Let's start with the rote method. Based on a top down idealistic philosophy, the teacher explains how to do a mathematical problem like how to add two numbers and two number (45 + 45 = ) explaining how to "carry over to the next column". Right. Perhaps the teacher would do several problems like this and then have the students do several like problems for themselves. If the students got the problems correct, the teacher could go on to a different mathematical problem like adding three number to two numbers (123 + 45 = ). What the teacher is trying to do is to get the students to understand the concept about carrying over to the next column of numbers. Most children get this concept--adding numbers together to get a total becomes second nature for most. This is the rote method of teaching whether it be mathematics or some other subject. If you want to make sure the students have the concept in long term memory you provide more problems for them to solve. Time on task is the phrase we teachers tend to use. The biggest problem is that by providing more problems for them to solve you introduce boredom to the ones that have quickly understood how to do it. We want kids to like mathematics (an Affective Domain objective) and not be turned off from it.
Okay then. What about the other method of teaching mathematics? The Discovery method. This follows the Realistic philosophy of using real things to learn and by allowing the student to "own the learning, i.e., discover the concept without someone else telling them. For example with this method a teacher might place forty five bottle tops on one desk and another forty five bottle tops on another desk. The teacher then asks the students to be sure of the numbers on each desk then ask the question how many total bottle tops are there? In all probability the students will count from the forty five on the one desk and continue counting numbers on the second desk. Time consuming and a high possibility of error in the county. Then the teacher asks the class, "Is there a better way to count these bottle tops?" Eventually the students would want to put numbers on the board and would "discover" carry over to the next column. Indeed if the discovery was truly earned, there would be no need to do larger numbers in addition except to "prove" it works. The "Aha" method. Supposedly the students enjoy gaining this knowledge, hence, the Affective Domain is involved. "Liking Math."
Studies have been going on for years on both of these methods in teaching mathematics. Several of my mathematics colleagues state that one method or the other is by far the correct method and I am led to believe that neither method has achieved stardom.
Recently the Issaquah School District had it's math teachers review both methods as well as review a number of textbook series to use in the district classrooms. Buying a large number of textbooks is an expensive proposition and the district doesn't want to get it wrong. As a principal I would be asking my follow principals in other districts "what do you use--how do you like it?" and the teachers would be going to summer school and asking the professors what is best? After all this, Issaquah School District made the unanimous decision to go with the Discovery method of teaching Algebra and Geometry. Good, let's get on with the teaching......except......... the State Superintendent of Public Schools said the other method was probably better. What? (see "Which Math Book to Use? A Passionate Debate Rages" by Katherine Long in the Seattle Times Newspaper 8/16/09)
Actually the real problem in all this mathematical debate lies with the colleges and universities who mandate that to get into their institution students must have four years of high school mathematics. They are the driving force behind all this. Bellevue (listed in the top 100 high schools) students want to get into the university of their choice (hopefully with a full scholarship) and so mathematics becomes the gate keeper. Issaquah (and Mercer Island and Everett and Lake Washington) school districts follow suite. They all want their high school graduates to be the best. It makes sense to me.
What doesn't make sense to me is that we are discussing which textbook to use. Haven't we forgotten something? Where are the computer programs to teach mathematics? Why not a teaching program that explains a mathematical concept to the student (viewer?) and have them do a problem. If they get it wrong the computer says "no, lets try it this way." For heaven's sake, we could have the rote method as well as the discovery method all rolled up on a CD....or a box set of CDs. We have some of the best programmers in this area who are doing nothing but working on game software. Why doesn't Bill Gate's Foundation on Improved Teaching put out a call for new software on LEARNING MATHEMATICS. Now someone is going to ask me why the publishers don't produce a CD on mathematics. The reason is that they make a heck of a profit on textbooks. CDs would not produce the revenue that the textbooks do. Simple. I really believe that technology at this point could solve a number of problems in the educational market. Good computer programs would allow a bright student to move ahead and hence not get bored. The slower student would be given more problems to solve but be given instant help when needed. Where am I wrong here? What am I missing? And perhaps a bright student would not need four years of mathematics. I learned my "statistics" in college from a computer program called "Statpak."
Okay universities. Produce a sample test of what you expect of entering students and we teachers will teach our students to do well on it and even do better. The ball is now in the Mathematics Departments at all the universities. What do you expect?
And software companies--pay attention. What our educational system needs is good CDs or web based programs that can teach mathematics. Wow! What a concept.
One more time--say it with me. "Thank you teachers for what you do in our classrooms!"