So what is meant by Counter Intuitive Mathematics?

Why do losses attract more attention that gains? Why do our brains stick with the original answer in a multiple-choice test? These are questions that were picking through my brain when we spoke about Counter Intuitive Maths. Such as the two ways our brains function answers such as System 1 and 2 which I will discuss further through my blog post.

Loss Aversion

As we look into the multiple perspectives explained by Liping Ma (2010) which shows the idea that people have different ideas and approaches to similar outcomes. It shows both the advantages and disadvantages to various concepts. This can be related to Loss Aversion and the idea that most people do not like to lose anything that they own as it conveys negative emotions and fear. It can be seen that there are multiple perspectives in the idea of loses attract more positivity than constant gains. For example, we are far more upset about losing a £20 note rather than gaining one. This is because our idea of loss produces a stronger emotion and we gain the feeling that we do not want to give something up (Heshmat, S. 2018).

This video explains the concept of Loss Aversion in a simple form which makes us realise that a loss can be more painful than a gain. A similar example we can expand on could be Netflix giving you 4 free months trial which you are happy about but then eventually they cancel, and give an upsetting thought of having to give it up and lose out on something you like to have.

Both our brains running

When we look into the question of “Why our brains stick with the first answer in a multiple-choice test”, this is through the terminology of System 1 and 2. These systems are processes which are carried out in our brain where one system is automatic, and the other is thought processed. System 1 occurs with our automatic responses once we see a question and put down our initial thought. System 1 proposes the idea without us going into further depth of the concept of the question (Kahneman, 2011). This can be linked to our memory bias as this system suggests we can recall studying this kind of question before and the content of the answer is remembered. It can show a context effect as the person/pupil sitting the test have memory of being in this same situation and recalls the same information that has been learned before (Cohen, 2012).

System 2 involves thought processing and decision making to occur in our brains. This system allows us to take time to think over the question and the outcome of the answer, for example whether it would be beneficial or not to change the original answer in a multiple-choice test or not. Quite often in these situations our brains look at a question initially but after further reflection it shows the main point of attention (Kruger et all, 2005).

An example such as “Emily’s father has three daughters. The first two are named April and May. What is the third daughter’s name?”. Have a think about it! When we look at this question our intuitive answer (system 1) is going to be ‘June’. This is often people’s initial thought when they read the question as they are trying to follow the pattern of the question and have fallen into the trap. Where-as the considered question (System 2) is actually ‘Emily’. The third daughter can only have this name as it states in the question that ‘Emily’s father has three daughters’.Image result for system 1 and 2 thinking

Overall, I have thoroughly enjoyed looking at this topic in our module as it gives a deepened insight as to how our brains process questions in different ways. There are many debates about whether it is more efficient to stick with your original answer or change it in a multiple-choice test. In my opinion I have never thought about this discussion before but reflecting on experience I would always stick with my original answer although my views have changed after research. Kruger et all (2005) states that research has shown that those who change their answers have shown an increase in test results and regardless of your system 1 answer which was shown in the example your first instinct is not always correct.

My perspective about counter intuitive maths relates to Ma (2010) and her explanation of fundamental maths and how we view equations through the 4 properties which all relate to the way of counterfactual thinking. Such as connectednesswhich provides a link between the 2 systems and through deep understanding of the question we are able to represent the correct answer in the end. Multiple perspectiveswhich was discussed through the discussion of loss aversion and how many people have different approaches to situations such as some would rather gain a £10 note rather than losing it. Through the idea of considered questions relates to Basic Ideasand how we are guided towards the depth of the real mathematics behind the question and are aware of the simple strategies needed to convey the final answer. Lastly Longitudinal Coherencecan relate to the knowledge of those who may sit these multiple-choice tests and use all their knowledge gained and take opportunities to use their mathematical progress to put down the correct answer.

Cohen, H. (2012) What is memory bias? Available at:http://www.aboutintelligence.co.uk/memory-biases.html(Accessed 25 October 2018).

Heshman, S. (2018). Psychology today. Available at: https://www.psychologytoday.com/gb/experts/shahram-heshmat-phd(Accessed 25 October).

