Kingsley Educational Pty Ltd |
Back: Published Papers | Home |
Throughout this Web site: = Australian; = Christian; = new/updated; = Special. = Early (age 3-9); = Middle (age 7-15); = Senior (age 13-18); = Parents; = Everyone |
Sadly many of us were brought up believing in ourselves - what WE could do, what WE enjoyed etc. And we balked at the hard work of arithmetic and discovered that we "couldn't do" maths! So now we dread teaching our children. Deep down we know our fear will be exposed and we'll have to either overcome it or pass it on - what a choice! We have only poor role models and no teaching to do a better job. if we don't break the cycle, our children might have the same problem!
Once you have decided to squarely face the problem of mathophobia, you'll probably find that the rest is not as difficult as you thought. Preschool readiness can be achieved naturally with very little effort and no expense (bar what you want to spend). Primary school maths requires a normal 20 to 40 minute maths class (at school or at home) PLUS A FIVE-MINUTE Drill. The maths session is for understanding, manipulating, seeking patterns, applying and generally grasping concepts. The DRILL is the bit that only one-to-one tutors (eg parents rather than classroom teachers) can do efficiently, and that will provide the challenge and fulfilment that parents deserve.
In other words, even though you may use a curriculum, you need to know where you're going - know your educational goals. in maths they are generally clear. First you learn the basics - how to count, and comparison words like less than, more than, bigger, smaller. Then you learn to add in ever increasing proportions and with different units. This leads directly to subtraction, the reversal of the process. Long addition leads naturally onto multiplication, starting with small numbers (3+3+3+3+3=5x3=15). The children will soon see that multiplication is a short-cut to avoid long mental computations, and then they are ready to start learning their multiplication tables (don't start earlier). When each table is thoroughly learned, it can be reversed to begin on the division road. By moving in small logical steps you will soon cover long multiplication, long division, various units (eg money, weights, etc), fractions and percentages. None of these bits will be fearsome if you patiently wait for (ie work on) complete mastery before moving on.
And that's all there is to primary maths! If you don't believe me check any curriculum and see what I've left out. You will see lots of pages of this and that, but when you look at exactly what concepts are being taught, you will note that despite some variations in terminology, the curriculum is merely basic arithmetic. The inefficiency comes in because the book can't tell when your child has grasped it, and either goes on too long or not long enough.
Picture it: the student enters with a vague awareness that they skimped their way through addition and never quite mastered sub and multiplication, and as for division, well However they are full of hope that they can put it all behind them and learn some new concepts and maybe they'll do better with them. However it doesn't work that way and they get their first few examples wrong due to arithmetic errors. They try harder, but there is too much to concentrate on. The concepts get more and more difficult, and then comes the test. Guess what? They fail. But there is no time for much revision because the next chapter awaits. Much the same happens with it and the next and the next. By the time they get another look at that first topic. it's the following year and all they can remember of it is the frustration and failure.
This awful prospect can be totally short-circuited by ensuring a solid grasp of foundational arithmetic and by choosing an efficient secondary program with plenty of built-in revision. If you are locked in to a program that doesn't keep learned concepts fresh, consider one day per week of mixed maths where you choose examples from all previous sections. (I think I just saved you $30/week on maths tuition: you CAN do it!)
Preschool "maths" includes counting, matching and sequencing. counting is actually three totally separate skills: reciting the number words in order; reading the numerals eg. 3 is three, 15 is fifteen; and one to one correspondence ie. pointing to one object at a time and counting in sequence to reach a total. Counting up to what? you may ask. It doesn't really matter - I would be concerned if a six year old was unable to count to ten but otherwise, when they can count to twenty, extend that to a hundred and so on; build their understanding to count by two's, tens and fives etc. Subtract. Keep on giving them broad experience with counting and delight in your child's development.
Preschool maths is also matching - find two of the same size, colour or whatever. Point to bigger, smaller, taller, fewer etc. A preschooler should be able to identify shapes (square, hexagon, cone etc.), the sequence of events (eg. first wake up, then dress, then go out), the elements of a clock and a calendar, and number concepts, including zero.
