Peano Axioms 2015 Never more important

Learning the Peano Axioms has never been more important than in 2015 and the years to come.  The Peano Axioms mean that when we do elementary math at any time, we are thinking in terms of foundations of math and logic. They remind us of ordered pairs, sets, functions, recursion, definition by induction, proof by induction, and so on.  These are the basic concepts of math, logic and programming.  So we are always getting a reminder of these every time we use numbers when we learn the Peano Axioms.

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Learning different multiplication methods is a sound idea

Schools are sometimes ridiculed for teaching different ways of doing multiplication.  However, if this was so bad, then how can we teach applications of multiplication at all?  Isn’t another method to do multiplication an easy application of a first method?

If a student can’t apply one method of multiplication to another, then how can he apply it to something else?

What allows us to apply multiplication or even addition to some application?

What allows it is that the application satisfies the laws of addition or multiplication so that it behaves the same way as addition or multiplication does.

So to understand that we can apply multiplication to a problem, we have to be able to show that the problem satisfies the properties of multiplication.

For whole numbers, this means showing that

n*0 = 0

n*m’ = n*m + n

For addition, we have to show that

x+0 = x

x+y’ = (x+y)’

For successor, we have to show that x’ satisfies the 5 Peano Axioms.

If we don’t show or even understand these proofs or that they are required, then an application is really applying rote learning.

One reason it is hard to apply one multiplication method to another or translate them, is that the definition of multiplication of natural numbers in terms of repeated addition is not formalized.

n*0 = 0

n*m’ = n+n*m

These two equations define multiplication in terms of addition.  To apply this to area, we have to show that areas have this property, for whole lengths of the sides.  This is a matter of intuitive geometric understanding at the earlier grades.

If we structure math ed using the Peano Axioms, we can understand why people can’t easily go from one method of multiplication to the next. The reason is they didn’t learn the definition of multiplication, or its properties.  That is, they didn’t learn its structure. So they can’t apply the structure even to another method of multiplication, and recognize it has the same structure.

When we use the two equations to define multiplication, it is easy to verify or at least make plausible that another method of multiplication or an application such as area satisfy the same two rules.

This teaches abstraction and concepts.  If this is not done for natural numbers, it is difficulty for the student to form any proper concept of fractions.  Fractions are a mathematical structure with data and rules.  The data are natural numbers and the rules are methods of manipulating natural numbers.

So if the understanding of structure of natural numbers is completely absent, fractions are learned by rote if at all.

We can tell if structure of natural numbers is absent by testing if the student can apply or relate one method of multiplication to another.  If they can’t, they won’t understand the math structure of fractions either.

Rote learning experiences diminishing returns in math. As one ignores structure at one level, naturals, it becomes harder to learn the next level, fractions. After fractions, there are infinite decimals.  There is also algebra.  If showing two methods of multiplication have the same properties is not possible to the student, they will fail at algebra other than as rote learning. This shows up when they can’t do algebra word problems. They didn’t learn the structure of numbers let alone algebra, so they can’t translate a word problem to symbols.

Teaching math structure has to start with counting and relating counting to the 5 rules of counting by one, ie the Peano Axiom.

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Structure and abstraction fight the math is random drills feeling

Many students and teachers complain that math is a sequence of random drills with no connection or structure.  What is the plan? How does it fit together?

The answers to these questions comes from structure.  Structure is how parts of math fit together.

Abstraction is the architecture of the structure of math.  The nodes and a pair of nodes as a link are abstractions.

Natural numbers are a topic in math.  The structure of the natural numbers is given by the Peano Axioms.

Induction is itself an abstraction.  Hermann Grassmann recognized that induction is the key abstraction to define addition and multiplication of rational numbers and prove the properties of these functions.

Dedekind then rebuilt the structure Grassmann started and filled in gaps.  Peano then picked out some results in the middle of Dedekind’s book and made those a starting point as a set of axioms that would capture the essence of natural numbers.

The structure of the natural numbers as a sequence shows up over and over again.  There are other ways to constructively set up such a sequence such as the von Neumann structure of natural numbers.

These ideas lead to a better understanding of fractions as ordered pairs of natural numbers.  Understanding the abstractions in natural numbers is fundamental to understand a fraction as an ordered pair of natural numbers.

This takes away the randomness and substitutes meaning instead. The meaning is the abstraction.  The abstraction is key to the structure.

We have removed the applications.  The applications are not the meaning of natural number.  They are possible as applications because the abstractions of natural number apply to that situation.

We understand an application by recognizing that it is suitable to the abstractions in the natural numbers. That includes induction.

The motivation is the insight of how it fits together.  The motivation is going from seeing it as random drills to seeing it as a structure that has parts that fit together.

The applications can illustrate the abstractions and structure of natural numbers.  This way they teach the natural number abstractions and structure.

If we do applications willy-nilly, it is likely that the wrong concepts are taught.  Ones that are pre-Grassmann.   Thus emphasizing applications can lead to wrong teaching that creates problems as abstractions like fractions as ordered pairs are taught later.

