Calculation without Understanding

[coberst]

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

 

[Washington]

Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding.

In grade school we are taught to manipulate numerals (symbols) not numbers (concepts).

Absolutely not true in my case, and my schooling took place within a large metropolitan public school system.

 

[keith99]

Not true in my case either. But I don't count. I've been told I described how I arived and answers to Arithmetic problems when I was in elementry school and my mother (who was a math wiz and taught math both before and after that time) was so impressed that they were teaching such methods that she talked to my teacher about it. Only to find out the teacher had no idea what she was talking about. I honestly don;t remember this, and can not claim I invented anything. I think I just built on tricks I had been taught at home.

 

[coberst]

Absolutely not true in my case, and my schooling took place within a large metropolitan public school system.

You were very fortunate to have such a good education.

 

[Washington]

You were very fortunate to have such a good education.

And it's disheartening to think that it wasn't a matter of course everywhere.

 

[No Swansong]

Not meaning to jump on the bandwagon but it wasn't true of my public school education either. If this forum is any indication it seems that the authors premise may be faulty.

 

[Foolish_Fool]

It varies wildly from school to school and teacher to teacher. My HS had a fairly horrible math department. Most of the course work relied heavily upon graphing calculators and the memorization of formulas. I learned more about how to work the calculator and memorization tricks than I did about what was actually going on. It wasn't until my calculus AP course with a crazy russian teacher who barely spoke english that somebody finally made sense of the squiggly lines dancing all over my paper.

 

[MoonlessNight]

My calculus students do seem to have a problem in this area. At least if I am interpreting the original post correctly.

I take it as saying not that people are bad at all math, or that they are constrained to using only calculators at the expense of even basic arithmetical skills, but rather that when they use math they have no idea how or why any of it works. For instance, with Newton's Method it is fairly simple (relatively speaking) to just apply it and know that it approximates where you get your zeroes. But to understand that it's using better and better linear approximations is something that is at another level, and ideally where people should be.

The second I give any problem where you don't have the exact definition of a function, and thus are restricted to getting things like the derivative from the actual meaning of the term, the amount of effort students need greatly increases.

I'm not trying to be too hard on them because it takes a while to make the transition to thinking about math in that way, and for most of them they'll never really need to. But certainly I think that it's a product of the education system that they have trouble understanding concepts, but much less trouble using techniques.

 

[coberst]

MoonlessNight

I can remember when in grade school that our teacher would, on rare occassions, assign us "thought problems" to do for home work rather than the mechanical "follow the algorithms" problem. We went crazy with anxiety.

 

[Eudaimonist]

Early in our institutional education system we learn arithmetic. We learn to add, subtract, multiply, and divide. We learn to calculate without understanding.

I didn't, but then I went to good private schools.

I think we may need to accept that much schooling is aimed at the LCD, and such people might not be capable of dealing with math on such an abstract level. And so the educational system does what it can to impart just enough mathematical knowledge to allow people to survive in modern society.

Of course, for those people who are capable enough to deal with abstractions, education should at least be made available for their needs.


eudaimonia,

Mark