# In Pursuit of Mathematical Art

In a recent input in Discovering Maths, we had a chance to create some mathematical art and my favourite of these was a curve of pursuit.

(MathsMasterOrg (2011))

This is created by choosing a regular shape, making a mark an equal distance from each corner and joining these dots to create a slightly smaller shape.  Then repeat…and repeat…and repeat…

The finished product is almost an optical illusion – the straight lines appear curved and it is hard to believe it is created purely by drawing straight lines.  I think this is a useful activity to try with a primary school class because it provides an opportunity to develop a specialist knowledge of fundamental mathematics.  A curve of pursuit can reinforce the basic idea (Ma, 2010) of the properties of shape by challenging the perception of what a square normally looks like as well as concepts like rotation.

The intersecting lines and the amount of patience required to draw this reminded me of an activity I did with the class I taught on my recent placement.

Zentangle is described as a relaxing way to create a piece of art by using simple lines within a shape to create ”tangles” and each ”tangle” is filled with a repeated pattern.  I had chosen to try this activity during a particularly hectic few days for the class for a bit of mindfulness and I also recognised an opportunity in the intricate patterns for the children to work on their fine-motor skills.

There are no rules in Zentangle – something I thought the children would be pleased to hear but they were very hesitant to start without being told exactly what to do.  We looked at some examples together and soon after, they all set off ready to get creative.  Haylock and Thangata (2007) made sense of this reaction for me in discussing how when mathematical concepts are involved, shape in this instance, children are normally concerned with giving the ”right” answer – thinking divergently.  However, without realising, I was asking them to think convergently in this activity which meant they needed to allow for more flexibility and openness to see what would happen, for want of a better phrase.  We all really enjoyed it in the end as an art lesson.

Throughout this module, I have become more aware of opportunities to exploit the fundamental concepts of mathematics and I wondered if I could combine these two activities to give more mathematical purpose to Zentangle.

Admittedly, I would increase the measurements next time to give myself more space and I probably spent far too long doing this

BUT

I do think this hybrid of Zentangle/curve of pursuit is a good way of exploring the properties of shape whilst also considering Curriculum for Excellence’s principles for curriculum design (Scottish Government, 2018).  It incorporates personalisation and choice by encouraging the children to choose which regular shape they would like to use to create a piece of art as well as the patterns they fill it with and coherence as it could be used as one of a series of lessons looking at mathematical patterns in artwork.

References

Haylock, D. and Thangata, F. (2007) Key Concepts in Teaching Primary Mathematics. Los Angeles: SAGE Publications Ltd (SAGE Key Concepts). Available at: http://libproxy.dundee.ac.uk/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=268633&authtype=shib&site=ehost-live&scope=site [Accessed 26/11/18]

Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.) New York: Routledge.

Scottish Government (2018) What is Curriculum for Excellence? Available at https://education.gov.scot/scottish-education-system/policy-for-scottish-education/policy-drivers/cfe-(building-from-the-statement-appendix-incl-btc1-5)/What%20is%20Curriculum%20for%20Excellence [Accessed 27/11/18]

# The Daily Mile-matics

Throughout this module, we have often been asked to reflect on our own experiences of learning mathematics at school and my memories are almost exclusively textbooks in the classroom and being excited that you had filled all of the squared paper in your jotter.  Fast forward 15 years or so to ‘Miss Duncan’ teaching mathematics on placement, I tried to include a lot of stimulating concrete material to introduce concepts and always asked the children how we might use the mathematics outside of school but didn’t actually use the mathematics outside of the classroom with them.  In this blog, I want to look further into some of the benefits of planning outdoor mathematics lessons as well as the challenges and give an example of how I might incorporate fundamental principles of mathematics in future lessons.

A benefit of any outdoor learning session is that it allows for a slightly different relationship between teacher and learner away from the, sometimes stressful, classroom.  This is supported in Maynard and Waters’ (2007, p.260) discussion about outdoor learning which states ”while the teachers indicated that they were unsure whether or not they should change their pedagogical approach when working outside, several indicated that they felt more relaxed outdoors.”  In the outdoor lessons I took part in during my recent placement, I witnessed a lot more teamwork and peer support among children who would normally be at opposite ends of the classroom in their normal maths groups.  So, outdoor learning may provide more opportunity for children to build important transferable skills and even improve relationships with peers and teachers than the more constrained nature of the classroom.

