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Maths Paper

Teaching With Love, Not Fear

One of the most enjoyable and fulfilling teaching tasks you have ahead of you as parents is helping your children achieve mastery in maths. It may also be one of the easiest! You use it every day of your life and therefore have never had a chance to forget it like you may have forgotten other facts you learned. Maths is so straightforward to teach: there is only one right answer and it only has to be learned once. Maths reflects God's Nature as no other discipline does - His order, His infinity and yet exactness, the patterns within patterns, the simplicity and complexity all at once, the clear choice of right and wrong. The learning of maths also displays our attitudes very clearly - whether we are diligent, careful, obedient, or whether we lust after entertainment, easy gains, or our own way. Thus it makes clear what is not so easy to discern in other areas of child training. For all these reasons you should take a concerned interest in your child's maths mastery.

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!


First, we need to overcome our fear! "Perfect Love casts out fear." Pray, love God, confess your fear (and laziness as a child confronted with arithmetic?) and hand the whole area over to God. Focus on your child, his future, his need for basic maths, and your desire for him to be confident and capable in these basic areas so he can progress to higher thinking. Then think how he will reach that goal. There is no easy way. Yet no step on the path is insurmountable on its own. It just requires consistent guidance. Are you unselfish enough to provide that, for his sake? Will you guard against his desire for self-gratification - keep him at it when something else seems more appealing at the time - so that he does have an appealing future?

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.

Efficient Teaching

In this paper I will concentrate on the forgotten art of drilling, with only some comments on the maths class. if your child is at school, you may need only to concentrate on this daily "zap," except on those occasions when something more pops up. If you are taking full responsibility for your children's tuition, you will probably gain more understanding of the efficiency of maths courses and feet more confident about eliminating pointless "busy-work" which achieves no educational benefit, after drilling.

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.

Secondary Maths

When arithmetic is mastered the rest of maths is fun. There are lots of interesting puzzles that they can learn to solve in a variety of ways. The only thing to avoid in secondary maths is systems that compartmentalize the various maths disciplines to the point that students forget previous work. The kind of program that does a chapter on algebra, followed by the geometry work for the year, then measurement, indices, Pythagoras, vectors and all those others offers the reluctant mathematician little hope.

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!)


All preschoolers' "studies" can be done in everyday life. There is no need to buy lots of expensive books, gadgetry (though, of course, some of it is very nice) or read up on the latest "thing" in the women's magazines. To the contrary, a good preschool education depends solely on you - how much you talk, read and do with your child (this is a totally separate topic - ask for our paper on Early Education, due out around August) if this is not already clear to you.

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?

Primary Arithmetic

Primary school maths is basic "arithmetic": addition, subtraction, multiplication and division. It is strictly sequential - misunderstanding of addition will lead to problems all down the line so there is NO point in progressing for any reason until that is grasped. Speed is not essential at first but if it doesn't develop steadily there may be a shortfall in previous learning. Don't be impatient!

Teaching Addition

Once counting is achieved, addition is a logical extension. We have six in our family, today a family of four is visiting - how many forks will we need on the table? I borrowed three books, you borrowed six, he got four. How many do we have to return? Lots of practice with real life examples, or made up ones, with or without manipulatives, help your child to see the relationships and conceptualize the solutions.

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.

Knowing When To Stop

That may be all you do for that day. Fine! Never get to the point where either of you get frustrated of impatient. Five minutes per day of this intensive work is plenty for most. Some can go longer but pushing too hard means you lose what good you've already done. You may also cause a long-term fear or sense of frustration with Maths. Enthusiastic children do better with two, occasionally three, five minute sessions than with one longer one.

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.

Daily Routine

Each day you revise the previous several sessions' work. First the most recently learned "family" alone and in order (in this example the +2's). Then that family shuffled. Then shuffle in the +I's and finally add the +0's so that you now have 27 cards to revise. This may be too much all in one breath, so do three groups of nine, timed separately and add the three times together. The groups of nine will be in no particular order - just shuffled and dealt out as if playing Snap! Eventually your child should be able to do the 27 in fifteen seconds but you decide when would be the best time to proceed - remembering that your child will not be fooled or benefited by moving on too quickly. A true sense of achievement comes from achieving. Adding further facts will not reinforce the previous ones but merely confuse them.

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!