Good teaching materials based on the Peano Axioms are the only way to prevent teaching the wrong concepts of natural numbers.

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Same 5 rules over and over again in Peano arithmetic

The same 5 rules of the Peano Axioms repeat over and over again as we develop elementary arithmetic of whole numbers. The 5 rules apply not just to the natural numbers, but can be restated to apply to the head sets as well.

So the sequence 0,1,2,3,… has the 5 rules.

  1. Start node.
  2. Out of each base node a single arrow consisting of a base node and tip node.
  3. The start node is not a tip of any arrow.
  4. No node is the tip of two arrows.
  5. If a path contains the start node and the tips it points to, then it contains all the nodes.

The sequence of head sets also follows these 5 rules

0

0,1

0,1,2

ie the head sets of natural numbers.

So do the sets

1

1,2

1,2,3,

ie the head sets of the counting numbers.

The 5 rules apply to the unit fractions, i.e. to the sequence

1/1,

1/2,

1/3,

They apply to any sequence 0/n, 1/n, 2/n

They apply to any sequence   m/1, m/2, m/3,…

..

Showing this teaches meaning to students.  This has not been pointed out by math profs or math literature or math education.  This is because they don’t do a good job explaining Peano Axioms.  This is because they don’t fully understand it.

Math textbooks put the easy stuff in the problems.  They put the think stuff in the problems. So the textbook writers don’t write the explanations of this.   You can’t improve what you don’t write.  Which is the result of the practice of putting a lot of steps and easy stuff into math problems instead of the text. The textbook writers never write the explanations and never improve them.  So they don’t get developed.  This is why the above points have never been articulated before.

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Head sets v tail sets the unknown controversy

Head sets of natural numbers starting from 0 are sets like

0

0,1

0,1,2

etc.

Tail sets are all the numbers starting from and after that number.

0 to infinity, including 0

1 to infinity including 1 but not 0

2 to infinity including 2 but not 0, 1

etc.

Dedekind in his 1888 book teaches tail sets first because the logic is easier. However, for students head sets are more concrete. Modern axiomatic set theory books teach head sets first, calling them initial segments. However, those books typically don’t do as good a job isolating head sets as a concept and teaching their properties.

If math profs understood their own literature from Dedekind onwards, they would consider whether it was better to teach head sets first or tail sets first. However, they are unaware that Dedekind teaches tail sets first and modern treatments like von Neumann construction teach head sets first.

This tells us that even math profs don’t understand this very well and don’t understand the historical literature. This then carries over to math profs not knowing how to teach math majors or even math grad students this material. If math profs don’t even know that Dedekind taught tail sets first and axiomatic set theory books teach head sets first, it shows they themselves don’t know Peano axioms very well.

This carries over to the whole teaching of pre-service math teachers in college. This is done by math profs who don’t know math foundations very well or how to teach math foundations even to math majors/math grad students. So they don’t know how to teach the substance to pre-service math teachers. They also don’t know how to teach the pre-service teachers how to teach it to the students.

The same applies to the textbook makers or to worksheet makers or to Khan Academy or other video makers. At least until Khan starts copying from my webpage. Which seems the only way that anyone will ever use it.

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Logarithms and calculators

Base 10 logs

log 2 = .301

log 3 = .477

log 5 = .699

log 7 = .845

How to remember these

log 2 = 2 * .15

log 3 = 3 * .16

log 5 = 5 * .14

log 7 = 7 * .12

So it starts at .15, goes up to .16, down to .15 at log 4 and .14 at log 5 and .13 at log 6 and .12 at log 7.

These are approximations. Using a calculator like Google can check how accurate they are.

Just remember .16 for 3. Then count down by .01 in either direction per whole number. This approximately works up to 10. You can straight line interpolate between whole numbers using the same slope. You can then work on improvements to this rule.

http://www.karlscalculus.org/l6_3-2.html

http://oldmathdognewtricks.blogspot.com/2012/05/my-version-of-log-wars.html

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Why good theory must inform pedagogy

If we understand the theory first and then rework it into slogans for children to learn, then we give them a path to understanding. If we invent slogans first and they don’t line up exactly with theory, then there are logical gaps and misdirections. These confuse the student and hinder an understanding of logical structure.

When logical structure is not developed this way, even without their knowing it, they are not ready for fractions or algebra.

It is very hard to work out rules of counting and understanding the counting of small sets. If we count a herd of cows in a different order do we get the same number? This is a hard thing to work out correctly why we do. One tends to assume the result in naive explanations.

To work out why we get the same number if we count a set in a different order requires theoretical understanding. This has to be written down exactly correct, or as best as possible.

Once this is done, we can then work the correct logic into slogans, chants, puzzles, games, etc. that teach this logical structure.

If we just invent the slogans and chants without understanding the actual logical structure, then we will create gaps and confusions. These accumulate and show up in walls that can’t be scaled at fractions and algebra.

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