Much like morning break and lunchtime, an outdoor lesson is a chance to get some fresh air and expel some energy before returning to the classroom with more focus. This is echoed in the reasoning for introducing ‘The Daily Mile’ which encourages teachers and children to walk, run or jog for around 15 minutes each day to improve health and wellbeing.

NB: not the last mention of The Daily Mile… stay tuned.

However, as a student teacher, I can certainly say that outdoor lessons were a bit daunting. Fear of the unknown, or something like that.  It seems I’m not alone as Dillon et al (2006, p.107) cited teachers’ lack of confidence and limited time and resources as barriers to outdoor learning in our schools today.  Understandable, right?  Well, the good news is that these fears are already being somewhat alleviated by more exposure to outdoor learning early in teacher education.  The General Teaching Council for Scotland’s (2012) updated standards for provisional registration now state that students must demonstrate that they can use a variety of teaching approaches and resources before qualifying as a teacher – outdoor learning, for example.

So, how can we incorporate more outdoor learning in mathematics?

Back to ‘The Daily Mile’.  The class are already outside for at least 15 mins – it would be a good start to firstly make the most of this time for the children to start to develop a profound understanding of fundamental mathematics (PUFM)However, as Bilton (2010, p.4) states ”merely taking indoor equipment outside does not constitute quality provision.”  Not to worry – the best part is that the resources are already there waiting for us.

In our ”Maths & The Great Outdoors” input, we were able to estimate the height of a tree by measuring the same distance on the ground.  We modified the method somewhat by (gracefully) looking at the tree upside-down and creating the 45° angle by continuing on from where we could only just see the top of the tree to the grass.  For those of you who didn’t have the pleasure of seeing this first-hand, the start of this video shows how a question like ”how tall do you think that tree is?” leads to discussions and problem-solving opportunities using basic ideas such as the properties of the triangle and how concepts such as estimation and finding an average help us answer the question (inter-connectedness).  This is just one example of many new ideas I have for future lessons, thanks to my participation in this module.

In conclusion, it is clear that outdoor learning has benefits and I don’t think anyone really disagrees with that.  Instead, what holds us back is the fear of trying something new.  New = good!  We know that teachers and children benefit hugely from learning in a ‘new’ environment, even if that environment is the school grounds they walk across every day,  We have discovered so far in Discovering Maths and as cliché as it sounds, maths really is everywhere.  It is simply not enough to ask the children where they might use their new-found knowledge, we must support them in using it so that they truly understand what it is they are learning about and why it is so important.

References

Bilton, H. (2010), Outdoor learning in the early years. 1st ed. New York: Routledge.

Dillon, J., Rickinson, M., Teamey, K., Morris, M., Choi, M.Y., Sanders, D. and Benefield, P., (2006) The value of outdoor learning: evidence from research in the UK and elsewhere. School science review87(320)

GTCS (2012) Standards for Registration Available at: http://www.gtcs.org.uk/web/FILES/the-standards/standards-for-registration-1212.pdf

Maynard, T,  Waters, J. (2007) Learning in the outdoor environment: a missed opportunity?. Available at: https://www.tandfonline.com/doi/abs/10.1080/09575140701594400#aHR0cHM6Ly93d3cudGFuZGZvbmxpbmUuY29tL2RvaS9wZGYvMTAuMTA4MC8wOTU3NTE0MDcwMTU5NDQwMD9uZWVkQWNjZXNzPXRydWVAQEAw

# ”4 for £1!”, ”buy 2, get the 3rd free!” Think fast! Or slow…

As a part-time store assistant in a popular discount store providing ”top brands at bottom prices”, my colleagues and I spend a fair amount of time filling up baskets of items considered to be ”Star Buys” with bright, yellow stickers to attract the attention of shoppers as they walk through the door.  On the (rare) occasion I decide to do an extra shift on the tills, I’ve lost count of the amount of customers who have claimed they ”only came in for a couple of things” before asking me to look out a couple of bags and a small trailer to accommodate all their bought goods.  These customers are likely to have succumbed to the power of the sticker and bought some things impulsively contributing to the estimated £144,000 we each spend during our lifetime on these sorts of purchases. (The Independent, 2018)

Our shopping lists are often ignored once we get into a shop full of these types of promotional offers therefore my question is how does mathematics link to what we call ‘impulse buys’?

