First of all I apologise for the long silence. I was teaching as a substitute special-ed teacher. While it offered me a fascinating new look into the Finnish educational system, albeit from the narrow viewpoint of a single school, it did take up my time. I will be writing about the observations I have made this year visiting schools and teaching there at a later date as they still need some processing.
Today I want to write about opening up worlds for your children. I almost wrote that it is about teaching young kids, but in fact, this is not what I want to propose and, when concerning my own child, it doesn’t really work so well. But this difference in approaches and world-views is something that needs to be addressed first.
Most of us in the Western world have an idea about teaching and learning that derives from schools. And what is more important that it often derives from what schools were like when we were young instead of what they are now. Few of us are lucky enough to be able to see what they are actually like. My observations in Finland is that they are better. I have understood that this is not the case in all places. But this is where our understanding of learning and teaching comes from, so if I talk about teaching math to a child, many will have a mental image of a classroom, a teacher in front of a class, a workbook and calculus. What I have in mind is something vastly different.
What I have in mind is open to be used in any way that suits your child or the children in your care. It is about opening up possibilities, introducing to phenomena that you find fascinating. It is about paying attention to the world from a mathematic viewpoint.
Now, if one is scared of math (thanks school!) this can sound very intimidating, but I promise you it is not. I will not ask you to do anything you find uncomfortable, as that feeling will pass on to a child, so please don’t do anything like that. What is needed most of all is openness and willingness to see math not as calculus, trigonometry or algebra, but to see it as discovery, adventure, making sense of the world and noticing details. And for this, the vary basic skills in calculus are sufficient. If you can recognise numbers, do addition and subtraction with numbers between one and ten, understand what multiplication and division mean, and differentiate between basic geometric shapes (triangle, circle, sphere, cube etc.), you are good to go. And actually, I am not asking you to take action based on this text. If you decide to do so, please let me know how it is going. If you find other ideas to what is presented here, please send me a line.
I am writing about this as this is something I do with my own now 4-year old. I am not perfect at it and on most days I forget this whole thing. But for her it is better that way, because she has a very cunning way of noticing when I get into my teacher mode. That is usually also when her excitement dies out. So I am learning and that is also why I am writing this. One learns best when teaching it.
So now that we know what I am talking about, the next question of course is why. Why should a parent or a caregiver open up the world of math to a child? Isn’t that the job of the school system? Yes it is, but even though we talk about schools trying to include more individualised elements into learning, they are far from being able to be fully individualised. Their way of teaching may suit your child or it may not. Also, and even more importantly, schools tend to focus on calculus, trigonometry, and algebra, and on giving the tools for solving problems, and understandably so, because to be able to calculate well, one needs plenty of practice, but that is not all math is. Still when looking at it from the viewpoint of the pupil, it may look like it. But if they see even before they go to school that math is not about just doing additions, but finding solutions, then, hopefully, the teaching will not put their interest out so easily. Also, while I am a fervent defender of equality and every child’s right to learn and their right to access to quality schools, I do also believe in trying to help mine with anything I can. Then the question is, why not. Math is nowadays nearly required in all professions that bring in big money and it is needed in many professions that previously did not require it. Even cleaners need to be able to dilute cleaning agents properly. Math is everywhere in our society. It is solving problems and making new openings.
So how? There are many ways and I am inclined to believe there are as many ways as they are people doing the discovering. But I’m about to give some ideas to get started with and to build upon based on the child and the parent or caregiver.
This first thing you can do with a baby and any child from that on up. This is about noticing and saying out loud the amount of things. Instead of saying: “honey, look at the dogs”, say instead “honey, look at the three dogs”, or five birds, or two flowers etc. A baby’s and young child’s brain is very malleable and they pay attention to what their caregivers pay attention to. By noticing and voicing the number of whatever’s, their amount, you are showing that child that the amount matters and it is something to pay attention to. Another thing that also happens is that the word signifying that number and the amount that is referred to by that word are connected so when the time comes to look at figures like 2, 3, and 7, their consciousness already is familiar with what two looks like making the connection to the figure easier to make as the concept is familiar.
The second thing that I think can be done with toddlers at least and anyone older is to count out loud. A parent can count out loud steps the baby walks or the time it takes for them to get dressed, or how many Dublo’s form a tower the child just made etc. One can play hide-and-seek and count to ever larger numbers. One can count almost anything. Around this time, in toddlerhood and older, a parent can also read books about numbers and amounts, which is a third thing. These teach the figures we associate with amounts. These are all vary basic skills: being able to recognise amounts and quantities, being able to know what comes next after a particular number and recognising the figure.
