Calculation without Understanding Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding. This mode of education follows us throughout our formal education system. We learn to develop answers devoid of understanding. We do this because, in a society focused upon maximizing production and consumption, most citizens need only sufficient education to perform mechanical type operations; that is perhaps why our electronic gadgets fit so well within our culture. If we think about this situation we might well say that this form of education best serves our needs. It is efficient and quick. However, beyond the process of maximizing production and consumption we are ill prepared to deal with many of life’s problems because we have learned only how to develop answers that are “algorithmically friendly”. In grade school we are taught to manipulate numerals (symbols) not numbers (concepts). We are taught in grade school not ideas about numbers but automatic algorithmic processes that give consistent and stable results when dealing with symbols. With such capability we do not learn meaningful content about the nature of numbers but we do get results useful for a culture of production and consumption. We have a common metaphor Numbers are Things in the World, which has deep consequences. “The first is the wide spread view of mathematical Platonism…[it] leads to the metaphorical conclusion that numbers have an objective existence as real entities out there as a part of the universe…Given this metaphorical inference, other equally metaphorical inferences follow, shaping the intuitive core of the philosophy of mathematical Platonism.” Quotes from Where Mathematics Comes From by Lakoff and Nunez
Perhaps where you came from that is true but here I understood what math was from the very start of schooling. My teachers knew the importance of showing how math worked by doing actual things with real objects around us. We needed to count the cars and trucks then find out how many were of each percentage wise. Many other ways we learned math in primary schools. When it came to algebra , that's when they lost me though for there weren't any real time ways of making me understand that maze of letters and symbols with numbers.
Two things: First, in order to be able to understand something, you have to be able to calculate it first. Unless you're actually synthesizing knowledge (i.e. conducting research), then there is very little hope that you can understand something fully before you calculate it. An example is Feynman diagrams in quantum field theory. No one really understands what they are, but you learn how to draw them, and you learn how to calculate them, and you have the rest of your career to contemplate what they actually represent. Also, how much understanding do you want? Many analogies can be drawn between english and math---both are a language, and both have logical sets of rules (although english is less logical than math by FAR). How well do you have to understand the structure of the english language to be able to use it? How well do you have to understand the structure of the english language to become an author, or a poet, or an english teacher? Really, the number of people who use english proficiently are far less than the number of people who really understand the structure of the language.
I think that our educational system fails us all when we learn only what to think but never learn how to think in our schools and colleges.
I would say the opposite. One of the great advantages of a formalism, an abstraction that can be used to calculate even if you don't really understand what you are calculating, is that it can be used to bootstrap your understanding. You get something, and you have to figure out what it means translated back. You get a feedback - formalism comprehended yielding new directions of inquiry.