After an input in Discovering Maths, I believe that it is much to do with the theory of loss aversion and how clever marketing appeals to one of our two systems of thought.

Loss aversion
This term describes the human need to prioritise avoiding loss and the feelings that come along with losing something over gaining something supported by Kahneman and Tversky (1979) who state that ”losses loom larger than gains.”  In the context of promotional offers in my work, the potential loss in question is the potential of missing out on a good deal.  Products featured in the ”star buys” section are portrayed as the week’s best deals but are not necessarily new products and can be found in the main part of the shop.  By placing them in a section of the shop in which the products change on a weekly basis, venders are introducing a sense of urgency to customers as they appear to only be available for a limited time.  When researching this effect, I came across a theorist named Timothy Brock (1968) who found that scarcity increases value (or desirability, in this example) and I can understand why phrases like ”limited time only” are used so heavily in marketing.

Thinking fast and thinking slow – impulse buys

Kahneman (2011) believes that our behaviour is a direct result of co-operation – or lack of – between two different parts of our thinking process.  System 1 wants to give a quick, impulsive answer based on intuition at first glance whereas System 2 takes a little longer to consider what’s going on to make the best decision.  Ideally, System 1 would wait for System 2 to have a look before deciding on what action to take but in certain situations, System 1 is so over-powering that we act before System 2 has had enough time to convince it to have some self-control. (as pictured)

With this in mind, I put this into the context of ‘Star Buys’ again;

• System 1 & 2 enter the shop (not a bad joke, I promise…)
• System 2 has the shopping list and advises System 1 to walk past ‘Star Buys’ because there will be a wider range of products in the main section of the shop
• System 1 is already distracted by the yellow stickers and throws a few (impulsive) items in the basket – ”I need this!” or ”What a bargain!”
• System 2 might be able to regain control to consider what products are best value but the impulse buys are likely to remain firmly in the basket.

1-0 to System 1.

On a more considered trip round the shop, System 2 would be able to employ basic ideas of mathematics to work out what product was best value by comparing price per litre, for example.  A shopper would also need multiple perspectives if working with a budget – perhaps prioritising within a list so that there was enough money for the most important items.  It is quite clear that clever marketing is the ‘Achilles’ heel’ of most shoppers and it is not hard to see how shoppers can get carried away and end up out of pocket.  Of course, everyone likes a bit of retail therapy now and then but for the sake of our bank accounts, let’s hold off until System 2 has given the thumbs-up.

References

Brock, T. C. (1968). Implications of commodity theory for value change. In A. G. Greenwald, T. C. Brock, & T. M. Ostrom (Eds.), Psychological foundations of attitudes (pp. 243- 275). New York: Academic Press.

Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263-291.

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

After learning about the cognitive and affective domains with regards to learning in an input in my Discovering Maths module, I have been thinking about how maths groups based on ability and their seemingly neutral names may be affecting the way children feel about mathematics.

Do ability groups still have a place in a modern classroom? Are there other options?

## We should have ability groups because…

It is almost guaranteed that in a typical class, there will be children from very varied backgrounds – some nurtured and encouraged to learn but also some children who have extremely difficult home lives with little to no encouragement and support,  This is reflected in a child’s attainment through their learning journey and unfortunately, there is a significant ”attainment gap” between those living in the least deprived areas and the most.  (Scottish Government, 2018)

So, because there is an attainment gap between children in most classes, surely we need to group children by ability in order to provide enough support and to differentiate within lessons?

## We shouldn’t have ability groups because…

The children know what they really mean.  Whether their group is named ”The Triangles” or ”The Circles”, there is evidence to show that children see right through these seemingly harmless and neutral names – right through to ”top” or ”bottom” group.

An article published on independent news website The Conversation delved into how children feel about ability groups and quoted one child as saying “I’ve always been last in every maths group … I’ll just be low now in my next school, too.” (The Conversation, 2016)

This relates directly to our learning about the relationship between a child’s cognitive and affective domains. (Hiebert, J., & Grouws, D. A. (2007)

In other words, content knowledge and feelings respectively.

