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Topic: How would YOU simplify arithmetic?
Replies: 41   Last Post: Jul 11, 2012 12:56 AM

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 Jonathan J. Crabtree Posts: 355 From: Melbourne Australia Registered: 12/19/10
Re: How would YOU simplify arithmetic?
Posted: Jul 10, 2012 2:11 AM

Thank you for the subtle prod Joe :^)

As mentioned before, for the 'wider spread' consensus amongst the consumers of arithmetic education (children under the age of ten) simpler arithmetic means easier to understand, do and get correct.

IMHO, the four relevant pillars of children's (primarily concrete) arithmetic are:

1 - Logic
2 - Language
3 - Patterns
4 - Nature

LOGIC
The axioms of integer arithmetic need to be logical and self evident to children. Commutativity and associativity of natural addition and multiplication are fine. Yet mathematics could/should be directionless as well as dimensionless and so subtraction may be better viewed as an additive inverse that preserves the properties of commutativity and associativity.

LANGUAGE
The language of arithmetic needs to mirror 'concrete' models that can be performed by children. Similarly, the grammar of word sentences, need to be correctly translated into mathematical expressions and equations.

A mathematical approach can be applied to language to minimise the mental overhead required to master base ten.

Currently the one to one mapping from language to symbols is deficient.

An expression may require either two nouns and a verb for a geometric plane product, yet when units are introduced, an equation needs noun, verb and adverb.

The use of the '-' and '+' signs as both verbs (take away and add) as well as adjectives (negative and positive) result in confusion. As mentioned, I use a model of bumps and holes for integers. While -7 - -4 confuses most adults in society, 7 holes take away 4 holes leaves 3 holes confuses nobody. I will expand on this later.

PATTERNS
Patterns identified such as base ten and the use of zero as a number need to be reinforced to leverage the benefits of arithmetic as a simplifier.

The intra-numeric base ten logic of Esperanto requires only 11 words to count to 100. By comparison, Hindi requires 100. Mapping the logic contained within Esperanto to English is the equivalent to the Asian counting model of ten one ten two etc.

'The exception that proves the rule' should only be a cop-out for humanities teachers. For mathematicians, the exception breaks the rule and hence the rule no longer exists.

Until a teacher can explain why -3 x -2 = 6 arithmetic may as well remain voodoo magic to many children. Quoting Ogden Nash is not acceptable!

NATURE
Natural outcomes consistent with observed laws of nature will empower children to accept arithmetic as being 'real world' before it morphs into the abstract and symbolic realms. Things should not break just because I change my point of view, as happens when right-left becomes part of arithmetic pedagogy.

HYPOTHESIS
The body of knowledge that passes for arithmetic has been developed in a haphazard manner over thousands of years.

The reality is that the brain of a child is much more than an amazing supercomputer. The fact that billions of children's brains have failed to master arithmetic reflects the fact mathematicians have failed children.

Children failing to learn = mathematicians failing to teach OR arithmetic not being as simple as possible and no simpler (ASAPANS).

Either the math profession has failed, or the model of arithmetic available to its intended audience (children) has not yet been completed.

Mathematicians fail children, yet who deserves the F?

I have ideas to share and genuinely hope people reading this thread will not be too cynical or closed minded to help make arithmetic simpler so more children can get correct answers to numerical calculations at both integer and rational levels.

I do not have a sophisticated mathematical vocabulary and while my logic is solid, I hope by sharing my ideas, collectively we can make some progress.

I am here to learn and share, not teach.

So to those interested, I will kick start this via another post.

Thank you,

Jonathan Crabtree