Aperiodic Tilings: A collaborative and interactive Maths-Art exhibition and workshop.

Written by Senior lecturer in Mathematics Education, and co-organiser of the Aperiodic tilings exhibition, Charlotte Webb.

In this post, I will reflect on the Aperiodic tiling exhibition and workshop and offer the reader suggested activities and resources to use at home or in the classroom with learners.

An event inspired by, and in honour of Professor Uwe Grimm

In June 2022, the Open University hosted a collaborative art exhibition and interactive workshop inspired by the life and work of Professor Uwe Grimm, who sadly died in October 2021.

You can visit our online gallery here: Aperiodic tilings online exhibition | School of Mathematics and Statistics (open.ac.uk)

Uwe’s friend and colleague Ian Short wrote the following passage:

Uwe’s research career began in mathematical physics and later moved into mathematical biology, combinatorics and the theory of aperiodic order, which was the mathematical study of the artwork displayed in the exhibition. He was a world leader in aperiodic tilings. Uwe and co-author Michael Baake wrote and edited the beautifully illustrated series of mathematical books Aperiodic Order, which are invaluable resources for researchers in the field. Some of the images from the first text in this series were displayed at the exhibition and are available to view on the online gallery. These same images were used by Uwe to convey the beauty of mathematics to school students and the public in many outreach events.

Two of Uwe’s designs on display at the exhibition

Reflecting on the exhibition and workshop, there were three key features which made this event unique in my view: Collaboration, Creativity and Community.

Collaboration 

From the very start this event was collaborative in nature. The planning committee itself consisted of academics from different fields in mathematics and education, each bringing our own perspective and drawing on our experiences. We worked closely together to organise the three strands: a mathematical conference, an Aperiodic tiling inspired exhibition, and an interactive workshop for school children and members of the public.

The exhibition and linked conference were unique in bringing together mathematicians, engineers, artists, musicians, and educators from across the UK and beyond, all with a common interest in Aperiodic tilings and patterns. This resulted in a rich and diverse display of artworks and interactive workshop materials. For example, Liam Taylor-West, an artist-in-residence at the school of mathematics for the University of Bristol, brought musical installations based on aperiodic patterns which students were able to interact with and discover aperiodic wavelengths. Another artist, Natalie Priebe Frank, who describes herself as a mathematician studying self-similar tilings, brought handmade mobiles and ‘Self-similar Self-portraits’, demonstrating her mathematical research in a visual and accessible way.

A student visitor interacting with Liam Taylor-West’s light and sound installation: Wavelength.

Self-similar self-portrait by Natalie Priebe Frank

 

 

 

 

 

 

 

 

 

 

Feedback from another of the mathematicians who displayed artwork in the exhibition read: ‘I was greatly impressed by the range and depth of art displayed. For me, this was the largest collection of aperiodic art I have ever seen. For my own exhibit, a collection of items collected over my 25 years involvement in the field of aperiodic research, I received much appreciative feedback.  I was so pleased to return from the conference talks to see students playing with the aperiodic tiles I brought, making beautiful patterns.’ 

Finally, the workshop element allowed collaboration during the exhibition. Adult and school-student visitors worked together to solve geometric puzzles and tiling challenges, and to create collaborative art inspired by Islamic geometry with artist Maryam Smit for our live art wall (pictured below).

Students working together on an Aperiodic tiling puzzle

Maryam Smit and the start of our live art wall.

Uwe had inspired cross-curriculum collaboration throughout his career, having moved himself between disciplines. Most recently he had founded the NOVMAT (Novel superior materials based on aperiodic tilings) project, which brought together engineers and mathematicians to explore the potential applications of 3D aperiodic structures, including in acoustics, aerospace, automotive and medical engineering. Earlier in his career he had worked with artists and mathematicians to create outreach resources for the 2009 Royal Society exhibition.

One of the posters created by Uwe and his collaborators for the Royal Society exhibition.

A colleague of Uwe commented that this event was ‘a lovely and fitting memorial to Uwe, an amazing confluence of artists, mathematicians, outreach enthusiasts and their work, and a rare and incredible opportunity for all who could attend’.

 Community

In addition to bringing together the already established community of mathematicians, physicists and engineers involved in the research field of aperiodic order, the exhibition brought together a community of Islamic geometric artists, many of whom had never met in person, local school children, OU academics, and the Milton Keynes Islamic Artists Heritage and Culture group (mkiac.org). OU academics had run a workshop at the MKIAC Art in the Park festival earlier this year, inviting members of the public to create tiles for an Islamic geometry inspired tessellation, which was displayed at the exhibition and Art in the Park visitors were invited to view their work on display.

A community of Islamic geometric artists brought together through this event: Samira Mian, Clarissa Grandi, Maryam Smit, Richard Henry, Hasret Brown.

Artwork created at MKIAC’s Art in the Park on display at the exhibition

With around 700 visitors across the two days, this special event allowed two-way engagement between members of the public, artists and academics, where visitors could ask questions and interact with the artwork and puzzles on display. In particular, we worked closely with students at Kents Hill Park secondary school, providing them with learning materials to work on before the exhibition and inviting them to contribute artwork to the exhibition itself. Further details about their involvement can be read below.

