M1Maths

Raising Kids to be Good at Maths

Kids can sometimes find maths hard in school. If they get behind in getting the hang of early primary-school mathematical concepts, they tend to having trouble picking up new ideas that build on those concepts, and so just get further and further behind.

The resulting confusion, lack of success and failure in tests inevitably result in disliking maths, and disliking it results in avoiding it where possible, and in not being willing to give serious thought to new concepts and problems during maths lessons, something essential for understanding. This all guarantees ongoing failure.

Kids who have a basic understanding of concepts like numbers, fractions, measurement, money, graphs etc. when they first meet them in school are much more likely to find the ideas they meet to be natural, logical and obvious. To them, maths will be just common sense. They will enjoy it and be willing to think hard about new ideas and problems and will be interested in learning more. This will guarantee ongoing success and maybe a life as an engineer, economist, teacher or whatever, rather than a life in mundane, unsatisfying, low-paid jobs.

Below are some suggestions for things that any parent can do, whether or not they are good at maths themselves, to help their kids be comfortable and confident with mathematical ideas before they meet them formally in school.

Counting

Children's first foray into maths is learning to count. There are two main stages to this: the first is being able to recite the numbers in order; the second is being able to count a set of objects. Ability to count objects always takes longer to develop than the ability to recite the numbers. Children who are beginning to count objects can point at the objects in turn while reciting the numbers, but the two activities tend to go at different speeds. It can help initially if the objects being counted are moved from one place to another as they are being counted and the child learns to count one step with each move.

Once a child is getting the hang of counting objects, finding things in the environment to count can reinforce the skill.

Estimating Numbers

Once the basics of counting are developed, it can be good to ask kids to estimate the numbers of things, like the number of toys in a box, the number of stairs in a flight, the number of slices of bread in the loaf etc. This can be made into a fun competition where the child and the parent both guess and then the child counts to see who was closest. Of course the parent needs to guess a bit wild sometimes so that the child wins at least half the time.

Measuring Length

Once kids can count reasonably, they can start doing simple measurements using informal units like steps or stick lengths. They might see how many spoon-lengths it takes to go from one end of the table to the other. At first, using multiple spoons to stretch the length of the table then counting the spoons is easier. Later, they might advance to using a single spoon and counting the lengths as they go.

Once they have the hang of measuring, estimating the distance before they measure can be introduced in the same way that it was for counting.

Formal units like metre and centimetre can be introduced once kids have the idea of measuring lengths. To a small child, a metre is about the biggest jump-pace that they can manage. They can measure the room or the garden in metres. For the distance to the shop, it might be easier for the parent to pace it while the child counts. For an adult, a normal walking pace is about 80 cm; the largest comfortable pace is generally about a metre. Of course, at this stage, accuracy is not essential, we are just trying to develop the concepts.

Centimetres can be introduced by finding or making objects a centimetre in length, maybe finger widths or pieces of cut up paper. Laying the object to be measured onto centimetre grid paper and counting squares is an intermediate step before the introduction of measuring devices like rulers and tape measures.

Using rulers, tape measures and the like is a more difficult idea, especially if the device is marked in say metres and centimetres or centimetres and millimetres or centimetres and inches. Using simple devices marked in only one unit, e.g just centimetres, should be introduced first.

The idea that 100 cm make a metre, 1000 m make a kilometre and 10 mm make a centimetre can be introduced, though without the expectation of converting between units.

Time

Telling the time on a digital clock just involves reading the numbers. It is good to have an analogue clock (with hands) in the house so kids can learn to tell the time on that too. Apart from anything else, telling the time on an analogue clock requires more understaning of hours and minutes and the relation between them and the length of the day. If the clock has a second hand too, that can add further understanding.

When learning to tell the time on an analogue clock, it can be helpful initially to ignore the minute hand (and second hand if there is one) and just read the hour hand. Even removing them if possible can help. Kids then can see if it's two o'clock or just past two o'clock or getting towards 3 o'clock etc. without being confused about which hand does what. Once they are good with that, the minute hand can be introduced as a way of being more accurate, e.g '10 minutes past 2' rather than 'a bit after 2'. For this, a clock with the minute divisions marked can make things easier, especially for times like 12 minutes past 2. If there is a second hand, that can make things more accurate still, but that takes quite a bit of mental processing and probably shouldn't be attempted too soon.

