## Ultimately math foundations has lower cognitive load

Math foundations may come out awkward and difficult, but it can be worked with in its verbal formulations to lower the cognitive load.   Math foundations tells you how to remove excess until what is left is the concept.  Math foundations has already found that line.  The job of math education is to rewrite it with lower and lower cognitive load. Because we know the line of what has to be included from math foundations, we can keep fiddling with how we say it until the load is lowest.

If a set contains 0 and contains the successor of each element, then it contains all natural numbers.

A set is closed under successor if it contains the successor of each element.

If a set contains 0 and closed under successor, then it contains all natural numbers.

A set is successor closed  if it contains the successor of each element.

If a set contains 0 and is successor closed, then it contains all natural numbers.

If a tick on the number line is in a set, then the tick is green.

If 0 is green and each tick immediately after a green tick is green, then all the ticks on the number line are green.

If a tick is green it is a member of the team.

So if zero is on the team and each tick on the team is immediately followed by a tick on the team, then all ticks are on the team.

So if zero is green and each green tick  is immediately followed by a green tick  then all ticks are green.

Or we might say it:

So if zero is green and each green tick  is immediately followed by a green tick  then all ticks on the number line are green.

Or we might say it:

So if zero is green and each green tick  is immediately followed by a green tick  then all ticks on the number line are on the team.

Or we might say it:

So if zero is green and each green tick  is immediately followed by a green tick  then all ticks on the number line are in the set.

We know what it is we need to teach from math foundations. So we play with our lesson plan until the lesson plan teaches that.   Math foundations gives us permission and guidance on what to cut from the lesson plan but still convey the message.

Shorter lessons that convey the message are often easier to learn and retain.  They are also easier to review.

One of the great benefits of math foundations is to create short texts that we can go back to.  Of course, if we are opposed to using a text, we lose that advantage.