Category Archives: Maths Elective

Demand planning

Demand Planning

Although a three-hour slot was put aside for this workshop it absolutely flew by. We were all so absorbed in how we were going to buy next and what was going to make us the most money etc. we finished with over £50,000 profit! Pretty good considering we started with £5000.

The mathematics involved in this task was endless. We worked within the financial quarter and started with April-June. Not only did we have to work out how many products we were going to buy, we had to first look and see what kind of products would make us the most money. Tinned beans (pack of 4) could be bought at the bargain price of 25p but sold for a whopping £2. From there we had to then decide how much of our budget we many tins of beans we would order. This process was repeated for maximum of 5 products in the first instance. When we had chosen 5 or less products, Richard gave us the percentage of products that were sold for that quarter. Cue more maths. We had to multiply the number of products we had bought by the percentage and then multiply that amount by how much they were sold for. And then repeat for maximum of 5. And then move on the next quarter. Now we knew that some of the products did not sell as well as other we could make wiser decisions and also incorporate new products as we still had some old ones left over.

This went on and on and I absolutely loved this task. We were all so engaged. Richard did put in a few trip ups for us – for example all beans had to be recalled -which for one group was a nightmare as they had spent almost all of their money on beans because it was going to make them the most amount of money and then they were left with not very much at all. Also, the wafers had a shorter sell by date so any leftover products had to be disregarded.

Richard also explain at the end of the year if we had any left-over products, the suppliers would buy some of them back. This allowed any products which we had previously anticipated to sell at a higher percentage to then be bought back and not leaving us at a total loss. He explained stores such as Home Bargains and B&M buy products with shorter sell by dates from suppliers who know they may not sell them in time at the profit they would like. Therefore, allowing these discount shops to keep the prices extremely low. The suppliers do not really have an option here as they may lose out of money anyway if they decide not to sell them to the shops at a low cost.

Although the mathematics involved seemed complicated, we used calculators and were really working out basic percentages. What we were doing was relatively simple – to us. To get to this point we had to have sound knowledge of basic arithmetic, ability to use a calculator to work out percentages, budgeting, problem solving, such as time of year to buy products i.e. probably not as wise to buy selection boxes in April etc.

Profound Understanding of Fundamental Mathematics (PUFM)

Multiple perspectives applied here as with working out how much you have made from a quarter there were two options. Calculating how many ordered by the percentage sold, then calculate the cost or calculate how many ordered by the cost then by the percentage sold. We did it the first way as it just seemed easier but both ways would have been fine and perhaps the second way would have been less confusing for some.

Basic ideas – the basic mathematics was not difficult but the thought and problem solving behind this.

Connectedness – there were so many topics pulled together in order to complete this task we had to combine a lot of previous knowledge. To be able to get to this point to carry out this task, if doing this with an upper stages level, it would be beneficial of the teacher to have revised some of these previous topics in the run up to this.

Cross curricular opportunities and wider world connections

The demand planning task allowed for opportunities of awareness of “financial awareness, assessing risk and making informed decisions”, all of which are stated in the Experiences and Outcomes of Curriculum for Excellence (Scottish Government, undated, pg. 1). Taking beyond basics and connections within mathematics, by predicting increase and decreasing in prices drawing on other knowledge out with mathematics demonstrates the Experience and Outcome mentioned above. Although the basics and fundamentals within this task were mathematical, due to the context, it did not seem this way, showing how this gives many more opportunities to link to other curricular areas.

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. London: Routledge.

Scottish Government (undated) Principles and Practice: mathematics. Edinburgh: Scottish Government. Available at: http://www.educationscotland.gov.uk/learningandteaching/curriculumareas/mathematics/principlesandpractice/index.asp. (Accessed: 10^th November 2016).

Multiples perspectives

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Multiple perspectives

Ma’s Multiple Perspectives of not only having the ability to adopt different strategies but to know the advantages and disadvantages of them to differentiate for the students is something of difficulty to ask for teachers. Although I do see the importance of multiple perspectives, sometimes there is just an easier way of doing this and over complicating things for children obviously has its disadvantages. Some teachers may use that type of attitude as a cop out however the following examples prove to me the importance of a teacher to know and when and where to apply multiple approaches.

