Explanation: a Variation sequence

{Blank} So hello. This is a movie about variation theory or at least we are trying to demonstrate some of the aspects of variation theory with some sections of the Max Maths books, which use Singapore approach. So the tricky thing with demonstrating variation is that, of course variation takes place by having a sequence of tasks and something is varied along the way of this series of tasks. And that of course makes it quite tricky because variation actually is demonstrated best by actually making this whole sequence or completing this whole sequence of tasks.

So for example here we have a couple of pages from the Max Maths books about adding and subtracting near multiples of 10 and 100, and I think some of the aspects already look quite familiar compared with previous weeks. We can see elements of the Concrete-Pictorial-Abstract approach of course with here a little bit of, here a little bit of the pictorial approach going on. But what I now want you to focus on is actually the the numbers that are being used so it starts off here. Let’s use our knowledge of counting tens to add 9 what is 34 plus 9 we can count up 10 and then adjust buy 1.

In one of the weeks we actually did this together in the Concrete- Pictorial-Abstract section. And then this is as you can see, this page 39 in the book. So we’re introduced to this in such a way that, and using arithmetic in this case, it’s aimed at algebra, in such a way that certain features stand out. And it’s quite clear, I think, that the focus of this section was about these multiples of 10, these multiples of a 100, so that you actually will recognize them easily. Of course, what you could do and that aspect, and then rather than going…

having random examples and having random tasks that you need to complete, variation theory there really is an attempt to actually make it more meaningful, to make the sequence more meaningful and making sure that certain aspects are varied. So here for example going up to 99. So first we had 10s and then we have the 100s, is one of them. But also notice that there is this aspect of addition over here, so here we have more something about addition. here we can see that there is actually some subtraction. So you already see some variation in the way that tasks are sequenced and are represented. And then it goes on… we’re now at page 41.

Again, we see some subtraction going on here; we see another subtraction and with the tens going on we see a lot of pictorial presentations as well. Then it goes to the last page where again you can see 98, 99, we see 19, we see here 11 which is close to 10 of course and then we even have a 97, which of course is plus a hundred and minus 3. And you can see this throughout these tasks, so the way that these pages, a section at the moment. But just to leave you with a thought. What you could do of course is also add a number that is actually not so much close to 100 or close to 10.

And then you get into the tricky position that you need to decide whether it actually is worthwhile to go up to 100 or not. So let’s say you can see here 27 or you see 298, you see 97 over here, so let’s say you have the sum 274 plus 96. You could of course ask yourself whether it is useful to go up to 100. I would say it still is; I would we still do 274 plus 100 minus 4 etc. But what if it’s 95, what if it’s 94, is it easier to add a hundred and then subtract 6 or is it easier maybe to add 90 and then add four more?

I would say that the point isn’t that you always go to the multiple, the same multiple all of the time, but that you vary these numbers. So that in the end you make sure that this varied, this variation theory, this varied sequence of tasks that you ask students to do, that it keeps on building a lot of practice on the one hand but on the other hand also builds this recognition of certain patterns of certain numbers that are more meaningful than others. And then I think you really are onto something and I think that is one of the essences of variation theory.