Another thing that is important is estimating time intervals. How long does it take to walk to the shops? for the toilet cistern to fill up? to play a song? etc.

Money

Cash is going out of fashion to some extent, but it is still around and handling it is a good way to get the hang of money concepts. Pocket money in cash is very helpful. And all the better if it is calculated based on numbers of chores completed etc. so that the amount paid isn't whole numbers of dollars, but involves cents as well. Once kids are used to handling the notes and coins, digital money that involves the same calculations can take over.

Playing shops - with real or pretend money or even digital money is helpful, as are games like Monopoly that involve handling and calculating money. The game 'Millionaire' (M1Maths > Extras for Teachers > Fun and games > Millionaire) is designed to help students deal with money and, particularly, to use percentages and other fractioons in the context of money. This is appropriate in late primary school.

Fractions

They say 4 out of 3 people have trouble with fractions. While that's not quite true, fractions are something that many students struggle with through much of their schooling. Ask anyone who understands them, and they can't really see what the problem is. They seem to give a lot more trouble than their degree of complexity warrants.

Schools do tend to treat them fairly formally and abstractly and this can result in some students not seeing how they connect to real things. What they learn is then just memorised rather than being understood. And of course, things that are just memorised are then just forgotten.

This whole situation can be avoided if children are familiar with the meaning of fractions, expressed in the various ways, before they meet them formally. If they have a good understanding of what they are, the techniques for converting, adding, multiplying etc. will make sense, will be logical and obvious, and will not be forgotten.

Familiarising children with fractions isn't difficult and there are many opportunities without having to make any special effort.

'Are we there yet?' usually gets a response like 'Not Yet' or 'Nearly' or 'Wont be long'. But a response like 'Were about 60% of the way there' (or '3 fifths' or '0.6') is much more educational. How much of the cake is left? How much of your dinner did you eat? You have to eat 9 tenths to get dessert. There are lots of situations where the answer could involve a fraction rather than something like 'mostly' or 'hardly any'. Kids can be made quite familiar with the meanings of the vaious ways of expressing fractions without having to do anything like converting or adding them. With this familiarity, converting and adding will be common sense when they do meet them.

For parents who aren't as confident as they might be with fractions, looking at Module N1-2 of M1Maths might help. Also, don't worry too much that some of the things you say might be wrong. Mentioning fractions even if not completely accurately, is much better than never mentioning them. And the kids might really enjoy telling you you got it wrong. That is a wonderful confidence booster for them, even if it might not be so much for you.

Probability

When playing games with dice or cards, kids can be introduced to the idea of how likely something is. For instance, are you more likely to get an 8 or a 3 if you roll two dice, are you more likely to pick a spade or an ace from a pack of cards.

The probability of picking an ace from a pack of cards is the fraction of times it would happen if you tried it a very large number of times. All 52 cards are equally likely to be picked and so, in the long term, all would come up about the same fraction of the times - about 1/52 of the times. There are 4 aces and each will come up 1 time out of every 52 in average. So aces will come up 4/52 of the time on average. We say the probability of picking an ace is 4/52. On the other hand, the probability of picking a spade is 13/52, so there is a better chance of picking a spade.

This encompasses the basic ideas of probability that kids will meet in high school and maybe upper primary school. Though it might seem reasonably obvious, kids who aren't exposed to this idea when younger do tend to have trouble with it. The basic definition of probability as 'the fraction of times something will happen in the long run' is an important base for all further understanding and it can be a very helpful thing for kids to be comfortable with this as soon as they have a sufficient understanding of fractions. In fact this can be another avenue for getting them familiar with fractions.

Finding the probability of picking an ace relies on the fact that there is no fifference between any of the cards that will make any one be more likely to be picked than any other. This allows the probability to be calculated exactly. In formal terms, this is 'calculating probability using indifference'.

