Category Archives: 3.1 Teaching & Learning

Enquiry and Planning in Social Studies

Following the input about enquiry and planning with children, I decided that I would like to look further into enquiry-based learning and explore the importance for using this learning strategy.

The input started with looking at planning. We were given a whole-school topic on weather, in which we had to plan activities for each level. Dinkele (2013) states that planning is key to a successful lesson in social studies. Therefore, when planning the activities that could take place, we kept in mind the progression across each stage and what would need to be taught in early level before the children could progress into first level etc.

The experiences and outcomes that we chose to focus on were:

SOC 0-12a ‘While learning outdoors in differing weathers, I have described and recorded the weather, its effects and how it makes me feel and can relate my recordings to the seasons’

At early level, the activities that could take place would focus on identifying weather and learning words to describe the different weather, possibly using songs to do so.

SOC 1-12a ‘By using a range of instruments, I can measure and record the weather and can discuss how weather affects my life.’

At first level, now that children would know how to identify and describe weather, the activities could now move towards using instruments to record the weather. For example, making windmills and seeing how fast they spin in the wind and using temperature gauges in the classroom to record the temperature.

SOC 2-12a ‘By comparing my local area with a contrasting area out with Britain, I can investigate the main features of weather and climate, discussing the impact on living things.’

At second level, the activities can begin to look out with the local area. The children could research what crops are grown in Scotland and what crops are grown in a different country (i.e. Spain) and then go on to discuss the differences and why these differences may occur.

(Scottish Government, 2009)

First, we need to look at what enquiry-based learning is. Catling and Willy (2009) describe it as encouraging children to ask questions and search for answers. It builds on the children’s prior knowledge, understanding, values, beliefs and preconceptions about the world, develops their curiosity and supports them making sense of the world for themselves (Pickford et al, 2013). Using enquiry-based learning is not about just passing information between a teacher and a learner. It is about using the knowledge children already have about the world, from experiences within and out with school, and using these experiences as a basis to build the child’s learning upon.

Enquiry-based learning is very heavily based on the work of Vygotsky and his theory that knowledge is not transmitted directly from a teacher to pupil; but rather children learning about the world actively (Roberts cited in Catling and Willy, 2009; Dinkele, 2013). Vygotsky theorised that learning and development is first meditated between a child and a more knowledgeable other (in this instance the teacher) which later moves through a process called internalisation (Dimitriadis and Kamberelis, 2016). Vygotsky furthered the theory by explaining that this is not a one-way transmission of knowledge from the teacher to the learner but is an appropriation in which information is taken in to develop new skills in different ways (Dimitriadis and Kamberelis, 2016).

Through the input and wider reading, I feel much more confident with my understanding of enquiry-based learning and certainly more confident with ability to use this learning technique when I go out on placement and when I have a class of my own.

References

Catling, S. and Willy, T. (2009). Teaching Primary Geography. Exeter: Learning Matters.

Dimitriadis, G. and Kamberelis, G. (2006). Theory for Education. New York: Routledge.

Dinkele, G. (2010) ‘Enquiries and Investigations’ in Scoffham, S. ed., Primary Geography Handbook. Sheffield: Geographical Association. pp 95 – 103.

Hoodless, P., McCreery, E., Bowen, P. and Bermingham, S. (2009). Teaching Humanities in Primary Schools. 2nd Edition. Exeter: Learning Matters.

Pickford, T., Garner, W. and Jackson, E. (2013). Primary Humanities: Learning Through Enquiry. London: Sage Publications.

Scottish Government (2009). Curriculum for Excellence: Social Studies – Experiences and Outcomes. [Online] Education Scotland. Available at: https://education.gov.scot/Documents/social-studies-eo.pdf [Accessed 16th October 2018]

Maths in Music

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I have never played an instrument in my life or classed myself as musically talented. My brother plays the bagpipes and it amazes me how he can look at the notes and knows what it means. My musical abilities go as far as playing the recorder in primary school. So, when I found out we had a musical input, I was a bit apprehensive with what to expect.

We started off by listing some things that we think link maths and music. We came up with:

Rhythm
Scale
Chords
Tuning

It was quite difficult to think of the relations between maths and music without having a musical background, as I did not know but about music anyway. Du Sautoy (2011) said “rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry.”

