Discovering Mathematics – Tales of a Newly Qualified Teacher

February 18, 2016

Timetables Break Through

I have recently been looking into pattern and how this helps with learning different multiplication tables (instead of just rote learning – although yes I do think children should be able to memorise their multiplication tables it is not as easy for some as to just memorise and chant them out straight away) that children struggle with and Hopkins, Gifford and Pepperell (1999). seem to have great ideas that I have never come across whilst teaching or being taught my timetables!

Hopkins et al. (1999, p.26-8) focus on the 3-8 timetables due to the fact the 1,2 and 10 timetable are relatively easy to learn. They suggest using a staircase pattern (for an example see below) for the 3-8 timetables:

For the three timetable – Hopkins et al. suggest using multilink cubes/cuisenaire rods, sides/corners of triangles or the segments of fingers.
For the four timetable – the legs on animals or tables, corners of squares.
For the five timetable – (although this is relatively easy table to learn as the pattern for the five timetables the last digit alternates between the 5 and 0) Hopkins et al. suggest using 5p coins or hands to build the staircase.
For the six timetable – use hexagons, egg boxes or 6 chairs around a table to build the staircase.
For the seven timetable – use heptagons to build the staircases
For the eight timetable – use spider/octopuses legs or octagons.

All of these are also great links to learning shape – learning the name of the shape to the amount of sides/corners it has.

Any of the above can be taken and put into this example:

However, the nine timetable is not easy. Well it is with the help of your hands. If you go along your fingers by the number of times you wish to multiple by nine put that finger down and you get the answer with the tens on the left hand side and the units on the right and side.

S&D

Reference:

Hopkins, C., Gifford, S. and Pepperell, S. (1999) Mathematics in the Primary School: A Sense of Progression London: David Fulton Publishers

December 5, 2015December 6, 2015

Finale of Discovering Mathematics

My essay is submitted. The last workshop is over and it’s the end of the discovering mathematics module. To round off the module, I thought I would post an end of module blog post.

Was the module what I expected?

Discovering mathematics wasn’t exactly what I expected if I am being honest. I expected it to be a lot more directed at our future in the education profession (even with it being an elective module – I still thought this because it was an education elective) and I sort of thought it would be like our maths inputs from Teaching Across the Curriculum module in second semester of first year.

Did the module disappoint me?

Not at all – even without it being what I expected it to be. The best thing about the module had to be Richard’s excitement and enthusiasm. There isn’t many people who can take demand planning, turn it into a game and have everyone in the palm of their hands.

What did I gain from the module?

Before the module, I thought I knew a fair bit about mathematics. After the module, I realised I only knew the mathematics I was taught in school and mainly from secondary school which I perceived as pointless during and after standard grade maths – there was no relevance to my life which is exactly why I didn’t finish higher maths.

During the module, I realised just how relevant maths is to our day to day life (for everyone) and society. However, I still stand by the fact I think standard grade and higher mathematics is pointless… Who even uses standard deviation really?

The mathematics we discovered throughout this module was maths and music (who doesn’t listen to music?), the mathematics behind pineapples, the mathematics outside the classroom: it is the type of mathematics that is simply all around us. But it is also the mathematics we are never taught about at school. This module has given me a fresh outlook on why we need mathematics which leads nicely onto the next question I had to ask myself…

Where/when will I use this in the future?

The fresh outlook on mathematics that I have gained throughout this module has made me realise just how little relevance there is in learning mathematics out of a textbook for children. Examples in textbooks go on about little Jimmy buying 30 Watermelons (really?).

This module has given me multiple ideas of how to bring relevance into mathematics when I teach – even without the module being directed at teachers.

A particular lecture that stood out for me, for teaching in the future, was the maths and astronomy lecture by Dr Simon Reynolds. I think the majority, if not all, children learn about space when they are in primary school and I have never thought about the space in space. Yes, teachers tend to leave out learning about the space in space – the irony. Dr Simon Reynolds spoke to us about the size of the planets, pictures normally used to convey the planet sizes compared to one another but never about the distance between each planet. This will definitely affect my teaching when I teach a class about space.