One of the many positive outcomes of the event was the impact on the students of visiting a university site. Students commented that they were pleased to have the opportunity to visit the OU, which is situated only a few hundred metres from their school. Many students were surprised that staff and students were on their doorstep, highlighting the importance of this kind of community outreach for schools and colleges.

Creativity

The event was a hive of creativity, as can be seen in the online gallery, but this went beyond just viewing the beautiful works of art displayed in the exhibition. Participants were creative in the way they solved challenges – such as our giant soma cube and tetrahedron puzzle – having to try different approaches, and were able to create their own artwork, working closely with some of our exhibiting artists. Some student visitors commented on how they felt about taking part in the workshop activities: ‘I felt creative, and I thought hard about each activity’, ‘I felt excited, amazed, wowed, challenged when I tried the puzzles.’

Students working on Isometric patterns with artist Samira Mian

Prior to the exhibition, Kents Hill Park students were asked to complete a Pen-puzzle challenge, based on the famous Penrose tiling, and to create their own mathematical artwork inspired by this tiling using their own choice of colours and decorations. These collaborative pieces of artwork were displayed at the exhibition and are shown below.

KHP students’ Penrose inspired designs

The students reported feeling: proud, happy, excited, surprised, and confident when seeing their art displayed at the exhibition. They commented on the enjoyment of working together to create mathematical artwork and the communication and teamwork needed to solve the challenging puzzle, putting the tiles together correctly.

 

 

Many students also commented on the unexpected nature of the event and activities, suggesting they had not previously considered the links between mathematics and art. Their teacher commented: ‘Many of our pupils were surprised to learn that the exhibition was all based around maths and this really opened their eyes to maths as a much broader subject than perhaps the curriculum currently allows for. They certainly were given the opportunity to see maths in a very different way. The links between maths and art/music were obvious in the exhibition which allowed the pupils to explore the links between the subjects’.

Join in!

Create your own Penrose inspired artwork like the students of Kents Hill Park School. A PPT resource and printable templates for this activity can be found via the online gallery.

You can also create your own Islamic geometry inspired tessellation, similar to that created by the visitors at Art in the Park (link to resources below).

Share your creations with us on Twitter: @OUMathsStats – we would love to see what you come up with!

Further reading and exploration:

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Meet the Maths Education Tutors

It has been a while since we introduced our tutors formally on this blog and, in that time, some tutors are sadly no longer with us, and some new tutors have joined from other areas in the field of maths education.

Barbara Allen

My school teaching career was in middle schools in Worcestershire where I specialised in mathematics. I developed an interest in girls’ attitudes to mathematics and that became the focus of the dissertation for my MEd. My PhD focuses on Pupils’ Perceptions of Mathematics Classrooms and found the ways that pupils think they learn most effectively.

In 1994, I moved to the Open University as a Research Fellow and in 2000 I became the Director of the Centre for Mathematics Education. I continued as the Lead Academic for Mathematics Education until my retirement in May 2017.

I have written on a large number of OU modules from Access to Masters Level. For some reason, I always ended up writing the sections on fractions!

I am the co-author of the children’s book series The Spark Files and the writer of the children’s radio series The Mudds starring Bernard Cribbins and Mark Benton. Now available on iTunes!!

My main hobby has always been playing the clarinet. I play in Bewdley Concert Band and also play alto saxophone in the Wyre Forest Big Band. Now that I am retired, I am learning to play the xylophone and threatening to learn the drums. I also volunteer at Bewdley Museum and work with school groups that are learning about WWII.

Thabit Al-Murani

Hello, I am Thabit and I am an associate lecturer teaching on the ME322 course.

I have been involved in maths education for 25 years. Over this time, I have been a teacher, head of department, researcher, and more recently I started my own freelance business offering maths education consulting and specialised tutoring. My work has offered me the opportunity to live and work in several countries including the US, Sweden, Australia, Malaysia, and the UK.

I have a DPhil in Mathematics Education and my research interests are variation theory, the teaching and learning of algebra, and SEN mathematics education.

Ian Andrews

Hello, I’m Ian and I joined the OU as an Associate Lecturer in 2022.

I studied Maths and Statistics at university and once completed, I worked in data analysis in London for several years. Wanting a change of career, I started a PGCE in 2007 and have been a maths teacher ever since!

In 2015 I joined an 11-18 comprehensive secondary school in West Sussex as Head of Maths and member of the senior leadership team. I have also worked with the Sussex mathshub and lots of locality schools to improve maths in Sussex.

In 2019 I joined AQA (alongside my full time head of maths role) as Chair of Examiners for GCSE maths, GCSE statistics and L2 Further maths. This work involves running training and events for AQA and helping set grade boundaries for their exams.

I really enjoyed learning again when I took my PGCE and in 2013 I completed an MA in Education studies. Engaging with academic research whilst being a teacher has been hugely beneficial to my practice and my department.

When not teaching I enjoy music, playing football, running and acting as a taxi service for my 2 children.

Nick Constantine

Hello everyone, I am Nick Constantine associate lecturer for the Open University, I tutor on ME620, ME321 and ME322.  I also tutor on MU123 and MST124. I have been working with the OU for 9 years but I also used to tutor on the PGCE course from 2000-2002. I have had a very mixed career. My first degree was Astronomy and Astrophysics at Newcastle University, I then joined the Royal Navy as aircrew for a little while. I left way back in 1989 and did several ‘gap’ jobs before retraining as a Mathematics teacher and PE teacher.