The next thing we looked at was the Fibonacci sequence within music. Having previously looked at the Fibonacci sequence within art, it was interesting to see it appear in music as well. I have written about the Fibonacci sequence in my last blog post so I won’t go into too much detail in this blog. There are 13 notes in an octave, scale is composed of 8 notes, the 5th and 3rd notes of the scale form the basic ‘root’ chord and are based on whole tone which is 2 steps from the root tone, that is the 1st note of the scale (Meisner, 2014). This links to the Fibonacci sequence (0,1,1,2,3,5,8,13…). A piano keyboard scale goes from C to C with 13 keys, 8 white and 5, also split into groups of 3 and 2. This 13th note as the octave is essential to computing the frequencies of the other notes (Meisner, 2012).

Here is the piano scale.

We then had a shot playing the instrument ourselves, which at first, I found a bit difficult, but after a few tries, I began to get the hang of. Getting the opportunity to play the instrument allowed me to experience it for myself and improve my understanding of notes and rhythm, which allowed me make connections between maths and music.

Finally, we looked at tuning an instrument. If a guitar is tuned perfectly, the notes that are strummed out are in perfect Fibonacci sequence. This is, however, not the same with a piano. If a piano was tuned in Fibonacci sequence, it would not sound right to the ear. Instead, it has to be tuned with a man made adaption on this sequence. This video does a great job of explaining why a piano cannot be tuned using the Fibonacci sequence.

It was not until taking the discovering maths module that I realised how many different areas maths connects to. This reinforces the ideas of Ma’s (2010) ideas of longitudinal coherence and multiple perspectives. I will take what I have learnt form this module into my practice as a teacher, showing the pupils how maths interconnects with other subjects, whilst creating some cross curricular lessons involving maths, music and art. I thoroughly enjoyed this input as it was practical and I liked getting to have a shot of it for myself.

References

Du Sautoy, M. (2011). Listen by numbers: music and maths. [Article]. Available at: https://www.theguardian.com/music/2011/jun/27/music-mathematics-fibonacci [Accessed 27th November 2017]

Meisner, G. (2014) Music and the Fibonacci Sequence and Phi. [Online]. Available at: https://www.goldennumber.net/music/ [Accessed 27th November 2017]

Minutephysics (2015) Why It’s Impossible to Tune a Piano[Online]. YouTube. Available at: https://www.youtube.com/watch?v=1Hqm0dYKUx4 [Accessed 27th November 2017]

Tindal, C. (2017). Maths in Art and The Fibonacci Sequence. [Blog]. Available at: https://blogs.glowscotland.org.uk/glowblogs/cbteportfolio/2017/11/24/maths-in-art-and-the-fibonacci-sequence/ [Accessed 27th November 2017]

Maths in Art and The Fibonacci Sequence

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In our input with Anna, we learnt that there are many different aspects of maths within art. It was very interesting to learn about and see for ourselves.

The first activity that we took part in was drawing in the style of Mondrian. Piet Mondrian (1872 – 1944) was one of the founders of the Dutch movement De Sijl (The Art Story, n.d, b). De Stijl was an avant-garde style, that was appropriate to all aspects of modern life, from art to architecture (The Art Story, n.d., a). Mondrian was best known for being simplistic, using lines and rectangles to create his paintings.

Composition with Large Red Plane, Yellow, Black, Gray, and Blue (1921)

Here is one of Mondrian’s abstract paintings.

I then had a go at drawing my own Mondrian inspired piece of work. I started off by drawing two parallel lines about 9cm apart. I then began to draw lines within the first two lines, both vertically and horizontally, which created rectangles. I then coloured in some of the rectangles that were created and this was the result:

Here is my own interpretation of a Mondrian style of artwork.

The next thing we looked at was the Fibonacci sequence. This sequence starts with 0, and is when the two previous numbers add up to the following number, so it goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…etc. When this sequence is drawn out in squares, it makes spiral, also known as the ‘Golden Ratio’. This video looks at Fibonacci’s sequence and how it is found in nature. It also talks about the golden ratio and its relevance to Fibonacci’s sequence.

The golden ratio is the number 1.618033, also known as phi (Φ). This number is found by the formula (a+b)/a = a/b . This relates to Fibonacci’s sequence, as ratio of each successive pair of numbers in the sequence approximates Phi (1.618. . .), as 5 divided by 3 is 1.666…, and 8 divided by 5 is 1.60 (Meisner, 2012, b). Phi was first used by Greek sculptor and mathematician, Phidias (Meisner, 2012, a). It was Euclid that first talked about Phi as taking a line and separating it into two, in a way that the ratio of the shortest segment to the longest will be the same ratio of the longest to the original line (Bellos, 2010).

The golden ratio was first called the ‘Divine Proportion’ in the 1500’s. Many people believed that by taking measurements of your body, and then using the formula of phi, you were beautiful if the result was 1.60. We had a try at this ourselves by measuring:

Distance from the ground to your belly button
Distance from your belly button to the top of your head
Distance from the ground to your knees
Length of your hand
Distance from your wrist to your elbow

Then we had to divide:

Distance from the ground to your belly button by distance from your belly button to the top of your head
Distance from the ground to your belly button by distance from the ground to your knees
Distance from your wrist to your elbow by length of your hand

Some of the results i got were 1.7 which is very close to the golden ratio, and others were way off with results of nearly 2.1.

Many Renaissance artists used the golden ratio in their paintings to achieve beauty. For example, Leonardo Da Vinci used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background (Meisner, 2012, a).

Image result for the golden ratio in the last supper da vinci

Here is how Da Vinci has used the golden ratio in two of his most famous paintings.

The golden ratio and Fibonacci’s sequence is also a prominent feature in nature, as mentioned in the video before. Here are some more examples:

Fibonacci’s sequence in the bones of a human hand.

The golden spiral in a hurricane.

Image result for golden spiral in milky way

Even our own galaxy that we live in is made up of Fibonacci’s sequence.

I found this input incredibly interesting and will definitely be taking what I learned into the classroom. This input has definitely benefited me in not only seeing the link between maths and art, but also with nature, science, and much more. . Moving forward, I can see the possible cross-curricular activities that can be undertaken with maths, linking it with art or science, by creating our own artwork using Fibonacci’s sequence or looking at how plants grow in particular ways that link to the golden ration. Overall, I thoroughly enjoyed this input and can’t wait to look at it again.

References

Bello, Alex (2010) Alex’s Adventures in Numberland London: Bloomsbury

Meisner, G. (2012, a). History of the Golden Ratio. [Online]. Available at: https://www.goldennumber.net/golden-ratio-history/ [Accessed 24th November 2017].

Meisner, G. (2012, b). What is the Fibonacci Sequence (aka Fibonacci Series)?. [Online] Available at: https://www.goldennumber.net/fibonacci-series/ [Accessed 24th November 2017].

Memolition. (No Date). Examples Of The Golden Ratio You Can Find In Nature. [Online] Available at: http://memolition.com/2014/07/17/examples-of-the-golden-ratio-you-can-find-in-nature/ [Accessed 24th November 2017].

Robb, A. (2017) Maths and Art. [PowerPoint Presentation] ED21006: Discovering Mathematics [Accessed 24th November 2017]

The Art Story. (No Date, a). De Stijl Movement, Artists and Major Works. [Online] Available at: http://www.theartstory.org/movement-de-stijl.htm [Accessed 23rd November 2017].

The Art Story. (No Date, b). Piet Mondrian Biography, Art, and Analysis of Works. [Online] Available at: http://www.theartstory.org/artist-mondrian-piet.htm [Accessed 23rd November 2017].

TheJourneyofPurpose (2013) Fibonacci Sequence in Nature [Online]. YouTube. Available at: https://www.youtube.com/watch?v=nt2OlMAJj6o [Accessed 24th November 2017]

Profound Understanding of Fundamental Mathematics

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If someone asked me two months ago what ‘profound understanding of fundamental mathematics’ was, all I could tell you it was something to do with maths. PUFM sounds very confusing and complicated, when in fact it is quite the opposite. Over the course of this module, my understanding of this statement has developed greatly, and I now feel much more confident in my abilities of teaching in a classroom setting.