Hey, presto! The end of prep maths!! Easy, eh?
Part of this should be done by examining number-fact families. Adding one to a number is just like counting. Adding zero is easy, too (once the child understands the concept of zero, which is not easy). If you were to write all the numbers 0-9, plus one and plus zero on 19 cards, how fast could your child flick through them and say the totals? How about, all shuffled? - that's harder. What about on the next day - could they improve their time? Now - adding 2 is not much harder than adding one - just count two forwards in your head. Make eight cards for the facts 2+2 to 2+9 (you already have 2+0 and 2+1). Do them in order. Then try them shuffled.
Watch for signs of saturation: droopy eyes, slowing down, mistakes. Your smiles and encouragement and their own awareness of growing achievement will be highly motivating. Let them finish like that - not frustrated.
Each session will be as long as you can both stay enthusiastic - 5-10 minutes is fine. Fill the rest of the maths time with practical maths eg. measuring for cookery, playing games (eg. Uno, but almost any will enhance maths skills of matching, scoring, sorting, comparing values etc.), word problems, text book exercises, using manipulatives to solve puzzles or worksheets, setting the table, pairing socks, sorting the wash, arranging flowers or toys - all requiring logic and mathematical judgment.
If you seem to be taking too long, spend less time testing and more with the manipulatives, games and activities. You will see your child growing in confidence and speed, and know when it is time to resume your five minutes of intensive work together. Don't become impatient!
Next, doubles (3+3, 6+6, 8+8) are fun - many children already know them. This adds six more, bringing the total to 40. Twenty-five seconds is good going now!
Now you can introduce the concept of neighbours - they live next door, to each other: 3 and 4 are neighbours. If 3+3 = 6 and 4+4 = 8, then what will 3+4 be? Half way between 6 and 8, of course: 7! You now add 3+4, 4+5, 5+6 and 6+7. Of course you can practice with 0+1, 1+2, 2+3, 7+8 and 8+9, which are already learned, but now you are teaching a different mental route to the same problem and solution. Hopefully they'll never forget 7+8 or 8+9 again! Neighbours shuffled in with their previous work makes a total of 49 facts - only six to go! The forty-nine can be done in five random families in 25-30 seconds.
The +7 set can be practised with the aid of a calendar. For example ask, "Today is the 6th; what will next (Tuesday) be?" Finally the last two (3+5, 3+6) just have to be learnt and the whole lot should take about 30-40 seconds - the faster the better. Make them a nice certificate when they achieve this milestone and pat yourself on the back for a job well done. If you can please let us know how long it took so we have more experience from which to guide others.
When consolidated through practice, application etc. you build on their basic addition skill, thus - if 3+5 = 8, then what is 13+5? 83+5? etc. if 7+3 = 10 then what is 700+300? If 8+5 = 13, what is 28+5?
Some final comments on Domino Drill: You don't have to purchase anything if you don't want to - just make flash cards or dominoes from card or wood. Domino tiles are very nice to handle and store and there are lots of great games you can play with them as a family too. If you don't know any, just ask us!
If your child gets nervous from being timed, just watch to see how long they "um and ah" before saying an answer. It's not hard to tell when they are hesitant and need more practice. Try to note which facts trip them up, and if there is a consistent pattern, teach just that fact for a while.
The times above are derived from the experience of a number of families, including ours. Therefore, they are possible, but you decide what is reasonable for your children. We aim for about two facts per second. with a little grace. It is possible to go a little faster but some of us are more tongue-tied than others.
It's great if you can get the larger numbers too - we can help you with 8-, 10-, 12- and 20-sided dice. If you look for them elsewhere, beware of dice with dots rather than numerals which are harder to read quickly; and of poor contrast in the background colour (eg. "pearl sheen" and "crystal gem" dice, as well as being more expensive, are harder to read than the plain opaque ones).