My teaching career followed the standard path up to Deputy Head/acting Head but I always attended many mathematics training weekends with the ATM.  I also used to attend the MEI further mathematics conference in Nottingham for a few years. I was a Head of Mathematics in a 13-19 high school in Northumberland from 1998-2004 and enjoyed the process of organising and planning activities that reflected the fundamental philosophy of the OU ME(x) modules.

I also studied for a Master’s in Education from 2000-2002, one of my dissertations was ‘conjecture and proof in the most able’ (ME822), I really enjoyed designing my own research project and had a wonderful class to try some tasks with.  For me, if you can change the language of the mathematics in the classroom from a didactic controlling language to an atmosphere of questioning, conjecturing with learners and investigating relationships then you are really at the top of your game!

I now work part time as a teacher and OU lecturer.  I also work as a running and yoga coach and operate a small business where I organise retreats and workshops for private groups in Europe and in Scotland. Other hobbies are reading, radio 6 music, cooking and doing Maths problems!

Tom Cowan

I have been an Associate Lecturer with the OU since 2008 when I worked on the MEXR624 summer school each year in Bath.  When that ended I was lucky enough to be offered a chance to work on an earlier maths education module. I currently tutor on all the modules which we offer in the Mathematics Education suite of modules at Level 3 (ME620, ME321 and ME322) and also E209 – Developing subject knowledge for the primary years.

I completed my Master’s Degree with the OU in 2010 so remember what it was like to study at a distance and cramming in study whilst juggling other things in life.

My full-time role is as the Programme Lead of an initial teacher education programme at the University of Plymouth. I support the education and development of new Primary teachers on the BEd and PGCE – looking after those students with a specialism in mathematics.  Prior to this I was primarily involved with working with Secondary and Primary schools to support them with mathematics in challenging inner-city schools around Manchester and Salford.

I’ve never really left education and have found my next challenge in aiming to complete my Doctorate in Education in 2026! Hopefully I’m well on my way to becoming Dr. Cowan!

When I have some spare time, I enjoy going to the theatre, supporting Liverpool FC and Widnes RLFC and work as an officer with the Boy’s Brigade which keeps me in touch with further voluntary work (I did say spare time right?)

Jeffrey Goodwin

As well as being an Associate Lecturer for ME321, I also tutor on ME620. I first worked as a tutor for the OU in the 1980s on EM235 Developing Mathematical Thinking and returned to my current role of Association Lecturer in September 2014.

I was a classroom teacher for 10 years, working in Secondary and Middle schools. I moved into the advisory service in 1980 as Head of the Hertfordshire Mathematics Centre. I worked in Initial Teacher Training and running CPD courses for teachers; being head of mathematics education at Anglia Polytechnic University. For four years from 1986, I worked for the National Curriculum Council on a curriculum development project: Primary Initiatives in Mathematics Education (PrIME). I have always had an interest in assessment and testing and in 1998 was appointed to establish and lead the Mathematics Test Development Team at the QCA. We developed the end of key stage tests and other optional tests for all three key stages. I was in this role for eight years and then moved to become Head of Research at Edexcel and then Pearson Research and Assessment. In 2010, I became an independent consultant and worked with schools on making changes, particularly engaging with Japanese Lesson Study. For four years I was the Programme Director for the King’s College London MaST course.

I have seen it as important to make a professional contribution to education. This has involved being Secretary of the Mathematical Association, a member of the Royal Society Mathematics Education Committee and Chair of the coordinating committee for Primary Mathematics Year 1988. I have also been chair of governors of a primary school.

I have two main research interests: the role that Lesson Study plays in the profession development of teachers; and, a member of the research team at the UCL Institute of Education looking at The Nature, Prevalence and Effectiveness of Strategies used to Prepare Pupils for Key Stage 2 Mathematics Tests, a project funded by the Nuffield Foundation.

 

Suki Honey

I have been an associate lecturer with the OU since 2003. As well as being an AL for the OU, I also work for Plymouth University. I was a mathematics lecturer on the BEd and PGCE primary ITE programmes for 10 years, and am now a researcher in pedagogy.

Prior to working with Plymouth University and the OU, I taught mathematics at various secondary schools in Plymouth, Cardiff and Tring. As much as I love working with adults in HE, I really miss the younger students. They have such a wonderful outlook on life and mathematics that it keeps me on my toes. They also make excellent participants in my research, and I have worked with some amazing girls and young women on their experiences of being maths learners.

When I’m not working (that fleeting moment just after I finish marking ECAs), I can usually be found sitting cross-legged doing some mathematical origami or deeply engrossed with my nose in a book, cup of tea and some custard creams always close by.

 

Pete Kelly

I taught for fifteen years, first in London and then in South West England, before moving into higher education, and am now a researcher and tutor in comparative education and mathematics education. Long ago, I studied with the Open University, and so understand the benefits of online learning and some of the challenges our students face.

 

Angie McConnell

I’m Angie McConnell and I am currently a tutor on ME620 and ME322. I have been an OU tutor since 2005.

