Second Lecture Math100

8/13/2019 Second Lecture Math100

1/15

The Whole Numbers

The most basic objects in mathematics are the natural or whole numbers, 0, 1, 2, . . ..

Here are the introductory remarks in the chapter on Developing Early Number Concepts

and Number Sense in one of the best selling mathematics methods texts in the UnitedStates:

Number is a complex and multifaceted concept. A rich understanding of number,a relational understanding, involves many different ideas, relationships, and skills.Children come to school with many ideas about number. These ideas shoud be builtupon as we work with children and help them develop new relationships. It is sad tosee the large number of students in grades 5 and above who essentially know littlemore about number than how to count. It takes time and lots of experiences forchildren to develop a full understnading of number that will grow and enhance all thefurther number-related concepts of the school year.

This chapter looks at the development of number ideas for number up to about 20.These founcational ideas can all be extended to larger number, operations, basic facts,and compuation.

Then a list of three big ideasis given:

1. Counting tells how many things are in a set. When counting a set of objects, the lastword in the counting sequence names the quantity for that set.

2. Numbers are related to each other through a variety of number relationships. The

number 7, for example, is more than 4, two less than 9, composed of 3 and 4 ans wellas 2 and 5, is three away from 10, and can be quickly recognized in several patternedarrangements of dots.

3. Number concepts are intimately tied to the world around us. Application of numberrelationships to the real world marks the beginning of making sense of the world in amathematical manner.

Note that in the U.S. book there is a clear view that whatever the concepts of numberchildren come to school with, they are valid. These ideas should be built upon as wework with children.

Additionally, it would appear that whole numbers are regarded as somewhat myste-rious objects in the United States imbued with profound depths, ideas and relationships.But the big ideas about number focus first on counting, then there are unnamed rela-tionships, and finally there are unnamed connections to the world around us.

Underlying the approach in the U.S. is the following formal notion of what a whole

1


2/15

number is:

We say that two sets, A and B, have the same cardinality if there is a one-to-onecorrespondence between the elements ofA and B.

We say the cardinality ofA is infinite if there is a one-to-one correspondence between

the elements ofA and the elements of a proper subset B A. If a set is not infinitethen it is finite.

The cardinalities of the finite sets are the whole numbers.

Hidden in this is the notion ofan equivalence relation. This is the process that goes fromsame cardinality to the set of cardinalities.

By contrast, the introduction of numbers in the high achieving countries is quitedifferent and involves far less sophisticated set theory. In particular a major point is madein those programs that counting properly is only a use of numbers, not a defining property

of numbers. This distinction will turn out to be profoundly important.

Lessons on Numbers in a Solid U.S. Text

These lessons and those on the next page are from the Kindergarten program.

2


3/15

3


4/15

4


5/15

These are from the first grade program.

5


6/15

In mathematics instruction in the high achieving countries such as the former ironcurtain countries, China, Singapore, Korea, and Japan, instruction also starts in their firstlessons in the first grade with an introduction of these numbers and their basic properties.But the approach is strikingly different.

Here is the first part of the introduction to the book for the first Methods Course in

the Hungarian Mathematics Education curriculum:

When a parent takes his child to first grade he is usually happy to tell the teacher thatlittle Tommy can count till 100 already. Pride is a good sign, because it signals: itsimportant for the parent to see his child develop not only physically, but intellectuallyas well. Sometimes it soon turns out (and the sooner the better) that the correctlynamed numbers by this same child have no content; the child knows the numerals,but not the concept of numbers. He can say the words in order: one, two, three ...thirty eight, thirty nine, forty, forty one ..., but he cannot really tell which one ismore: 5 apples or 7 apples, or have a clear picture about the order of size, equalitiesor the contents of the numerals.

We need to add quickly that its not a problem: building the concept of numbersis the task of the school. If a child doesnt possess the concept of numbers whenstarting school, then he will learn it with the teachers help; its not too late in thefirst grade. The only thing is that the child who is ahead, will need to do somethingelse, than the one who is just learning. (Its not an easy job to adapt to differentneeds, but developing starts with learning about the different levels: its importantfor the conscious developmental process.) Concept building has many components:it consists of forming many thoughts that can be separated in theory, whereas inpractice they appear together, reinforcing each other.

