Origins of Counting or Tallying

A Possible Start

Students need to know how to count. A knowledge of how counting might of began with tally marks on tally sticks (or walls) may provide a context.

Tally marks on a stick may have given the earliest way for an individual to keep track of how many objects possessed or owed.

One to One Correspondence between tallys and a set of objects

As long as there is a one-to-one pairing, matching or correspondence between the tally marks and the objects being tracked, the owner assumes none have been lost or gained. The tally marks altogether describe and visually count or keep track of how many objects there are.

For example, a shepherd may put a tally mark on the tally stick above for each sheep that enters a pen, and later on verify as the sheep leave the pen that there is still one mark per sheep. That keeps track of the sheep and ensures none have been lost nor gain. Births and predators are assumed to account for all changes in the correspondence. If the tally marks are all alike as they are made, the order in which sheep leave the pen may be different from then one the entered. All that is important in concluding that no sheep has been gains or lost is that each sheep on exit be in one to one correspondence with a tally mark and that all tally marks are used.

In pure mathematics, two sets are said to be have the same cardinality (count) or to be equipollent when and only when there is a bijection or one-toone-correspondence between them. A bijection in brief is a one to one pairing between the elements of one set and the elements of the other, so all elements in each set belong along to one and only one ordered pair. Tracking sheep or marbles with tally marks on a stick (or paper) provides a bijection between the set of tally marks and the set of sheep.

The shepherd in keep track of his sheep may have tally marks on several sticks. In that case, the sheep may be allowed to exit the pen via several exits. Again as long this exiting gives a one to one pairing between the sheep and tally marks, the shepherd and we assume none have been lost. The sheep could be tallied in the first instant by entering the pen via several entrances or in several groups. It is possible to allow the sheep to enter grouped in one way and to leave grouped in another way. And if we tallied the entering and leaving each day using different tally ticks, and different groupings of the sheep and tally sticks, the different tallies and the sheep should be in one to one correspondence.

Working Definition of Whole Numbers: A set of tally marks on a stick gives a whole numbers. So whenever we speak of a whole number, we think of a set of distinct tally marks on a real or imagined stick.

Assumption

Tallying or Enumeration Assumption: If a set is tallied (counted) in two different ways, with or without the use of grouping, the set of tally marks for one way will lie in a bijection (one to one correspondence) with the marks in the other way.

In other words, we assume any two ways to count the elements of a set will result in the same number. Later on, we will see that the equality of two ways to count or measure what is a set or region leads to properties of arithmetic.

First Tallying or Enumeration Assumption: For a person who can count, the number of tally marks will be the same regardless of the order in which the elements of a set are tallied or counted.

Remark for Students of Pure Mathematics: The following gives a set theoretic justification of the unique tallying or enumeration principle

Two sets are equipotent when and only when there is a one-to-one map from one onto the the other.   The latter notion works for both finite and infinite sets as well.

Theorem: If there is a one-to-one map f of a set U into a proper subset W of itself then the set is not finite:

Proof   If U has N elements and we can find a one-to-one labeling of those N elements u of U given by a map c(j): [1,N] -> U. Now d(j) = f(c(j)): [1,N] -> W, a proper subset of U. Therefore, there is an element s in U not W. Pick such an s and Put d(N+1) = s. Then the extended map d: [1,N+1] -> U has a range with N+1 elements in U. The foregoing shows that if we count N distinct elements in U with the aid of the mapping c then with the aid of the mapping f, we can also count N+1 elements as well using another function d. Therefore for all whole numbers N, there is no bijective map of a range of whole numbers 1 to N onto U. That is what we mean by saying the set U is finite.

Contrapostive form of theorem: If U is a finite set (that is, is in bijective correspondence with a set of whole numbers 1 to N) then then any one-to-one map of U into itself must be surjective.   - if it was not then U would be not finite, that is infinite.

Unique Tallying Theorem: If P is a finite set with two maps c: [1,N] --> U and d:[1,M] --> U are bijections of two possibly different intervals [1,N] and [1,M] then N = M.

Proof: If N < M then for each j in the interval [1,M], there is a unique u = d(j) in P and hence a unique k = c^-1(u) = c^-1( d(j) ) = f(j) in [1, N] and hence in [1,M] Therefore f:[1,M] --> [1,M] is an injection. Yet [1,M] is finite. So the injection must be surjective and hence N = M.   The alternative possibility M < N is treated similarly. Q.E.D.

Note each of the maps in the Unique Tallying theorem represents a way to count the number of elements in the set P

Collecting the tally marks into groups provided a counting or numeration. Today, we go further. In decimal notation we use digits 1 to 9 to as marks for single object or a group of them, and we use place value to keep track of how ones, tens, hundreds, thousands etc there are in a count. So we track groups of power of ten instead of individual elements of a set or sheep in a pen. None the less, tallying goes on. And when we are counting elements of a set, we assume the final count or the decimal for it will be independent of how and in which order the set elements are tallied, counted or numbered.

Decimals

Decimal notation provides a compact form of tally marks for tallying or counting or describing how many objects there are in a set. The decimal view of numbers and number theory may be read parallel to the general number theory pages.

