Monday, March 28, 2011


How to teach algebra to young children

Too many books and articles about algebra suggest that this is a highly complicated subject, and cannot be mastered until a student has a good grasp of all the complex arithmetic rules, including long division and lengthy multiplication. I was rather horrified to find that the (American) home education curriculum course my sons used for a few years did not even begin algebra until the high school level.
Yet algebra at its simplest (pre-algebra, if you prefer) can easily be understood by a child of about six or seven who does not have the accuracy or concentration necessary for complex arithmetic. In the UK, complex arithmetic is taught in schools much later than it is in America, but simple algebra is introduced in the early years, in concrete forms that give young children the important concepts so that they do not become afraid of equations and unknown letters when, later on, they reach more complex algebra.

Introducing algebraic concepts at home

Find a large bowl of fruit. Or toy cars. Or lego bricks... whatever comes to hand. Count out two apples (or red cars, or long red bricks). Then count out another two of the same. How many are there? Obviously there are four apples (or cars or bricks).
Two apples plus two apples make four apples. You can draw a picture of this, to represent it visually. Draw an old-fashioned set of scales with two sides balancing. Put two groups of two apples on one side, and four apples on the other. Or - if your child understands basic arithmetic symbols - draw two little apples, and a 'plus' sign, and two more little apples, then an 'equals' sign, and four apples.
You have just introduced your child to the algebraic equation:
2a + 2a = 4a
Try the same with bananas. Or oranges. No matter what objects you use, two of something plus two more of the same will make four of that object. The supposedly simple arithmetic '2 + 2 = 4' is actually a shorthand for the algebraic expression. If we have the algebraic concept, the arithmetic is obvious. Unfortunately too many children are introduced to the arithmetic without having a clear concept of the algebra, and then struggle to make sense of those symbols. '2' is a shorthand for '2a' which is a shorthand for 'two of anything'.
Now count out two bananas, and then two cherries.. What do you have? Two bananas and two oranges. (If your child insists you have 'four pieces of fruit', acknowledge this is true, and try instead with two bananas and two toy cars). If you add on another banana, you have three bananas and two cherries. Or two cherries and three bananas, but it's not five bananas nor is it five cherries. This is showing your child the concept that
2b + 3c = 3c + 2b
Unless you know a value for b or c you cannot simplify this any further.

Use diagrams to illustrate algebra principles

Draw diagrams to demonstrate these principles, until your child is completely comfortable with them. Show another set of scales, but this time the bananas are considerably heavier than the cherries, and they won't balance. This shows the algebraic inequality:
2b > 3c
If your child wants to make them balance, guess how many cherries would balance a banana. If you have real balancing scales, then try this out. Perhaps it would be 20. This gives the equation b = 20c. Ask your child what would balance two bananas, and see if he can work it out. You will probably find that he knows immediately that he needs to add on another 20 cherries (or however many it is) even if he does not know what 20 + 20 is. The algebraic concept is simpler than the arithmetic:
if b = 20c
then 2b = 20c + 20c
Don't try to rush this. Work with your child's ability and interest, and take opportunities to demonstrate the way that algebraic concepts work at a simple level. Don't try and 'teach', but let him experiment. A real balancing scales makes this easier, but drawing may work just as well, and can go ahead without having to buy vast amounts of fruit! When this kind of opportunity is built into everyday life, many children will learn the rules of algebraic addition and subtraction without any struggles. Treat it as a game, and algebra will be fun. As it should be!
If your child is interested in algebra, and has a good understanding of logic, you might like to see my page on simultaneous equations. But that's a hard concept for a younger child, so you might want to leave it until they're at least 10 or 11

Friday, February 4, 2011

Bagaimana Kanak-kanak Belajar Matematik


Manusia mula belajar sejak dari peringkat bayi lagi. Bayi yang baru lahir mewarisi berbagai-bagai bentuk gerakan pantulan. Contohnya apabila sesuatu objek menyentuh bibirnya, bayi terus menghisap objek tersebut tanpa perlu diajar kepadanya. Ini bererti wujud satu atur cara genetis semula jadi dalam diri bayi untuk menyerapkan kewujudan sesuatu objek. Hari demi hari, maklumat dalam ingatan bayi semakin bertambah selaras dengan peningkatan fizikalnya daripada seorang bayi ke dunia kanak-kanak seterusnya menjadi dewasa. Maklumat-maklumat yang diterima akan dipecah-pecahkan menjadi pengetahuan dan memperkembangkan keupayaan kognitifnya.

Menurut Mohd Daud Hamzah (1996), kanak-kanak mempelajari matematik melalui kegiatan seharian tertentu. Ada beberapa aktiviti yang membantu kanak-kanak memperolehi konsep-konsep awal matematik iaitu aktiviti padanan (matching), penjenisan (sorting), reguan (pairing), dan susunan aturan (ordering).

Padanan ialah kegiatan memilih sifat tertentu dan membuat perbandingan. Penjenisan pula adalah kegiatan memilih sifat umum di kalangan bentuk-bentuk. Reguan merupakan kegiatan menyatakan keselarian objek-objek secara satu lawan satu. Manakala susunan aturan adalah kegiatan meletakkan perkara sepanjang satu barisan. Walau bagaimanapun, terdapat beberapa teori daripada pakar-pakar pembelajaran bagaimana kanak-kanak mempelajari matematik dan jenis matematik yang boleh dipelajari pada peringkat yang berbeza dalam perkembangan kognitifnya. Antaranya ialah Teori Piaget, Bruner, Gagne dan Dienes.

Dipetik daripada: