I use mathematics manipulatives a lot to teach mathematics at home. For very young children, who are very tactile and visual, the best way to approach maths, which is a pretty abstract concept, is to go from concrete towards abstract. The Singapore Maths is based on this concept. First, you introduce the concrete. After that, from concrete you move on to pictorial. Finally, from pictorial you move to the abstract, which are the numbers. Mathematics manipulatives fit into the first step of the whole process.
What are mathematics manipulatives? Essentially, they are tools you use to teach mathematical concepts concretely. A very simple and common manipulative is counters. Counters can be anything and do not have to be fanciful ones you buy from educational stores. You can use normal household items, such as beans, ice-cream sticks, straws, bottle caps, toothpicks, cotton buds, buttons, etc. Basically, they are anything that you can count. They are used to teach simple counting to very young children so that they know that 1, 2, 3, 4….are not merely sounds of numbers that you rote recite, but they actually mean something. They represent quantities. Visually, they are able to see that 1 is less than 2, 2 is less than 3, 3 is less than 4, and so on, and the way you arrange the counters will not change the actual quantity. They can also be used to teach very young children how to sort and classify according to colours, shapes and other attributes. Later on, they can be used to teach addition, subtraction, multiplication and division. They can also be used to teach odd and even numbers, as well as patterning and graphing. In other words, they are simple yet flexible, with wide ranging use.
These days, you can find all kinds of manipulatives in education stores. There are practically manipulatives for every kind of topic and situation. Some are really fanciful and, of course, expensive. The sales people in the stores will try to convince you that you need all kinds of stuff to help your child learn better. However, don’t be surprised if they are not very good at explaining how the manipulatives works. I find that the simpler the manipulatives, the harder it is for the sales people to explain what the manipulatives are used for (beyond the obvious). Yet, in my experience, the more useful manipulatives tend to be the simpler ones. This is good news because it means you really don’t have to spend a lot of money to buy a lot of fancy stuff. Try to resist the urge to buy “themed” ones no matter how attractive and fun they look, e.g. supermarket themed counters in the form of canned food with food counters inside complete with a cash register and shopping trolley and whatnots. While you may improvise and use simple toys, such as dinosaur or animal figurines, as manipulatives, it is usually not a good idea to buy manipulatives that blurs the distinction between a teaching tool and a toy. Besides the problem of the kid being distracted by the play factor from actual learning, such manipulatives are usually limited in their use. You get better mileage out of something basic and simple that is flexible enough to be used for a wide range of topics at a fraction of the price.
Here are some of my favourite manipulatives. I actually have quite a few manipulatives due to the education tools business I used to run a few years ago. However, there are only a few that I use the most.
Snap Cubes (Or Linking Cubes)
Snap Cubes or Linking Cubes are very versatile and can be used to teach a wide range of subject. You will see this in Singapore Primary School mathematics textbooks. You can practically use this right through to Primary 6 level, so you do get a lot of mileage out of this. Some of the topics you can use this to teach include : patterning, sorting, grouping, addition, subtraction, number bonds, multiplication, division, ratio and fraction. Using the Model Method to solve maths questions is a very big topic in Singapore Maths, and these cubes can be use to create a concrete form of the model.
These cubes usually come in 10 colours, 10 cubes for each colour, making up 100 pcs altogether. You can, of course, buy more according to your needs.
The the colours are useful for visual discrimination. E.g. 2 of blue and 3 of red makes 5. This is a good way to teach number bonds, part-whole model (Primary 1 syllabus) and simple addition.
Base 10 Blocks
If I can only own one set of manipulatives, I think this will be it. Base 10 blocks are so essential for the teaching of Place Value, and place value is such an integral part of maths, you cannot go wrong getting a set of Base 10 blocks. You can also see this a lot in the primary school maths textbooks. You use Base 10 blocks to illustrate that 10 ones make 1 tens, 10 tens make 1 hundred and 10 hundreds make 1 thousand. It is the same as the Montessori Golden Beads. The kids learn why they have to “borrow” or “carry over” when they do simple addition and subtraction. This is called “re-grouping” and is in the Primary 1 syllabus. (I remember when I was young, I was only taught the “method” of “borrowing” and “carrying over” without explaining why. Basically, we only needed to memorize the steps and execute as necessary. These days, maths are taught so differently and children are not expected to compute like a calculator but to understand the concept behind the steps.)
