...just invert and multiply!
This seems to be one of the fundamental philosophical questions in math education. Do you teach tricks and rules, or do you let students explore and construct their own knowledge? Does it matter what type of students you are working with? Behaviorist, Constructivist, Socio-Constructivist?
I, personally, believe in Socio-Constructivism. The idea is that the teacher creates a well-structured pathway for learning that takes students through the levels of understanding. Students start off being given basic facts or information that they will need, and then are given time to work (individually or collaboratively) on expoloration or inquiry based activities. Through this process, they begin to generate conjectures and create a quasi-mathematical understanding. Then, through class discussion and direct instruction, the teacher helps correct misunderstandings and formalize the knowledge (i.e. algorithms, processes, etc.) The drawback to this, of course, is that it takes a lot more time - both to plan the materials, and actual class time. Is it worth it? That's the real question. I believe it is, but I think there are those who disagree.
One of the classs that I have developed is called Numeracy, in which we put freshmen who test below 7th grade level when they arrive (this is typically 70 - 80% of the class). We start off at basic operations and place value concepts - this takes the entire first semester. The second semester is all fractions. We spend a few weeks working with fraction circles, drawing fraction bars, using reasoning, etc. to compare, order, and evaluate fractions. We work on determining if a fraction is closer to 0, 1/2, or 1 whole. We figure out how you can compare 7/8 and 8/9 by reasoning. Then we spend 6 weeks adding and subtracting, with manipulatives, with pictures, and finally, with the algorithm. Then there are 6 weeks dedicated to multiplication and a few more for division. Yet at the end of all this, I still have a large number of students who haven't learned to work with fractions fluently. Sometimes, there is the temptation to teach the rules and then practice them to death, but in my heart I believe this won't work. Plus, if you blindly memorize the "flip and multiply" rule, you haven't really learned anything about division and you won't be able to apply that knowledge to other situations (i.e. Algebra). Also, if you develop no context for your algorithm, you have no way of knowing if what you are doing makes any sense.
I think students have been trained to want algorithms in math. They resist exploration. A noisy class will quiet down and get to work when a worksheet is put in front of them - why? Even if it is not being graded, they will rush to try and get answers down, regardless of whether they are learning or not. Like lemmings, they seem compelled to "finish the worksheet". It boggles the mind! "Mr. Greene, just tell us the easy way! Stop asking us questions!" How many times a day do I hear that in Numeracy? Students know that drawing fraction circles, for example, will help them solve a problem, but they would rather ask me for help, or just skip the question. When I force them to draw a picture, they begrudgingly do so, then look at their picture and say, "Oh, that's all you want us to do? That's easy!" and proceed to answer the question with little difficulty. Yet, on the next question, the process will repeat itself. Patience is definitely a learned skill!
I am happy to say that, at the end of Numeracy, most of the students have learned that, when adding or subtracting fractions, you don't add across!
Any thoughts on the matter?
Monday, June 26, 2006
...just invert and multiply!