Ma (2010) defines PUFM as “an understanding of the terrain of fundamental mathematics that Is deep, broad and thorough.” There are four principles in teaching and learning that represent a teachers understanding of fundamental maths in the classroom: interconnectedness, multiple perspectives, basic ideas and longitudinal coherence.

Interconnectedness –

Ma (2010) defines interconnectedness as when “a teacher with PUFM has a general intention to make connections among mathematical concepts and procedures.” Interconnectedness occurs when the learner is trying to make a connection between the mathematical concepts and procedures. Here, it is the teachers job to stop the learning from being fragmented, and help the students to develop the ability to make links between underlying mathematical concepts, and showing how maths topics rely on each other.

Multiple Perspectives –

“Those who have achieved PUFM appreciate different facets of an idea and various approaches to a solution, as well as their advantages and disadvantages” (Ma, 2010). Multiple perspectives is when a learner can approach mathematical problems in many ways, as they understand the various perspectives when taking on a maths questions, and are able to look at the pros and cons of the different viewpoints. It is the teachers job to provide opportunities for the learners to be able to think flexibly about the thinking and understanding of different concepts in maths.

Basic Ideas –

MA (2010) says that “teachers with PUFM display mathematical attitudes and are particularly aware of the ‘Simple but powerful basic concepts of mathematics’ (e.g. the idea of an equation.” Basic ideas is a way of thinking about maths in terms of equations. Ma (2010) has suggested that learners should be guided to conduct ‘real’ maths activities rather than just approaching the problem when practising the property of basic ideas. If a teacher is implementing this principle effectively, then they will not only be motivating the learner to approach the maths problems, but will also be providing a guide to for the learners to help them understand the maths themselves.

Longitudinal Coherence –

“Teachers with PUFM are not limited to the knowledge that should be taught in a certain grade; rather they have achieved a fundamental understanding of the whole elementary mathematics curriculum” (Ma, 2010). Longitudinal coherence is when a learner recognises that each basic idea builds on each other. When a learner does not have a limit of knowledge, it is impossible to identify the level or stage that a learner is working at in maths. the learner has instead achieved a holistic understanding of maths (fundamental). A teacher who has a profound understanding of maths is one who can identify the learning that has previously been obtained. The teacher will then lay the fundamental maths as a foundation for learning later on (Ma, 2010).

The four principles are vital to having a deep understanding of maths, especially when teaching the subject. I will make sure I have PUFM before I begin teaching mathematics, as without it, I cannot teach the subject as in depth and as thoroughly as possible.

References:

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

Maths, Play and Stories

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How could maths, play and stories intertwine? How could a child possibly be learning maths whilst playing with their friends, or getting read their bed time story? Maths is all around us, even where we least expect it.

Parents as Teachers

Parents may think the only way they can engage their children at home with maths is by helping them with their homework. Some parents may not even be able to help their children with their homework due to the maths anxiety that many adults suffer. As I have spoken about in a previous blog post on maths anxiety, the negative connotations that parents have about maths, can easily be passed down to the child, and create them to have their own worries about maths. It is vital that parents provide a rich learning environment for mathematics at home, so that children can be learning at all times and reaching their full potential. Studies looking at how often numeracy activities occurred at home found that they often happened less than once a day. They also found that the activities rated by experts as having specific numeracy focus occurred more infrequently than the activities with ambiguous numeracy content (Skwarchuk, 2009). Much of a child’s early mathematical development is enhanced through communication. This is why parent should ask open-ended questions to support and challenge their child’s thinking. For example, using varied mathematical language like bigger, smaller, fewer etc.

Importance of Play

Lev Vygotsky says that the learning of a child takes place in the ‘zone of proximal development’ which represents the between what a child actually knows and what the child can learn with support from those who are more knowledgeable. He also believed that the teaching of maths can be influenced by relating the subject to the child’s own experiences. This helps us to understand why play is such an important aspect of learning maths, as it allows the child to personalise it and be able to relate it to themselves. Saracho (1986, cited in Saracho and Spodek, 2003, p.77) explains that when children play, they confront social circumstances and learn to collaborate, help, share, and resolve social difficulties. Play is extremely important to a child’s learning for some of the following reasons:

Helps children to make connections in their learning
Allows child to experiment
Provides a meaningful context
Promotes social learning
Encourages perseverance

