MathsPaperSilk

where i ramble about all the things i love and try desperately to make them all link together

[mathspapersilk-blog]

On Ratio and Proportion

So, having finally started my PGCE, I now find myself dealing with all the problems I so desperately wanted to tackle as a TA. I knew that it wasn't going to be easy, but I find myself thinking so long and hard about how to construct the exploration of a topic that I'm spending about 4 days planning every lesson. Not time efficient... but the outcomes so far seem to be alright - not terrible, not disastrous anyway. (she says, after a starter and 1 lesson.)

I taught my first lesson this week, on ratio and proportion, to an incredibly bright young set of 11 year olds. It was scary, and daunting, especially witnessing the range of knowledge in the room. The thing that most stood out for me was that when I asked the students for definitions of proportion, one of the girls suggested that ratio was 'for every', and proportion was 'in every'.  Having not come across this definition before, this was a real shock as I thought I had anticipated everything the students would know. The rest of the lesson went on fairly well and those students seemed to cope, but even at the end, after my explanation of proportion (that proportion is about maintaining the same internal ratios), these students, when I asked them to write down a definition of proportion, still wrote down 'in every', for proportion.

Having been confused, I went and looked at where this 'in every' idea came from. It seems that in the primary curriculum, proportion problems are taught by this 'in every' approach - so, for example, if a ratio of red beads to blue beads was 1:4, the proportion of red beads would be 1 in every 5. That is, their idea of proportionality is the same as my idea of finding fractions within ratios.

I think there are positive things about the students knowing this - the 'in every' statement suggests something about applying the ratio/relationship to different scenarios. However, my main concern with it is that the 'in every' concept seems to only help if the internal relationship is between two very similar things.

Let me give an example: if you were to say that 1 in every 5 beads is red, that would be fine, and you could apply this linguistically to a situation where there were 10, or 15 red beads.

However, if we are talking about a proportional situation where, say, 15 bananas cost £1, then would you really say that there is "£1 in every 15 bananas..."? The "in every" statement takes the situation, and requires that it is succeeded by a very neat description of the whole. The 'in' part of it seems to suggest that you are inside a very describable something, somehow. 

I think that maybe this is a contradiction to ask the students about - and see if they can resolve it.. is there time in 1 hour? I am struggling to know what I can reduce down to a simple idea that concisely conveys what I am aiming at to students, as obviously what I have said is a bit technical. 

My thoughts at the moment are that I will introduce both definitions - the 'maintaining the ratio' kind of definition (taking one of the students wording), and the 'in every', and see if they can say how it would apply to it. This is the Malcolm Swan approach, I guess (cognitive conflict). 

However, I don't want to dash any of their confidence - I want to show that it is still relevant. I now just need to find a way...

[mathspapersilk-blog]

squaring the circle... why pi is needed in calculating the area of circles

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on decimals

The kids I work with seem baffled by them.

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. 

[mathspapersilk-blog]

Modular origami icosahedron - made by me.

[mathspapersilk-blog]

As a Maths Teaching Assistant I spend my days watching people teach maths to children, and then trying to fill in the gaps that the students missed. This is my third year, and I have finally become convinced that I would love to teach maths. I'm starting my PGCE in September and this is my platform for trying to develop my ideas.

So far, I appear to have a thousand messily organised notebooks/folders/pieces of scrap paper with ideas and ponderings on, but nowhere that I actually write in paragraphs, with a clear narrative. I am a list-maker and a mind-mapper but, having a degree in History, I need to put things down in a structured way.

When I'm not getting distracted by researching what was being taught at work that day, I spend my evenings attempting to study for my OU Maths degree (finishing up the equivalent of my second year, with the wonderful M208 Pure Mathematics course).

I have a degree in History, and I find this a source of conflict when it comes to mathematicians. I can concede that I might be a little over-sensitive about it. I adore both subjects, and they both provide me with completely different but complementary (I think) outlets for creativity and my love of researching things. Since I was tiny, I have loved both, and I spent my childhood years alternately

researching anything I decided was vaguely interesting (Copernicus/Whales/Stars/Spies/Codebreaking) in our giant Longman's encyclopedia and turning it into a project that overused crayons, and 

doing endless maths sums too long for an a4 piece of paper/setting up pretend banks and charging my dad 100% interest rates to store his spare change. 

I love my job. I love that I get to help kids that find maths difficult all day, and I love that I work with people who are incredibly inspiring. I get to see great ideas on an almost daily basis, and have the time to think about how I would tweak them to work for me. And the kids seem to appreciate my help, and while you can never quite be sure, occasionally I think I might be onto something. 

So. This is my platform for pondering about topics that seem to hold the kids I work with back. As a quick bit of context, I work mostly with C-D Borderline students, but this year I seem to be more with the E-D students and have been noticing, more than ever, a few common themes in where their misconceptions and difficulties lay. The main point is - something somewhere down the line hasn't worked for them, and so they need something new. 

Insanity is doing the same thing, over and over again, but expecting different results ~ Einstein

Here goes...

~ S.