# X marks the spot

So now we have come to the end of our maths journey, and I can honestly say that I have enjoyed this module very much. It makes you think of maths and everyday life in a different way. I think everyone would agree that maths plays a huge role in life, however, through this module we have seen just the extent of maths in places you would not imagine, and in many extraordinary ways. I have seen the whole class at points in this module confused, astounded and most of all in awe and wonder of the maths we are being shown.

The main text through this module has been Liping Ma’s (2010) book ‘Knowing and Teaching Elementary Mathematics’, which describes 4 main principles in having a Profound Understanding of Fundamental Mathematics: “Basic Ideas, Connectedness, Multiple Perspectives and Longitudinal Coherence” (p122). At the beginning of this module, these principles meant nothing to me, however, now I feel I have a good understanding behind them, and can identify them in real life through the inputs we have had.

Basic Ideas is where even the most complex maths has some simple mathematical principles at the heart of it. For example, the ability to add negative numbers has the basic principle of adding, however, the even simpler principle of a number line and knowing that negative numbers come before 0. While adding negative numbers is a higher order skill coming later in the maths curriculum, the basics of number lines are taught even before a child starts school.

Connectedness is where maths ideas are connected together. For example, being able to multiply, contains the ability to add as multiplying is just repeated adding. Knowing this connection between the two allows children to grasp the underlying principles behind what they are doing which, Ma argues, allows them to have a more solid understanding.

Multiple Perspectives is the ability to see how to solve a problem from different angles. For example, when solving 7+9+13 you could complete the sum in that order, or you could add 7 and 13 first, making the sum easier. Being able to comprehend that problems can be solved different ways allows you to choose the best one, making problems easier.

Longitudinal coherence is looking at the development of skills, described by Ma using the progression of the curriculum. When looking at measuring, children start by learning weights or lengths and what they look/feel like. They can then move on to learning about comparing objects and then converting units. This progression is linear, therefore describing longitudinal coherence.

As a result of this module, I find myself thinking more mathematically about life. At work, eating lunch, even putting up Christmas decorations I find myself considering the mathematical concepts involved. While I would not say that I understand every element of complex mathematics that I have been taught, I would say that the module has taught me things about maths that I had never thought about before, and has made me see life in a new way. A mathematical way.

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary edn). Routledge. Oxon.

# Music and maths

“Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry. As Stravinsky once said: “The musician should find in mathematics a study as useful to him as the learning of another language is to a poet. Mathematics swims seductively just below the surface.”                                                                                                                                          – Du Sautoy 2011

There has been some distinction in school subjects between those creative subjects such as music and art, and subjects which are considered more academic, such as maths and science. However, when we look at the relationship between music and mathematics, there are a substantial amount of links between the two.

Counting plays a huge role in music, especially if playing along with a backing track or other musicians in an ensemble. I personally joined the school orchestra in my sixth year of school as a percussionist, and found out these difficulties quickly. However, I believe that this is a prime example of Liping Ma’s “multiple perspectives”. There are large sections of the piece where certain instruments would not play, often written in sheet music as shorthand, such as in the picture right. This would indicate that there are 15 bars rest. If a piece of music has 4 beats in the bar (as indicated by the time signature, see picture left, where the number of beats is the top number), then this would indicate 15×4 beats rest, or 60 beats. A musician has a few option in how to approach this mathematically. They can work out that they are waiting for 60 beats and count these out, however, this becomes difficult if you lose count. The other option is to keep the numbers in their simplest form. I found it easier to count using your fingers, a skill which is often discouraged past early primary mathematics. Counting four beats and then denoting this with one finger, then counting another four beats and denoting this with a second finger was my strategy, which seemed to work well.

Looking back on this with the mathematically thinking I have acquired through this module, I can see many other mathematical qualities in this other than simply counting. The number of beats in the bar denote a base system, as I have discussed in a previous post. As I was counting in fours and then denoting this with a symbol (in this case a finger), I had effectively used a base 4 system in working out the timings, just as farmers denoted a particular amount of sheep with a stone in their pocket or a mark on a post.

