## 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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## Students failing algebra rarely recover

“Students failing algebra rarely recover”

Jill Tucker
Published 10:22 pm, Friday, November 30, 2012

If student’s don’t get algebra the first time, repeating it doesn’t work. The article specifically says teaching it again the same way doesn’t work.

The researchers challenged school officials to rethink when and how students are assigned to certain math classes and come up with new ways to teach the topics, especially to those who didn’t get them the first time around.

Take students out of the standard algebra curriculum and put into alternative logical foundations. They don’t get what algebra is about in the current approach. Current algebra teaching is procedural and it doesn’t form memories they can retain and use.

Current teaching in math is mostly memorization.
There are large gaps in the concepts. These gaps were filled in 19th century New Math of Boole, Grassmann, Dedekind, Peano.

However, this good new math was dropped in favor of an incomplete version that is mostly useless. 19th century new math was invented for arithmetic’s logic. this part is ignored in teaching new math and common core.

Common core takes Euler 1765 algebra and sprinkles a few words from set theory that have nothing to do and so are just ignored. Set, relation and function are not used in the way that Dedekind does. Is addition a function? Yes. What function is it? Can you write it in function notation?

x+y

f(x,0) = x
f(x,y’) = f(x,y)’

From this we can prove

f(0,y) = y
f(x’,y) = f(x,y)’

Are you completely lost? Think about it. Addition as a function is something you never heard of and never saw written before. However, this was developed in the 19th century to fill the logical gap in what addition is. yet modern teachers and students have never heard of it and are stupefied when they see addition written as a function.

Do you have any idea how to prove the second set of equations from the first set? This is fundamental to what addition is and why it works.

These two equations define addition

f(x,0) = x
f(x,y’) = f(x,y)’

Where ‘ means the next node in a chain from a starting node, 0.
This is counting along the chain. The f function here is the counting on function for the chain. this is addition. Addition is the counting on function for a chain of nodes on a line, i.e. satisfies the 5 linear chain axioms, i.e. the Peano Axioms.

Addition and multiplication of whole numbers are functions for counting along a chain of nodes. They depend on the chain of nodes being linear from a starting point. If we had a different geometry of the chain of nodes, we would get different functions for addition and multiplication by imposing these equations.

Let’s call f(x,y) as S(x,y) now. The S(x,y) function lets you move along the chain of nodes faster or in jumps like a Queen in chess.

The function S(x,y) has properties that make it what we expect of addition. Take S(x,y’) if you increment one of the inputs, the output increments by one. In fact,

S(x,y+z) = S(x,y)+z

If we increment an input by z, the output increments by z.

S(x,S(y,z)) = S(S(x,y),z)

Which functions solve this? Addition does. The addition function, S(x,y) satisfies

S(x,0) = x for all x
S(x,1) = x’ = S(x) for all x

Addition is a way to move along a chain of nodes by more than one node at a time. Addition is equivalent to going one node at a time.
Addition is repeated successor.

Learning to think in these terms is what algebra means and how it ties into arithmetic, a subject you may think you understand, but don’t. Those gaps were filled by Dedekind and others in the 19th century. My materials at this webpage and ebooks fill those gaps and are easier to understand than Dedekind or axiomatic set theory books.

This essay may seem cryptic to most. My materials in my ebooks help make it less cryptic. This is really what is going on with addition of whole numbers. This is what Grassmann, Dedekind and Peano showed when they filled in the logical gaps in Euler’s 1765 algebra textbook.

Yet current teaching in algebra is the same as Euler’s 1765 textbook. All the logical gaps found in the 19th century and filled in are ignored by modern teaching. They just sprinkle in the word set a few times and think this is the same as teaching the careful concepts in the Dedekind 1888 book. It isn’t.

Euler’s 1765 Algebra is a procedural text, i.e. a cookbook. Current math texts for K-12 are the same style of procedural cookbook. They don’t teach concepts and they don’t explain. They pretend to, but don’t. We know that from the Dedekind book and from axiomatic set theory books like Patrick Suppes.

We know some students can’t learn the recipes for algebra problems in current classes. Those classes try to go up a steep learning curve of procedural problems going to polynomial functions and rational functions including factoring.

This is not as important as understanding counting by one and how counting by one leads to the addition function for whole numbers. This is the logic of recursion and induction. This is mathematical thinking. This is what the struggling students need. This gives them insights that give them hooks to remember.

The best students can memorize procedural activities and repeat them on exams. They still don’t understand the underlying logic, but they can pass the procedural exams given. The books and materials are little more than rote memorization. But good students can slog through this amazing barrier of rote memorization material.

The struggling students can’t memorize an algebra book of procedures. So why not try teaching them the concepts of arithmetic using algebra and set theory? Then they might catch on to the meaning of what they are trying to learn.