Furthermore, in the future (a little bit closer than getting into my own classroom) I plan to take the information I learnt from Will Berry’s input on outdoor education and maths for my second year Learning from Life placement at Adventure Aberdeen and hopefully use this in relation to my placement as Adventure Aberdeen is an outdoor learning centre.

Where is my maths anxiety now?

We began the module with Tara Harper asking us to fill in an survey on how anxious we were about teaching mathematics and how we felt about mathematics (which I blogged about earlier), so I thought after going through the module it would be quite nice to reflect on this.

At the beginning my maths anxiety wasn’t overly high compared to a few of my friends anyway. I would say if I am being honest, it hasn’t really changed. I am still anxious to get graded on this module. I would however say that I do feel a bit more prepared for teaching maths in school now as I have many more ideas that I can use in the classroom (and I know my friends also have multiple ideas as well that they will hopefully share) and I can now approach maths in a classroom with ideas that will be a lot more relevant to a child’s life.

However, my friend who stated she had “awful maths anxiety” before this module now “definitely feels more confident with understanding more difficult maths such as (the) Golden Ratio and (the) Fibonacci (sequence)” and that this can all be put down to Richard. (Alexander, 2015)

Finally, do I recommend it for next year’s second years?

If this hasn’t already convinced you – whether you love or hate maths, whether you are great or just feel you can’t do maths – this module is for everyone.

Yes it isn’t directed at teaching but you are bound to take lots away from it whether you think you know a lot about maths or you know nothing.

There is no “difficult” maths. There isn’t really any sums involved. You are looking in depth at the ideas and principles behind maths (and no it is not as boring as that sounds) as well as how maths comes into society (it really isn’t just the traditional STEM subjects like I first thought – who knew there was maths in a pineapple?…).

Richard’s enthusiasm and excitement will see that it is another great module next year again I am sure.

November 28, 2015December 6, 2015

Maths in the Outdoor!

I have chosen to blog about maths in the outdoors not only for the relevance to the module, Discovering Mathematics, but also for the relevance for my second year placement where I will be doing Outdoor Education.

Before coming to university, I would never have considered myself an outdoorsy person. I did not grow up in the type of family that went camping or went hill walking at the weekend. However, I have realised my keen interest in learning from the outdoors. I thoroughly enjoyed Brenda’s input on the Swedish curriculum and how they focus mainly on learning and playing outdoors. I am always particularly excitable when there is anything to do with health and wellbeing or outdoor education on our timetable – my friends sometimes think I am mad wanting to be outside in the freezing cold weather that we are having right now but personally I could not wait for Will’s outdoor education lecture!

I have realised I have learnt best when I am actively involved in a task and not just remember facts to reproduce my knowledge in an exam and this is something I have taken away from this lecture before I even think about writing what we actual participated in for the lecture. I believe I am not alone in this feeling and that children need to be actively involved in their learning to even remember a lesson let alone what was taught. Outdoor education has the potential to inspire and involve children in an active learning task.

Anyway I have already gone off on a tangent. The Maths and Outdoor Education input.

From what I have already learnt from this module – maths is literally hidden all around us, including outside. Now, you are probably thinking yes I know that if you cut a tree in half and you can tell how old it is from the rings on the stub. However, there is so much more mathematical possibilities outside. I would never had thought the way a wave spirals as it comes into shore would involve mathematics. Yet, as I have already explained this concept in a previous post, “Creative Maths”, the spiral of a wave meets the golden ratio which links with Fibonacci’s sequence. Maths when you are stood outside it literally all around you – there has been mathematics concepts used for designing and creating any building you can see.

In this particular lecture, we looked at navigation in the outdoors. Something through doing my Duke of Edinburgh I thought I knew relatively a lot about – except I didn’t. I knew the basics and that was all. For the reason that learning navigation is something we rarely do these days – we have GPRS on our phones, Sat Nav’s in our cars. Is there really a need for it any more with the technology we have? Simple answer yes. Although there is a great level of convenience with having a technology item tell us straight away where we are going and how long it will take to get there. There is the slight issue that all of the technology we have relies on the device having power. Our phones rely on having internet connect. What happens if we don’t have this? What I realised in this lecture is there are very limited people that have looked at a map recently or even know how to read a map.