There is also an element of pattern and symmetry in music that is often overlooked. There are often repeated patterns or phrases in music, especially in minimalist music, which is made up of repeated phrases built upon one another. An example of this is shown below in the clapping music, where one phrase is repeated with slightly altered timing to create a minimalist piece:

I believe that this would be a good tool to use in a classroom with pupils in order to to reinforce their maths learning. This, I believe, can link with Ma’s principle of “Longitudinal Coherence”, which looks into how ideas and topics are developed. In using music to teach children about pattern and sequence, you show them how these skills can be applied, therefore giving the topic a relevance, which is extremely important for young children. This would be an interesting lesson in to try in future.

References

Du Sautoy, M. (2011). ‘Listen by numbers: music and mathsGuardian. Available at: https://www.theguardian.com/music/2011/jun/27/music-mathematics-fibonacci (Accessed: 25 November 2015)

# 12 is now 10

Image from https://www.dietdoctor.com/why-calorie-counters-are-confused

From the title, it would be understandable for you to think I had gone mad. In fact, quite the opposite. I had a wonderful lesson today exploring the joys of number systems and place value.  Although at first it was not as clear, it was exciting watching the numbers evolve and make sense in front of my very eyes.

The first thing that struck me about the lesson was looking at a counting system used by shepherds when counting their sheep in medieval times:

This system uses a base 20 system, where when the shepherd counted 20 sheep, they would put a stone in their pocket to signify 20. They would then start again until they again reached 20 (40 total) and another stone in the pocket, and so on.

You can see a degree of repetition in this system with 11 (Yan-a-dik) literally standing for one and ten. However, what perplexes me is why they have tags for numbers 6 through 9, as in their system they do not seem to be used again. They have no need to have numbers beyond twenty, as after this they begin again, so why not make 6 “Yan-a-pimp”? This seems, to me, a flaw in their system, which otherwise seems to be a sensible system for them.

As I had previously read about this system before the lesson, seeing it come up automatically hooked me. This I can relate back to the classroom, as if children, like I did, recognise what they are being taught, they will associate with it much better, and I believe will be more likely to take the information on board. I think this links to Liping Ma’s idea of ‘interconnectedness’ (p122) (Ma 2010) as relating new learning to what the children already know is likely to engage them and make them eager to learn.

My excitement grew when we were then told that “Yan Tan Tethera” was the name of a Scottish Country Dance and the crib (instructions) was put on the screen, almost like a foreign language to everyone else, but looked so familiar to me. It gave me a sense of homely warmth, that I can only imagine children feel when they recognise what is on the screen in front of them. This again links back to the idea that children are more likely to be engaged in the learning if they can relate to it. For example, talking about farming to children in the inner city will not resonate with them. I am hoping to try this dance out in my dancing group and will update my blog with it soon.

To us using base 10, or decimal, the idea of any other base system seems alien to us. The most common base system to us, apart from base 10, is binary. Binary is a base 2 system used by computers consisting of only two digits – either a 0 or a 1. Having done Standard Grade Computing, I am fairly familiar with how binary works and how you can make any number using this system. The way I know how to use binary is using a place value system, doubling the numbers each time. This is because in base 10 system you have place values of ones, tens, hundreds etc, where the columns are 10x as much. Because it is only a base two system, the numbers only double each time. There are some examples of binary below:

16+4 = 20

32+16+4+2+1 = 55

64+4+1 = 69

By adding up all of the numbers with a one in that column, you can work out what number the binary represents.

Image from http://gorpub.freeshell.org/dozenal/newdigs.html

With this idea in mind, it makes looking at other base systems much easier. For example, we then looked at base 12 systems, which is thought by many to be a more appropriate base system than base 10. Base 12 includes 2 extra single digits to replace what we know as ten and eleven (respectively shown in the picture left). These extra numerals are needed as the numeral “10” uses two columns. In the base 12 system there is a “12” column rather than a “10” column.  The video below explains this very well, however, instead of these digits above, she is using T and E to represent ten and eleven.