Friedrich Froebel was another theorist that also emphasised the importance of play in children. He viewed play as the ‘work of children’ and believed that children’s best thinking took place during play. During quality play, children are:

making decisions
imagining
reasoning
predicting
planning
experimenting with strategies
recording

Susan Issacs had many of the same views as Froebel. She saw the value in play as a way to allow children to explore their ideas and feelings freely. Through play, children can move in and out of reality, and whilst doing this, she encouraged them to be curious and express their feelings. But how does this relate to maths? Nutbrown (1994) said that “mathematics is never far away from young children’s actions.” All the different things that happen during quality play (as listed above) all link into mathematics. Predicting, experimenting, reasoning, making decisions are all needed in maths. Children using these strategies in play will help them to have a better understanding of the approaches they must use. It will also help them to understand why they are learning maths, as they can relate the problem solving and reasoning to their real-life experiences.

Maths in Stories

Another way that maths can be incorporated into a child’s everyday life is through White Teddy Bear With Opened Book Photo stories. Stories are something that are enjoyed by children, and can help eliminate that fear of maths that a lot of children have. They can be used to introduce new mathematical concepts, or to build on ones that are already known. By showing children that maths can be fun and interactive, they will be much more willing to engage.

Here is an example of a math story book ‘A Place For Zero – A Math Adventure’. This story talks about the number zero and place value.

Stories like this one can help children to connect with, and understand the concept being portrayed in the book. Having a visual aid will allow the children engage in the book, and listen to the story line. At the time, they may not even realise that they are actually learning mathematical strategies.

Maths storybooks are not the only way to teach maths through stories. There are many popular children’s books that can easily be adapted to fit a mathematical story line. For example, ‘We’re Going On a Bear Hunt’ can easily be change to ‘We’re Going On a Square Hunt’. Using a familiar text will allow the child to acknowledge how maths can be related to them, and used in their day to day life. It may also make the experience more comforting, by having something the recognise, especially for those that suffer from maths anxiety. You should always remember to match the book and your discussions to the mathematical abilities and development of the children in your class.

From the workshop and my own research, I now understand the great importance of play in developing a child’s academic understanding, by allowing them to freely explore ideas and express their emotions. Not only in maths, but in every aspect of schooling, play can help children relate it to their own experiences. When working in classrooms in the future, I will make sure to incorporate all that I have learnt about play and stories to make the learning as fun and interesting as possible, as I now know the benefit of them.

References:

Ehrhart, M. (2014) A Place For Zero A Math Adventure [Online]. YouTube. Available at: https://www.youtube.com/watch?v=76–wKA1yYQ&t=487s [Accessed 1st November 2017]

Nutbrown, C. (1994) Threads of Thinking. London: Paul Chapman Publishing Ltd.

Saracho, O.N. and Spodek, B. (2003). Contemporary Perspectives on Play in Early Childhood Education. Conneticut: Information Age Publishing.

Skwarchuk, S. (2009). How Do Parents Support Preschooler’s Numeracy Learning Experiences at Home?. Early Childhood Education Journal, [Online] 37. Available at: https://link.springer.com/article/10.1007/s10643-009-0340-1 [Accessed 30th October 2017]

Valentine, E. (2017) “Maths, Play and Stories” [Powerpoint Presentation] ED21006: Discovering Mathematics [Accessed 31st October 2017]

Is Maths a Language?

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“Is maths a language?” There are thousands of languages across the world – English, Spanish, German, French… all taught across schools and reinforced as being important to succeed in later life with jobs. So why is this attitude not portrayed with maths?

Math is a language that has been used thousands of years. It is spoken universally, and can be understood by all no matter the age, religion or culture. Sure, different countries may have different symbols or words for aspects of it, but the profound fundamental understanding of maths is the same no matter where you go. Paying for your shopping in a supermarket uses the same knowledge of maths whether you’re paying in pounds, rupees, yen or euros.

Some anthropologists suggest that the global language of maths was needed in order to trade. Many different countries were trading, and were not able to communicate with each other as there was such a wide variety of languages, so a universal language that could be understood by all needed to be implemented. Roman numerals were the most dominant number system used in trade. It was created on the base 10 system but was not directly position and did not include a value for zero (Mastin, 2010). The base 10 system is a system used today in every country, and our understanding of place value is based on this. It is thought that this system was introduced at least as early at 2700 BCE by the Egyptians (Mastin, 2010). This system is used widely and is an understood language across the world, even though it appears to have begun in Egypt.