A. More Arithmetic a must for algebra etc

D. Logic In Mathematics

G. Algebra with Take Home Value

I. Vectors & Functions

Decimal Lesson - Reference
Counting & Addition   (8 lessons)
Comparison to Subtraction (9 lessons)
Multiplication ( 11 lessons)
Long Division (12 lessons)
Decimals and Primes (8 lessons)
-Primes & Composites
-Primes Factorization
-Greatest Common Divisors & Multiples.
-Prime Factorization Aids (Learn how to find factors quickly)
-Prime Factorization Examples
-Counting & Generating. Factors.
-Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11.
Integers (12 lessons) Intro to Signed Numbers
Fractions (< 20 lessons)  Essential Skills & Concepts
Ratios & Fractions (3 lessons): Similarities & Differences
Units in calculations Fractions with Units

B. Basic Algebra

Solving Linear Equations
- in one unknown. Intro with stick diagrams?
the normal way & with good nttn. (the nttn that reappears in Gaussian Elimination. |
-in more unknowns: simultaneous equations essentially one unknown. the let algebra do the work view of word problems.
- still in more unknowns: Gaussian Elimination via substitution, by equality or comparison, by operations on equations.

C. More Algebra

Words before symbols: See if U like the lengthy chapters 8 to 12 in Volume 2, Three Skills for Algebra
What is a Variable. The answer here is a simple prequel to the modern mathematics viewpoint.

First, every rule & pattern U meet in math, logic & science will be used forwards and backwards. Get a head start with this theme by reading Chapter 14 in Three Skills for Algebra. Second, in the study of Proportionality Relations (3 dense lessons here) finding the proportionality constant gives an initial backward use of the proportionality formula.

Talking about words before symbols and the forward and backward use of formulas gives words to make algebra simpler & clearer.

If you can not read or write precisely, you will have difficulty in following instructions. One wordy remedy is given by chapters 2 to 5 in Three Skills for Algebra. Where does Logic or a geometric model for reason Appear in Mathematics? The answer lies in Euclidean-Geometry In North America, Euclidean Geometry disappeared from high school mathematics as it was too hard. The light treatment here is a possible remedy.

E. More Geometry

The Pythagorean Theorem. Chapter 17 from in Three Skills for Algebra uses algebra and geometry   to show why the Pythagorean equation for right triangles holds. Its forward and backward use is common exercise.. At a more theoretical level, the Pythagorean theorem leads the discovery that not all lengths can be fractional multiples of a unit length. That geometrically implies a need for and even existence of irrational numbers.
Analytic Geometry: Common Practices with Maps and Plans drawn to scale give coordinate-dependent base for senior high school development of similarity, trig, vectors and straight lines.
Complex Numbers: This lesson on Complex Numbers draws on Euclidean and Analytic geometry. Sbortcuts simplifiy trig identities, the cosine law; and   trig formulas for 2D dot- and cross-products.

F. Logarithms, Exponentials,
Roots & Powers

Logarithms, exponentials, rational and real powers for secondary students. This complete Operational Viewpoint. (Sufficient for the precalculus forward and backward use of compound growth and decay formulas in biology, physics, chemistry, personal finance, and calculus. To learn more, if you study calculus, see chapter 19 of Volume 3, Why Slopes and More.Math

In Volume 2, Three Skills for Algebra, chapters

identify methods useful in money computations, methods needed for calculus. Your teachers or other writer may present the same ideas with greater clarity and detail - A site to do.

H. Polynomial & Quadratics

Analytic Geometry: - Slopes and Lines - Take 1.   Take 2 appears in site section Maps and Plans.   Two views are better than one. I may combine them later. -In my school days, slopes appeared year after year.   This Why Slopes calculus preview on graphs of functions y = f(x) explains why. Enjoy.
Quadratics and Polynomials: - Operations on Polynomials: Meet a light and ultraquick geometric introduction to multiplication, addition and subtraction of polynomials. Then see how the foregoing combine to permit long division of polynomials. Compare Fractions with Units. Enrichment: A Plus: The Geometric introduction here gives or is almost identical to a justification for column methods in decimal arithmetic.
Geometric Derivation of the Quadratic Formula:   The account here gives a starter lesson for the more algebraically harder geometric-free derivation. If you study physics, chemistry or trigonometry, you will need to know about quadratics, their factorization and the quadratic formula.
Technical Value: The study of polynomials high school mathematics has technical value as part of the senior high school mathematics preparation for calculus. This simple account of Why Factor Polynomials   (Chapters 2 to 6 in Volume 3 .Why.Slopes.&.More.Math.) will give a context for the study of polynomials, their factorization, and sign analysis of functions, all in a way that should improve your algebraic thinking and reasoning skills.

Vectors in the Plane (2 simple lessons)
- Navigation with vectors or arrows
- Sum of Motions
- more lessons to be added later.
Operations on movement or vectors along the line and in the plane have value in mathematics in defining and implying the properties of real and complex numbers before the assumption of those properties as axioms. Vectors and their properties appear in physics, its mathematical description and formulation.
- Functions - Forwards & Backwards. Here is a full technical reference (24 lessons) for use in a calculus or precalculus course as needed. In it, the set viewpoint of functions expression of modern pure mathematics. comes from the set-based codification and
In the mathematics education reforms of the 1960s in North America, primary and secondary school mathematics were expressed in terms of sets. That expression has now retreated from primary and secondary school texts. But it still lingers on, and can be very useful, a source of clarity and precision, in the situations where it should be retained: Counting with the aid of sets and functions; the description of functions; the high school account of probability theory; and in the discussion or illustration of ideas in logic.

J. Pre-Calculus Skill Check

- Arithmetic Skill Check. In the calculus courses I taught 1983-89, too many students had weak skills in arithmetic. I would give and carefully correct these exercises to tell students what they needed to review and master.
- All the skills and concepts in Chapters 1 to 24 or Volume 2, Three Skills for Algebra: Look for those you do not understand and fill the gaps. Do so quickly while balancing this advice with your other duties. Good luck.

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest © 1995 onward by site author, Alan Selby, All Rights Reserved.

A Possible Start

Decimals

Number Theory & Practices Origins of Counting

Number Theory & Practices

Origins of Counting