As you can see above, decimal columns are multiplied by 10 each time, and binary numbers are multiplied by 2. This then translates to dozenal numbers  being multiplied by 12 each time, as you can see here.

There are some who have suggested that a base 12 system would make more sense, especially in terms of looking at fractions like 1/3 and 1/4, which are not as neat in decimal. This moves on to the fact that 12 is divisible by more numbers than 10 (1,2,3,4,6,12 rather than 1,2,5,10). 12 is a fairly common number in terms of maths, looking at time especially.

Image from http://dozenal.ae-web.ca/clock/diurnal_2

Note: these clocks are commercially available.

In conclusion, this is a confusing topic to get your head around, which is exactly what some children may feel when faced with what is difficult to them. It is important that we give children the chance to play with and explore the numbers, as we were given the chance to do, which really helped me understand the topic. Children should, in my opinion, be supported, but should be encouraged to be explorers in order for them to feel the sheer joy of what maths truly is.

References

Ma, L. (2010) Knowing and Teaching Elementary Mathematics. (Anniversary edn). Routledge. Oxon.

# What even is 6?!

Years ago, there were no numerals. No names to define amounts. This is because it simply wasn’t necessary until people began forming villages and such where numbers and numerals were needed as a means of comparison for trading. This in itself is difficult for us to imagine, growing up counting in an Arabic (sometimes known as Hindu Arabic or European) number system, with numerals 1, 2, 3, 4 etc. I should point out that there is a distinct difference between numerals and numbers.

Numerals – symbols we use to denote an amount i.e. when we use “6” , that is a numeral

Numbers – a quantity or measure of amount

Breaking this down further, there are digits. Digits are like the letters of maths – the single digits that make up a numeral. Maths is Fun shows this perfectly (link at bottom of blog):

Numerals are what we use to talk or write about amounts, whereas numbers are the idea that exists in our heads about quantity.

Our Arabic number system developed in India, as the video below shows:

Most languages today use these numerals, however, this was not always the case. The first number system was the Babylonian system in Mesopotamia, using not a base 10 system, but a base 60 system (however, there are elements of base 10 as the system uses the symbols in a tens and units format). Many early numerals are more like symbols to us, just as our numerals would be to the Babylonians – it is simply a case of what you know and what you are familiar with.

Even Roman Numerals, which we are more familiar with, are letter/symbol based. There are two theories about the origins of Roman Numerals (link at bottom). The first is based on hand gestures, just as children count on their fingers now (10 fingers = our base 10 system).

For example, 5 = V. The hand gesture is this (similar to a V):

10 = X. The hand gesture is this (a cross with the thumbs or hands):

The second theory is based on tally marks on tally sticks – a system used long before the Romans, especially by shepherds counting their sheep.

This same site also alludes to earlier number systems such as the Egyptian system. This is similar in ways, with a single line representing “one” however, they used lines for all digits up to “nine”, and pictures beyond these. An example of larger numbers is shown below (both pictures from Discovering Egypt website).

It is clear that there are many other number systems other than our own that have been used throughout history, and even today in Chinese and Japanese languages. I personally have found these fascinating to look at and I believe children would find this interesting too, looking at the unknown and how it relates to the obvious that we see and use every day. This study into different number systems all looks towards Liping Ma’s relational understanding. I would not teach these systems while children are still learning their first numerals (1, 2, 3 etc) but it would be an interesting activity with older children, showing them why we have the numerals we do, and where they came from. The activity we did this morning, creating our own number system (see my attempt on the right) was a thought provoking one. You grow up only knowing the Arabic numerals, that creating foreign ones is strange and alien. I think children would learn a lot from this type of activity, not just about number systems, but about how numbers are formed, written and constructed. Like I say, this is not an activity for early years children beginning their maths journey, but for children who are comfortable with the numbers they know and can be stretched further.

