I have just read a "tutors' manual" which deals with suggested strategies for helping high school students with those mathematics topics they most often have difficulties with. What I glean from this is that students are assumed to be helped effectively by giving them rules for how to deal with various situations.
The manual devotes many - and I say many - pages to rules and procedures by which students can get the right answer when asked to factor quadratic expressions. For example:
Factor: 2x^2 + 10x + 12
I recall from my classroom teaching days that some of my students found this topic more frustrating than others. The simple examples, such as the one above, were no problem; but for some reason the texts have all sorts of exercises that are not simple. I did then, and do now, ask myself the question:
Why are we doing this?
Is factoring complex quadratic expressions a worthy objective in its own right, or does it serve some further purpose?
If the purpose of the exercise is to solve quadratic equations, one must ask if factoring is the most sensible way to go about that; keeping in mind that no real-world problems lead to quadratic equations with integer coefficients - which are the only kind that can be factored.
I suppose the next question to ask is "What is the purpose of solving equations anyhow?"
Come to think of it, why learn algebra?
It seems to me that the answers to these questions should be stated "up front". They should be subjected to, and be able to withstand, critical analysis.
I have never heard any mathematics educator address these questions. If you have, I would be very interested in hearing about it.
My own answer to the last question runs something like this:
The purpose of learning algebra is to extend students' repertoire of techniques for solving appropriate and plausible word problems.
Maybe the wording could be improved; but this will do for now.
I say word problems in order to exclude symbolic "problems" such as 'find x' or 'simplify', etc. outside the context of some plausible real world situation. I view these as activities as abstract symbol manipulations. Whether symbol manipulations has any inherent value in its own right is a matter for debate.
It seems to me that the standard algebra and calculus curricula are examples par excellence of how, in School Mathematics, we can lose track of where we are going and why we are going there. Reading most textbooks and curricula descriptors and observing what kids actually spend most of their time at, it would seem that the focus in School Mathematics is on symbol manipulations rather than some sort of application.
In his book The Pleasure of Finding Things Out, Richard Feynman describes how he perceived Algebra as a child:
...a set of rules which if you followed them without thinking you could get the answer … if you didn't know what they were trying to do.
Here is my model for what I call the algebraic
process:
To illustrate with two simple examples, one from elementary algebra and one from introductory calculus:
Billy and Suzy have 5 apples between them. Billy has one more apple than Suzy has. How many apples do each have? |
If a rock dropped from a cliff is seen to strike the ground below after 10 seconds, how high is the cliff? |
Stage 1
Process: analysis: This word means to take things apart. To separate something into its components.
Result 1: question(s): What do we eventually want to know?
How many apples does Billy have? How many apples does Suzy have? |
How high is the cliff? |
Result 2: data statements These are unambiguous ways of expressing the essential data needed to answer the question.
The number of apples Billy has plus the number of apples that Suzy has add up to five. The number of apples that Billy has is one more than the number of apples that Suzy has. |
The stone is in free fall for 10 seconds. |
Stage 2
Process: translation: The data statements are translated into algebraic statements. In algebra we use letter symbols to represent numbers or values. In many cases we need additional relation statements For example, that the acceleration due to gravity is 9.8 m/s/s.
Result: relations: (footnote 1)
B + S = 5 B = 1 + S |
height = integrate (g * t) from t = 0 to 10 g = 9.8 m/s/s |
Stage 3
Process: manipulation The details of the various processes and techniques are discussed elsewhere. Symbolic approaches were developed as a matter of necessity at a time when calculating devices were not available; and their end result is a numeric solution to the problem. Numerical approaches are impractical when done by hand; but readily available software does either (footnote 2). Some people feel that that calculators/computers should not be used. (footnote 3)
Result: solution (numerical values for the variables)
S = 2 B = 3 |
height = 490 m |
Stage 4
Process: synthesis: The word synthesize means to put together or assemble from component pieces. The solutions and the original question are synthesized to produce a final result - the answer to the question(s).
Result: answer to the problem:
Suzy has two apples and Billy has three apples |
The cliff is 490 m high. |
Comments:
Analysis and translation requires human intelligence and knowledge. They require the ability to read, familiarity with the English language and its idioms, and they require familiarity with, and selection of, relevant properties of the systems under investigation. As such, analysis can serve as a vehicle for learning about relations between all sorts of real world objects and phenomena. I suggest that this activity has value in its own right - probably much more real educational value than the symbol manipulation which is the actual emphasis. Keep in mind, please that I'm talking about high school for "ordinary people" here; not those 15% who complete post secondary degrees.
Unless the student understands the problem in the sense that he can analyze it as I have described, then proceeding to the solution stage serves no plausible purpose.
There are several methodologies for obtaining solutions. The traditional algorithms were developed in an age when numerical methods were impractical; they have some severe limitations (footnote 4) and are not demonstrably superior from the pedagogic perspective.
On the assumption that what I have said in the preceding paragraphs makes sense, I believe we are permitted to ask:
Why is it that the majority of time and effort in School Mathematics algebra and calculus is devoted to symbol manipulation as opposed to problem analysis?"
and
Why do we teach symbolic, and not numerical solutions methods in School Mathematics?
and
Why is it that (Ministries of Education) seem not to be interested in the addressing these issues?
For that matter
Why is it that teachers don't raise these issues?
Footnote 1: Statements that show the relationship between two variables are properly called relations, not equations. Note that although setting up of one-variable equations in solving word problems normally precedes using two variables, it is much better to use two variables right away. For starters all single variable equations are better solved by trial and error. I have used this approach for years and have had no untoward difficulties with it.
Footnote 2: Here are many opportunities for enrichment by encouraging students to program their own numerical routines. There are many computer languages suited to this. Also access to mathematical software like Maple and Mathematica would encourage browsing mathematics, which by any criteria is a highly desirable activity.
Footnote 3:The essentials of "anti computer" arguments boil down to students must think and computer use discourages this. I disagree with the suggestion that traditional algorithms are more thought promoting than numerical computation tools.
Footnote 4: The calculus example given here is a frequently encountered and familiar example of how artificial things get. Rocks dropped off cliffs do not behave this way because of the reality of air resistance. There are no symbolic manipulations that will provide useable solutions to this problem. Computer based numerical approaches must be employed.