However, my connection with the OU stretches back a very long way – my first degree was from Liverpool University then in 1978  I started an OU degree and followed that with an MSc with the OU – all in Mathematics so I do understand what it is like to be an OU student. I taught for 10 years in a Secondary school then for 30 years in an FE college teaching a wide range of students but always Mathematics. The classes I loved were adult students, many of whom had a genuine fear of mathematics, and I am proud to say I converted many a ‘I hate maths’ student to an ‘it’s all right this maths stuff isn’t it?’ student.

In my spare time I love to travel and explore new places and countries and there’s usually an opportunity to do some maths. Sitting next to a young girl on a train in a remote part of Thailand I realised that she was struggling with her maths homework on quadratic equation. I offered to help and between us we sorted it. She spoke very little English and I speak even less Thai, but mathematics is a universal language.

I have two cats who are very helpful when it comes to marking TMAs and EMAs and love to walk across the keyboard as I am typing so if your feedback is returned with strange typos, blame the cats.

Rebecca Rosenberg

Hello. I have been working at the Open University since 2019, mainly on the development of the new Maths Education modules. 2022 is the first year I will be tutoring on one of these modules, and I’m really looking forward to putting all that hard work into practice!

Before joining the Open University, I worked as a maths education publisher, and before that I worked as a secondary maths teacher in Suffolk. I’m particularly interested in the way people talk about maths – both inside and outside the classroom; how do we form questions in maths lessons? How is maths discussed in popular culture and media?

In my spare time I knit, sew, cook, garden and binge-watch American high-school tv shows.

 

Jim Thorpe

I became a mathematics teacher through the accident of joining Bill Brookes’ PGCE course: suddenly I realised that much mathematical thinking could emerge from humble beginnings, numerical or geometrical, and realised that mathematics could make a major contribution to the intellectual and social development of adolescents if they were encouraged to function as young mathematicians within what John Mason calls a ‘conjecturing atmosphere.’

I have been committed to mathematics education for a long time, in the secondary classroom and then in a variety of ways supporting the work of mathematics teachers. My current occupation is mainly tutoring in mathematics and education, mathematics, and engineering for the Open University.

I am alarmed by much of what I see under the heading of teaching mathematics but remain unrepentant in proposing something richer than the all-too-frequent ‘training’ metaphor of communicating mathematics.

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Jittery motion of dust particles in water

This blog post was written by Jemimah O’Regan as part of a Summer Research Project under the supervision and guidance of Dr. Elsen Tjhung.

Jemimah O’Regan completed her BSc (Hons) degree in Mathematics and Statistics with first-class honours at the Open University in the summer of 2022. She felt that the Summer Research Project was a great learning experience, and found that it allowed her to apply and develop new skills.

The following blog post demonstrates an example of random processes in nature, which can be described mathematically using probabilities.

Consider, if you will, a particle placed inside of a box, which is then closed and filled with liquid. It is a tiny particle, so small as to be measured in micrometers, and it is subject to no forces other than the motion of the even smaller liquid molecules around it. If this box had a glass pane, and if one were able to see this tiny particle, one may note something curious. The particle will, of course, jostle and float as the water is poured into the box. Intuitively, though, one might expect that if one waits long enough, and keeps the box completely stationary, that the particle will eventually find a position to settle down in and stop moving. However, this is not what will happen. One could wait and wait, and yet this will not occur. This particle will continue to move.

In this simple case, it will not move due to gravity. There is nothing that the particle is attached to, no springs or strings acting on it and restraining or forcing its motion. There’s nothing obvious causing this motion, like the box being shaken or tilted, and it may leave one a little flummoxed that the particle refuses to settle.

Through examination, one may conclude that nothing larger than this particle is causing this motion, and one may even determine that it cannot be anything outside of the box making it move. Therefore, one may deduce, it must be something internal. Something very small, and an unobvious – uncommon – source of motion. The only thing inside of the box, other than the particle itself, is the liquid. The liquid which contains molecules even smaller than the particle – molecules which naturally move and collide in a manner that could be called random. When these molecules collide with the particle, they apply a force – ‘kicks’ – onto it, and those kicks push the particle in the liquid. These collisions happen all the time in other situations, but only really cause visible motion when the object being affected by the molecules is small enough to be displaced significantly by them.

This is the source of the particle’s continued motion, and the foundation of the concept of Brownian motion.

Brownian motion is, in a sense, an intersection between the fields of statistics and physics.

The particle’s motion can still be treated as a regular equation of motion, save in that it has a random component representing the random forces from the liquid molecules. As a result, its motion is not seen as deterministic, but rather, stochastic – that is, unlike many other cases in physics, one cannot guarantee exactly how the particle will move given the forces present on it.

Instead, due to the randomness introduced in the modelling, one may only draw conclusions on the motion the particle is likely to exhibit. This can be done using the concept of a probability distribution function from statistics, which provides the probability of (in this case) the particle being at a certain position at a certain time.

Say that the particle is at a certain position when one begins observation. At this point, which we take to be the beginning of the time scale, the particle is at its initial position with probability 1 – we know for certain where it is – and has the probability of taking any other position with value 0 – we know for certain where it isn’t.

After this first time point, however, the particle’s movement is not absolutely certain. It could start to move in any direction, it could collide with multiple particles and change directions and speed several times – as time increases, so too does the number of ways the particle could have moved since the beginning, and the number of positions it could end up in at the end of that time.