During placement there were a small group who did not understand BODMAS. This was a mixture of ability groups also so it was not down to a less able group not grasping this. We were doing brackets and adding really simple numbers in two brackets first then adding those together. I found it really difficult find a different way to explain what we were doing, especially as the majority of the class understood just as I had intended. Ultimately, they did show understanding and proved this to me using mini white boards and making up their own equations. However, I think lack of experience of having a different/multiple approaches to what we were doing contributed to both mine and the student’s frustration. I think especially because with something like BODMAS, as I mentioned in a previous post, there are rules surrounding the process which is as difficult to explain as grammar.

Another example during my school placement was a pupil whose parent, at home, was showing them how to do a mathematical problem differently to the teacher. The way they had been taught was not only resulting in incorrect answers they were also unable to explain their thinking process behind it. I cannot imagine how stressed this student must have been feeling as they perhaps their parent did know a better way. However, due to the incorrect answer and inability of explanation this caused great confusion because the teacher was progressing this topic further it caused more confusion (It would be helpful to myself and others reading this if I could remember the topic to pick it apart but I can’t – sorry!] Looking at this now, I see that perhaps the parent and teacher’s fundamental mathematics did not correspond which could simply have been from the language they were using right down to the understanding of the basics. Now I wasn’t there long enough to see this through and I am not sure how this would have been resolved. However, this does highlight the ability to adopt multiple approaches/perspectives is important but in order to be able to justify you must know the advantages and disadvantages of your own and other approaches.

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. London: Routledge.

Forcing Connections?

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Connectedness and basic ideas

Haylock states teachers should move away from the “notion of teaching recipes and more towards development of understanding” (Haylock, 2006, pg. 7). I believe this applies to both, student and teacher.

When learning multiplication for example it could mean that multiplication in its simplest form is repeated addition. Although most teachers would know this, the purpose of explaining these strategies in the first instance may take away any anxieties students may have when they see multiplication for the first time. So then connecting the basic ideas that multiplication is repeated addition, which they can do with ease, then moving to long multiplication will seem less fragmented. This is where teachers need to be careful in how they frame “rules” within mathematics (Devlin, 2008). Basic multiplication e.g. 4×4=16 is like repeated addition – 4+4+4+4=16. Therefore, is useful in the introduction of simple multiplication, however, moving on to multiplication of fractions, the “repeated addition” statement no longer makes sense (Devlin, 2008). This also corresponds with Haylock states about learning recipes. This also evident when introducing multiplication of negative numbers where the “repeated addition” no longer applies. Rote learning although may help with consolidating processes of how to do a particular concept, the meaning of why must also be in the mind of the teacher.

BODMAS

I think it is hard to connect something like BODMAS to the wider world. It’s almost like grammar, where there are rules that help you make sense to read and write something and that is it. This doesn’t agree with Haylock’s statement above, of moving away from teaching recipes because that is what BODMAS really is. There is a way to do something and it has to be in that order for it to be correct. However, when moving looking at concepts within BODMAS and multiplication in brackets which is a topic which I taught during my placement there is opportunity for making connections. Explaining to the children that the mathematics they were doing was not something new but it was the way (order) they were to do it in I think made the children feel at ease. I think it is very important that teachers not how know the connections but show positive attitudes when connecting topics.

Forcing connections

Ma’s view of connectedness is a teacher making and demonstrating connections between topics within mathematics. This is possible in a topic such as BODMAS however, making connections to the wider world is harder to make. I think children do need to see the connections between concepts and topics however certain areas, as proven with BODMAS, there cannot be a connection to the wider world so why force it?

Some teachers may feel under pressure to make connections with everything after reading Ma’s definitions of Profound Understand of Fundamental Mathematics (PUFM). I feel as long as they can see themselves and show the connections where possible and where necessary, without confusion the children. Making connections with every possible topic does not show PUFM. Making the decision when and where to show the connections shows the “profound” in my opinion.

References

Devlin, K. (2008). ‘It ain’t no repeated addition’ Available at: https://www.maa.org/external_archive/devlin/devlin_06_08.html (Accessed: 14^th November 2016)

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. London: Routledge.

Nature of time

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Time is something that I hadn’t really thought about as it just something that we have come to learn and know. It is probably one of the most important aspects of mathematics that directs our lives. Everything we do is according to a time and schedule.

It was a bit of a surprise to me that some aspects of time are man-made, such as second, minute, hour and week. However, days, months and years are not. The latter are natural aspects of time are shown from the Earth turning to turn light to dark (day), the moon and its shape (month) and changing of seasons and migration leading to recognition of division of seasons and therefore a year.