References:

Pierce, R.  (2015) ‘Numbers, Numerals and Digits‘. Math Is Fun. Available at: http://www.mathsisfun.com/numbers/numbers-numerals-digits.html (Accessed 3/10/16)

Roman Numerals and Roman Numbers (no date) Available at: http://www.knowtheromans.co.uk/Categories/SubCatagories/RomanNumerals/#VIII (Accessed 3/10/16)

# A Sixth Sense

Piaget believed that children were born with no cognitive understanding of mathematics, or “numerosity” – the ability to understand small quantities (Marmasse, Bletsas and Marti 2000). However, more recent research has shown that children of just a few months old understand very small quantities, distinguishing between 2 and 3 items, but not between 4 and 6, showing their understanding is not fully formed (Starkley et al 1990).This ability proves that infants are born with an innate number sense.

There are many different number systems, outlined in the article by Marmasse, Bletsas and Marti (2000), dating back to prehistoric times. When writing was invented, there became symbols which represented this ability to count. These symbols we call tags. The easier these tags are, the easier it is for children to learn to count. For example in China, 15 is spoken as “ten five” and in French, 92 is spoken as “four twenty twelve” (as in 4 times 20 and then add on 12). This system is helpful when it comes to teaching children place value.

Marmasse, Bletsas and Marti’s (2000) article also describes the 5 principles of counting, outlined by Gelman and Galistel (1978). It is these principles that young children must grasp in order to count accurately:

• One to One
• one counting tag for each item
• One, Two, Three NOT One, Two, Two.
• Stable Order
• counting tag order must be repeated and consistent (not as below)
• Counting 3 items – One, Two, Four
• Counting 4 items – One, Four, Three, Five
• Cardinal
• last tag represents the cumulative amount of items
• If counting the apples, you know that once you have counted 10, that the ten applies to the total of the apples, not the specific last apple you counted.
• Abstraction
• anything can be counted
• from age 2/3 children can count groups of mixed items
• Order Irrelevance
• doesn’t matter where you start counting from (left, right, top, bottom)
• This skill doesn’t emerge until age 4/5

Early counting involves using manipulatives i.e. fingers or toys. This then progresses onto verbal counting, removing these prompts with children counting in their heads, and eventually being able to recall facts from memory. For example, we all know that 3+7 = 10 without actually having to count on 3 from 7.

The part of the article that particularly resonated with me was where they discussed different teaching approaches. Marmasse, Bletsas and Marti discuss the two different teaching approaches as “traditional” and “constructivist”. These two approaches are much the same as Skemp’s instumental and relational teaching (as discussed in my previous blog available at https://blogs.glowscotland.org.uk/glowblogs/myeportfolioekma1/2016/09/23/316/).

It is clear that, while Piaget was partially correct in saying children are not born with any mathematical ability, numerosity is both innate and developed through learning experiences. There are some principles of basic mathematics that cannot be developed until around age 4/5, however, Piaget’s belief that it is not until age 8 that children develop a mature number sense has been clearly disproved by theorists such as Starkey. This is an important factor to consider in the classroom, and I will take this forward in teaching maths in school. Children are more capable than they are given credit for in mathematics, and I intend to use this knowledge and try more complex lessons and mathematical discoveries in future.

References

Marmasse, N., Bletsas, A. and Marti, S. (2000) Numerical Mechanisms and Children’s Concept of Numbers. Available at http://web.media.mit.edu/~stefanm/society/som_final.html (accessed on 1/12/16)

Starkey, P., Spelke, E.S., and Gelman, R. (1990). Numerical abstraction by human infants. Cognition, 36: 97-127.

# There’s Been a Murder!

In the clip below, Dan Walton (Teacher of the Year) is teaching his class Pythagoras. My memories of pythagoras at school mainly involve textbooks, much like most other maths topics in high school. We were taught the rule and how to apply it (instrumental understanding) and given textbook work to consolidate our knowledge. However, Dan’s approach to teaching this topic is very different.

Pythagoras is the rule that the squares of the shorter two sides with equal the square of the shortest side – a²+b²=c². Rather than simply teaching this rule to his class, he has them find out the answer. Some of the children simply draw the triangle to scale, giving them the correct answer, others must work out the rule using numbers. This is giving the children multiple ways to solve the problem and they then realise that the formula is more efficient than drawing triangles every time. By having the children investigate this rule for themselves, Dan instantly has the children engaged in the lesson.

