Summary
The article talks about the relationship between operational (a process) and structural (a product) conceptions of how abstract math concepts are formed. A structural approach focuses on higher-level thinking, whereas a conceptual approach is grounded on algorithmic skills. Although a structural conception is more abstract and more integrative than operational thinking which is sequential and computational, they are complementary. In addition, Sfard states that abstract concepts are formed in the following three phases.
(1) interiorization which helps you learn a new concept through a process (for example, "counting leads to natural numbers (p18).")
(2) condensation which allows you to combine multiple processes to make generalization
(3) reification which helps you "convert abstract concepts into compact ... new ... self-contained (p14)" ideas
Next, Sfard talks about why historical mathematicians found it difficult to reify the concepts of numbers and functions. The reason is that mathematicians considered and defined these concepts operationally before they did them structurally. Whenever a mathematician defined properties of a number or a function through some operational manipulations, someone else found a new abstract object from these processes and re-defined the properties. As a result, the development of these concepts became cyclic before they were fully reified in structural form. In modern times, mathematicians put a greater emphasis on structural development when studying math concepts.
Lastly, Sfard emphasizes the beneficial effects of having both structure (related to reasons) and operation (related to rules) on student learning. Without one or the other, students will have difficulty in developing good procedural and conceptual understandings of math.
Reflections
In my tutoring practice, I frequently notice that some of my students have a good conceptual knowledge of math concepts shown by their abilities to reason. But, they cannot demonstrate their knowledge on my practice tests and on their school tests. In contrast, some of my other students who achieve high scores on their school math tests do not seem to have a sound understanding of math principles. In the first case, the students whose goal is to understand the structural aspects of math concepts seem to suffer badly in the current education system which favorbly assesses learners' operational proficiency. Memorizing rules promotes operational skills but hinders reification of abstract concepts. That said, my students who learn math procedurally without reasoning skills really enjoy the rewards of their high grades on their school math tests. They always want me to tell them which methods work best for which math problems. Unfortunatly, the reality is that my students, especially in senior grades, come to me with lots of drills (basically, 10 - 15 questions on each concept) from their math teachers. These questions are no more than fillers preventing students from interiorizing, condensing, and reifying abstract concepts.
Relational understanding requires a lot of time and effort, whereas it is much easier and faster to get the right answer through procedural thinking. How can we find a balance between mastery of computations and relational understanding given the large amount of content to be covered across the high school math curricula?
It's interesting that you bring up the 10-15 question drills as a way of practicing methods, not as an example of developing deeper understanding of mathematics. I wonder if this practice of providing the same questions has been repeated in the hopes that the student completing these questions would eventually see a pattern among the many questions they need to answer.
ReplyDeleteFirst...... the font is tooooooooo small, would you mind using normal size?
ReplyDeleteSecond, math needs to be used A good math learner can master both structure and operation, but a "bad" math learner cannot understand at least one of the two knowledge because of various of reasons (See my article. It is amazing!). A student being capable of explaining concept but cannot do problems maybe due to lack of practice, but a student who can do problems but cannot explain concepts must have good imitation ability. What s/he needs to do may be do more readings (and there may be more effective solutions.).