Main contents of the forming of the concept of natural numbers:

Connection between numbers and reality;

Writing and reading numbers; place value form and numerical system form ofnumbers;

Magnitude of numbers

The many different names of numbers;

Properties of numbers, relations of numbers.

The forming of the concept of numbers begins way before the age of 6 and doesntend in lower grades. We will now examine the different content components from thechilds first experiences till the end of grade 4.

6


7/15

In this chapter we will first give a rigorous definition of the whole numbers as theyare understood by mathematicians. Then we will present the first lessons in the RussianFirst Grade books from the 1980s. The vocabulary used in the rigorous definitions is not,of course, available to first grade students, yet, by comparing the material in the actuallessons with the Peano axioms, it should be clear that using grade appropriate language these early lessons are setting forth the same properties, and in much the same way.

7


8/15

The Peano Axioms

The Peano axioms define the properties of whole numbers, usually represented as aset 0. The first four axioms describe the equality relation

(1) For every whole number x, x= x. That is, equality is reflexive.

(2) For all whole numbersx and y, ifx= y , then y= x. That is, equality is symmetric.

(3) For all whole numbers x, y and z, ifx= y and y =z, then x= z. That is, equalityis transitive.

(4) For all x and y, if x is a whole number and x = y, then y is also a whole number.That is, the whole numbers are closed under equality.

The first axiom says any whole number is equal to itself. The second says that if a wholenumber a is equal to a whole number b, then b is equal to a as well. The third can berephrased, using the second axiom, as saying that if two whole numbers are equal to athird, then they are equal to each other.

Later, when we talk about relations, particularly equivalence relations we will see thefirst three axioms again, in a different guise that will help us see why they are the keyproperties of equality.

8


9/15

The remaining five axioms define the key properties of the whole numbers. We startby assuming that there is at least one whole number, the number 0.

(5) 0 is a whole number.

Then we introduce a special map Sfrom the whole numbers to the whole numbers called

the successor function, that can be thought of as adding 1. Later in the course, when wetalk further about the underlying foundations for school mathematics we will discuss functions and their

properties in much more depth. For now, we note that a function or map from the setX to the setY issimply a rule that determines a unique element y Y for each element x X. We usually writef(x)for they Y that is associated to x, and sometimes we writef(x) =y ory = f(x).

(6) For every whole number x, S(x) is a whole number.

We obtain 1 as S(0), 2 as S(1), 3 as S(2) and so on. But to make this sensible we have tolist two properties that the successor function Smust satisfy.

(7) For every whole number x, S(x) = 0. That is, there is no whole number whosesuccessor is 0.

(8) For all whole numbersx and y, ifS(x) =S(y), then x= y . That is, S is one-to-one.But note that Axiom (7) shows that Sis not onto since 0 is not in the image ofS.

These two axioms together imply that the set of natural numbers has the propertythat it is in one-to-one correspondence with a proper subset, since Sis one-to-one and 0 is

not in the image ofS. We can see intuitively that if the set of whole numbers were finitethis cannot happen. Any one-to-one mapping of a finite set to itself must be onto theentire set. Consequently, our intuition is that the set of whole numbers must be infinite.

However, in mathematics, intuition is only an aid. All our terms must be preciselydefined, so we DEFINEthe property of being infinite as follows:

Definition. A set Wis infinite if and only if there is a one-to-one correspondence betweenthe set Wand a proper subset ofW.

Now, we can unambiguously state that Peanos axioms (7) and (8) imply that the setof whole numbers is infinite.

These terms may not be familiar to everyone. One-to-one correspondence is a map or function froma setXto a setY that has the property that f(x) =f(x)if and only ifx = x. The functionx 2xis one-to-one on the whole numbers but the function x2 4x+ 6 is not one-to-one since the values forx= 1 and x = 3 are both3. The image of a function f is the set of those elementsy Y so that thereis some x X withf(x) = y. The image ofx2 4x+ 5 as a map from the whole numbers to thewhole numbers is all the whole numbers but0 and1.

9


10/15

There is one more Peano axiom for the set of whole numbers. So far, the axioms givea picture of the following kind

0 S(0) S(S(0)) S(S(S(0)))) . . .A S(A) S(S(A)) S(S(S(A)))) . . .B S(B) S(S(B)) S(S(S(B)))) . . ..

..

.

..

.

..

.

.. .