Kahneman, D. (2011) Thinking Fast & Slow. New York :Farrar, Straus and Giroux.

Kruger, J., Wirtz, D., & Miller, D. T. (2005). Counterfactual Thinking and the First Instinct Fallacy. Journal of Personality and Social Psychology. Available at: http://dx.doi.org.libezproxy.dundee.ac.uk/10.1037/0022-3514.88.5.725(Accessed 25 October 2018).

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States.London: Routledge

 

Creativity in Mathematics

When I first read the title of the first presentation I thought “No way? How can mathematics be creative?”.  This is when I began to realise that I did not “hate” maths in the way I thought I did because it is actually an extremely powerful subject which is used in everyday life which we do not realise.

Tessellation

As learned in our previous lecture, Tessellation is the repetition of shapes that fit together without any overlaps or gaps and for the shape of tessellate it must be able to exactly surround a point or a shape. It can be made through the use of 2D shapes although there are different types of tessellation which include regular and semi regular. The video below shows examples of how tessellation can be used to create patterns and how different 2D shapes can be used to show this.

 

Islamic Art

The key concepts of Islamic Art include texture, colour, pattern and calligraphy. It is often very easy to pick out a piece of Islamic art and is outstanding for someone to look at. This kind of art does not necessarily follow a religion but includes traditions of art used by the Muslim culture. Islamic Art provides meaning in its repetition and variation and shows the relationship between maths and art. The art avoids human and animal forms and instead uses different mathematic tools such as reflective symmetry and how many lines there are (BBC,2014).

Geometric Multiplication Circles 

These are made through geometry which is the heart of Islamic Art. These circles can be formed by using times tables to create patterns such a stars which I have attached below. The procedure works for example by using the 6 times table. 6 x 2 is 12 which simples down by completing 1 + 2 which leads to the final answer being 3. Similarly by working out 6 x 6 is 36 and therefore 3 + 6 equals 9 which is the answer. This allows children in classrooms to use creativity and enjoyment in their times table work and helps them develop transferable skills to work out the final answers. These digital root patterns which we can describe them as allow children the opportunity to learn new maths techniques without realising they are doing them (Warner, N/A).

Ma’s (2010) idea of connectedness shows that tessellation and creativity in art shows how individual aspects of maths can be linked together which can allow children to understand the concept of maths in a more efficient way. Also the idea of basic ideas, allows the idea of joining different aspects in maths together to structure a simple strategy for example multiplication, and how these quirky creative styles can be remembered by children.

Examples 

5×1=5

5×2=10=1

5×3=15=6

5×4=20=2

5×5=25=7

and so on…

6×5=30= 3

6×6=36=9

6×7=42=6

6×8=48=12=3

6×9=54=9

6×10=60=6

References

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

Warner, M. (n/a) Digital Root Patterns. Available at: https://www.teachingideas.co.uk/number-patterns/digital-root-patterns/  (Accessed 1 October 2018).

BBC (2014) Islamic Art. Available at: http://www.bbc.co.uk/religion/religions/islam/art/art_1.shtml (Accessed 1 October 2018).

Maths Anxiety

When the word “Maths” is mentioned my automatic response and thoughts are the words of “fear’, “dread” and “complete horror”. It is a subject I can’t remember being taught in primary school but however in secondary, it was an unpleasant subject. I have always had negative vibes about maths and always stuck by “if you don’t get it, you never will”.

The reason I chose this elective was to conquer my maths anxiety and gain experience and more confidence within the subject. Although my experience throughout secondary school was negative, the main reason being the influence of teachers and their ways of teaching, I ended up receiving one on one lessons with a private tutor. She was amazing and helped me power my way through the third year of sitting National 5 maths…which yes after 3 years I passed at a B down to her!

