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:
- 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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