Teaching Numbers and Operations: Strategies and Approaches

A.           Pedagogical Content Knowledge for teaching numbers

The aims of learning math are students able to apply the math concept in their life, they have to develop their thinking skill, and our job is designing the lesson plan which helps them in developing thinking skill, using tools, such as calculator, computer, etc. Like the other material, in teaching numbers we have to master the knowledge about pedagogy and the content itself, so we can deliver the material to the students in appropriate/understandable ways.

B.    Teaching the properties of Integer System, and Properties of Addition and Subtraction in Integer.

Let’s review about integer system.

-          Integer Set Notation is denoted by Z = {…,-2,-1,0,1,2,…)

The elements of M = {0, 1, 2} are called non-negative integer.

-          If n Є Z, then –n + n = 0is called as additive inverse property.

-          (a+b) Є Z is called as closed to addition operation.

-          (axb) Є Z is called as closed to multiplication operation.

-          a+b = b+a is called as commutative law of addition.

-          axb = bxa is called as commutative law of multiplication.

-          ax(b+c) = (axb) + (axc) is called as left distributive.

-          (a+b)xc = (axc) + (bxc) is called as right distributive.

-          a+0 = 0+a = a for every a Є Z is called as identity of addition in Z.

-          ax1 = 1xa = a for every a Є Z is called as identity of multiplication in Z.

C.           Teaching Scientific Notation

Our students have to understand and good in simplify the very large and very small numbers. A scientific notation is written by this form:

(N)(10)^k

For (N) is the number between 0 < x < 10, x is decimal and k is an integer.

Example: 512.45 = 5.1245 x 10²

                 0.0000000000000000456 = 4.56 x 10^-17

                 67,800,000 = 6.78 x 10⁷

D.           Samples of Teaching Five Processes in Teaching Numbers

-          Teaching Mathematical Problem Solving

Problem solving is not only used for applying the concept, but also for developing math skill, training the strategies, etc. most of students feel the problem solving as the most difficult. Here are some suggestions for problem solving.

1.      Read the problem carefully; make sure that you understand all words.

2.      Read again the problem until you get what are being questioned, what is to be found, etc.

3.      Using diagram or chart to make it clearer.

4.      Choose the meaningful variable to represent an unknown quantity.

5.      Looking for guidelines which given.

6.      Form an algebraic equation from the languages.

7.      Solve the equation.

8.      Re-check the answers.

-          Teaching Reasoning and Proof in Numbers

Our students must have to recognize reasoning and proofing as fundamental aspect of mathematics. Also, they have to investigate mathematical conjectures, using various types of reasoning and method for proofing. One of the method is proofing by induction.

Example: Proof by mathematical induction that: 1+3+5+7+9+…+(2n-1) = n² for every natural number n.

First: Assuming that the statement is true for n=k
1+3+5+7+9+…+(2k-1) = k²                           …(1)

We will show that for n = k+1, that is:
1+3+5+7+9+…+(2k-1)+[2(k+1)-1] = (k+1)² …(2)

From equation (1), we got the true statement:
1+3+5+7+9+…+(2k-1) = k²

Add both sides by [2(k+1)-1] and we got:

1+3+5+7+9+…+(2k-1)+[2(k+1)-1] = k² + [2(k+1)-1]

= k² + 2k + 2 – 1

= k² + 2k + 1

= (k+1)²

It’s proven that equation (2) is true, so by mathematical induction the statement 1+3+5+7+9+…+(2n-1) = n² is  true for every natural number n.

-          Teaching Mathematical Connection

Our students have to know and apply the math concept into our daily life.

Example: A man who wants to go to hospital has 3L gasoline; the distance from his house to the hospital is 48km. If 1L could reach 12km, so will he arrive to the hospital successfully?

Answer: 1L = 12 km

               3L = a

By crossing multiplication, a = 3x 12

                                        a = 36 km

so, he only reaches 36 km, instead he have to pass 48km, then hi is not successfully to go to the hospital.

-          Investigative Approach in Learning Numbers

In this activity, we have to help our students in investigating and find out the formula.

Example: Find out the formula for arithmetic sequence if the 1^st term is -3 and the 3^rd term is 9.

n-th term	value	The differences
1st	-3	6
2nd	3	6
3rd	9	6
4th	15	6
5th	21	6
6th	27	6

From the problem, we have the 1^st term is -3 and the 3^rd is 9, so we know that the differences is 12. Because of the sequence, so between 1^st and 3^rd term must be 2^nd term, the differences 12 must be divided for the difference between 1^st and 2^nd, and between 2^nd and 3^rd. Then 12 : 2 = 6, now we have common difference, so that we can find the 7^th term.

E.           Teaching Numbers Using Technology

-          Using Calculator

Calculator is very well-known because everyone can use it easily. The cost of calculator depends on the smartness of calculator itself. More expensive, more “complete” calculator that we have. Calculator helps students to calculate the complicated quantification, so we have to harness it for our teaching and learning due to NCTM recommendation in 1998.

-          Virtual Manipulative

It’s an application for the computer, so our students can explore it more and more just by mouse. It’s like Maple, Geogebra, Carmetal, Calculator 3D Shape, etc. It really helps students to visualize something that feels complicated. If they cannot visualize by their own mind, so they can use this to visualize it. As a teacher, we have to harness this to help us in teaching and learning.