By using a real life example (golf hole), Dan is showing the children the context of this learning in the world outside of the classroom. He gives them the option to solve the problem any way they want, allowing them to choose between the two methods they have explored. All of the children opted for the formula method, showing their ability to select an appropriate method for solving a problem.

When Dan then moves on to working out the length of a shorter side, he has a small piece of paper showing the rules of pythagoras, but does not tell the children anything about what they have to do. This investigation and discovery really has the children engaged and involved in the lesson in a way that a textbook cannot do. The children are working out what they must do – they are taking control of their own learning. By giving the children the opportunity to do this, Dan is helping them with long term problem solving skills. The children learn that the answer is not going to be given to them, they must work it out themselves.

The climax of the lesson, drawing on all the pythagoras they have learned, is a murder mystery problem. There are multiple pythagoras questions which they must solve in order to find out who the murderer is etc. This is an incredible way to find out if the children have taken in all that has been taught. If the children can find out the answer, then they understand the rules of pythagoras that they have been taught. Dan even hides some of the questions around the school to bring some energy and movement to the lesson. There is no doubt that this is substantially more rewarding for children than sitting at a desk with a textbook.

While pythagoras may be a more advanced topic than those of primary mathematics, I believe that many lessons can be learned from Dan’s lesson. Children should be encouraged to explore numbers and mathematics, rather than being told a rule and applying it. If they can work the rules out for themselves, not only are they much more likely to remember it, but they are more likely to enjoy this learning. While we cannot allow primary children to run around the school looking for questions, it is important that we allow them to have active maths, even taking them outside as a class into the playground to solve some problems will be much more engaging and exciting than a normal maths lesson. Dan also does not share the learning intention with the children at the start of the lesson, again, promoting discovery and exploration. I believe that this is even more important in a primary classroom as allowing children to find out what they are learning themselves shows that they have actually understood what is being taught.

In my future maths lessons, I want to incorporate some of these techniques, especially taking learning outdoors. Maths does not need to be sitting in front of a textbook. Maths should be a wonderful, exciting discovery of the whys and hows of numbers – an idea all too often overlooked in the primary classroom.

# Lost in Number Translation

Skemp suggests that there are 2 types of understanding: instrumental and relational.

Instrumental understanding – Not quite about playing the piano, but I think the analogy applies. Instrumental understanding is like knowing the notes to play, but not knowing the tune – in maths terms, it is about knowing the rules, formulas and processes to get an answer, but not knowing the underlying concepts. This method is easier to understand and children get to see the results of their learning quicker, giving them a sense of success that children, like all of us, are excited by. These reasons are why it is understandable that teachers use instrumental understanding in the classroom. If there are upcoming tests or exams, it is quicker and easier to teach this way if it is a subject they simply need to know. A teacher may also feel that the children have not developed skills that are needed to understand relational thinking, the other type of understanding described by Skemp.

Relational understanding is a more complex affair, however, the long term effects are substantially worthwhile. It is about knowing why you are using a certain rule and the concepts beneath the strategy, i.e. why two negatives make a positive. This approach is more adaptable to new tasks and is easier to remember in the long term. Relational understanding is knowing all of the connections across mathematical topics.

Skemp uses the analogy of a town to explain the difference between these two approaches. You can walk through town knowing your route from A to B and a few other routes nearby to get to the essential places you need (instrumental) or you can create a “cognitive map” of the town in your head, knowing all the routes and which is best for your journey (relational). If you can master the second approach, then you will never be lost.

References

Skemp, R. (1976) Relational Understanding and Instrumental Understanding. Mathematics Teaching. Available at https://alearningplace.com.au/wp-content/uploads/2016/01/Skemp-paper1.pdf (Accessed 23/09/16)

# “Tough guys don’t do math. Tough guys fry chicken for a living.”

Stand and Deliver (1988) is a film following teacher Jaime Escalante as he teaches a class of mainly Hispanic students in a fairly impoverished school/community. He has a radical teaching style much different from the norm of teachers across the globe, where he frequently makes fun of children and has a ‘no tolerance’ policy for anything less than perfect.