Europeans were still using Roman numerals in the 13^th Century, but found that they were difficult to work with when trying to divide or multiply. This is when Italian mathematician Fibonacci introduced Arabic numerals into Europe. These are the numerals that we know and use today to represent values of numbers. The difficulty of the Roman numerals led to merchants and bankers embracing the simpler Arabic system (Maths Careers, n.d.). This number system eventually spread across the globe, as the inclusion of zero meant that so much more could be done.

Here is a great video from Dr. Randy Palisoc, talking about maths as a language. This video also touches on maths anxiety, and how looking at maths as a language can help to eradicate the anxiety and fear around maths.

References:

Mastin, L. (2010). Egyptian Mathematics. [Online]. Available at: http://www.storyofmathematics.com/egyptian.html [Accessed 23rd October 2017]

Mastin, L. (2010). Roman Mathematics. [Online]. Available at: http://www.storyofmathematics.com/roman.html [Accessed 23rd October 2017]

Maths Careers. (No Date). A Universal Language. [Online] Available at: http://www.mathscareers.org.uk/article/universal-language/ [Accessed 23rd October 2017]

TEDx Talks (2014) Math isn’t hard, it’s a language | Randy Palisoc | TEDxManhattanBeach [Online]. YouTube. Available at: https://www.youtube.com/watch?v=V6yixyiJcos [Accessed 23rd October 2017]

Maths Anxiety

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I always enjoyed mathematics all the way through Primary, and never once felt anxious about participating in it. Even during the early years of High School, I still had no worries about maths and liked going to classes. This all began to change around about 3^rd/4^th year, when the maths became harder and more stressful. Trying to memorise formulas just to pass my exams without fully understanding how or why I was learning this is where my maths anxiety stemmed from. This anxiety is still something that I carry with me now.

I decided to choose ‘Discovering Mathematics’ as my second year elective, as I wanted to try and further my understanding of maths and try to get over my maths anxiety. In one of our first few workshops, maths anxiety was one of the topics that was discussed. Hembree (1990, p.45) describes maths anxiety as “a general fear of contact mathematics, including classes, homework and tests.” This anxiety of maths can have both physical and psychological effects on students. These include headaches, muscle spasms, shortness of breath, dizziness, confusion, mind blanks, incoherent thinking and many more (Arem, 2010, p.30).This anxiety of maths can cause the pupil to become disengaged in their learning, as they lose some self-esteem, and in turn, the anxiety increases. I found it really interesting that maths anxiety can be considered a diagnosable condition, as I always felt that it was just my own fault for not being great at maths.

For most children, this anxiety continues into adulthood and can affect their confidence in tasks such as paying bills and handling their finances. If as kids they did not learn the basic mathematics as a result of maths anxiety, this could potentially affect them for the rest of their lives. This anxiety can also be transferred into their own children, giving them a negative impression of maths. Many parents will be unwilling or unable to help children with their homework, which will also greatly affect the child.

Maths anxiety in teachers also greatly affects the student’s performance. The teachers on the Maths and Science Survey (TIMSS) it is shown that Scottish P5 pupils are scoring below international average, and S2 pupils are scoring well below the international average. Even the highest achieving pupils in Scotland scoring well below the international average, which is very worrying figures for the country (IEA, 2008). As a teacher, I want to try and get over my maths anxiety, after seeing how greatly it can impact on your class’ performance, and help the children that do suffer it to see maths in a different light.

This video by TED-Ed does a great job of explaining what maths anxiety is and ways in which it can be helped.

References:

Hembree, R. (1990) ‘The nature, effects and relief of mathematics anxiety’, Journal for Research in Mathematics Education, 21.

IEA (2008) Trends in Mathematics and Science Survey 2007. Lynch School of Education, Boston College: International Association for the Evaluation of Educational Achievement.

TED-Ed (2017) Why do people get so anxious about math? [online]. YouTube. Available at: https://www.youtube.com/watch?v=7snnRaC4t5c [Accessed 4th October 2017]