I believe I will follow her steps throughout my teaching career and teach children many different solutions to figure out the correct answer to allow children to pick what strategy works best for them. As explained by Boaler (2009)teachers must not dictate a child’s level of work and assign them to low levels based on early years testing or statistics. Children must be given work that is challenging and interesting so that they gain the confidence they need to improve at maths. It is all about keeping up with the curriculum and recognise that children learn at different stages and rates unlike the way my secondary teachers seemed to teach. By allowing children to experience and try out solutions which suits them best, it gives pupils a sense of control and ownership through their own work leading to positive self-worth.

When we think about Mathematics, is it just me that thinks… Algebra, Pythagaras and Standard Deviation. Well after research and actually thinking into depth about everyday life maths, it has made me realise maths is everywhere!! When you buy your new jeans out of new look? Weighing your ingredients for baking a cake? Or even cutting and sharing your domino’s pizza with your flat mates? This is why everyday maths must be explained and shown to children as soon as they come into primary education, as they are likely to enjoy and engage with the subject if it has a meaning towards it. It is important that we as teachers or parents allow children to have the experience of hands on maths such as them wearing a watch to tell you the time or helping calculate the shopping list (Eastaway, 2010). It is important children develop the basic maths skills so that they are able to use their knowledge for everyday tasks and simple problem solving. As said by Ma (2010)“Mathematics use and value different approaches to solving problems whether they are arithmetical properties or relationships”.

Maths anxiety occurs through parental involvement throughout a young age if parents are not engaged in the children’s maths homework or activities. If parents play a role model in making maths exciting and hands on, then children are more likely to see the positives towards the subject. Parents tend to think that maths is a school-based subject and therefore avoid the content of it but by them demonstrating a lack of every maths, the children are going to be led in the same way (New York times, 2017).Maths anxiety which was first identified in the 1950s is said to affect at least a quarter of school children alone (Guardian, 2012).This article explains the importance of recognising this fear within children as this concept of maths anxiety is said to be similar to the fear of spiders or snakes in some people. We must not pressure our children into any forms of assessment or testing and allow children to work through reasonable stages of the subject through their own progression. We must not assume or expect children to get the correct answer the first time because it may take a while for them to become familiar with what stages they must carry out to achieve the final answer (Eastaway, 2010). It is important to encourage the children to be proud of their mathematical skills and focus on the understanding rather than the idea of setting standards.

In conclusion through the introduction of maths anxiety and based on experiences from myself and others I am very excited looking forward to becoming engaged in this module. I hope it I will be able to help overcome my maths anxiety and allow me to gain the confidence to correctly teach children throughout my career to have a positive approach towards the subject.

Below I have attached a Ted Talks video which I enjoyed watching and felt inspired to conquer my maths fear after watching as Dan Finkel states the way maths should be taught. He starts of by saying many people either claim they love/hate maths and why that it is seen as a “can or can’t do” subject.

Not knowing is not failure, it is the first step towards understanding”.

References

Boaler, J. (2009) The Elephant in the Classroom. London: Souvenir Press Ltd.

Eastaway, R. (2010) Maths for mums and dads. London: Square Peg.

Ma, L. (2010) Knowing and Teaching Elementary Mathematics – Teachers’ Understanding of Fundamental Mathematics in China and The United States.London: Routledge.

The Guardian. (2012) “Maths Anxiety: the numbers are mounting”. Available at: https://www.theguardian.com/education/2012/apr/30/maths-anxiety-school-support (Accessed 30 October 2018).

The New York Times. (2017) “Fending Off Math Anxiety”. Available at: https://www.nytimes.com/2017/04/24/well/family/fending-off-math-anxiety.html (Accessed 30 October 2018).

Video- TEDX Talks (2016) “Five principles of Extraordinary math teaching” Dan Finkel. Available at: https://www.youtube.com/watch?v=ytVneQUA5-c (Accessed 30 October 2018).

Organising the classroom for learning

In relation to Organisation and Accessibility, we see that classroom organisation and management is key in relation to the aims, requirements and curriculum plans within the classroom. The organisation within the classroom can increase the effectiveness of the teachers and their strategies and it will make the children find it easier to learn. It is important that there is organised space within the classroom as children need freedom within the class to move around, however if this is not in order it is more likely for pupils to cause disruption. Such as having carpet space within the classroom where the teacher can have a discussion with everyone in the classroom and it gives the teacher a sense of control.

