I, on the other hand, kind of love them. I like that they can represent the teeniest, tiniest things. I love that some of them go on forever, that you find them in pi and e and surds. I even like the fact that some of them have matching fractions.
But for the kids, it's the ol' 0.1 versus 0.01 conundrum. Sometimes if i remind them that they wouldn't add 10 and 1 together to make twenty, they are fine, they go for it, but more often than not, this only works on paper, where some of them (not all) can set it out in columns to help. When they try to calculate adding decimals in their heads, it all gets a bit squiffy.
Mental maths is detailed as a key daily practice in the primary strategy, but I wonder if because "tenths" and "hundredths" sound a bit like "tens" and "hundreds", the similarities between the words added to some students' confusion. However, I can't help but think, that if students want to work in their heads, maybe we'd be better off teaching them how to do these sorts of things in their heads. The only problem is, when it comes to the exam, they've got to be able to write it down.
Piaget argued that until the age of 12, children do not have "formal operational" abilities. It is only as they hit puberty that they start to develop the capacity to reason abstractly. Based on how all children develop at different rates, if Piaget is correct, then are kids that are already achieving below expected levels in Maths being hindered by the emphasis on writing things down, over understanding/reasoning in their heads/using manipulatives?
Obviously this is a very basic analysis of Piaget's ideas and their importance in teaching mathematics, and I need to look into this more. But I have noticed that if given a physical object that represents what we are talking about, and being explained to how to use it properly, some kids seem to achieve more, or feel more confident about doing the work.
I think my idea with decimals therefore is that kids need to be thinking in their heads and with their hands rather than being taught how to do something on 'paper'. They have been taught on paper for years and it hasn't helped some of them.
I was wondering about when in students education ideas such as place value and decimals are introduced and it turns out it is around Year 4, and and around levels 3 - 4. So while many have Level 4s or 5's, their understanding is not thorough enough to move on successfuly to higher levels of attainment.
Flicking through the internet on a way to introduce decimals, I have come across the idea of using poker chips for place value. It seems that they could be used effectively, with the language of 'trading' and 'exchanging' to understand smaller place value. If they can understand this then they might be able to mentally think about the ideas easier - the difference between 0.1 and 0.01 for example, and then think about WHY we write it down as we do. Place value is an abstract idea, where numbers have values in relation to other values - poker chips could easily replicate this.
I am not sure on the specifics of how I would teach this, but I do think this could be helpful if kids are clearly repeatedly making mistakes with their decimals. They could be taught to add decimals and subtract decimals in this way also, and then shown how it translates to a paper format, using chunking on a number line and then using column addition perhaps finally to help. It seems that this more closely replicates how students are taught to add and subtract initially at school.
The challenge is to encourage students, especially the older ones, that it's not babyish to use objects. All kinds of experts use models to help them understand what is going on - architects, engineers.
I think this could also be useful for the idea of multiplying by 10 and 100 and 1000 - this language of exchange and trade, and physically acting it. (if we multiply 0.1 we have ten poker chips, that we can exchange for one other one which means 1). Having the colours to think about in their head could help those that struggle with dyscalculia etc. I have seen some great lessons where students stand up and move up and down in a line holding mini-whiteboards to show multiplying and dividing by powers of ten, and I think that actually standing up and moving helps ingrain this idea, but I wonder if the difficulty is with not understanding why they are doing it, and that perhaps poker chips will make this idea of place value more clear.
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This is not the clearest exposition of my thoughts on this, and it needs to be looked over, thought out more, and more research done in areas that I'm certain I don't know nearly enough about, but it's a start.