Procedural math for factoring polynomials is not as important as conceptual math of how addition of whole numbers is defined as a function using recursion. That is indicated above in the equations for f which is renamed S. My materials here and in ebooks do this at a slow pace with many problems. They are the slowest paced set of material to teach this subject. They also have the most examples and worked out problems for Peano Arithmetic.

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## If your students can’t do the associative law proof they have gaps

If your students can’t do the proof of the associative law of addition of whole numbers, then they have gaps in their knowledge.  Moreover, they don’t know they have them.  They will think they know common core math principles but will have hidden gaps in their understanding.  Those will catch up with them in fractions, decimals, sets, and algebra.

When you can teach them the associative law of addition, you will have mastery of the math behind common core math.  So will they.  This will make fractions a snap.  It will also get them over using a letter to stand for a variable.  This is supposed to happen in first grade in common core.  So you better know it well to explain it to first graders.

The younger the student you teach something to the better you need to know it. This way you can come up with the right answers to their questions. They have the same questions you do or math grad students do, and you need to learn the right answers. Those are in the Peano Axioms.

Hermann Grassmann, Dedekind and Peano proved that the only way to fill the logical gaps in Euler’s 1765 Algebra was with the Peano Axioms.  The same applies to common core math today.

Handwaving is what Euler did at critical points.  Handwaving by you the same way may end up with your students being befuddled for good reasons. The wrong answers don’t teach. That is why Grassmann was not satisfied with Euler’s 1765 Algebra book and found the right answers.  Dedekind and Peano finished the job of putting the right answers in a proper structure.

Now you just have to learn it and teach it. There is no other way to teach the concepts of elementary math because these are the concepts of elementary math.  Without Peano Axioms you are doing rote learning at critical points without realizing it.  This creates conceptual gaps that can open up like sink holes in fractions, sets and using a letter to stand for the unknown.

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## High stakes testing means self esteem bias will get you fired

Whether you have self esteem bias over your knowledge of common core math or your students do, it can be dangerous to your job.  This is why you should learn Peano Axioms and basic proofs of the associative law of addition of whole numbers.

A job is a terrible thing to waste, especially when it is yours.  Don’t waste it, use it to teach your students common core by learning Peano Axioms yourself first.

Hermann Grassmann in 1861 showed that the meaning behind school math is proofs by induction and definition by recursion.  This is what is behind common core math. So learn it.  Learn the proof of the associative law of addition of whole numbers. Both you and your students can learn that.  Learn it by heart.  It is doable.

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## Recursion is the key to common core math and math finance

Both common core math and mathematical finance depend critically on recursion.  This is one reason that a math finance guy is a good guide to learn common core math.

Green’s functions in finance are a method for doing recursive calculations.

I have had over twenty years experience in teaching recursion and basic finance on Wall Street, to actuaries, computer programmers, and so forth.

They and you would do well to start by learning the proof of the associative law of addition of whole numbers using recursion.  The same techniques can then be used in probability and in pricing financial instruments.

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## Math teachers beware of self esteem bias in common core

Math teachers should realize they think they know more than they do.  We all do.  Students do, teachers do, profs and textbook writers.  This is why we need homework and tests to remind us we don’t know as much as we think.

My books on Peano Axioms have more simple examples on the Peano Axioms then exist in other materials combined.  (A little self esteem bias there.)

Unfortunately, it may be true.  This is because textbook writers and university math webpages underestimate the difficulty and hand holding needed to learn the Peano Axioms and apply them to addition of whole numbers.

That is why my books contain so many examples, solved problems and other problems on the very start of Peano Axioms.  These will teach you the 5 rules for counting by one, ie the Peano Axioms and how to apply them.

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## Teachers should learn Peano Axioms to prepare for common core math

Teachers should learn the Peano Axioms to prepare for the common core math.  This is almost a single stop answer on how to prepare for common core in math.

Learning the Peano Axioms and the development of arithmetic using them will teach

1. 5 Rules for counting by one.
2. How counting by one becomes addition by recursion.
3. How repeated addition becomes multiplication by recursion.
4. How the initial segments or head sets of the natural numbers are formed by set recursion.
5. The basic parts of set theory needed for counting, addition and multiplication.
6. How to think mathematically.
7. How the rules of algebra are the same as the rules for numbers.
8. How the logical gaps in Euler’s 1765 Algebra book are filled by 19th century math foundations.
9. Real explanations of math as opposed to pseudo explanations.
10. The origin of the associative, commutative and distributive laws for numbers and for algebra.

These topics are covered in this blog and in my books.

See links above for more information.

Aim to learn by heart how to prove the associative law of addition for whole numbers.  This is within the grasp of teachers and is a significant milestone that Grassmann made in 1861.  Grassmann was himself a teacher and he was the one who first proved the associative and commutative laws of addition of whole numbers.

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