As time increases, the particle will explore the area of the box. Above is a representation of what the data over many such ‘runs’ of the physical situation and the probability distribution function at three times might look like for a particle constrained to move only in one dimension when the origin is zero – at small time values, the particle will most likely still be close to the origin, but as time increases, the particle’s probability of straying further away from its original position also increases. Similar reasoning holds for two dimensions.

There are many ways one can take this further – one can make the physical situation more complex (for example, by attaching the particle to a spring), or one could simulate the particle’s motion using computer software and analyze the data, one can find the trajectory of motion that the particle is most likely to take under certain conditions, and so on.

Motion is ubiquitous in life, and Brownian motion shows that it can occur even in situations one may assume would be stationary.

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Flatland as social satire: Women’s status in Victorian times and the push for educational reform By Xiang Fu

This blog post was written by our summer research student Xiang Fu, who was supervised by Andrew Potter and June Barrow-Green. This blog post focuses on the status of women in mathematics in Victorian times, and today.

Xiang Fu completed her Open University mathematics with statistics bachelor degree with first class class honours this July. She has commented that she absolutely enjoyed the study and appreciated the opportunity of the summer research project.

Flatland: A Romance of Many Dimensions by Edwin A. Abbott was first published under the pseudonym of A. Square, in 1884. It quickly gained popularity as a science fiction story for the introduction of higher dimensions; but also it is a satire criticising social inequality in Victorian Britain. The description by Square, the protagonist from Flatland, that women were regarded inferior to men was so vivid that the author was denounced by contemporary book reviewers as a misogynist. Abbott, writing in defence of A. Square in the preface to the second edition in 1885, had to explain that, as a historian, he had “identified himself (perhaps too closely) with the views generally adopted in Flatland and […] even by Spaceland Historians” (Abbott, 1884a, p.9), and in fact he believed that the straight lines (women) were superior to circles (the priests forming the top of the social caste) in many respects.  This essay will focus on how Abbott, using Flatland as a satire, exposed the injustice of women’s situations in Victorian times and Abbott’s effort to improve education, especially that for women.

Edwin A. Abbott (1838—1926) was a graduate of Cambridge, and worked as the headmaster of the City of London School (MacTutor, 2005).   Abbott’s written works covered various topics. Flatland was inspired by mathematician and science fiction writer Charles Howard Hinton (1853—1907), who was interested in the fourth dimension, and coined a word – tesseract – for the four-dimensional hypercube. Hinton, who was married to Mary Boole, the eldest of the five daughters of the renowned mathematician George Boole, taught Mary’s sister Alicia about tesseracts; and Alicia edited one of Hinton’s books about the fourth dimension (Chas, 2019). The fourth-dimension concept offered a great stage for drama and art in the late 19th century and first two decades of the 20th century (Ibanez, 2017).

Since women were viewed as disadvantaged in their intelligence level, they were excluded from Oxford or Cambridge until the 1860s to 1870s, and could not earn a degree from these two privileged universities until 1920 and 1948 respectively (BBC, 2019).  Female German mathematician Emmy Noether (1882—1935), who made a significant contribution to invariant theory, was discriminated against at different stages of her career due to her gender and Jewish origin. Like Alicia Boole, Russian mathematician Sofia Kovalevskaya (1850—1891) had to be tutored privately (Flood & Wilson, 2011, p.167).

In addition, the diminishing of women in the 19th century was supported by the widespread idea of the ‘missing five ounces’ of the female brain, though modern science has not found any significant difference in the functions of the brains of different genders (Eliot, 2019). In Flatland, one’s geometric properties decides one’s intelligence.  This can be understood as an analogy with the widespread idea of biological characteristics, such as the circumference of the head (measured with cephalometer) being supposedly linked with brainpower. French psychologist Gustave Le Bon (1841—1931) allegedly believed that women were the “most inferior forms of human evolution”, hence they “excel in fickleness inconstancy, absence of thought and logic and incapacity to reason” (Quotefancy, n.d.). Gustave Le Bon’s evolutionary theory about genders echoes Darwin’s conclusion that a man can attain “a higher eminence, in whatever he takes up, than women — whether requiring deep thought, reason, or imagination, or merely the use of the senses and hands” (Darwin, 1896, p.12).

That women held lower social status in Victorian Britain is mirrored in the narration about the social hierarchical system in Flatland by Square:  in Flatland each resident is a polygon; and one’s social status and intelligence are positively correlated with one’s number of sides and the equalness of one’s angles. The more sides one has, the higher one is positioned in the social pyramid. As a square, the protagonist is a middle-class mathematician. However, women are put in such a low rank that they are excluded from the two-dimensional hierarchy: women are nothing but “straight lines” (Abbott, 1884a, p.21); hence they are “needles” (p.25) from sideways and shrink to a point from the very front or back. Since women “have no pretensions to an angle”, they “are devoid of brain-power, and have neither reflection, judgement, nor forethought and hardly any memory” (p.27). Therefore, women have no “hope to elevate from the caste”; and “the very laws of Evolution seem suspended” (p.29) in their disfavour. Women are subject to emotions, not logic, and they identify others by feeling.

