Peano Axioms help deal with classroom boredom problem

The following blog raises this problem

It links to this one

The Peano Axioms can help deal with boredom and lack of interest because they provide logical structure that the student can understand.  They create a story.  The natural numbers have a beginning with the Peano Axioms.  We then work out order comparisons for naturals that are not consecutive.  We have the ability to form functions recursively.  Using this we define addition.  We prove its properties such as associativity.  We do the same with multiplication.  We then construct place value notation and the place value algorithms.

We then move onto fractions as functions on natural numbers with limited domains that are a multiple of the reduced denominator.  We then define rules for adding fractions as functions, multiplying them, subtracting and dividing.  This lets us then abstract fractions as ordered pairs of two elements with the same rules.  These we can prove obey the same types of laws as natural numbers and a solvability for division.

As we do this, we introduce and use letters to stand for numbers.  We learn how to state a mathematical proposition using letters and to prove it.   We understand arithmetic from a fundamental point of view.  This is inherently exciting.

Even mathematicians find the Peano Axioms and proofs on them interesting.   Whereas another random algebra problem is not.  These are boring to mathematicians and to students.  So get rid of them to the extent possible.

Peano Axioms is a story from axioms to definitions to theorems and proofs.  It goes all the way to the standard algorithms and to fractions.  It then sets up irrational numbers.  As a story, it is inherently more interesting than random algebra problems one after another.

Theorems are stories.  They have a beginning, the theorem, a middle setting up the proof, and a conclusion when the proof is done.  Stories are more interesting than random algebra problems or random arithmetic problems.

Going through the definitions and theorems develops the same skills as doing algebra problems.  It does it as part of telling the story of the logical structure of arithmetic.  This keeps the attention of mathematicians and will also keep the attention of students.

The successor function provides the key that holds together the natural numbers. If we never mention it, then we leave out what holds the story of the natural number together.  So the students become bored.

The prime notation for successor is itself a type of manipulative.  It is a very handy notation that can be played with.

Right Addition Identities

x+0 = x

x+y’ = (x+y)’

This is the definition of addition.  From these properties we need to prove

Left Addition Identities

0+y = y

x’+y = (x+y)’

This is an exciting game to learn to do.  It is of obvious importance, as opposed to most algebra problems.  It is a riddle that seems of ancient origin and of profound importance.  That makes it fun.

When we learn to go from the right addition identities, the first pair to the left addition identities, the second pair, we have learned something important. We understand something meaningful.  This contrasts to most algebra problems.

The problem has a challenging and meaningful nature to it suitable for a project.

After writing this, I randomly searched on an old entry from dy/dan and found this. “Vic Mackey: Teacher of the Year”

This entry is about making a lesson or class a story to hold interest.  This is what Peano Axioms does built in.  It makes math a story.  It has a start, a direction, encounters problems and overcomes them.  These problems are order comparisons for non successive numbers, defining recursive functions by appending an ordered pair at the end of what is defined, defining addition, proving its properties, going onto multiplication.

The Dedekind 1888 book is the story of natural numbers in this order.  This includes the recursion theorem as just appending ordered pairs.  It is all laid out in order.   It is the story of the natural numbers and it is fundamental to making the natural numbers fascinating.  The natural numbers seen in this way are deeply meaningful and connected to our basic concepts of existence and time.


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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