When Good Teaching Leads to Bad Results - Schoenfeld

As a teacher of mathematics, Schoenfeld's (1988) article " When Good Teaching Leads to Bad Results: The Disasters of " Well Taught" Mathematics Courses" made several significant points which are as pertinent to Mathematics Education today as they were when the article was first written. The issues of what constitutes actual learning in mathematics and which approach (traditional or reform) provides the best instruction for that learning are, for me, ideas I have pondered as a mathematics teacher today.
I struggle with providing students experiences which cover the curriculum but also develop understanding and I know that I am not alone. Many teachers express the same sorts of frustrations with the current program. This article has helped me to see the bigger picture and has put into words what, up to now, has been a collection of vague notions about why we struggle so much with math instruction. Schoenfeld's discussion of the delineation between "becoming competent at performing the symbolic manipulation procedures... and grasping the underlying mathematical ideas..." (Schoenfeld, 1988 p.3) hits at the heart of problems we and students face with the current math program that is offered in Newfoundland.
Are we developing competency with understanding, can any one program provide students with all that they need? The 'problem solving' approach to mathematics that has been toted by our school district and our Department of Education should provide students with enough experiences that true understanding is gained. Yet, many teachers and parents believe that we are turning out students who do not have the 'basic skills' in mathematical procedures let alone 'understanding'. What has been referred by many as ' math wars'. It does not seem that we are providing what students need. How much of this is due to teachers of elementary math being 'non-specialists' in this field? The majority of us are not mathematicians in any sense of the word. We did not complete a math degree, but instead completed the minimum number of courses required for our Bachelor of Education degrees. Whatever the reason or reasons it is evident that we should at least be aware of the misconceptions about math that our instruction may be causing our students to develop.
Schoenfeld, highlighted a set of 4 'beliefs' that students in his case study had developed about math and the misconceptions these beliefs caused. I can relate to much of what he has stated because it represents my experience with math as a student and teacher. I believed, perhaps without realizing it, that "students who understand...can solve the math problem in five minutes or less". This was the very reason I gave up so easily when confronted with math work in my own school days. This is the reason I continuously tell my students that not knowing the answer right away does not mean you can't figure it out. I also always believed that only mathematicians really understood math, and if I reflect on this point I have to say that this was a belief I held right up until I read this article. Now I am questioning my thinking surrounding this idea. I am not totally convinced that there isn't a good deal of truth in "only geniuses are capable of discovering, creating or really understanding mathematics" as my statements about mathematicians vs math teachers in previous paragraphs attests to. Still this has certainly given me food for thought.
This article has highlighted for me what some of the critical issues in mathematics education are. It has provided me with a framework for many of the conclusions I have come to as a mathematics teacher. I know that many students do experience success with the math curriculum to one degree or another. I also must conclude that in my experience it is true that students see themselves as " passive consumers of others' mathematics' and that connections between concepts and solutions to problems are not always demonstrated. Making connections between process/procedure and understanding the 'big ideas' behind the concept are simply few and far between.
My own son has always received very high marks in mathematics all the way to the twelfth grade where, we hope, he will continue to do so. So as I read through Schoenfelds article I decided to ask him to complete the problem that Wetheimer ( 1959) asked elementary school children to solve. I showed him the fractions as seen on page 5 and asked him if he though he could solve this. " Yes, sure I can" was his reply. However, when I asked him how he would solve it he gave a step by step procedure that included adding and dividing the numbers involved. When I prompted him to use what he knows about operations and fractions to find another way to get the answer he couldn't procede. He was stunned when I pointed out that there was no need to calculate because the relationship between fractions and division and the fact that inverse operations cancel each other out. Then, of course, he wanted to prove himself, " Give me another one" so I did. I gave him pencil and paper and asked him " If a bus holds 36 passengers how many buses would be needed to get 1128 soldiers to training. Again he quickly divided and told me 31, you can imagine his chagrin when I asked how the remaining 12 soldier were going to get where they needed to be. I believe him to be very mathematical and yet he did not make the needed connections. I can't help but wonder that if Wertheimer's problem came from 1959 and the problem from the third National Assessment of Educational Progress (Carpenter, Lindquist, Matthews and Silver,1983). was given more than 20 years ago how much progress have we made in our approach to teaching mathematics?