All aspects of time coincide with the sun and the moon. To me, I know that it is a new month because another day has passed and my calendar tells me so. But without a calendar of a digital clock telling the day how did people before this invention tell the time. The oldest artefact found and thought to be 35,000 years old, called the “Lebomba bone,” had 29 lines scratched in to which could represent the recording of a lunar cycle (Bellos, 2010).

There are various theories about how the 24-hour day developed. The fact that the day was divided into 12 hours might be because 12 is a factor of 60. There is also reason to believe finger-counting with base 12 was a possibility – the fingers each have 3 joints, and so counting on the joints gives one ‘full hand’ of 12. Another theory based on Egyptian time was that the 24-hour day was broken down to 10 hours of sunlight, 10 hours of darkness, 2 hours of dusk and 2 hours of dawn. However, how did they account for time at night as one of the oldest ways of telling time was a sundial. We looked at the history of water clock which again seem so simple yet the sophistication of this invention is amazing considering the period of time.

Time is not something I had put much thought in to before and this occurred to me that it is because the foundations of it were taught at elementary stage (Ma, 2007, pg. 124). We are so used to telling the time in 60, but that really is quite a hard concept to understand as a student. During my placement I was teaching hand of clocks and just the language such as “quarter past, half past and quarter to” are quite difficult concepts to understand. Not only do you need knowledge of how to divide something by two and four, it becomes more difficult where using fraction on the number 60, which is easy enough for you and I because we know it so well. However, when breaking this down to teach and trying to reframe my methods was something I found difficult as it is just something that I’ve come to know so well and use every single day.

Profound Understanding of Fundamental Mathematics (PUFM)

I think many of Ma’s principles of PUFM can be applied to the concept of time. However, defining “telling time” is very difficult as there are so many stages and blocks to build on.

Basic ideas – recognising numbers in a sequence that is different to the base 10 system we use for everything else. Children may not be able to tell the time but they can recognise the numbers on a clock.

Connectedness – Following simple routines for break and lunch times e.g. knowing that it is nearly lunchtime due to positioning on the clock. To moving on to connecting language from other topics such as fractions e.g. “quarter past”, “half past”.

Multiple perspectives – time becomes very confusing switching between 12 and 24 hour clocks so knowing when to it is necessary for children to be able to differ between the two is also important. Children are exposed, even more so nowadays in our digital world, to digital clocks telling 24-hour time so it is important the difference is taught as soon as possible without causing confusion.

Longitudinal coherence – I think this principle plays a vital part when learning, understanding and teaching “time”. Being able to tell the time is not the only part children need to understand. The ability to understand a routine or a timetable are difficult concepts to grasp along with “telling the time”. The level of intellect used in problem solving for structuring and deciphering timetables is optimum at elementary level. Teachers who keep in mind why they are laying foundations of basic ideas of time exhibit longitudinal coherence and showing responsibility out with their horizontal teaching.

What have I learned?

As I’ve said, the mathematical concept of “time” does direct the world around us in all that we have ever done. I have taken from granted my own ability to tell the time and use it to direct my own day. My new knowledge of how some aspects of time were developed has given me a deeper understanding and confidence for teaching this topic in schools in the future. Connecting the language between topics e.g. quarter past for time, which I taught on placement would have been greatly beneficial to appreciate and understand at the time. Children often think they are learning isolated topics and I’d have said, before delving in to this module, I would agree slightly. Based on, that a lot of mathematics I think I learned is no longer of any use to me. Deciphering and making up my own timetables in school may have been a boring subject but I did not realise the learning and “time” it would have taken me to get to that stage now helps me organise my own day.

What do I want to find out?

Day lights saving is a fairly new concept that was introduced in the last 100 years or so what that’s all I know of the why. It is something that will affect all of us – in the winter we get an extra hour in bed – win! However, understanding why and what happened before this was introduced would be something further to research.

References

Bellos, A. (2010). Alex’s Adventures in Numberland. London. Bloomsbury Publishing Plc.

Ma, L. (2010) ‘Knowing and teaching elementary mathematics’. London: Routledge.

Rogers, L. (2011). A Brief History of Time Measurement. https://nrich.maths.org/6070 (Accessed: 14^th November 2016).