If you have, say, a 12-sided dice, you could revise single "families" eg. the +9's by throwing it for a random number to add to nine.
My children did dice drill for a long time (I think about two years with all the various operations) before tiring of it. We used lots of different novelty dice, including spinning ones which added variety. The medium doesn't matter - it's the learning that does, so if one tool starts to bore them, switch to another. We used different dice for subtraction than for addition, to reduce confusion.
1. Imagine a number line. So to find the answer to 9 - 4:
How many steps to get to 4? Or:
After 4 steps where do you land?
2. Imagine columns eg: | 17 -7 10 | 14 -3 11 | 18 -6 12 |
3. To subtract 8 or 9, use the adding strategy in reverse ie. to subtract 9, count 1 forward from the units, I back from the tens and for 8, count 2 forward from the units then 1 back from the tens.
4. If the difference is 1 or 2 (but the answer isn't obvious eg. 14-3), the answer will be 9 or 8 respectively eg. 13-4=9, 17-8=9, 13-5=8, 15-7=8
5. Go to the nearest 10, then the rest of the way:
13-5: 13 - 10=3, 10 -5=5; 3+5=8
15-6: 15 - 10=5, 10 - 6=4; 5+4=9
11-7: 11 - 10=1, 10 - 7=3; 1+3=4
Say a string of numbers and have your child multiply them by nine as fast as possible. Then, as always, intermingle x0, xl and x2, one family at a time until all are mastered. Continue in this way, adding families in to the known bunch gradually and only after all previous ones are known.
Take the square numbers eg. 7x7. What happens if we subtract one from one 7 (making 6) and add it to the other (making it 8)? Answer: 1 disappears from the result. 7x7 = 49; 6x8 = 48 (one less than 49!) Try it with others: 4x4 = 16; 3x5 = 15 (one less than 16!) 9x9 = 81; 8x10 = 80 101x101=10201; 100x102 = 10200.
We can write that this will happen, whatever the numbers you choose, by calling our original number "x" so that x x x = x^{2}. Then we say take 1 off the first x and add it to the second x so we get an (x-1) and an (x+l). Multiply them (x-1)(x+l), and you get one less than x^{2} ie. x^{2} - 1. This is in fact the algebraic concept called the Difference Between Two Squares.
So we've introduced basic algebra, changed it from a fearful to an exciting topic and taught some more multiplication facts, all in what appeared to be a diversion. Keep revising the squares and (x^{2} - 1)'s until they're off pat.
Fifty six is seven eights
Shut the doors and lock the gates
Six times seven is forty-two
If they all come, what will we do?
Seven fours are twenty-eight
Hurry now, it's not too late.
Eight times four is thirty-two
Ants and flies, we don't want you!
Say it over and over and when it's known, say it without the underlined parts.
Once you've covered all four operations, you will need to maintain the speed by drilling or testing regularly (eg. fortnightly) for quite a while.
Do your children know how to test Divisibility? They should know 2, 3, 4 and 5. If not, tell them. Then let them work out how to test for divisibility by 6, 9, 15, etc.
When they know how to convert a fraction into a decimal, show them the relationship between the sevenths: one-seventh = 0.142857 repeating, two-sevenths = 0.285714 repeating, three-sevenths = 0.428571 repeating etc.
They can multiply any number by 11 very quickly. They now know that 6x11=66 etc. Show them that 63x11=693 and 41x11=451 and they will soon see the pattern. Even with large numbers eg. 1025x11=11275. You get this by putting down the first digit (1). Then you add the first two digits (1+0=1) for the second digit of the answer. Then the second and third digits (0+2) for the third answer, third and fourth (2+5) for the fourth answer and finally put down the fourth digit (5) at the end of the answer.
The six times table has some internal patterns eg. 6 multiplied by 2, 4, 6, 8 = 12, 24, 36, 48. Notice that the units digit is the same as the multiplier and the tens digit is half of it.
Multiplying large multi-digit numbers need not always be done by long multiplication. Often you can use either factors or the Distributive Law to short-cut your computations eg. 62x49= (62x7)x7 and 97x34= (100x34) - (3x34).