. .together with a number of circular sequences V, S(V), S(S(V)), etc. with

S(S(S( S(V) ))) =V

for some iterate ofS. The axioms to this point do not specify how many elements thereare in the whole numbers that are not in the image of the successor function, nor thenumber of circles, just that Sis one-to-one, which means each element not in the imageof S leads to an infinite tower as sketched above, and the towers or circles for differentelements do not intersect. The final axiom, Axiom (9), specifies that 0 is the only wholenumber that is not in the image ofSand that there are no circles.

(9) IfK is a set so that

0 is contained in K and

for every whole number x, ifx K, then S(x) K,

then K contains every whole number.

This axiom is usually called the axiom of induction because one of its consequences isthat ifT is a property that can be true or false for whole numbers then if it true for 0,

and if we can show that the truth of the statement for any whole number n implies thatthe statement is true for S(n), then the conclusion is that the statement is true for everywhole number.

Here is an example. We note that 1 = 12, 1 + 3 = 22, 1 + 3 + 5 = 32, and we wonderif the sum of the first k + 1 odd whole numbers is the square (k + 1)2. We can verify thatthis is true for all whole numbers if we can verify that it is true for the first which it issince 1 = 12 and verify that if it is true for k then it is also true for k+ 1. In this casethe argument is that the truth for k is exactly the statement that

1 + 3 + 5 + + (2k+ 1) = (k+ 1)2

is true. Then the statment for k+ 1 is(1 + 3 + 5 + + (2k+ 1)) + 2(k+ 1) + 1 = (k+ 2)2.

We have to show that the truth of the statement for k implies the truth for k +1. Becausewe assume the truth of the statement for k we can rewrite the statement for k+ 1 as

(k+ 1)2 + 2(k+ 1) + 1 = (k+ 1 + 1)2

= (k+ 2)2.

10


11/15

But this is a true statement since we know, from algebra, that (a+b)2 = a2 + 2ab+b2,and substituting k+ 1 for a and 1 for b gives (k+ 1 + 1)2 = (k+ 1)2 + 2(k+ 1) + 1.

There is one more concept that is tied in to these axioms. We say that the wholenumber n is greater than the whole number m if and only ifn is not equal to m, but isthe image ofm under some iterate ofS. We write n > m as a shorthand notation for n

is greater than m.

Likewise, we say m is less than n if and only if n is greater than m, and we writem < n as a shorthand notation for m is less than n.

From the definitions, if m and n are any two whole numbers then they will satisfyone and only one of the three statements m= n, m < nor m > n.

11


12/15

The First Lessons in the First Grade Russian Text

Here the basic notion of successor is introduced in the guise of more or less. A numberis more than another number if and only if it is in the image of an iteration of thesuccessor map on the original number. Then names are introduced.

12


13/15

The symbols for greater than, less than, and equalsare introduced as well as the namesof the first three numbers and the sums and differences for them.

13


14/15

The numbers through five are introduced and the students are asked to do a number ofproblems. Note that the focus is on the mathematical concepts. There are virtually nowords present except for essential vocabulary. It is also worth noting that all the keyconcepts are introduced using very small numbers - numbers that are almost hardwiredin our brains.

14


15/15

After these problems, the next lessons introduce the rest of the numbers between 6 and 10in a very similar way. Note that subtraction is developed for the same number set at thesame time as addition is for the numbers to 5. Subtraction is also handled simultaneouslywith addition in the lessons to 10.

Mathematically, this makes perfect sense because subtraction is defined in terms ofaddition: a b = c is a true statement if and only if b+ c = a. So subtraction is justaddition where one of the addends is not necessarily known.

Of course, we have not really defined addition for whole numbers to this point. How-ever, this can be accomplished by using the successor function and induction. The keyinductive statement is that 0 + n= n for each nas the start of an inductive definition ofaddition. Then, if we assume that the value of k+n is known for each n we define thevalue of S(k) + n as S(k+ n). Using this definition we have to show that the standardproperties for addition hold. These are

(1) k+ n= n + k for all whole numbers n and k, and

(2) k+ (s + n) = (k+ s) + n for all whole numbers k, s and n.

Exercise.

(1) Can you prove statement (1) above in the case wheren= 0?

15

Date post:	04-Jun-2018
Category:	Documents
Upload:	thich-toan-hoc
View:	223 times
Download:	0 times

Second Lecture Math100

Documents