Number systems and Connectedness

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Richard said to us that in order to look at some of what we would look at in Discovering mathematics, we would have to forget everything we know. And he was right. One of our lectures we looked a place value and binary and my mind boggled. So much so that Richard went over binary in greater detail in another lecture – thank you!

I never knew the existence of any other number system until this module. Well, I knew time was not a base 10 however as I said in a previous post, time along with a base 10 system and place value a base 10 system is taught from the very beginning of school. In the use of units, tens, hundredths and so on. One of the first things we learn with numbers, and that I have taught my daughter I to count to ten (even though technically a base 10 system is 0-9, but who learns to count this way?) I do this every day with her, from counting going up stairs and reading books. 1-10 and so on is the way it is, she does not know why, and I did not think about it in depth until this module either.

The idea of different base systems because of how easy I think the one we use is seems bizarre. We looked at “yan, tan, tethera…” a base 20 system Lincolnshire shepherds used to count their sheep. If a shepherd had more than 20 sheep, he would record one cycle of 20 by putting a pebble in his pocket or marking a line in the ground and start again (Bellos, 2010). So five marks or pebbles would represent 100 sheep. This base 20 system works well for what it was used for – counting sheep. If a base 10 system was used for this example, then there would be a lot of notches or pebbles to carry (depending on the size of course) and could possibly become more confusing. However, using this base 20 system for anything other than herding and counting sheep, does not seem the most sensible option.

Bellos suggests the trick of a good base system is that the base number needs to be large enough to be able to express numbers such as 100 with ease. Obviously this why a base 10 system works so well in his option but I also think the base 10 system works so well as it is in tuned with the human body. I still use my fingers occasionally to count. But if we weren’t born with 10 fingers – who knows if we would be using a base 10 system or not.

Binary

So looking at a base 20 system wasn’t too difficult but then we moved on to binary. Cue the utterly puzzled feeling and look on my face! In the first lecture I didn’t really get it and we quickly moved on. I’ve gone through my whole life not knowing what binary is and how it works so I wasn’t too concerned at the time. But, that’s not what this module is about is it! Cue BBC Bitesize website which explained Binary is a base 2 number system that uses 1 and 0 and is processed by computers! WHAT!? Yep, back to not ever knowing or needing it or wanting to look at it again. Thankfully Richard did show us the YouTube video below and showed us a different table to what he had shown in the previous lecture and alas, as I said at the start of this post, in order to understand, we need to unlearn all that we know about numbers. As there is only two digits that can represent values in binary (0 and 1) this what I found hardest, the difference between number and numeral. We are so used to the number 1 meaning 1 and 2 meaning 2. I still don’t feel confident enough to explain how binary works but I can use this table myself and have included an image of my handy work! (I just hope it is correct!)

I don’t suggest going in to great detail of different base number systems with children but perhaps delving in to them to make them aware that we should count ourselves lucky that the one we use is easy compared to “yan, tan tethera”. But, the fact that binary is used our digital world and Code Club was something was popular at my placement school, it is important for at least as a teacher to know one of its existence but to be able to explain it in simple terms. I’m still not entirely keen on Binary, due to it being so different from all I’ve ever known. However, appreciating how difficult it was to get my head around certainly gives me an understanding of how learning something new for a student is and how easily fragmented it could become.

Connectedness

And of course, all of this links to Ma’s “connectedness” of Profound Understanding of Fundamental Mathematics (PUFM). In order to understand different number systems, I had use prior knowledge to link different mathematical concepts together. That is why it is so important for elementary stage to have the simpler “basic ideas” instilled within students so that they can be used instinctively when learning new topics. However, it is the duty of the teacher to show the connections between what they have learned and how that knowledge is implicit to learning new topics and for their future. I feel that by learning about different number systems and binary in particular, I was able to draw on previous knowledge and with that in mind apply it to a new concept which together become the “unified” body of knowledge that Ma describes (Ma, 2010).

References

Bellos, A. (2010). Alex’s Adventures in Numberland. London. Bloomsbury Publishing Plc.

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. London: Routledge.

Beauty of numbers/art

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The golden ratio is also known as Phi (Φ) and goes (a + b)/ b = b/a

Using Fibonacci’s number sequence to calculate the Golden Ratio works for all of the numbers after number 3 in the sequence. 1,1,2,3,5,8,13,21,34,55

Example:

(2+3)/3=1.6666666
(8+13)/13=1.612384

To anyone who did not have the opportunity of having this equation explained and demonstrated to them may be sitting there thinking that looks rather complicated. I like to think of it as where by you divide any number by the one before you will always get 1.6…. Much simpler – to me anyway.