I had never thought that I would have considered map reading to be fun. Once we had gone over the basics and everyone understood how to read a map. Will made it into a game – who could get to the next place the fastest. He would be given a set of the 6 point grid references (point A) for the starting position and a second set of 6 point grid references (point B) for where we were going – we had to find out the degree we were “walking” in on the map from point. We had to find who could find the degree the quickest – now we are a group of university students who got very into this and very competitive, very quickly. We all wanted to win.

No one was particular paying much attention to the fact we were having such fun reading a map. It could have kept us entertained for ages. Now with a group of primary fives – potentially it may need to be simplified a bit but I cannot see any reason why they wouldn’t act the same way. It gives them a chance to learn to read maps and actually enjoy it.

I feel this is something I could easily use in my future practice. I could easily take a group of children who have learnt to read maps and allow them to use estimation (another fundamental mathematical principle) to work out how long it would take us to walk from point A to point B using this chart below and compare it to reality of how long it did take us to walk and if we managed to do it in the correct direction the compass told us when we looked at it on the map.

I feel this is a beneficial and relatively easy way to get children engaged, outdoor, actively learning about map reading skills and take it away from constantly looking at a screen for directions and relying on a piece of technology to get us where we need to be.

I am thoroughly looking forward to getting outdoors in my future practice but in the near future for my learning from life placement – I hope to have the opportunity to either put these skills I have learnt into practice or learn even more about it and how it can influence my future practice.

November 16, 2015December 6, 2015

Maths but not as we know it…

What I have realised recently in my Discovering Maths module is that maths is subconsciously all around us in society especially in areas I was totally unaware of. I would never have realised that art and maths had so many links. There is obviously examples such as the ratio for mixing paints, however this module has opened me up to realise that maths is in the art and day to day life around us. I have already blogged about tessellation and how maths links to Islamic art previously. This alone made me think that maths at school was never creative. However, there is more links to maths and creativity!

A pineapple – has a link to maths through the Fibonacci sequence. The Fibonacci sequence begins with 0,1 and then adds the two previous numbers together so 0,1 then adds 1 onto the sequence, 0,1,1 then adds on 2 to the sequence and 0,1,1,2 then adds on 3, then 5 and so on and so forth. What has this got to do with a pineapple though? A pineapples spiral is linked to the Fibonacci sequence. The layers of spirals increases by two each time, linking to the Fibonacci sequence, which provides the more aesthetically pleasing look of the spiral.

The Fibonacci sequence appears in a lot of nature around of such as spirals in waves, in pinecones, in sunflowers, the list is endless. None of these things I would have begun to interpret into maths before. However, I could see myself using these as intriguing maths ideas in my classroom getting children to look more closely at the Fibonacci number sequence. I would allow the children to have a look at Maths, Art, a pineapple to see if they could figure out the hidden mathematics trying to figure out the pattern before explaining Fibonacci’s sequences and then looking at ideas like the pineapple, to see how it fits into the Fibonacci sequence now they are aware of it.

Some of the ideas that Anna taught us in the lecture I would definitely use in my classroom. Before I tell you which one was the Fibonacci sequence of these two drawing I will allow you to look and decide which you think is more appealing.

CSee4YRXAAACXMP CSee4YMWwAANV9p

The one on the left is before we learnt about maths, art and the link with the Fibonacci sequence and the right is after. The left is just drawing random squares in a square and the right is squares just using numbers from the Fibonacci sequence (for example 1 by 8 or 3 by 3) which is supposed to give it a more exquisite look.

After following the Fibonacci sequence I believe that the Fibonacci sequence may sometimes be hidden into art or subconsciously produced by artists and nature around us but it definitely does, when you are specifically following it, make art seem more attractive. Although, it also took me a lot longer to count the squares to draw it. I feel this would be a more engaging task within maths activity that I could easily use with my middle to upper stage primary school than simply doing number patterns and sequences as well as a great cross curricular link to art.

This has also drawn my interest more into art, as normally I am quite a numerical person and prefer to be surrounded by figures over pictures but the link is something I feel I could delve deeper into to develop my knowledge in art, which is currently quite limited, but still be interested in how the picture came to be and how the Fibonacci numeral system influences it.