The “romance” narrative in Flatland develops with the key action in the second part of the book: Square is enlightened by a sphere from Spaceland, a higher dimensional world than the 2-dimensional Flatland. The sphere embarks upon a mission to Flatland to explain to Square about the existence of Spaceland. Square extrapolates his new knowledge of dimensions and insists that there must be a 4-dimensional terrain. However, this is rebuked by Sphere. The square offends both the sphere and the noble class in Flatland by challenging their superiority and privileges derived from the hierarchical system. The ruling class in Flatland sends Square to jail for his heretical proclamations about Spaceland.

The resistance from Sphere or from the governing class in Flatland to the acknowledgement of the existence of a higher dimension is assumed to come from their fear – the fear that the hierarchical system may be undermined, or even overthrown – leading to the loss of their social supremacy. It is the same kind of fear that drives the priests in Flatland to suppress the Colour Bill, an historical event detailed in the first part of the book. Starting from an unknown pentagon, a ‘chromatist’ – he painted himself and the new fashion spread quickly and widely in Flatland. When the lower-classed polygons begin to colour themselves too, the new colour culture in Flatland blurs the clear-cut social status based on being able to recognise by sight one’s number of sides and the equalness of one’s angles. The belief flies from mouth to mouth that “Distinction of sides is intended by nature to imply distinction of colours” (Abbott, 1884a, p.47). The lower-class polygons put forward the concept of social equality and asserted that there was not much difference between them and higher-class polygons. The Colour Bill was presented at the all-state assembly of Flatland, proposing that Priests would be painted in the same way as Women. The Colour Bill movement led to civil war and was ultimately suppressed.  The existence of the fourth dimension and the events of the Colour Bill revolt ridiculed Flatland’s (and therefore Victorian society’s) stiff social pyramid by exposing it as unjustified and illogical, which had been ingrained for generations as ‘natural’. The democratic movements in Flatland echo the fights for equality in the French Revolution, campaigns for political rights for working-class men in the 1840s, and fight for women’s rights  and the suffrage movement from the 1860s (Jann, 2008).

Flatland attracted attention from contemporary critics immediately after it was published. It was reviewed, in the journal Science in 1885, as “an amazing story” based on the “transcendental mathematical concept” of higher dimensions (Comment and Criticism, 1884). Robert Tucker in Nature recommended this humorous book for readers when they had a “leisure hour from their severer studies” (Tucker, 1884, p.77). Though puzzled and distressed with the “geometrical romance” of Flatland, (estimating only six or seven people in the US and Canada would enjoy reading it), the author of a review in the New York Times admitted that it made some apparent sense “in an appeal for a better education for women” (New York Times, 1885).

Though, as described in Flatland, women were confined by domestic duties and excluded from formal academic education, change involving women’s social status was happening in the 19th century. In 1846, the British government launched a teacher training program to take people older than thirteen years old into a teaching apprenticeship (Intriguing History, 2011). The paid teacher training scheme allowed women to enter the teaching profession, though female teachers received lower payment than their male colleagues. The suffragette movement, which fought for women’s voting rights, started in the 1860s. The Manchester Society for Women’s suffrage was established in January 1867. In 1866 and 1867, petitions promoting women’s suffrage were presented to the parliament by John Stuart Mill MP (1806—1873), who supported equality between the genders. However, all these petitions were turned down and women’s suffrage was not secured until August 1928, when Parliament passed the Representation of the People Act 1928 and British women won suffrage on the same terms as men.

In 1884, Abbott had his book Hints for Home Training and Teaching published to help parents who did home schooling with their children: boys and especially girls, since girls had much less chance to be admitted to the universities.  In the preface of the book, Abbott claimed that more educational opportunities then available for women “justifies the belief that in the next generation mothers will take a large part in the teaching and training of the young” (Abbott 1884b, p.12).  Abbott was one of the leaders of the local teacher training organisation. His effort in education reform was praised by the major female educators in Victorian times (Banchoff, 1991).  Thomas Banchoff (b. 1938) Professor Emeritus of Mathematics at Brown University argued that Abbottt was a strong believer in the equality of rights between men and women, especially in education (1991).

Flatland was written in the genre of science fiction (then called a ‘scientific romance’), introducing the mathematical concept of higher dimensions. By belittling women as one-dimensional line segments while the men were two-dimensional polygons, Abbott revealed the unfair adversity Victorian women were born into and had to live with. It is important to remember, however, that Flatland is a social satire. With his books and efforts in pushing female’s education reform, Abbott contributed to the fight against the social inequalities between men and women.  More than one hundred years on, the struggle for gender equality continues. For example, there still persists a gender wage gap (Advani et al., 2021); and in 2019, only 24% of the STEM professionals in the UK were women (Dossi, 2022). When reading Flatland today, enjoy the mysteries of the fourth dimension, but don’t forget the hidden message which strives for gender equality which is still as relevant today as it was in 1884.

 

 References:

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The men who invented zero

This blog post was written by Roshani Senior, an OU mathematics graduate who undertook a 3 month virtual internship with the school of mathematics and statistics. Roshani’s placement was focused on developing external engagement materials based on mathematics.

In this blog post, Roshani writes about under represented voices in mathematics, which is a particular interest of hers. She reflects that as a school student of Indian origin, she learned about  European mathematicians but it was not until she left school that she learned about the many contributions of Indian mathematicians. She believes that mathematics should be taught in an inclusive way, sharing the experiences and achievements of mathematicians from across different cultures.

This blog focuses on the contribution of two Indian mathematicians to the concept of zero.