Next, and fourth, is basic calculus of addition and subtraction. While this is not about learning to do calculus at a very young age, it is more about teaching the concepts of addition and subtraction and familiarising the child to the terminology. With this, the younger the child is, the more concrete this needs to be. They cannot be abstract about this. Their brains are not ready for that. I started with my daughter when she was three years old and we calculated with fingers and we still do. This was a game we played for instance in the car that she would as: “mommy, what is three plus two?” and I would count with my fingers and she would count with hers. Now we have reached two digits so my fingers are not sufficient anymore for everything, but then we use hers as well. Addition is fairly simple to understand. With subtraction we also used fingers and I would literally show the numbers. For example 3-1: I had 3 fingers up in the left hand and 1 in the right and that 1 finger in the right hand would come and take down one finger in left hand and then she could count how many were left. Subtraction is harder to grasp and can take a bit longer.
This is basic calculus and the important thing here is not to push it. It is so easy to feel good about having a child do calculus at the tender age of 3 (for the parent as often the child really couldn’t care less), but doing more than the child wants has the severe danger of putting the child off the topic altogether. As long as they are engaged and interested, then do continue, but the moment they start to lose interest, it is best to stop. At least my child does not like to feel pushed or being taught to. It is more about doing it every now and then and when ever the child wants and there is a natural opening in what you are doing together. That is enough to introduce the concepts and the more you can do this connected to playing together and having fun, the more these concepts, and math in general, are connected with those feelings safety, fun, joy and interest.
And here comes the important bit, which is showing that math can solve problems. This is the big one, because it shows what it is for, the “why” of it, which is the fifth idea. So instead of just adding up quantities that pop into someone’s head, why not count the number of bread slices are needed for breakfast if mommy usually eats two and the child eats one. Or how many slices of cheese are needed. or how many shoes leaves the house that morning. Or if daddy eats one egg then how many are left. We can count practically anything and that is what most of us use math on every day even though we don’t always think about it. But verbalising this process helps introduce it to kids and they pick it up by following us.
Once this process is familiar, then one can go to topics such multiplication and division. Also here it is far more important to get a grasp of the concept than it is to do it abstractly with the numbers. So for instance the example of cheese slices on bread slices. One morning we had four pieces of bread and I asked my daughter that if we put 2 slices of cheese on each, then how many slices would we need. This is basic multiplication 2×4. Of course she didn’t know, but made a guess of four if I remember correctly. I counted the number of slices I cut from the cheese and she let me know when there was enough and then we counted them on the slices of bread and came up with 8. After I just verbalised the math bit of two times four is eight. The more concrete it is the better and the more it is based on inquiry and curiosity about solving a problem the better.
Also division works the same way. My daughter wanted to have 12 scrunchies in her head. I calculated for her that if we do it in two ponytails then it is 6 scrunches in each. But how many would it be if we did three ponytails. And she divided up the scrunchies correctly into three evenly sized groups. Even a baby can tell if there is an equal amount or more or less so quite a young child can solve these kinds of problems.
The sixth idea I wish to share is play. This can be done for instance with geometry. One can get started at first by noticing and verbalising what each shape is. Then the fun begins. I ran across this game where you could draw almost anything using basic geometric shapes. So why not play a game where you cut out basic shapes from a cardboard and explore with your child what kinds of things you can create with them. Or pick a few shapes, like a circle, a triangle and some lines (or anything for that matter) and see how many different shapes you can make with them. This, finding many solutions, is a really good exercise to do, because in math there often are more ways than one to solve a problem and this is another aspect that gets too little attention in most school classes. Or you could play with the ideas of bigger and smaller to see how you can best hide items under different sizes of containers or which can be placed into which. Or you can play with water and familiarise the child with how much water can be placed into different sizes of containers and how many smaller sized containers full of water can empty their contents into the bigger one.
In conclusion, math is a big part of our everyday lives. By being conscious of it and verbalising it, we are teaching the concepts to the children that so attentively notice everything we do. We make it meaningful and relevant. We show them that math is more than 3+5. We can make it fun, engaging and interesting. By being conscious of this and using it by verbalising the math we do in our heads, we are giving the child something that can be achieved later on, but what seems to come naturally through practices like this. Try these out, use them in any way you want, adapt and personalise as always and have fun with math with kids. It is the best intro they can have into something that is everywhere in our societies and everyday lives.
Copyright text and images: Satu Korhonen. You are free to try these experiments out, use them in your teaching. But instead of copying the text or images, link back to this page.