November 16, 2015December 6, 2015

Deadly Maths

I could probably guess that maths has ever seemed to be a life or death situation before to many people. I also hadn’t seemed like that to me before our Discovering Mathematics input with Dr Ellie Hothersall. Nevertheless, to doctors that is exactly what maths is. This is another place and job in society that I never thought of linking mathematics to. However, being a doctor means using mathematics to potentially save someone’s life and using mathematics incorrectly means they could potentially make a fatal mistake for their patients.

Doctors use maths every day and every day the maths they do affect the patients they treat. Dr Ellie Hothersall taught us just how much maths doctors us on a daily basis. They are constantly monitoring patients and plotting all of the information they take into several forms of graphs to make sure the patient’s health is improving.

However, one of the most important aspect of their job that they use maths for is prescribing our medication to us. As patients we trust that our doctors are prescribing us the correct amount of medication and telling us correctly when to take the medication and how much medication to take at any time. This can take a lot of mathematics to work out when and how much of a drug to take.

Michael Jackson’s doctor was found guilty of involuntary manslaughter for giving the world famous singer an overdose of an intravenous sedative (Graham, 2013) (Unknown, 2011). Jackson died in 2009 (Wikipedia, undated) after allegedly being given too much intravenous drugs including Propofol (“a fast acting hospital sedative” (Unknown, 2011)) which his doctor gave to him to help him sleep. The overdose of drugs – if it was not on propose which has never been proven – would have been a miscalculation of a prescription from his doctor leading to Jackson’s death.

This shows a fatal miscalculation of drugs can lead to someone death. This has happened many times and to ordinary people as well – a doctor in America was charged with murder after killing three of her patients when she prescribed fatal overdoses of drugs they were already addicted to (Gerber et al., 2015).

Although this is not normal practice and these show extreme examples – it goes to show how essential mathematics is to a doctor in their day to day practice. Without their ability to do mathematics there could be a lot more deaths due to overdoses.

There is goes to show that maths is a crucial aspect of the medical profession and it is critical that doctors understand mathematics to continue to prescribe us with medication because if the medication is prescribed wrong it can be a fatal error.

References:

Graham (2013) ‘No I didn’t Kill Michael. He….King of Pop Available at: http://www.dailymail.co.uk/news/article-2512469/No-I-didnt-kill-Michael-He-did–massive-overdose-using-stash-What-really-happened-night-Jackson-died-Dr-Conrad-Murray-doctor-jailed-death-King-Pop.html (Accessed on 10/11/15)

Gerber et al. (2015) California Doctor Convicted of Murder in Overdose Deaths of Patients Available at: http://www.latimes.com/local/lanow/la-me-ln-doctor-prescription-drugs-murder-overdose-verdict-20151030-story.html (Accessed on 10/11/15)

Unknown (2011) The Drugs Found in Michael Jackson’s Body After He Died Available at: http://www.bbc.co.uk/newsbeat/article/15634083/the-drugs-found-in-michael-jacksons-body-after-he-died (Accessed on 10/11/15)

Wikipedia (undated) Death of Michael Jackson Available at: https://en.wikipedia.org/wiki/Death_of_Michael_Jackson (Accessed on 10/11/15)

November 5, 2015December 6, 2015

Why do we use a base 10 number system?

Easy peasy question many people might think. We have ten fingers so it can be a tool to help us count. However, how many teachers told you as a child to not use your fingers to count? I remember hiding my fingers under the table from around primary five upwards to count.

What about topics that we use in maths that don’t use a base ten system? Time. Time uses a base 60 for minutes and seconds. Time uses a base twelve system for the hours. Why don’t we count in any of these base systems? This is why time can be very difficult for a child to learn. When counting they have only ever used a base ten system and then when we introduce time anything familiar goes out the window. They are now introduced to counting in 60’s for each minute, for each hour. Another student on this elective discussed whether or not we should still be teaching time because of this (Dunne, 2015) – her opinion gives for an interesting read.

So apart from having our fingers as a tool I thought I would have a look into other base systems and the advantages of using a base ten system in this post.

