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Author Topic: Is it sensible to teach people how to calculate things without understanding it?  (Read 2860 times)

Offline coberst

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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
« Last Edit: 19/01/2009 09:30:30 by chris »


 

Offline Chemistry4me

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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.
What do you thinik should be taught at school then?
 

Offline coberst

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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.
What do you thinik should be taught at school then?

I think that our schools teach us what to think, which is perhaps necessary.  I wish that they would add some significant effort in teaching us how to think.  Also it appears to me that our schools leave us with severe learning handicaps that must be overcome before we can become self-learners.
 

Offline Chemistry4me

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You mean 'think' as in? Teaching us about life skills? Making our own decisions? Or what?
 

Offline coberst

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You mean 'think' as in? Teaching us about life skills? Making our own decisions? Or what?

I am talking about Critical Thinking, which I consider to be art and science of good judgment

CT is an acronym for Critical Thinking.  Everybody considers themselves to be a critical thinker.  That is why we need to differentiate among different levels of critical thinking.

Most people fall in the category that I call Reagan thinkers—trust but verify.  Then there are those who have taken the basic college course taught by the philosophy dept that I call Logic 101.  This is a credit course that teaches the basic principles of reasoning.  Of course, a person need not take the college course and can learn the matter on their own effort, but I suspect few do that.

The third level I call CT (Critical Thinking).  CT includes the knowledge of Logic 101 and also the knowledge that focuses upon the intellectual character and attitude of critical thinking.  It includes knowledge regarding the ego and social centric forces that impede rational thinking.

Most decisions we have to make are judgment calls.  A judgment call is made when we must make a decision when there is no “true” or “false” answers.  When we make a judgment call our decision is bad, good, or better.

Many factors are involved: there are the available facts, assumptions, skills, knowledge, and especially personal experience and attitude.  I think that the two most important elements in the mix are personal experience and attitude.

When we study math we learn how to use various algorithms to facilitate our skill in dealing with quantities.  If we never studied math we could deal with quantity on a primary level but our quantifying ability would be minimal.  Likewise with making judgments; if we study the art and science of good judgment we can make better decisions and if we never study the art and science of judgment our decision ability will remain minimal.

I am convinced that a fundamental problem we have in this country (USA) is that our citizens have never learned the art and science of good judgment.  Before the recent introduction of CT into our schools and colleges our young people have been taught primarily what to think and not how to think.  All of us graduated with insufficient comprehension of the knowledge, skills, and attitude necessary for the formulation of good judgment.  The result of this inability to make good judgment is evident and is dangerous.

I am primarily interested in the judgment that adults exercise in regard to public issues.  Of course, any improvement in judgment generally will affect both personal and community matters.

To put the matter into a nut shell: 
1.   Normal men and women can significantly improve their ability to make judgments.
2.   CT is the domain of knowledge that delineates the knowledge, skills, and intellectual character demanded for good judgment.
3.   CT has been introduced into our schools and colleges slowly in the last two or three decades.
4.   Few of today’s adults were ever taught CT.
5.   I suspect that at least another two generations will pass before our society reaps significant rewards resulting from teaching CT to our children.
6.   Can our democracy survive that long?
7.   I think that every effort must be made to convince today’s adults that they need to study and learn CT on their own.  I am not suggesting that adults find a teacher but I am suggesting that adults become self-actualizing learners.
8.   I am convinced that learning the art and science of Critical Thinking is an important step toward becoming a better citizen in today’s democratic society.


Perhaps you are not familiar with CT.  I first encountered the concept about five years ago.  The following are a few Internet sites that will familiarize you with the matter.

http://www.freeinquiry.com/critical-notes.html

http://64.233.161.104/search?q=cache:mkodBBrpMg0J:www.criticalthinking.org/TGS_files/SAM-CT_competencies_2005.pdf+critical+thinking+multi-logical&hl=en&gl=us&ct=clnk&cd=11

http://www.chss.montclair.edu/inquiry/fall95/weinste.html

http://www.criticalthinking.org/resources/articles/glossary.shtml

http://www.doit.gmu.edu/inventio/past/display_past.asp?pID=spring03&sID=eslava


 

Offline Chemistry4me

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The fourth and fifth links didn't work when I clicked on them. Do you consider yourself as a CT, I presume that the whole point of you starting this topic was to a)persuade adults to become CTs b)CT should be taught in schools.
Am I right? Was there anything else you wanted to get across?
 

Offline coberst

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The fourth and fifth links didn't work when I clicked on them. Do you consider yourself as a CT, I presume that the whole point of you starting this topic was to a)persuade adults to become CTs b)CT should be taught in schools.
Am I right? Was there anything else you wanted to get across?

I am a retired engineer who managed to get an MA in philosophy in the middle of my engineering career.  Studying philosophy was a life changing experience.  One course I took was Logic 101, which I thought was something every high school graduate should learn, at least in a "lite" form.

Many years later I became aware of CT and studied it and decided that CT is philosophy lite and should be taught to every high school student.  Any normal person can and should be taught CT.


 

lyner

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Kids don't learn the basic life skills by being taught in an 'intellectual' way. They learn the first few years' worth without trying at all. It happens by repetition of simple tasks - crawling, playing with toys /  pebbles and by exposure to language.
Having studied my children and grandchildren I realise that most of their progress is hard to spot. They have a habit of presenting one with faites accomplis: suddenly they can do and say a whole swathe of stuff which, the day before, they didn't give you a clue that they could do. Mostly they don't show what they have learned until they feel confident enough to show it off.

Many times I have realised that my 'intellectual' approach to teaching them something has had very little effect - except, in the long run, when they have acquired the habit of 'formal learning' partly through my influence ( and that of others). By that, I mean that I've demonstrated the worth of a bit of self motivation and effort.

There are many educationists who have tried to 'improve' on the old methods of primary and secondary education. They have based their views on a limited number of students (take Piagiet, for instance, who's source of data for his cognitive levels was from a handfull of children from middle class families).

Many of them seem to have missed the point about kids' attitude to and methods of learning. For 17 years, now, I have been teaching Secondary Science and a fair bit of Maths. Repetition and rote learning are fantastic ways of getting basics learned. The early rules of arithmetic and algebra are based on very very hard concepts. If you tried to get Primary aged children to appreciate Number Theory before they are familiar with 6X7=42 they wouldn't get it at all. The only reason for learning Maths, for a young learner is to get the approval of the teacher and a gold star. There is no other possible reason they would want to.
Kids tend to like facts - not ideas. A blanket statement, I know, but I am referring to 'Piagietian' concrete level thinking (which is as far as the majority of humans operate). I would love to know how any young mind is supposed to work at a higher level if it hasn't already acquired a sound body of concrete factual knowledge and a set of mechanical skills to enable it to do some manipulation of ideas and achieve CT.

We can't argue against the idea of CT - it's got to be the way forward. But it can't be delivered at the expense of the boring stuff first. Kids who haven't learned their tables - or equivalent- won't be capable of any critical thinking because they won't have any manipulative tools.

You can't interpret a Piano Concerto until you've thoroughly learned your Scales.
 

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