With use of resources, it is important that teachers have access to a range of different resources to enable children to have the experience of hands on and practical work. However, simpler resources such as paper, pens, scissors etc, is something that should be accessible for children to use as and when they need it. On the other hand, back to relation of organisation, teachers must take into consideration if other resources such as computers, gym space or art resources are available to the children. Another main everyday source is the use of whiteboards such as interactive whiteboards, which can be great opportunities to work as a whole class and allows individual children to take part.

In terms of displaying and presentation of work, it gives children the sense of inclusion within their classroom as they feel a sense of identity within the classroom and that their work is on display for everyone to see. Having strategies and topic tips on show throughout wall displays, it gives children the opportunity to refer to these if they were stuck and needed some support to get their answer.

Effective class rules and routines are important as each lesson is likely to have similar patterns to grasp the attention of the children, so they are used to the structure in which the way the teacher delivers the information to the class. The purpose of effective class rules is to control the behaviour management within the classroom and provides children to engage with their learning in the correct way. It is important that as a student teacher, as we take on the role of the class from another teacher, we must consider that not all the teacher will respond to the same rules and regulations, so it is important to display your rules often and keep reminding the children unto these rules are put into place.

Allocating activities is an effective way to give children a sense of responsibility and take on lead roles within the classroom, rather than the teacher doing it all. It teachers the children to take turns and the concept of sharing within the pupils of the classroom, as every child is allocated different jobs each week as it changes around. It is not only easier for the teacher, it allows children to develop many skills such as their time management, organisation and even communication with other people within the school environment.

The process of delivering the curriculum can be put across in many ways and not always by the well-known ways of the learning intentions and success criteria’s, there are serval exciting ways in which different curricular areas can be addressed to the pupils. Such as the use of resources within the class, allowing the children to work independently, with a partner or in groups, which is a more appealing way rather than explaining the basic structure of the curriculum.

Health and Wellbeing

The key messages from Dr Suzanne Zeedyk and John Carnochan’s videos both focus on how important it is for brains to develop from such a young age. The brain development in the first 3 years of a child is essential to ensure each child grows relationships with the people around them and know which way to response to them.  It is important as teachers to consider the excitement or on the flip side, worry, of early year brain pathway as it defines the way our brain will function in adulthood. Such as a child who is being put in a situation to witness domestic violence within their household, the child;s brains is having to learn how to cope with threatening behavior, which means they are unable  to learn in any other way. This can lead to children having high levels of stress hormones such as Cortisol as they are having to learn to adapt to the environment they are living in. A stressful home life can influence the children in many negative ways such as poor development of life skills, relationships and can have a large impact on inequality.  As teachers, it may be difficult for us to have that 1 to 1 strong bond with children if they have not built any close relationships with family in their early years. By having a lack of engaging contact and spending quality time with parents, this can have a negative impact on the children when they begin to enter the classroom for the first time. To ensure good brain development it is essential that to at least the age of 3, the child has some relationship with an adult, which may just be simple as smiling to the child. By ensuring this, it means there will be consistent good links with the teachers, which will allow them to feel confident of their presence in the classroom.

Reflections gathered from the Sugar Crash video, the importance of sugar was shown and how it can have a negative influence on not only children but adults also. The effects of sugar can increase many health related diseases such as diabetes, coronary heart disease and decreases the life span among us. Through the video, by being shown the amount of both added and hidden sugar in foods, it is a real eye opener as a trainee teacher, as it explains the figures in the rise of Child Obesity within our society today. As teachers it is essential to encourage exercise in our gym lessons or after school classes, to ensure these levels of child obesity drop rapidly rather than rise. Other activities such as walk to school week and healthy tuck is another effective way to encourage children to take note of the levels of sugar they are consuming, and that having a treat is acceptable as long as they are able to have a balance with physical activity to burn it off. We must have a positive impact on the children ourselves to play the lead role in inspiring children to have healthy food and routines in their childhood, to ensure this continues onto adulthood and encourages good eating habits within the children.