As already discussed briefly in a previous post (“Our Number System”) there have been numerous different number systems which count in different base systems. The Sumerians “developed the earliest known number system” which had a base 60 system. The base 60 numeral system came from two tribes merging together as one used a base 12 numeral system and the other used a base 5 system and the lowest common multiple of both systems would be 60. Their symbols for their number system are below:

A base 60 system would perhaps make teaching time a lot simpler than a base 10 system. It would make sense for the minutes and the seconds but would also be very confusing still for the hours. On the other hand, this system has a lot of symbols which could be quite challenging for children to learn.

However, a base ten system does have a number of benefits. The base 10 system allows for simple explanations of hundred tens and units etc. Using a base two system such as the Arara tribe in the Amazon would get very repetitive and confusing rather quickly but on the other hand using a base 60 system it would take a long time until you exchange it for another to start again. A base 10 system has the benefit that it is big enough to not be repetitive but small enough that you are not continuously counting before exchanging.

This has given me a profound understanding of why our base system has come about. The benefits of the base ten system and why this is simple, as well as complicated when it comes to things like time or angles, for children to learn to count. Although there are many different types of number systems that are not just out normal base ten system or the roman numeral system that we teach children which I could use to broaden children’s knowledge, in my future career, on the different types of number systems such as showing them the yan, tan, tethera… video below as a different type of number system (a base 20 system) used for counting sheep more commonly used in the early 20^th century but still used by farmers now.

References:

Dunne, J. (2015) Is it time to scrap time? Available at: https://blogs.glowscotland.org.uk/glowblogs/jennysjourney/2015/10/25/is-it-time-to-scrap-time/ (Accessed: 4/11/15)

October 7, 2015December 6, 2015

Our Number System

During our Discovering Mathematics lecture yesterday, we were looking at the origins of number systems and how other number systems such as Roman Numerals work.

We were given the task to think of the other numeral systems and in groups create our own system. Our group originally started just playing around with dots which became a lot more when we tried to appeal to children. Our simple dot system became like smiley faces (like the emojis used on smartphones which children seem to really enjoy expressing themselves through), named by minions (because of the insane obsession with them currently) and our last section, for any number after five, our numeral system resembled the Munduruku tribe numeral system where after the number five they had “many” where as we used “plenty” after our fifth dotted symbol.

Richard then posed us with the question – how would you order ten drinks at the bar? If you say plenty – how do you know you will get exactly ten drinks? But there is a solution to this problem Richard, sums. Multiplication, addition.. If you want ten drinks, simply order kevin drinks twice… You then have exactly ten drinks without needing a number for ten. The Fundamental mathematics Richard…

October 7, 2015December 6, 2015

Can Animals Count?

This question seemed to hold a lot of debate in our recent Discovering Mathematics lecture and it intrigued me. Before the lecture, I was very narrowed minded on this issue and thought no of course not. However, there has been some pretty convincing arguments against my opinion which for anyone with the same opinion as me I am going to have a look into to broaden my mind a little.

The biggest influence that made me think that animals possibly could count was Ayumu the Chimpanzee who could correctly identify the order of number 1 through to 9. He could do this by just a few seconds looking at the numbers, which were in a completely random order, before they were covered up. Ayumu could also still correctly order numbers 1 through to 9 even if there were numbers missing from the pattern. This made me think that Ayumu could count to nine.

Although there were some pretty strong arguments against this as well. The fact that 30 students and 1 lectures who can all count could not do the challenge made me think does this chimp just have a great memory? Is the chimp really counting or does he just remember the patterns through rigorous training? However, the idea that he could do this even though there was no logical pattern to the numbers and there were just random digits between 1 and 9 on the screen then he was able to put them into a numerical order showed that this could have been related to the idea he knows the shapes of number 1 through to 9 and could put them in an order. There must have been some cognitive process going on – either counting or something similar to counting – to show that he could put the numbers in the correct order without every number being there. You can watch Ayumu impressively memorising where the number were and in the correct order below and have a shot at the same challenge below to see if you could manage it. Just remember 30 students and 1 lectures couldn’t do it together! Below is Ayumu showing you his skills and here is the Ayumu Counting Challenge Game link.

