes we over-react and provide too little symbolic work.  We have course which emphasize ‘functions’ but never use basic function notation [f(x)], let alone variations such as ‘sin(x)’.  An irony is that most technologies that students use for our math courses (calculators, apps, web sites) generally use function notation.

Maintaining a strong focus on procedures and correct answers encourages a PEMDAS-mentality, causes problems for us later, and (I would suggest) limits student motivation to learn mathematics.  Think how much better it might be to have a balanced approach, where the key principles are:

  1. Meaning
  2. Properties
  3. Choices
  4. Application
  5. Extension and feedback on prior steps

Some of my colleagues have said that students should “do mathematics” in math classrooms, though they are mostly talking about step 4 (application).  I also believe that students should “do mathematics” in every math class by using all levels (1 to 5 in my list) with all topics.  If we are not willing, believe that student’s can’t, or think that we do not have time … well, then we should question whether we are really committed to teaching that mathematics.

Most of our collegiate math courses are overly ‘full’, not too full of topics but too full of wasted effort.  We focus so much on “simplify” and “solve” in the basic courses that students use the PEMDAS-mentality; of course they won’t remember most of it, and of course they can’t apply ideas to other contexts — we are training them to just process the symbols.

So, if you have been wondering what I would have us do to replace “PEMDAS” for basic expressions, we should focus on four items:

  • Meaning of each expression
  • Inherent priority of each operation (a generally predictable list, based on level of abstraction)
  • Properties for the type (meaning) of expression
  • Choices for this expression

It is almost useless to know that a student can correctly calculate “8 – 3(5)”.  Value comes from knowing that there are multiple procedures to correctly calculate “6(2x) + 8(4x)”.  It is also almost useless to know that a student can correctly solve “12 – 5x = 7″.  Value comes from knowing that we have choices for that equation and for “8(x – 2) + 4x = 48″.  [I also suggest that PEMDAS itself is both incorrect and incomplete.]

Mathematics did not become so valuable because we know how to correctly arrive at an ‘answer’.  Our work is indispensable because we can present alternatives, and in some cases one of those alternatives provides great benefit to people, companies or societies.  That is ‘doing mathematics’, and is the type of experience I want for our students … whether in pre-algebra, pre-calculus, or anything else.

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Math Paths Workshop for Michigan Community Colleges

When:  June 17 – 18, 2015

Where: North Central Michigan College (Petoskey, MI)

Here is the flyer: MI workshop flyer_final_2015_04_14

To register, use this link: http://www.cvent.com/d/trqzrd

 

Toward a Modern View of Mathematics

We face many opportunities in the coming years, in our professions of mathematics and mathematics education.  Will we seize the opportunities, or merely survive with the least efforts that avoid the largest problems?

As professionals, we know that mathematics is a collection of sciences dealing with quantities, shapes and relationships.  We have allowed one of these sciences — calculus — to dominate the mathematical experience of our students, and often only have students study other mathematical sciences after a mastery of calculus (even when there is not conceptual connection).

Now, I realize (as we all must) that calculus deserves a prominent location in undergraduate mathematics.  Not only are the concepts and methods of calculus used in a variety of fields, but the study of calculus allows students to experience some of the greatest achievements in science (and see the beauty as well).  I would like more students to learn calculus.

However, we lack balance in our curriculum.  The vast majority of undergraduate mathematics courses are part of the path to calculus, where the content is (loosely) based on what is needed to learn calculus.  The fact that this path is not effective and needs a new design is a related but separate conversation.

Many recent conversations have amounted to “calculus/calculus-path OR statistics”, with the refrain “people can actually use statistics”.  I question the accuracy of that statement in many ways, but more importantly — are there no other areas of mathematics that have a modern practicality?  Do we really believe that life begins after calculus … that study of other areas must be delayed?

Graph theory is ‘hot'; much of our modern technology is related to this work.  Is there a reason not to include a basic understanding of graph theory in undergraduate mathematics?  The work of graph theory seems accessible.  How about basic number theory and ideas of cryptography?  Discrete mathematical ideas? Matrices and numeric method?

Forum 5 of the Conference Board of Mathematical Sciences (October 2014) focused on mathematics in the first two years of college, with a prime motivation coming from the book “Mathematical Sciences in 2025″ http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025   As people talked about the vitality of mathematics, my question was (and still is):

Do we integrate any of these topics or concepts into basic college mathematics, or do those courses continue as single-minded diversions into mathematics that nobody cares about?

Many of you have a deeper understanding of the mathematics described in the “2025” book.  What I recognize is that our students are (in general) prevented from seeing any topics or concepts related to current mathematical research until after the first two years.  Perhaps we can not avoid that condition; however, I think we can include multiple mathematical sciences within the basic mathematics courses our students take.

I hoe that mathematical diversity is coming to a math course near you.

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The Four-Year-Myth as Misnomer

Like many institutions, my college is embarking on a ‘guided pathways’ mission; I am fine with that, and will be working on one team involved.  One basis, a rationale of sorts, is a report called “the Four Year Myth” from Complete College America (see the ‘field guide’ post earlier today).  The ‘myth’ report is available at http://completecollege.org/wp-content/uploads/2014/11/4-Year-Myth.pdf

The basic idea of guided pathways is to eliminate two apparent causes of excess credits (and time) — remediation and options in courses.  Remediation is blamed for much of the excess credits and time, so we are supposed to mainstream and contextualize; we can (and should) do some work this way, though I am sure that we need to have developmental-level math courses.

Eliminating options is supposed to reduce extra credits taken for a bachelor degree.  The idea is that students start the process with a major, and only take courses on that program (path).  If the major is not known, the student is assigned to a ‘meta-major’ where they take courses very likely to apply to their final goal.

I see two parts of this mythology that lack validity.  First, student goals … the presumption is that students know enough about their goals that they will almost always end up in the correct program or meta-major.  This presumption contradicts some very strong occupational data from the past 20 years: today’s occupation or goal is tomorrow’s memory, when we have a new goal or occupation.  I see no evidence in our students that they will end up in the correct meta-major or program at a high rate, when starting college.

Second, the ‘four year’ modifier is an external thing.  Few people in education call that a ‘four year degree'; we say “bachelor degree”.  It’s true that the requirements for a bachelor degree can be completed in 4-years of full-time study if the goal is stable and there are no problems (initially or along the way).  Even in ancient times (way back in 1970) these conditions were not universal.  Today’s students arrive at the college door with a high probability of problems, many of which are not related to ‘remediation’.  There is a higher risk of instability in goals as well as bumps in academic performance along the way.

This last analysis leads me to say that the “four year myth” is a misnomer.  A myth needs to be held as true, and this one is not held as true in general.  You might notice that claims like this tend to come from certain types of ‘reform agents’ (see that field guide again).

College completion is a fantastic goal, and we in mathematics have a special responsibility in this work.  Our students deserve better.  However … basing a program on a shaky mythology supported by data cherry-picked to support that position is not the best we can do.  We need to do the hard work of identifying all of the major causes for not-on-time completion, categorize and prioritize the ones that can be improved, and develop plans that deal directly with the causes.  [Guided pathways is likely to be an improvement, though minor.]

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