Real Jaime Escalante with Edward James Olmos, the actor who played him in the film. Picture courtesy of http://www.olmosperfect.com/stand-and-deliver.html

Escalante, born in Bolivia in 1930, became one of the most famous educators in America, moving there in pursuit of a better life. As shown in this film, he taught disadvantaged pupils, whom some teachers had given up on completely, and managed to have a handful pass an extremely difficult Calculus test. Escalante’s success has earned him many accolades, including being entered into the National Teachers Hall of Fame in 1999.

It is interesting to analyse Escalante’s teaching style in the film as it is far from conventional. He does not accept poor behaviour, and even why one student, Angel, walks out of the class, he merely says goodbye rather than chasing after him. Escalante is more concerned about the children who want to learn and believes that it is a privilege to be in his class. The children respect their teacher and want to do well for him. While his teaching style may be considered to be scaring his pupils into working hard, I believe he does this with a true sense of compassion and care for the future of these children. This caring nature is highlighted in the scene where Escalante gives Angel 3 textbooks – one for class, one for home and one for his locker – as he did not want to be seen carrying them around. This compromise can be useful in the classroom as it encourages conversation between teacher and student, rather than a one sided lecture from the teacher, which can cause the students to disengage.

When looking at specific teaching points in the film, it is clear that there are lessons to be learned from Escalante’s teaching. For example, a quote from the film that resonates with me is “Students will rise to the level of expectation”. It is crucial that teachers have high expectations for their students as it shows a confidence in them that they may not have themselves. I hope to remember this along my teaching journey.

Another example is how to help the children understand what he is teaching them, he uses language and problems that make sense to them – he gives the problems a sense of relevance. One example of this is when he uses numbers of girlfriends in a problem (see the below clip).

Using girlfriends instead of other objects such as apples or pens engages the children, making them laugh and therefore creating a relaxed classroom atmosphere where the pupils are confident in answering questions. Another example is where the pupils ask the real life benefit of learning these things and their teacher takes them on a trip to show them exactly how that maths is used in the real world. This idea links to the principle of relevance within the Curriculum for Excellence (Scottish Executive 2004, p15).

It is important that we teach children why they are learning things. While this seems to be an accepted part of the curriculum, maths, in my opinion, seems to miss this principle. In a broader sense, you can often hear students, especially at high school, saying “Where’s the point in this? When will this ever help me in real life?” This shows a distinct lack of teacher communication with the children about why they are learning these topics. In terms of more basic level mathematics, learners need to know what all these different rules and formulas actually mean. Pupils could be given rules and memorize them, which tends to be the case, however, they have no true understanding of why. This may be the case because many teachers themselves do not know the reason why, as they have been taught just to memorize the rules, therefore getting trapped in an endless cycle of knowing but not understanding.

I believe that I myself fall into the above category. While I enjoy maths and teaching it, I would say that I do not have a depth of knowledge in why maths happens. I hope to look into areas of mathematics in more depth in order to improve my knowledge of maths and my ability to teach it to a strong degree.

References

Biography.com Editors (2014) Jaime Escalante Biography. Available at: http://www.biography.com/people/jaime-escalante-189368 (Accessed: 15/09/16).

Scottish Executive (2004) A Curriculum for Excellence. Available at: http://www.gov.scot/Resource/Doc/26800/0023690.pdf (Accessed on 15/09/16)

# A Journey of Discovery

I have chosen this elective because of my love of maths. I have had a real enthusiasm for maths and numbers from a young age and have always been fascinated by the way numbers are connected and can move and change.

In this elective I am looking forward to learning more about maths and where it comes from, and especially how maths is used in school. I have found that children in primary school have a love of maths and enjoy learning about it, in general. However, when we get older, usually reaching high school, that love is gone. Textbooks chase out the enjoyment in maths.