Many more of these short-cuts to more complicated arithmetic are available in Short-Cut Maths at $12.50.
Because if you didn't have maths class, they couldn't learn all this in five minutes a day! Maths class is for meeting new concepts, becoming acquainted with them in various aspects, using manipulatives, finding patterns, working things out, standing in awe of the God Who ordered our world so well. Drill time is when you distil all that learning and commit it to memory - for instant recall. In maths class you look at 3 and 4 and observe that the answer is always seven. In drill you make certain that when you see 3+4 you instantly think "7."
2. When should I start drill?
Basic concepts must be understood first. When a child does not know what + (plus) means, there is no point in demanding addition fact recall. If you have done the domino-style teaching of facts in families so that there are mental pathways already mapped out in his brain to reach an answer, you build on that to speed up the recall and revise it with drills - domino, dice, verbal or any other.
3. What if my children get bored with their drill?
Firstly - just because something may be boring doesn't mean it's not worthwhile! Many things in life are, on the surface, not appealing eg. getting up for work each morning, tying shoelaces or washing nappies. However, they are necessary and beneficial. Drill is in the same category. Children can be taught to enjoy what must be done, rather than to do only that which is enjoyable, and this will be of great benefit to them. The assumption that all teaching must be made interesting is partly to blame for the fast-lowering standards in our schools.
Secondly, are you doing the drill for too long? Each session should be very short - 5-10 minutes. An eager student could have two sessions, but more does not help - you will actually go backwards. If you don't watch the clock, watch your child very carefully: do not allow fatigue, impatience or frustration to develop.
If neither of the above are a problem, change tools. Switch say from Holey Cards to dice or dominoes; if you've never tried verbal drill, that might make a fresh change; there are also cassettes with the facts set to music to sing along with for a complete change of pace. The method is not important - keep the goal in mind and keep the progress coming!
4. For how many years will this go on?
You can expect 3-6 years of drill to be required for each of your children to cover all the facts (addition, subtraction, multiplication and division) but before you start wondering if you'll be able to keep up the momentum, consider how much more time you will spend on their meals, clothes etc. Consider it a personal, quality, mother to Child time of encouragement and upbuilding. Delight in your child(ren) - enjoy seeing them develop, mature and build character. if you do one of the written forms (Calculadder, Holey Cards etc.) you can do drill for all your children together.
Children begin their speed work about Grade 2 and continue daily until they master the four operations up to 99 at least. Once this is achieved, only weekly or fortnightly revision is required. You do need to be prepared to keep at it, but if you cannot for some time. at least they will have benefited from the time you did sow into their formative years. (However, their declining mastery may drive you back to it!
5. My children are in upper primary/lower secondary and have done okay in school, but I've noticed they still count on their fingers and have to think hard about multiplication (and they still get it wrong!) Where should I start?
At the beginning. Maths is strictly sequential and you cannot build upon shaky foundations. Many people think they or their children are quite numerate but have never checked their speed. Without instant recall, all higher maths will be nerve-wracking and tenuous. This is the maths they will use every day of their lives - they may as well learn to use it efficiently now - they only have to learn it once! If your child really is already quite competent, he'll fly through the early addition, without feeling any stigma but having to return to it after starting higher up would be harder! Don't be impatient, nor allow them to be - just be encouraging.
6. Why do so many school teachers say drill is bad?
Educational thinking unfortunately goes through fads and fashions, all based on philosophical differences. To assess whether a "new" method is "good" or not, you need to look at the ideas and worldview behind it, as well as whether it works in the long term.
A Christian worldview says there are right and wrong ways of doing things; that God knows everything (so we depend on Him as the ultimate source of knowledge); that you can learn the right way by listening to wisdom if you don't want to experience all the negative consequences of unwise decisions.
Humanistic thinking deifies mankind and worships the power of the human intellect to discover or invent solutions (hence the exalted respect for science and human creativity). The end points of these two views (belief systems) are totally opposite each other.