Indifference can't always be used though. The probability that a matchbox will land on its edge cannot be calculated this way because there is a difference between the six sides that make some more likely to land face-up than others. We can only estimate the probability of a matchbox landing on its edge by trying it a few times. This is called 'estimating probability by experiment'. If we toss the matchbox 10 times and it lands on its edge twice, then we can estimate that the probability of landing on its edge is 2/10 (or 0.2. or 20%). This is probably not all that accurate because if we do the 10 tosses again, we might find it lands on its edge 3 times, or maybe zero times, giving probabilities of 0.3 or 0. If we toss it 100 times and it lands on its edge 12 times, then we can be reasonably confident that the probability isn't 0.3 or 0, but it might still be 0.15 or even 0.2. The more times we toss it, the more reliable the result will be. In theory, to get an exact result, we would have to do an infinite number of trials and that is impossible, so experiment only ever gives an approximate probability.

Probabilities of different results when two dice are rolled can be calculated by indifference, but it is more difficult than with the cards. Experiment can be used instead to get a rough answer. It will quickly be seen that an 8 is much more likely than a 3.

Even experiment can't always be used. What's the probability that Mum will notice the stain on my bedroom carpet? We can't use indifference there and we can't try it a lot of times. All we can do is guess. If the stain is very obvious, we might guess 95%; if it's only just visible we migh guess 10%. Everyone can put a probability on there being life after death. To an atheist, this might be 10%; to a committed Christian, it might be 95% or even maybe 100%. Guesses of probability can be very subjective, but are still meaningful.

Probability is maybe the branch of school maths least well understood by teachers. Even textbooks often don't do a good job. If students have confidence in their own concepts of probability, this can be very helpful.

Statistics

Getting kids to look at and interpret various kinds of graphical displays (graphs) can be beneficial. Schools tend to teach kids how to read certain types of graphs (like line graphs, bar graphs, pie charts etc.). But graphs found in newspapers, textbooks and online are much more varied. The most basic and essntial concept of graph reading is not how to read any particuar type of graph, but rather, how to take an unfamilar type of graph and to work out how it works and how to read off the information it contains.

Some examples of non-standard graphs can be found in the M1Maths modules S1-1 and S3-2. Other examples can be found in books etc. Googling 'Graphs' will provide plenty more.

Another concept worth developing early is the idea of average. At first, it should just be seen as a typical amount or an amount that's not unusually large or unusually small. Examples that might be used include the average height for a 7-year old, the average age of people in a group etc. Only once this idea is well established should calculation of the average (mean) be embarked upon. Median and mode can wait until even later.

Rates

While driving, talk about what 60 km/h means and ask quetions like 'How far would we go in 2 hours at this speed?' and 'How long would it take to go 300 km? 30 km?'. When shopping, if bananas are $4.99 a kilo, maybe round it to $5 and ask how much for 2 kg, 1.5 kg and so on.

Rates are really just common sense and answering questions like this tends to get kids to see them as such and thus to be able to deal with rate problems with no special knowledge.

The important thing about a rate is that it is the amount of something for each one of something else, e.g. the number of kilometers for each one hour or the number of dollars for each one kilogram. That's all the 'per' or '/' means.

Ratios are often reated differently, but they are the same except that they are not always for each one. If the ratio of boys to girls is 2:3, then there are two boys for each three girls. How many boys would there be if there are 12 girls? 12 is 4 lots of three, so there will be 4 lots of 2, i.e. 8 boys. Just like rates, ratios are just common sense and no special techniques need to be learnt.

Problem Solving

I have always had a bad memory. Ask me to remember a mathematical formula or method, and I usually forgot it fairly quickly. So, to have any success with maths when I was young, I couldn't rely on memorising methods. The only way I would be able to solve the same problem later would be if the method for doing so was common sense and obvious.

Though most people have a better memory than me, most students forget mathematical algorithms and formulae fairly quickly. It is partly for this reason that schools generally test students only on what they have learnt in the previous few week.

Getting kids to use common sense to work out their own way of finding answers to questions involviing maths is much better for them in the long run than getting them to memorise techniques. There are a few things that have to be memorised, like the fact that 6x7=42 and that sin = opposite/hypotenuse, but the less students can rely on memory and the more they can use common sense and problem solving skills, the better their maths will be in the long run.

Problem solving gets students to practice using common sense to find their own ways of working things out. The best mathematicians are always the best problem solvers.

