Is green has lower cognitive load than belongs to a set

A tick is green has lower cognitive load than a tick belongs to a set.  Belongs to a set has higher cognitive load than belongs to a team or is on a team.

The tick zero is green has lower cognitive load than the tick is a member of the set G.

All ticks are green has lower cognitive load than all natural numbers are elements of the set G.

This is because these ancient formulations are fashioned to our minds, or our minds fashioned to them.  So these formulations are easier to understand and don’t require translation.   On the team, friendly, not friendly, these are words and concepts that have to be automatic and instantaneous.  So we should search for such words and phrases to lower the cognitive load.

When we start from math foundations, we know what the abstract message is. So we can find the concrete parallel version that does not sacrifice the abstract meaning.

This simple form is suitable for children at younger ages to understand.  It is easier for them to understand than some wrong formulation of the concept or a mixture of two concepts.

Friendly, non-friendly, on the team, safe, etc. are concepts children learn young.  So if we use such formulations we can convey the meaning of mathematical induction as a version of on the team or safe or trustworthy or true.  Because they can understand them, they can remember them.

In contrast, a confusion of many concepts can not be simplified down to the bare minimum.  If we mix length and fraction together as a big bundle concept, then we can’t pare it down, we are stuck with a bundle.

When we have an isolated concept, we can pare it down.  If we stick two different concepts together like length and fraction, then our ability to find an ancient formula in language that does both of them at the same time is small.  We have made the problem of finding a simple verbal formula too difficult to achieve.

Math foundations isolates each concept in its own separate form.  This is what abstraction does for us.  We can then work on that single idea and find the best way to express it for younger ages to grasp it immediately.  This may take time, experimentation and the contributions of many people.   But the task is made light by using math foundations to isolate each concept in its own formula, even if it is awkward at first.  Then we can keep trying alternative verbal formulas until we get one that is easy to grasp quickly, i.e. has low cognitive load.

Once we find one, others will come to us.  We will also see how to make other definitions or theorems or proofs easier to grasp, lightening their cognitive load and increasing the understanding and retention of younger students.  This can shorten the lesson and leave them time to play.  During that time, the unconscious mind can work the concept without being overloaded by the confusion of many concepts that should be distinct and taught in separate lessons.

There is a diagram floating around that suggests pushing different forms at them at the same time.   But one thing understood well at one time is an alternative goal to have.  That guides us to lighten the load by a simple formulation of a single concept that uses ancient word forms to convey the meaning instantly.  Abstraction is an essential part of finding that isolated single concept and expressing it correctly if awkwardly at first.  The benefit of proceeding this way is to find the shortest and easiest path through arithmetic and algebra.

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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