Aryabhata   

                                                                        Brahmagupta

The move from zero as merely a placeholder by the Mayans and Babylonians – a tool to distinguish larger numbers from smaller ones to a digit of its own was established in India by a man named Aryabhata in the 5th Century. A mathematician and astronomer, Aryabhata contributed multiple mathematical concepts, crucial to maths as we know it today, including the value of pi being 3.14 and the formula for a right-angled triangle. The prior absence of zero created difficulty in carrying out simple calculations.

Gwalior, India

Following this in the 7th century a man known as Brahmagupta, developed the earliest known methods for using zero within calculations, treating it as a number for the first time. The use of zero was inscribed on the walls of the Chaturbhuj temple in Gwalior, India. Carved into a wall the numbers 270 and 50 can be seen today and have been established as the second oldest recorded zeros in history. The city of Gwalior was designed so that the gardens around the temple were large enough so that each day the gardens would produce enough flowers to create 50 garlands for the employees of the temple. When the temple was built this was inscribed on the walls and it is this 50 that can be seen, annotated almost as we would write it today. 

What we now call zero in English, Brahmagupta named “shunya” or “sunya”, the Sanskrit word for emptiness or nothingness.

Aryabhata and Brahmagupta wrote their works in Sanskrit, an ancient and classical language of India. Their use of numbers would have looked quite different to what we use in English now. However, Sanskrit had a large

influence in how the English numeric system is written and so there are quite a lot of similarities.

Numbers 0 to 9 in Sanskrit 

 

 

Within Indian culture there is an idea of one having a “nothing” or a void inside of yourself. Long before the conception of zero as a digit, this philosophical concept was taught within Hinduism and Buddhism and practised through meditation.  The ancient Hindu symbol, the “Bindi” or “Bindu”, a circle with a dot in the centre symbolised this and was what probably led to the use of an oval as the symbol for the Sunya. It has been suggested that this cultural and philosophical influence on the concept of zero is what allowed India to develop what previous civilizations did not think of.

 

Brahmagupta was also the first to demonstrate that zero can be reached through calculation. He wrote these rules in his book the “Brahmasphutasiddhanta”. He was therefore able to make another important leap – in the creation of negative numbers, which he initially called “debts”. Brahmagupta placed small dots above numbers to indicate they were negative, unlike today where a minus symbol is used. The use of negative numbers was shown in “Brahmasphutasiddhanta”. Brahmagupta also demonstrated their use to produce the quadratic formula. and demonstrated rules for calculations involving both negative numbers and zero.

 

His rules were as follows:

Addition and Subtraction with zero and negative numbers:

  • When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.
  • A debt minus zero is a debt.
  • A fortune minus zero is a fortune.
    Zero minus zero is a zero.
  • A debt subtracted from zero is a fortune.
  • A fortune subtracted from zero is a debt.

Division and multiplication with zero and negative numbers:

  • Positive or negative numbers when divided by zero is a fraction with zero as denominator.
  • Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator.
  • Zero divided by zero is zero.
  • The product of zero and a debt or fortune is zero.
  • The product of zero and zero is zero.
    The product or quotient of two fortunes is one fortune.
  • The product or quotient of two debts is one fortune.
  • The product or quotient of a debt and a fortune is a debt.
  • The product or quotient of a fortune and a debt is a debt.

When Brahmagupta attempted to divide 0 by 0, he came to the result of 0. However, most modern mathematicians would argue that 0 divided by 0 is undefined, or an “indeterminate form”. Despite this outlier, the rest of Brahmagupta’s grasp on the number zero is exactly how we conceptualize it today.

 

The concept of zero gradually moved East into China. Then West to reach the Middle East. And finally, over half a century from its conception, the zero made it to Europe, where its importance was finally recognised by the Western culture that previously frowned on the idea of nothing, referring to it as meaningless or even satanic. In 1200 AD, Italian mathematician Fibonacci, a man who has been considered the ‘most talented western mathematician of the middle ages’ wrote of Indian Mathematics and their use of zero:

Despite the number zero having quite literally no value, its concept has allowed mathematics to develop into what it is today. Its curation led to the three pillars of modern mathematics: algebra, algorithms, and calculus. The use of calculus (the mathematical study of continuous change), which the zero is crucial for, has allowed engineering and modern technology to be possible. The use of zero and one within the binary system is what made computing possible. So, without the invention of zero much of what we know today would not have been possible. The device you are reading this on would not have been able to be invented, if not for Aryabhata, Brahmagupta and India’s fascination with the idea of nothing.

 

 

 

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Introducing MathsCity. 

This blog post was written by Staff Tutor and Senior mathematics lecturer Katie Chicot. 

MathsCity marks a first, major step towards creating a first National Mathematics Discovery Centre for the UK. MathsCity opened to the public on 4th October. For more details visit https://mathscity.co.uk/

MathsWorld UK:

MathsWorldUK is an educational charity working to advance the public understanding of mathematics so that everyone, regardless of age and background, can access and delight in the fun and power of maths. The mission, ultimately, is to create in Leeds, by 2028, the UK’s first Mathematics Discovery Centre, a world-class visitor attraction celebrating the maths at the heart of the patterns and structures of our world. MathsWorldUK is led by Open University staff tutor Dr Katie Chicot

A cultural intervention

We know that a person’s culture is having the strongest impact on their study choices. A cultural intervention is needed to address the UK’s lack of relationship with or culture of maths.