Try to get kids to work out things that they haven't been taught a method for. If they can do it once, they will probably always be able to do it. Solving problems is a skill which requires practice and confidence. Confidence is important in getting them to persist in the face of initially not knowing what to do. To ensure that confidence grows, make sure that most of the problems the kids attempt are ones they will have success with.

There are lots of things in everyday life that lend themselves to getting kids to work something out. But there are some problems in M1Maths > Modules > Skills > Problem Solving. Note that much of the skill of mathematical problem solving can be learnt through non-mathematical problems - puzzles and the like.

To reinforce my point, here are a couple of true stories from my experience.

The first involved two of my children, Jake in Year 2 and Melissa in Year 4. The car had broken down and waiting for someone to come and help us. I asked Jake what 1000 minus 60 was. He put up some fingers, mumbled some words and then said '940'. I later asked Melissa the same question. (She wasn't there when I asked Jake.) She thought for a moment, then said 'I need a pencil and paper.' I provided them and she wrote:

  1 0 0 0

-       6 0

She took 0 from 0 and wrote 0

  1 0 0 0

-       6 0

            0

Then she tried to take 6 from zero, thought for a moment and said 'You can't do it'. That was it for her. I asked her if she thought there was an answer and she said 'Yes, but I can't work it out'.

It occurred to me that, in the two years between Year 2 and Year 4, the main thing Melissa had learnt in maths was that there is a right way to solve every problem and, if you can't remember the way, you can't solve it. This hadn't yet happened to Jake.

The second story involves a Year 8 class I took for maths at Springwood State High School one year when I was doing regional advising work. Barbara, the head of maths allowed me to spend the whole year developing the kids' problem solving skills without explicitly teaching them any methods. At the end of first semester, Barbara suggested we do the same will all the Year 8 maths classes. The next year, in the Year 9 division of the regional Maths Team Challenge competition, Springwood got 4 of the top 5 places.

Geometry

There's a lot of very esoteric stuff taught in Geometry. Very few people will ever need to know that the diagonals of a rhombus bisect each other at right angles. However, describing position is a useful skill. Giving kids practice at measuring and/or estimating position using coordinates (distance in two directions at right angles), using distance and bearing, and using latitude and longitude is of value.

The idea of coordinates is helpful not only for describing position, but also later for graphing and algebra.

Size

The idea of size is often used very ambiguously in everyday life. We might say that the earth is 4 times the size of the moon. That is true in the sense that it's diameter is four times that of the moon. But its surface is 16 times as big and its volume is 64 times as big. A 100 kg man might be said to be twice the size of his 50 kg wife. But he's probably not twice her height. If he were twice her height and they were similar in shape, he would weigh 8 times what she does.

In specifying size, we need to be clear about whow many dimesions we are considering. For a piece of string or the distance between to objects, we are considering one dimension and the size is referred to as a length, measured in metres, centimetres, kilometres etc. For a block of land, we are considering two dimensions, length and width, and its size is referred to as its area, measured in square metres (a square meter being a square 1 m by 1 m), hectares, etc. For a fish tank or the air in a room, we are considering three dimensions, length, width and height/depth, and its size is referred to as its volume. Volume can be measured in cubic metres, cubic centimetres etc. or, more commonly in litres (a litre being 1000 cubic centimetres). Talking with kids about the sizes of things with different numbers of dimesions can help develop and reinforce these concepts.

The idea that, if you double the width of a square, you multiply it area by 4 and that, if you double the dimesions of a 3D shape, you multiply its volume by 8, can be hard to grasp without plenty of exposure. Drawings and models can be helpful. This concept will be essential later for understanding conversions of area and volume units. (See Module M3-4 for help with this.)

Algebra

Kids at school are taught a lot more algebra than most people will ever need in life. But it needs to be learnt to get good results in Maths. What's more, the way it is taught is often very abstract and detatched from the real world and day-to-day life.

The Algebra strand in M1Maths takes a more down-to-earth approach. Working through some of the algebra modules in M1Maths before each Year of high school can help to make algebra make sense. Working through the Level 1 modules before Year 7, the Level 2 modules before Year 8 etc. can be helpful. Of course, kids can go faster if they are interested.

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