Should formal math be taught to young children?

[Bartholomew]

Should formal math rather than practical math be taught from an early age to those children intelligent enough to learn? I am just getting started with formal math and logic and I wish I had been learning this stuff since kindergarten. It's all very simple, any bright preschooler could learn to manipulate symbols--I think it's easier than intuitive understanding of concepts. Maybe they wouldn't learn their times tables but they have calculators, so so what?

 

[loseyourname]

It is odd, isn't it? With all of the liberal arts and humanities requirements, we are told we have to take them, even though they have little practical value to anyone, because they make for a better-rounded education. But when it comes to math and science, they only teach us what is practical, without any of the well-roundedness.

 

[Bartholomew]

I'm not sure the way math is taught now at lower levels _is_ so practical. Formalism is very easy compared to intuitive understanding without formalism. Formalism has enabled great advances in math; it is practical. I believe that if someone learned formal mathematical systems from age 5 or so, he'd be much more adept at mathematics as an adult than someone who learned the conventional concept-based curriculum.

 

[loseyourname]

I'm not sure the way math is taught now at lower levels _is_ so practical. Formalism is very easy compared to intuitive understanding without formalism. Formalism has enabled great advances in math; it is practical. I believe that if someone learned formal mathematical systems from age 5 or so, he'd be much more adept at mathematics as an adult than someone who learned the conventional concept-based curriculum.

Knowing math isn't practical unless you're a mathematician. School-taught math is practical in the sense that they only teach you what you need to know to apply to other fields (algebra for grocery shopping, trig for basic physics, calculus for related rates). Little to no pure math is taught simply because there really isn't any reason to know it. It will become useful for those who go on to study math, and likely for physicists and engineers as well, but not for anyone else.

 

[Bartholomew]

I am saying that it is easier to learn formal logical systems than it is to learn abstract concepts. I think it would be _easier_ to learn arithmetic/algebra/trigonometry/calculus symbolically and formally.

 

[Crosson]

Bartholomew, if I understand your idea correctly it goes counter to what I think mathematics education is about.

Many students suffer in math and physics because they want to rely on formalism (easy) rather than understanding (hard)! Understanding is what it is all about.

 

[loseyourname]

I am saying that it is easier to learn formal logical systems than it is to learn abstract concepts. I think it would be _easier_ to learn arithmetic/algebra/trigonometry/calculus symbolically and formally.

Perhaps. Many people say that real analysis should be taught prior to calculus, and maybe number theory and even set theory before algebra, but doing so would require that extra courses be taken, and it likely isn't worth it for the common student, any more than it would be worth it for you to take literary theory before you took an English class, even if it would make the material easier.

 

[Philocrat]

Should formal math rather than practical math be taught from an early age to those children intelligent enough to learn? I am just getting started with formal math and logic and I wish I had been learning this stuff since kindergarten. It's all very simple, any bright preschooler could learn to manipulate symbols--I think it's easier than intuitive understanding of concepts. Maybe they wouldn't learn their times tables but they have calculators, so so what?

Formalism is the study of 'Natural Topologies', or Natural Forms and their underlying Normological Structures that form the main achitecture of propositions and thoughts. As you may well know, propositions and thoughts are snapshots of reality or natural world. Wittgenstein in Tractatus thought of them as pictures. These pictures capture not only overall structure of the world but also they capture the quantitative and logical structures as well. The Logical and mathematical forms of propositions and thoughts are revealed when you crack them open with a metaphysical hammer up to their innermost atomic levels.

It is a fundamental mistake to think that when you are learning a given mathematical and logical forms that you are automatically neglecting their most valuable and relevant empirical or practical details. Our Natural Langauge (NL) which is currently grossly misunderstood quantifies with remarkable precision all aspects of the world that it purportedly describes. This includes the logical and mathematical forms (or structures if you like), the empirical and practical details that may be conveyed by a combination of variables and constants. As Les has clarified, you need different types of maths for different purposes, and perhaps you learn what you need that is relevant to your chosen profession.

The world would be a better place, if we started teaching our youngsters logic (the tool for proper conduct of objective reasoning) from very early age. The BIGGEST disease that faces the world (infact since that advent of man) is that very poor understanding of logical forms and their derived logically precice symbols, as they are naturally embedded in our natural languages, has affected us in so many ways and is now costing humanity very dearly. When people cannot efficiently manipulate these embedded logical forms and symbols in conversations and routine communications they suddenly flare up in anger and frustration.

When they cannot win the argument in simple negotiations, in most cases, they turn depressively inward and the next thing they consider easier and quicker to do is to settle the negotiation with swords and guns instead of with cogent and carefully but fully deduced logically precise argument. You encounter this problem time and time again on disputes over land and property ownership in general. If you ask people 'why do you fight over a piece of land?', they will come up with all sorts of fictional statements that they want you to accept as carefully well argued reasons for laying claim to that piece of land. Countries who want to send good negotiators to international conferences and gatherings of communal values should start teaching their children logic from very early ages. Don't just assume that Logic is something you can easily learn and understand only when you grow older. Start teaching children how to debate from early age so that when they get to an adult age they would already be fully empowered with the ability to properly make their cases in public gatherings, local and international meetings, without feeling frustrated and overwhelmed with crushing sense of defeatism!

Logical and Mathematical forms are not separate from the formal structure of the world, nor neither are they in any way far removed from its underlying empirical and practical details. Langauge with which we construct propositions and thoughts automatically capture in greater detail the quantiative and logical structures of the world, the formal aspects being the foundation upon which they are built.

 

[Crosson]

It is a fundamental mistake to think that when you are learning a given mathematical and logical forms that you are automatically neglecting their most valuable and relevant empirical or practical details.

Pure symbolic manipulation, as bartholomew suggest, would automatically neglect the deepest points the material has to offer. Logic without intuition is not a very fun place, but our current world is primarily intuitive (as opposed to being logical) and it is decent; so remember where the extremes lie.

I don't think it is appropriate to preach glories of mathematics to people, or attempt to instill an appreciation for mathematics in our general students. Like many achievements of mankind that have been recorded into text, mathematics is easy to find if you have an inkling to look. I advocate freedom in the pursuit of knowledge.

 

[Bartholomew]

Well, you can't do much logic without intuition, unless you are a computer. Logic just provides a framework that your intuition can act within.