Another convincing, all be it strange and kind of cruel, argument that animals could count was that scientists now believe that ants count their steps back to their nests. Scientists glued on match sticks to the ants legs, leaving them with longer legs, or cut the ants legs, leaving them with stumps/shorter legs. The ants with the longer legs would walk straight past their nests where as the ants with shorter legs would not make it back to the nests. The scientists have put this down to ants having “internal pedometers” (which was first proposed in 1904) that they count the steps it take them when they leave the nest and they then go back using the same amount of steps to go back to their nest. Therefore, the ants with short legs would take the same amount of steps, but smaller steps, back to the nest and not make it back where as the ants with the stilts would take the same amount of steps and make it past their nests because of their new, longer legs. (Carey, 2006)

Another argument with two opposing sides to whether animals can count is the idea that mother ducks know when they have ducklings missing. One argument suggest that mother ducks can count. The idea that the mother duck knows that there is one missing as she does not have the same number of ducklings that she should. On the other hand, others argument that potentially the mother duck waits if there is a duckling missing is because she recognising the scent or features of a duckling is missing rather than knowing that there is a particular number missing.

Overall, I am now more convinced there is the possibility that animals could possibly count but there are also counterarguments that still support the view that animals cannot count.

Reference:

Carey, B. (2006) When Ants Go Marching , They Count Their Steps (Accessed: 7/10/15)

October 2, 2015December 6, 2015

Cross-Curricular Maths

After the Tessellation and Islamic Art maths lecture with Tara, I thought I would reach to the blog again.

Before starting this lecture, I thought shape in general for maths was quite boring. There was nothing interesting or mesmerising about shape when I was learning about it in primary school. However, tessellation was taking shape and doing something completely different.

Tessellation does not only involve shape but angles as well. I never realised that only shapes that when the points touch the external angles make 360 degrees would tile. For example four square edges would join together, each square internal angle is 90 degrees so four touching square angles equal 360 degree and they fit together perfectly therefore they have the ability to tile/tessellate. This also works for triangles (shown to the left), hexagons (shown below) and any quadrilateral shapes. However, it does not work for octagons and pentagons.

This is mathematics that is used by any tiler probably subconsciously. If I ever need to tile a room, I now know which shapes to use and which shapes to avoid! Although I do doubt if I will ever been tiling anything…

Using shape to make Islamic Art was fascinating. I have never thought of using shape to create art before in this way and it will definitely be a cross curricular activity I use when I go back into the class room. The nipple and paste idea to create new picture of tessellation is also another maths and art activity I will definitely consider using within the classroom.

This is the kind of art that inspire myself to get the children involved in rather than just another drawing or painting, it is something different for maths, art and religion that I would hope I engage children in all curriculum areas.

These are a few more examples of a shape that tessellate and a shape that doesn’t tessellate.

October 2, 2015December 6, 2015

My Maths Mindset

Just after starting th MathAnxiety (2) e Discovering Mathematics elections, we were asked to do a maths anxiety questionnaire and reflect upon our maths anxiety. Therefore, this is my blog post on my maths mindset.

I feel my maths anxiety is overall quite low just now. I am fine, and actually quite enjoy, to listen to maths lectures and tutorials. I do not mind doing maths assessments that I am prepared for and aren’t graded such as the OMA. However, I feel more anxious doing examinations that are graded – this isn’t just maths related. The last time I felt like this was in standard grade maths. I quite enjoy the relaxed yet challenging atmosphere my standard grade maths teacher created. This is the way I wish to portray maths in my own classroom. maths

However, I dropped out of higher maths after two weeks because of the anxiety that I would not be able to cope with the level of mathematics and the stress of higher maths. After two weeks, I was feeling extremely anxious that the mathematics was too difficult and I would not get the support I needed to pass the exam.

I feel that maths is approached with a lot less anxiety and stress in university – it is a lot more relaxed atmosphere which is how I hope to convey it within my classroom as a teacher. I remember in primary 7 having a teacher on their probationary year and being very scared to answer any maths question in his class in case I got the answer wrong and was told so in front of the entire class. I feel that show me boards like we used in the lecture today is a good way for only me, as the teacher, to see how everyone is getting on with the topic without shaming them in front of their peers.

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