Semester 1 Reflection

Reflect on one of the most important moments for your professional development in semester 1 and write a post about what you think you have learned from this critical incident and what the process of reflection is beginning to mean to you.

In semester one, when beginning the Working Together module, I felt nervous and worried about being put in our allocated groups in the process to complete our group presentation.  This put me on edge as I knew I did not know a lot of people on the course  and that my group was going to be with people I would not have the confidence to speak to. Although as the weeks went on I became more of a confident individual and had more courage to voice my opinions to the rest of my peers in the group. The group as a whole began as a slow start with not many people contributing in tasks because of lack of confidence which resulted in our group having lack of knowledge of the tasks we were meant to complete. Being in a collaborative group where all of the three professionals of Teachers, Social workers and CLD students came together, it showed the importance of each profession in relation to the link they have with each other. Through out agency visit to Action for Children and speaking to members of staff who are part of this agency, this can back up the fact of how the three professions link. Reflecting back on the agency visit, it appeared stressful to begin with as we did not quite have a main role lead in our group, so the progress of arranging the visit was slightly tricky but when it all came together and there was now a group leader by this stage, it became easier for us as a group as a whole.

Throughout the completion of group tasks and by analysing our agency visit, we seen that GIRFEC plays a huge role within the three professions and especially has a large impact on us as teachers today. GIRFEC explains that the child is the main approach of the theory and it is within our professions to keep the children in a happy, safe and friendly enviroment. (Scottish Government, 2012)

I began to feel an improvement in my professional development when we had to focus on working together in our groups during semester 1, with people who were randomly put in a group. I felt that in this situation it taught me that by working as a large group, it can become daunting at first but by over coming those so called “ice breakers”, we all began to voice each of our own thoughts which resulted well with great ideas for our final presentation.

Scottish Government (2017) Getting it right for every child (GIRFEC) Available at: http://www.gov.scot/Topics/People/Young-People/gettingitright (Accessed: 22 January 2018)

Resource Allocation Task

As part of our first “Values” workshop with Derek, our group of MA1 students were split into five different groups. Our aim was to create a “Freshers guide” including useful information a new student coming to Dundee University may like to have. I got involved as part of group 2, where we were given a range of stationary. Different resources included, card, coloured paper, pens, scissors, sticky notes and a few others.

By glancing around the room, it appeared that other groups had as little as a paper and a pen, and some others had bigger packs the same as ours.

Our group came up with the idea of a “Student Welcome Pack” which included the following, which every individual in the group designed themselves:

  • Map of the campus
  • Money off vouchers for food/shopping
  • Simple cooking recipes
  • Guide of Kitchen safety
  • Bus and train timetables
  • Personalised timetable
  • Union information leaflet

Further into the lesson, I began to notice our Lecturer Derek was giving out positive feedback to the front two groups only and never acknowledged groups 4 and 5 in particular. As I was part of group 2 who received a high amount of favouritism and comments, I did not notice unto we got the offer of biscuits and when group 4 and 5 began presenting. I feel that in relation to being a child in a classroom, because I was a child who was given a high amount of encouragement and praise, I did not feel the need to look around me at other pupils in the classroom. It was not unto our lecturer ignored the group 4 who were presenting and was more interested in his phone.

I really enjoyed this seminar and it was a real eye-opener in perspective to primary education. By giving each group a different amount of resources, it supported the idea that not every child is the same and may not have the best of support in their life compared to other children around them. As I am sure group 4 and 5 could tell you, children who may unfortunately be less well off, just learn to deal with the resources they have and make do to the best of their ability. This suggests the idea that no matter what background a child is from, is it in our power to show children that each and every one of them in the classroom must be treated the same. Each child needs persuasion and encouragement in their learning development to help succeed to their full potential so they do not have any feelings of neglect or feel lower than anyone else.

Overall, I think Derek demonstrated this lesson extremely well and left me feeling very open-minded about the situation in perspective of becoming a primary teacher. I look forward to the next workshop!