Recommended Reading
Grasping Mathematical Concepts in the Early Years and Beyond
Abstract: This article explores the natural method of learning basic mathematical concepts.
Consider first, learning in the early years. How do babies learn to walk? Does the parent merely show them how and they can suddenly do it? No, the baby achieves this great feat by observation, discovery, creative application, and the practice of continual improvement through trial-and-error. This is the real world. This is the natural way of learning.
We all learn step-by-step, we reach progressive levels of achievement, and we use that new foundation as a means of extending our learning even further. We cannot be forced to learn, but we have an internal ambition to do so. We are guided in what we chose to learn by those around us.
The greatest incentives to life-long learning are the good feelings we get when we achieve something. From our successes, we gain a confidence of our abilities to learn new things. We can now make even further progress.
Toddlers learn to count their toes and fingers.
Pre-schoolers learn to write the digit that matches the number of items counted. Some of the items they count are the same (or so closely similar that we do not differentiate): 5 apples, 4 oranges, 10 fingers. Some items they count are similar in some way but different in another way: 6 fruits but different types of fruits, 3 cars but of different colours.
Reception-year children learn "10's number-bonding": 0 and 10, 1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5, 6 and 4, 7 and 3, 8 and 2, 9 and 1, 10 and 0. They learn to count higher than 10, maybe up to 100, and even beyond.
Early primary school children become intimately familiar with the numbers from 0 to 100. They learn about groups: 2 groups of children are waiting for the bus - one group has 5 children and the other has 2; 2 teams of children playing football - each team consists of 7 children. They become familiar with the 10 x 10 number grid. They learn to count by 2's, 5's, and 10's. Counting up to 100 by 10's sure is quicker than counting by 1's. Repetition of patterned counting breeds familiarity.
They use a number line to add and subtract two numbers less than 20. They are taught to think of a number in terms of groups of 10 and units of 1. Adding and subtracting two 2-digit numbers is next. They notice that counting by 2's sounds similar to the products in the 2's times tables; and counting by 5's sounds similar to the products in the 5 times table.
Maybe we can count by 4's and that will sound like the products in the 4 times table. Can you count to 100 by 4's? Notice the pattern in the units/ones digit of the numbers: 4, 8, 2, 6, 0. Discovery is fun! Let's learn some more.
Counting by 5's was easier than counting by 4's. Why? When we count by 5's, the last digit only had 2 alternating possibilities: 0 or 5.
Write down the 2 digits in the products for the 9 times table. Do you notice any patterns? Did you notice that the 2 digits add up to 9? What other patterns do you notice?
We are babies, discovering to walk. We are children, discovering mathematics. We are learners, discovering life.