Domino Drill

You can make all this number family work easier with a set of double nine dominoes. Most of you will have a domino set on your shelf somewhere but it will probably be a double six set, ie. the largest number on each half is six, so there are 28 pieces in the box. Double Nine's go up to nine on each half, so there are 55 tiles and they neatly cover the addition facts in a handle-able hard-wearing tile. You can also play great games and have lots of family fun with them! They're not easy to find but we're hunting - cost is up to $40 in the game shops but we're hoping to find them at a much better price.

Adding Nine And Eight

When your child knows +0, +1 and +2 really well, move straight to +9 next! it's the next easiest! To add 9 to a number, say one less then add "teen" eg. 9+5 = four (one less than five) - "teen" = fourteen. 9+7 = sixteen; 9+3 = "two-teen" but we say twelve (you tell your child). Again, practice in order, then shuffled. Then when they are fast, add in the +2's, then the +1's and then do the whole 34 so far, in four random groups in a total of twenty seconds.

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!

Doubles And Neighbours

Adding 8 is not much harder than 9 - this time you count back two and add "teen" eg. 8+6 = 4-teen, 8+9 = 7-teen. This adds five more number facts (the other five are already in there). Practice with all ten of the +8's, then shuffle in the other facts as before. You will by now have been going for a few weeks and a growing sense of achievement should be helping both of you along - if you have remained patient.

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 Rest

Next, I like to emphasize the combinations which add to 10 because they are so useful to know. We've already done 9+1, 2+8, 5+5 so there are only 3+7 and 4+6 to do. Spend some time asking "3 plus what makes 10" etc. This is actually subtraction but it is so helpful to know that I think it worth emphasizing early. Shuffle the remainder of the dominoes or cards: 3+5, 3+6, 4+7 and 5+7 with the ones that add to ten and let your child simply answer "yes" or "no" - they do or do not add to ten.

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.

Building On Basic Addition

Once this speed is achieved it needs to be maintained. This means just doing the whole 55 again, once through, to make sure that the skill is not getting rusty. The frequency of such revision varies with age (eg. once a week for under eights, once a month for over tens and you adjust according to their success).

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.

Play Snap

At any stage you can revise the learned facts with games like Snap! Deal out the learned cards/dominoes and take turns as usual to lay one on the table. If two (eg. 9+5, 6+8) add up to the same number, the first to say "Snap" wins all the cards on the table. First to lose their entire stack loses the game, if you want winners and losers; but it's okay to play without - nobody picks up cards/tiles off the table.


Dominoes really ask for you to do pattern work - lay them out in a triangle (as at right) and look at all the "sets" you've arranged: the +0's, +1's, +2's etc. each going in two directions; doubles on the diagonal; and where are the neighbours? Look where the ones that add to 10 ended up; likewise find all those that add to four in another line. How many other patterns can you find? This sort of fun is done in the rest of your maths time - not in your 5 minute drill - but enhances the drill because it makes an association with interesting, family-involved, non-demanding yet challenging fun with your drill tool. the domino set.

Dice Drill

Once your children have mastered the various addition-fact families, one way to randomly check the maintenance of that skill is with dice. once again, you can use the dice you already own from various games. If you have some special ones, all the better. We have some games with six-sided dice containing the numbers 0, 1, 1, 2, 2, 3 or the numbers 7 - 12, instead of the traditional I to 6. Have you already guessed? We used the low numbers dice with a normal one when our children were in stage one: +0. +1 and +2. When the 3 came up on the special dice, it was amazing how few times the other had 3, 4, 5 or 6 - but if one did come up that wasn't learned we weren't fussed about them not knowing the answer quickly. Of course the high numbered dice is great for drilling the harder ones only.

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.

Dice Method

Choose two dice, (appropriate to what has been learned to date), shake, then PUT them on the table. Remove your hand, slap the table lightly, pick up the dice and roll again, but keep your hand over the new combination until the previous answer is called. if your child is still trying to work it out, tell him the answer. Write it on a sheet, get him to say it all (eg. 5+4=9), say it again yourself, then move on. The aim of drill is to improve speed and recall, to avoid calculating the answer. However, don't get impatient. It will come!

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.

Teaching Subtraction

If Addition is learnt well, then reversing the process should be easier. However, these strategies (introduced and practised in the maths lesson) give your child extra mental paths along which to quickly retrieve answers.