What is the point in all this. Well the Ratio 1.6… is known as the optimum ratio of aesthetic beauty. We tried this out on measuring our own bodies – from our hands to our full bodies to work out the ratio and if we are “beautiful”. This was a fun activity, giving us a real life example of the Golden ratio and how it works. Imagine my horror when I found my hands were not the optimum beautiful hands I thought they were and disproportionate to the rest of my body – perhaps my measurements were wrong!!! That or I have massive shovel hands!

The Fibonacci sequence and ratio occurs so frequently in our world and is applied across many areas of human endeavour including architecture and other forms of art.

During the Renaissance period where Leonardo Da Vinci painted The Last Supper and the Mona Lisa, further observation of these show the use of the Golden Ratio making them aesthetically pleasing to the eye. Having never been aware of the existence of the Golden ratio and how it makes up some of most famous paintings in the world I now feel slightly more knowledgeable and sharp of where and when to look out for it. I especially feel if I was teaching an art lesson on for example self-portraits I feel more confident in my ability to explain the ratio’s (perhaps not to use them in practice) but to at least outline and explain the use and aesthetic advantages it holds.

References

Knott, R. (2010) Fibonacci Numbers and Nature. [Online]. Available at: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#petals (accessed 31/10/16)

Fibonacci

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I was a bit dubious before coming to this lecture and naively thought “what does maths have to do with art”. We had previously discussed the Golden ratio in previous lectures and I had not linked the two. My ideas, having only heard what the golden ratio was, was it made buildings look symmetrical – but again I thought that was perhaps to do with the structure etc.

We first looked at the Fibonacci sequence. Did he (Fibonacci) invent this sequence or merely discover its existence?

It goes, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 and so on.

The rule of this number series is the next term is formed by the sum of the two previous terms.

(There other rule is if you divide each by the number before it you will get 1.6 – sounds complicated and I will talk about this in another post).

This sequence seems simple enough however, we watched a video of how it explains rabbits and it was, to me, rather confusing. However, if you remove the part of where I watched the video I understand. The sequence is what it is and it makes sense to me. The rule is very simple to follow and it makes mathematical sense.

Golden Spiral

The Fibonacci number sequence is found in natural objects and phonema. For this to make sense we drew the golden spiral based on the sequence above following written instructions of drawing squares but the number of squares followed the Fibonacci series. From the example you can from the squares drawn you can create a spiral.

This spiral is found in many natural objects and phenoma including:

So back to my earlier questions of did Fibonacci invent this sequence.

“We don’t invent mathematical structures – we discover them, and invent only the notation for describing them”. (Tegmark, 2014, p259).

It has been proven to us through many different examples such as the curve of a wave, nautilus shells, sunflowers, pineapples, the reproduction of bees and rabbits. The list goes on and on and now that I’ve seen it in so many different places and forms I will be on the look out. However, the sequence is recurrent according to Bello and the cyclical nature of it explains why it comes up in many natural life forms as they grow by a process of recurrence also (Bellos, 2010).

References

Bellos, A. (2014) Alex’s Adventures in Numberland. London. Bloomsbury.

Knott, R. (2010) Fibonacci Numbers and Nature. [Online]. Available at: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#petals (accessed 31/10/16)

Tegmark, M. (2014) Our Mathematical Universe.

Can animals count?

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It does make evolutionary sense as why they would need to do this. Mathematics helps us direct the world around us so why not animals too?

But what really is counting?

“To name or list (the units of a group or collection) one by one in order to determine a total; number”.

So can animals understand numerosity?

This seems a bizarre question and not one I have put much thought in to before. During our lecture with Richard he proposed a few ideas on how some people thought different animals could count.

The first example was Clever Hans – the horse who could apparently count. It seemed bizarre at first. Hans’ owner would point to number and Hans would stamp his hoof however many times. This animal could not only understand the numeral but the number (concept) that represented the numeral – amazing right? This was eventually debunked by psychologists who found that Hans was merely following subtle cues by his owner so knew when to stop “counting”.

During the lecture I thought there were some degree of ability of some animals to count. My logic was thinking about an animal, a lion for example, would know how many cubs it had. If one was missing, the lion would know that two wasn’t enough and off they would go to look for the third.

