In our last article we looked at what algorithms are and why they are so important to maths education. I promised that I would go into some specific algorithms for homeschool maths, but before I can do that, I need you to understand the underlying principles that form the foundation of a good maths education.
Once your child is well on the way to mastering basic counting (up and down by ones), it’s time to introduce the ideas of addition and subtraction. Addition is normally pretty easy for most kids to understand. Subtraction not so much. At least, that’s what most teachers will tell you. But here’s a little secret: if you use Sumstix you can piggyback the teaching of subtraction onto addition so that your kids will learn it without even realising it!
Sound good? Want in? Good… Here’s how! First you’ll need a set of Sumstix. If you’re not a Milestone Maths user and don’t have a set, checkout the resources in our shop for some of the best value ones around. Not ready to buy just yet? No drama! Check out the free download at the end of this post!
Using Sumstix and Fact Families to Teach Subtraction
Once your kids are familiar with the colours and different sizes of the sticks, you can lead them to realise that they can make patterns like the following:
This can start happening BEFORE they have even learned to count. Having the geometry and colour scheme of these patterns in second nature will give your kids a huge head start in the addition and subtraction game!
Before your kids know the numbers associated with the sticks they can already get a ‘feel’ for the language of addition and subtraction with statements like, “a green plus a pink equals a yellow.” To show subtraction you can build the pattern and then take the pink stick away while saying, “Yellow take-away pink equals green.” You should also show them that it doesn’t matter if the pink comes before or after the green stick, they will always be the same length as a yellow stick when placed together!
Now, once your kids have learned their numbers and they have learned that a brown stick represents the number 8, the yellow stick can stand for the number 5, a green stick can stand for the number 3 and a red stick can stand for the number 2, you can replace the colour names with numbers. The end result will be that each Sumstix pattern can represent up to four equations (sometimes called “number facts”). We call the set of four equations “fact families.” The two patterns in the graphic above reveal eight basic “number facts” in two “fact families”:
Actually, if you start with the Sumstix in prep or kindy, your kids may already nail all of the sums (and subtractions) to 10 before they even get to the numbers! You almost won’t have to teach this level of addition.
Now, your kid may not 100% understand this when they first see it, but this “fact family” concept is very, very handy for a couple of reasons. First, it means that they really only need to remember one fact (the trio of numbers involved in the fact family, eg 2,3,5) in order to be able to solve four different problems! This is highly efficient thinking. Secondly, this thinking is also at the heart of algebraic reasoning so you are setting the stage for high school maths right from prep (or kindergarten)!
Number bonds: moving beyond the concrete materials
At Milestone Maths, we don’t want to cripple your kids by overusing manipulatives, so in Level B (Year One) we also introduce the concept of number bonds at the same time as tying number fact families to the Sumstix patterns. Number bonds are basically a graphical way of representing a number trio that represents a fact family. They are kind of like diagrams of the Sumstix patterns:
Read more about number bonds and get a free set of flashcards here.
From Level D we replace Sumstix with bar models. Bar models are diagrams that look a little like illustrations of Sumstix patterns but they do not rely on colour and they are rarely drawn to scale. The thinking that the kids used with number bonds to 20 and Sumstix in Level B leads directly and very smoothly on to bar models and number bonds into the hundreds in Level D. The kids use these simple tools and the same reasoning that is becoming second nature by this stage to solve equations like 25 + x = 90 and 58 – x = 30 with relative ease. Of course, at this early level we don’t use a variable like x for the unknown but we probably could without too much drama. The foundation is well and truly laid for true algebraic thinking!
At this point you’re probably wondering, how exactly do we teach the kids to solve problems like 90-25 and 58-30? Enter the algorithm!
Putting it all together
As I mentioned in my previous post, algorithms are a hotly debated topic in mathematics education circles. Just when you think that the dust is settling on the argument of whether or not algorithms need to be taught at all, you find yourself choking in the algorithm wars of, “my way is better than your way!” At Milestone Maths we take a common sense approach on both fronts.
First, mental maths methods are taught and encouraged right alongside pencil and paper algorithms. Why? Because both contribute to the development of important learning and critical thinking skills. Crucially, each method develops a different set of thinking skills that are applicable to different real world problem solving scenarios. So both deserve to be taught.
Secondly, when there is more than one algorithm that can be used to efficiently perform a calculation, we teach at least two of them. (For some processes there’s literally dozens of approaches. Our aim is to give the child some flexibility in their thinking without overwhelming them so we generally stick to only two or three methods per process).
In our next article we’ll get down to the “nuts and bolts” of addition and subtraction algorithms. Heads up! If you’re over 40, you’ll probably get a shocking surprise when I tell you the truth behind the subtraction algorithm popularly taught in schools today!
A free gift for you!
If you’d like to try some of the ideas mentioned here involving Sumstix but don’t want to commit to purchasing a set just yet, here’s a little present for you: a printable set of Sumstix. Totally free. No email address required. We would be over the moon if you to sign up to our mailing list at the bottom of the page, though, so we can keep you in the loop when we release more freebies 🙂
To access the freebie, just press the button and it’ll fly through cyberspace at the speed of light (literally!) directly to your device!