The Christian believes in teaching (by a wiser adult to an unwise student), memorization, right & wrong etc. The Humanist believes in each child discovering for himself (re-inventing the wheel?) by creative (ie. unrestricted) thinking, trial and error etc.
In maths, the Christian will be shown concepts, learn them, then understand them and use them; the Humanist must discover them all alone, and rediscover them whenever he needs them. There is no room for memorizing because this assumes a Truth underlying experience which transcends individuals. This philosophical Grand Canyon means that humanistic educational institutions disapprove of drill. However, neither the long term fruits of such philosophy and methods. nor Scripture, allows us any luxury - the world is the way God made it: not as the Humanists wish it to be!
7. What about real life maths? We try to do only what's relevant eg. measuring for cookery or craft, predicting requirements, scheduling etc. We try never to teach "subjects" in isolation, as if they don't relate to real life.
You are taking on an immense task each time you tackle a domestic chore! I prefer to break up tasks into their components. As most babies don't begin talking in full sentences, but say nouns, verbs etc. in isolation, gradually adding other parts of correct speech, so I believe we should teach other skills piece by piece, logically and sequentially. We always demonstrate the full outworking of the skill in everyday life, but teach it one piece at a time. Think how you would teach someone how to swim, build a dolls house or grow vegetables - each aspect is usually taught and practised separately, in sequence. before the total is put together.
Real life maths opportunities should of course be seized upon, but individual skills like adding and fractions also need enough practice in correct sequence to attain mastery. Once mastered, complicated mathematical tasks are exciting and your child can step back from the tedious mechanical skills acquisition to get a total picture of the problem at hand, rather than get bogged down by the various arithmetical computations, all of which seem mountainous hurdles.
8. But shouldn't children be taught to think, instead of regurgitating facts?
We agree - word problems and applications are extremely important. Don't neglect them! But the place for these is in maths class (or life in general). There is still a need for drill. Just as you cannot write an instruction manual for car repairs without understanding details of how the car works, how grammar and spelling etc. work how the reader might read and receive what you write, and much more, so you can't even hope to predict requirements for your next craft project without basic arithmetic.
If you want to think things through you need tools. You can easily get sidetracked or make errors if you never learned to multiply. The bigger your vocabulary, the higher the thinking skills you can develop. Arithmetic is like a bigger vocabulary. With it you can short-cut the mental gymnastics required in word problems or complex examples, and the logic will be less cluttered. It works! Drill, remember, is required to bridge the gap between understanding and instant recall - it supplies heaps of directed practise.
9. Should I allow my children to use calculators?
Not until they have learned to do it manually, well and then only for long. complicated tasks. They will seldom be required in primary school. There are some games that are fun on calculators, but this is a recreational activity, not an educational one. The things required of primary school maths students should be mastered manually. Even secondary students doing complicated mathematical tasks with the aid of a calculator must be required to estimate the expected answer so that they aren't trusting the machine too much - after all, flat batteries and human frailty can cause ridiculous errors!
10. We don't have a computer. Will our children be disadvantaged?
I don't think so. Computers can be worse than the endless worksheets for wasting time - and they're addictive! Interestingly, the countries with the most computer advances do not have the most computers in their classrooms. A good basic education is a better foundation for life than mere experiences of computers.
Of course, there are good computer programs about which do very nice things and if you do have a computer you may find some useful educational aids to use in your total curriculum, but even then I would limit it, so as not to neglect the basics. The well taught young adult will easily pick up the basics (and the rest) when required.
Before using a computer program, take a long hard look at how much effort, time etc. it takes to learn how much material. Also look at the "entertainment value" and what else is being taught - what do the programs teach about kindness to others, problem-solving (violence?), care of the environment, goals in life?? Consistent doses of bad philosophy may undermine everything else you are trying to achieve.
The best preparation for a future life with computers is to learn touch typing correctly (despite all the hype and hoopla about getting rid of keyboards).
Back: Published Papers | Email us |