I then thought about my own daughter, who is two years old. She can you tell you that she is two but not two years. She can recite numbers 1-10 then 11, 12, 13, 14, 17, 19, 20, 20 20 – she loves saying the number 20!

She has a new book which counts down from 10-1 with lady birds. This is something she is struggling with as we have always practiced counting from 1-10. Again, we have recently been practicing jumping in puddles citing 3-2-1 blastoff. However, she always says 1-2-3 blastoff. So yet another confusion for her.

So can she count? Yes, to an extent. She can recite the numbers.

Does she know what the numbers mean? No, not yet. She cannot yet assign numeric value to a group of objects yet.

She understands that if she insists on taking three teddies to Tesco and one goes AWOL (or mum has hidden it) she knows that one is missing. One not being the correct word here – the exact one. She knows that something is not right and what she had is now not there. She doesn’t know that three is suddenly two. Not the concept of number anyway. But, what she does know that she had Peppa, George and Iggle Piggle and recognizes which one is missing. (A similar analogy to the lion example I had thought about).

This brings me back to my own argument about whether animals can count their young. They know that each of their cubs represents a number and if one is gone – I think they would know which it is.

Lions can count to a degree – what I found what slightly more complex than I originally thought above. Below is a video which explains that lions can count based on the number of roars they hear. The results were that if their roars out numbed the roars played to them by two they would go ahead and fight/defend themselves.

So what about Ayumo the chimpanzee who could count.

As a class we tried to the tests that this chimp did and were not very successful. Perhaps with some practice (and a reward like Ayumu received) we would improve on this. When watching this it did seem more like memory of number rather than understanding numerosity.

I did find another clip however, which shows a chimp counting dots thus demonstrating they do understand numbers and numerals.

https://www.youtube.com/watch?v=VCy7M6ueSpE

References

http://www.thefreedictionary.com/counting (accessed on 18th October 2016)

Maths and play

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Maths and Play

In our lecture we discussed some reasons of importance of play when teaching mathematics. Some of which included:

Enabling children to experiment
Providing meaningful context
Promoting social learning
Encouraging creative and flexible thinking

All of these reasons would be helpful in any area of learning so why are we so focused on making maths fun?

Maths is so deeply embedded in our society that if from an early age if it seems too difficult, boring or irrelevant and a person switches off from it then this may have consequences for their future. Therefore, the early experiences need to be positive ones, as do the latter, however in order to stamp out any anxieties especially, the early experiences must be positive ones.

Are we disguising what CfE wants all learners to do which is “opportunities for communication, discussion and explanation of thinking” (Scottish Government, 2009, p 189) by calling it play or is what is described above is actually play. Reading that quote does not first bring play to mind for me. However, I think sitting at a desk, being didactically taught new concepts is exactly how our curriculum is designed not to be taught. As a former maths textbook enthusiast (as a pupil not as a student teacher) this is the complete opposite of what is described in CfE so play or whatever you think it should be called has its place as was proved to me in some maths lectures we’ve had for this module so far.

In one game we had to describe to our partners a shape that was on a card. This proved to be quite difficult to explain but easier as the person trying to figure out the shape. I first started by describing whether it was a 2D or 3D shape then could move on to how many faces (sides) the shape had. Haylock and Manning suggest the importance of experiencing this type of game (naming, sorting) are components of young children’s play which lay foundation for genuine mathematical thinking through the classification. This type of game could be used with children to consolidating different types of shapes e.g. square and triangles and then used as a building block to understand new concepts such as equilateral and isosceles triangles or being able to distinguish between 2D and 3D shapes.

Another task in a different lecture we were asked to complete was to find a formula for how many snaps of chocolate to break a bar in singular pieces of chocolate. The thought of doing this on my own was a bit daunting especially when the word “formula” comes up. You think it will be complicated.

However, as we got to work together as a group on this task, had props to help us and it was relatable as we could all imagine the blocks we had were actually chocolate this was a relatively easy task.

I think the four areas of Lipping Ma’s profound understanding of maths are promoted in these games or come through at some points.

Interconnectedness (how mathematics topics depend on each other). Simply as, in order to understand and describe 3D shapes we needed to know the properties and characteristics of 2D shapes such as squares, rectangles, circles.