My experience may be limited. Nevertheless, it is almost a given that if a kid cannot understand the ‘carry over’ and ‘borrow’ part of addition and subtraction, using Base 10 block to explain will help. For me, I made up a little story based on Max Lucado’s children book character, Punchinello and the Wemmicks, to go along with the use of the Base 10 blocks. It makes it easier for the boys to get it. I also printed “Houses” to go with the Base 10 Blocks.
I think most people tend to dismiss these pattern blocks as unimportant, “play” sort of stuff. I know I did. In actual fact, these are very important especially for preschool age kids to develop visual spatial discrimination. I didn’t know how important this skill is for maths until I attended a talk on Singapore Maths! Besides learning about the attributes of various geometric shapes, these can be used to teach symmetry, fractions, patterning, sorting, geometry, etc.
Doing things like this :
This is Primary 1 Maths. This skill is not confined to lower level maths only.
I included this because I really like them, not because it is a very important, must-have items. Transparent counters are more versatile in that they can be used like normal counters, and at the same time, they can be used on things like Hundred Board to teach concepts like odd and even numbers, skip counting, etc.
This can be really useful for teaching fractions. I have them because I had them as stock last time. However, you don’t really have to buy such a set. Even if you wanted a fraction set like this for the teaching of fractions, you can easily get templates of it off the internet, print them on colored stock card, laminate and cut to use. Just google terms like ‘Fraction Circles Templates’ and you will find some you can use.
How to Use Maths Manipulatives?
I know that many people are a bit turned off by manipulatives because they think it is difficult to use and they have no idea how to use them. The fact is that you don’t have to be a trained teacher to know how to use these tools and it is not difficult to learn how to use them. Some of it is quite intuitive, and as you use them more, you will get the hang of it and also start getting ideas how you can use them to teach other things.
There is one thing you cannot do with manipulatives :
You cannot throw the sets at your kid and expect your kid to go figure out how to use them and you expect these manipulative to be magical cure-all that will cause your kids’ maths results to improve by leaps and bounds this way.
When I was selling these educational stuff, one of the things that parents told us all the time was that they have bought such things before and their kids had no interest in using them at all. When probed further, we discovered that this was exactly what they expected : chuck these expensive, brand new “learning aid” to the kids and expect them to use them. When we tried to explain that “you’ve gotta use it with them”, we got looks that probably meant “Are you kidding? I have better things to do!” or “Isn’t this the job of the tutor?” or “This is beyond me.”
It is not difficult to use maths manipulatives but you have to start using them. If you do not know how to use them, simply google terms like “uses of [manipulatives]” and you are likely to get answers. There are also books that teach you how to use various maths manipulatives. Here are some places where you can start :
1. Maths Solutions – Hands on Help
This is a pdf of how to use manipulatives in the classroom, with some information on the benefits of manipulatives and how to use some of them.
2. Ultimate List of Manipulatives Printable
A whole list of things you can print!
3. Tips for Manipulatives Use
A list of pdf files giving tips on how to use various manipulatives.
4. Developing Number Concepts Book 1 : Counting, Comparing and Pattern, by Kathy Richardson
An excellent book that is full of maths teaching activities that use manipulatives to teach maths concept. It also has lots of templates at the back of the book which you can use. Great for teachers and parents alike.
5. Developing Number Concepts Book 2 : Addition and Subtraction, by Kathy Richardson
A follow-up book from Book 1, covering Addition and Subtraction, again using manipulatives.
6. Developing Number Concepts Book 3 : Place Value, Multiplication and Division, by Kathy Richardson
What about Virtual Manipulatives?
There are now lots of websites with virtual manipulatives. Virtual manipulatives are essentially virtual form of whatever concrete manipulative you see. Personally, I do not see the point of virtual manipulatives because the point of manipulatives is for hands-on, tactile, multi-sensory learning. Virtual manipulatives are essentially 2D and only involves the use of the hands (which only move the computer mouse) and eyes.
With everything going high-tech these days, one can argue that children are learning things differently nowadays. E.g. instead of reading text, they are watching videos and music more. The theory on learning and brain function is constantly evolving with time and at this point, nobody can be sure whether things are going to be better, worse or just different. Brain function and learning is a whole different topic altogether, which I will not attempt to write about. Call me old-fashion but I am not too keen on all the electronic stuff, especially when young children are concerned. I think at the moment, there are enough evidence to support the reduction of screen time for young children. So if you ask me, I would go for the real stuff than virtual stuff any time.