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

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

Teaching Multiplication

Multiplication must be learned in maths class, usually by addition of several numbers eg. 3+3+3+3=? Make sure your child thoroughly understands the concept before drilling it. Again, when you teach, do it one "family" of facts at a time and ensure unhesitating recall before proceeding. One possible order you could introduce "families:" x0, x1, x2, x9, x5, squares, x3, (squares -1), x4, x8, x7.

Zero, One Or Two

x0 is easy: all the answers are nought. xl is easy too - but intermingle them and see how they go! Then add x2 which of course is the same as Doubling in addition.

Nines Are Next

When they are able to throw answers back at you on all those, jumbled together and randomly arranged, teach them x9 next. Yes - you guessed it - it's the next easiest. To multiply a number by 9 (eg. 4), simply subtract 1 from it (thus 3) and subtract that number from 9 (so 6) . Now put the two together: 36 is the required answer. Try it again with say 8. 8x9 = (8-1 = 7) and (9-7 = 2) giving 72.

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.

Fives, Squares & Threes

After x9, x5 is easy and also very useful for work with money and clocks. You already know the pattern with x5. Next the squares are good to know - they are used so often and worth emphasizing (4x4, 7x7 etc). Then add x3, telling your children we've already done 45 - so-o-o many! Only 10 to go!! If you haven't already, have your child chant with you: one 3 is three, two threes are six, etc. and also just count by threes: 3. 6, 9, 12

A Taste Of Things To Come

Tell them that when they get really good at maths facts, they'll be able to get on to the really interesting maths like algebra!! Say it with enthusiasm - it is a coveted reward for good arithmeticians! If you can carry it off, it will motivate them both now and later. We've always done this with our boys, and whetted their appetites with little snippets like this:

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 = x2. 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 x2 ie. x2 - 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 (x2 - 1)'s until they're off pat.

The Last Few

Only four facts to go - 4x7, 4x8, 6x7, 7x8. Learn them. 56 = 7x8 (5, 6, 7, 8): show them the sequence of digits. Make up a rhyme to remember those last few eg:

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.

More Later: Ten, Eleven, Twelve

I'm sure you'll be able to build on all this - add the x10, x11 and x12 tables (for the x12, show the similarities with the x2 eg. 7x12=84 and 7x2=14; in fact 7x12 = (7x10) + (7x2) which is another algebraic sentence to show if you wish). Then begin division and keep going.

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.

Mixed Drills

Holey Cards

After each operation (+ - x ÷) is mastered you could try Holey Cards. These are little cards with the maths facts 0+0 to 9+9 printed randomly. under each sum is a hole for your child to write their answer onto a jotter sheet or piece of scrap paper behind. They're quite difficult to achieve in the specified two minutes, but inexpensive ($6 the set) and easy to administer. There is one card for each operation and each has 100 questions.

Blumenfeld-Style Drill

Another way is to simply write the facts out on a large sheet, cover the answers, read them out and have your child say the answers before you get a chance to uncover them - a great game for when facts are nearly learned. In this way, they see, hear and say each fact.


An excellent, though slightly more expensive drill system is Calculadder. These colourful worksheets take your child right through their primary school work, building that bridge from understanding to instant recall. There are six books, each containing 12 copies of 16 drill sets or "Learning Vitamins" (a tiny dose regularly does a wealth of good!) It is easier for children to "achieve" their targets than with Holey Cards, but then, you'd want some advantage after spending so much more money. You can use them for all the family day after day, year in, year out, and you really will see them progress (they come with a copyright release for families so you can make extra copies as necessary, and are also available as photocopy masters for both families and schools). They are particularly easy to administer and correct - a real bonus for busy mothers who do not wish to compromise educational goals.


Occasionally vary the routine with something altogether different. Ask your child to tell you something unreasonable, like:

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.


1. If children are learning all this in five minutes a day, why have maths class as well?

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).


There is no good reason why your children should dislike, fear or fail at maths. other subjects may be difficult to learn and remember, but maths is like the straight and narrow - there are no choices to make. Step by step you go - always forward. With just a few minutes of your full attention daily. you give your children hope for the future. In ten years' time when they seek employment and others can't add up, their teachers will be nowhere in sight, but you will know that you were able to make a difference that counts "and your children will rise up and call you blessed."
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Ps 78:4
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