As shown in this diagram, a child’s development has affective and cognitive domains which both need to be nurtured in order for them to thrive academically and socially.

The quote above tells us this might be a child who’s affective domain has been attacked by feelings of negativity and inadequacy in mathematics which, unfortunately, might hinder his academic progress as he moves into secondary school.

Surely, as educators, we wouldn’t want any child to have such negative feelings towards their ability in a subject?

## So, are there other options?

YES.  Plenty, in fact.  Across the world, schools have come to different conclusions about how to create the best learning environment for their children.

Maths Mastery resonated most with me as I could easily link it to Liping Ma’s 4 fundamental principles of mathematics (2010);

1. Inter-connectedness (understanding how concepts are related)
2. Multiple perspectives (ability to adapt and use alternative methods)
3. Basic ideas (foundational knowledge)
4. Longitudinal coherence (developing understanding through time)

The following link explains Maths Mastery in more detail but in a nutshell, it has the whole class working together on a particular concept until everyone is confident, albeit with some mistakes along the way.

https://www.tes.com/news/maths-mastery-makes-setting-according-ability-irrelevant

Although I haven’t seen this approach in action, I can almost picture a bustling classroom filled with children helping each other and sharing their own ways of working which would undoubtedly support their affective development more than just accepting that they are destined to be in the ”bottom” maths group.

Perhaps there are aspects of both approaches that will be useful in my future classrooms.

I’m looking forward to finding out.

References

Dweck, C., Walton, G. M., Cohen, G. L., Paunesku, D. and Yeager, D. (2011) Academic tenacity: Mindset and skills that promote long-term learning. Gates Foundation. Seattle, WA: Bill & Melinda Gates Foundation. Available online at: http://www.scribd.com/doc/118972490/Academic-Tenacity-Mindsets-and-Skills-that-Promote-Long-Term-Learning Accessed 24/09/18

”Children put in the bottom maths group at primary believe they’ll never be any good”, The Conversation.  Available at http://theconversation.com/children-put-in-the-bottom-maths-group-at-primary-believe-theyll-never-be-any-good-54502 (accessed 22/09/18)

Ma, L., (2010) Knowing and teaching elementary mathematics (Anniversary Ed.) New York: Routledge. (Chapter 5, page 122)

Scottish Government (2018) Scottish Attainment Challenge. Available at https://education.gov.scot/improvement/learning-resources/Scottish%20Attainment%20Challenge (accessed 22/09/18)

# Resource Allocation

”Use the resources in the envelope to create a useful tool for a new student at the University of Dundee”.

The instruction sounded pretty straightforward, but after the task, I had an enhanced understanding of the variables that affect how easy it is to complete a given task.

Our group was split up into 5 smaller groups and each given an envelope.  Our envelope was jam-packed with materials – our idea was to make a weekly planner with colour-coded timetable and small map of campus with reminders posted to keep our student on track.  We could have made a few of these with our surplus supplies!

We were praised for our creativity, our organisation and presentation whilst being offered help throughout the task.  Some of the other groups weren’t so lucky – they seemed to have less resources and did not receive the same attention, their projects were almost scoffed at for not being as colourful or detailed.

Needless to say, we felt a bit sheepish and shook our heads quietly when the tutor asked us if we noticed the other groups had less, had we thought about offering up our discarded items.  It had been so easy for us to achieve the task because we had more than we needed.

We discussed how this links directly to inequality within a classroom and agreed that our role as future teachers is to endeavour to spread our time and attention throughout the whole class to develop each individual.  We may not all stand on a level playing field but every individual deserves the opportunity to achieve their full potential.

# Welcome to your WordPress eportfolio

Welcome to your ePortfolio. This is where you will document and share your professional thoughts and experiences over the course of your study at the University of Dundee and beyond that when you begin teaching. You have the control over what you want to make public and what you would rather keep on a password protected page.

The ePortfolio in the form of this WordPress blog allows you to pull in material from other digital sources:

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You can just about pull in anything that you think will add substance and depth to your writing.