Multiple perspectives: In a group setting we were able to approach mathematical problems in different ways. This was shown in the formula challenge with chocolate. With six in a group it was difficult to for one person’s idea to be the same as everyone’s. The ability to look at this problem with a flexible approach was important but something that we all did. This again links to what is promoted in CfE as mentioned above where emphasis on explanation of thinking is important.

Basic ideas (or principles) e.g. ordering, place value, commutativity, etc. The mathematical language we used to describe the properties of the shape links to basic ideas described as we need to know the basic characteristics in order to describe a shape.

Longitudinal coherence: that one basic idea or principle builds on another. i.e. the understanding and properties of 2D shapes must be understood before moving on to 3D shapes.

The four ideas of Lipping Ma may seem boring (sorry!) when written out. However, experiencing maths games and linking it to these four ideas, they are more simplistic and obvious than I first initially thought. There were many opportunities for discussion and explanation of our thinking and we did have fun doing these tasks/games. I would not say that just by playing these games you will profound understanding of fundamental mathematics (PUFM). Lipping Ma describes PUFM having breadth, depth and thoroughness which goes much deeper than the four ideas. Although I was able to explain my own thinking and appreciate others perspectives, the ability to not only find a new way but to explain new procedures to a student who wasn’t understanding something is not something I feel I could do confidently at this stage. The thoroughness of my understanding of PUFM is not quite there yet.

References

Haylock, D. (2010) Mathematics explained for primary teachers. 4th edn. London: SAGE Publications

Ma, L. (2010) Knowing and teaching mathematics: Teacher’s understanding of fundamental mathematics in China and the United States. 2nd edn. New York: Taylor & Francis

Scottish Government. (2009) Curriculum for excellence, experiences and outcomes for all curriculum areas. Available at: http://www.educationscotland.gov.uk/Images/all_experiences_outcomes_tcm4-539562.pdf

What is mathematics?

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What is mathematics?

Though this question is impossible to summarise in a few words, these were my responses to this question from today’s input: $math-wordle$

“mathematics is solving problems with numbers”

“mathematics is universal”

“mathematics is everywhere” (This was proven to us by being tasked to highlight in a newspaper wherever we saw maths. The yellow highlight strewn across the front page was evidence enough).

Maths anxiety is not something I believe I have, however, being informed in our lecture that in order to be numerate, you must not only understand the mathematics you are doing, you must be able to explain your thinking. If I was confident in my own ability to do a maths problem and was put on the spot to show how I did it, I think I would feel immense pressure and embarrassment in case I was wrong. Showing me there can always be doubt and fear.

Before my first year school placement I felt confident about teaching mathematics and felt I had an understanding of the basics at least. Maths is a subject I enjoyed and was good at, at school until Higher level – but that’s a different story and I won’t go there now. However, during my placement, I taught an extensive amount of mathematics. During one lesson I won’t forget, I had a really tough time helping a small group understand the lesson objective. Repeatedly breaking down the success criteria for the children but this group just weren’t getting it. I thought the way I was explaining it was simple enough however saying it over and over again did not work. I realised the problem lay with me and my inability to explain it in a different way. Therefore, my confidence in my own ability plummeted.

The example above is why I am glad I chose this elective module as I need to better understand mathematical concepts, to improve not only my competence but re-build my confidence in a subject that I enjoy.

Our lecture on “what is maths? Why teach it?” was really interesting and dare I say fun! The task to work out how many snaps to break up a bar chocolate with 64 squares brought about many things which are so important to exploring maths: discussion, conversation, sharing language and most importantly, play. We were able to visualise and draw connections from this task to help us and we had help with props too.

I think we have got off to a great start in not only understanding but actually doing what mathematicians do. In the lecture we did all of the following without even realising it:

solve problems
investigate
explore
discover
use symbols, tables and diagrams
collaborate

Another task was to discuss which way we would calculate this problem:

“In a warehouse you can obtain 15% discount but you must pay 20% VAT. Which way would you prefer your final bill to be calculated: with discount first or with VAT first?”

I first dove in with working out the answer doing 15% off first then 20% VAT and vice versa and was not surprised to find that both answers were the same. It was interesting to hear other people’s ideas that the reasoning behind perhaps adding the VAT first was that so there was a greater number therefore when it came to the 15% discount there would be more money off. This was a great task to see how other people’s mind worked as, as I said I just dove straight in, however, others had more logical thinking, albeit the answers were the same, it shows that no two minds think alike.

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