MathsCity will act as a hub for a wider programme including maths activity clubs, evening talks and holiday activities. It will provide a valuable space for ongoing dialogue with young people and communities across the region, as we engage them in developing ideas and content for a major new discovery centre celebrating every aspect of mathematics.

MathsCity

This October we launched MathsCity (http://mathscity.co.uk/) a dynamic, interactive maths experience for children and families in a large unit in Trinity Leeds shopping centre.

MathsCity offers exciting maths experiences and challenges, designed as an immersive, exploratory ‘journey’ for a family audience and supported by skilled facilitators.

The aims of MathsCity are:

  • Over our first year of operation (October 2021 – September 2022), to reach 17,000+ people of all ages, backgrounds and cultures with inspiring activities, where they experience maths as playful, exciting and accessible
  • To build young people’s skills, confidence and interest in mathematics, particularly targeting those who are disadvantaged or excluded from such informal learning opportunities

Between 5th October and 20th December, we had 2381 general public visits and 464 school pupils visit MathsCity. The MathsCity feedback below has been posted on social media, either in tagged posts, comments, or reviews:

“We decided on the spur of the moment to book one last thing during the October half-term and we are so pleased we did. The staff were helpful, friendly and really kind. They took notice of the interests of them both. The children found it easy to talk to any of them and ask any questions they had. We spent over an hour and twenty minutes there and it was talked about all day. Would recommend it to anyone who wants to try something different. Hopefully this can become a bigger space one day!”

“Would highly recommend MathsCity – in particular, it is a great place to visit with people of all different ages. We loved it as adults, but we could see everyone from teenagers to toddlers having a brilliant time. A perfect choice if you have lots of different age groups in your party. We will definitely be returning soon!”

“Absolutely amazing place, would highly recommend to all ages.”

“Amazing few hours out, suitable for all. Great learning experience.”

“Wow!! What a place MathsCity is, in Trinity Leeds. We went today and spent 90 minutes doing all sorts of puzzles and learning through play. Our children were in their element and found it mentally stimulating, and whilst some bits were too hard, there was plenty to keep them entertained. There are around 30 puzzles and experiments for big and old to play on and the best thing is that if you don’t have a child and/or you don’t want to bring them, it is suitable for grownups too!!”

“What an amazing place and such a brilliant idea. We spent ages here, the perfect example of how learning can be fun.”

MathsCity was on all the regional news channels in a maths piece which is posted on the MathsCity Facebook page: https://fb.watch/95hPu1DM-G/

You can also get a feel for the centre via a great video made by a visitor:
https://twitter.com/LeedsPlusSocial/status/1451269462787829760

 

 

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Mathematics education assignments: working with learners during the pandemic

There has been so much change over the past year, especially in educational settings. At the Open University we know about distance and remote learning – it’s been happening for over 50 years! But even those students who did not normally have in-person exams or tutorials were still impacted in some way by the lockdown. We wanted to give an update on some of the things we put in place to ensure students could continue their studies during the pandemic.

At the end of March last year, students were preparing to begin their studies for two of our maths education modules that start in April: ME625 Developing Algebraic Thinking and ME626 Developing Thinking in Statistics. A key part of these modules is to work with a learner, and this is the basis for three out of four of the assessments. Working with learners means students on these modules must plan tasks, analyse learners’ work, and reflect on what the next steps could be. Analysing learners’ work involves recognising mathematical thinking and using ideas and frameworks from the module to discuss the mathematical activity. These are considerations that students consider when studying mathematics education modules under normal circumstances. But then there was March 2020…

The learners that OU maths education students work with can be of any age, with any level of previous maths experience. They can be adult friends or family members or school-age learners that the student lives with or knows. Some of our maths education students work in schools themselves and will often work with learners at the school as part of their assignments. So, when schools closed to non-key-worker children, and social distancing measures were put in place, there was the potential for some students to be unable to successfully complete their assignments. When we realised this would be the case, we put our heads together (remotely) to come up with a solution that would afford students the opportunity to analyse and reflect on a learner’s work in an authentic setting, but without the need to interact with them face-to-face. There were two options for those students who could not work with a learner in their household bubble:

  • Work remotely with a learner, via video conference or telephone
  • Complete an alternative assignment that does not require the need to work with a learner.

Last year there was still a significant number of people who were not familiar with using video conferencing software, especially to teach and learn with! We created some supporting videos that demonstrated how students could work with a learner and, importantly, suggestions for viewing and saving learners’ work so that it can be annotated and analysed as part of the assignment. These included sharing a whiteboard, creating a homemade visualiser, annotating on a shared screen and options for working asynchronously (not ‘live’). These supporting videos are available within the ‘Resources’ area of the maths education modules.

The second option was a bit trickier. Where there was no possibility of working with a learner in person or remotely, we needed to provide an alternative assignment that could be marked using the same assessment criteria as the original, and that was equivalent in terms of difficulty.  Across all the assignments that include working with a learner, a common element is writing a reflective account of the learners’ work. In doing so, students can demonstrate their ability to evaluate and reflect on the ways learners responded to a particular task, paying attention to module ideas and frameworks such as specialising and generalising, freedom and constraint and doing and undoing. This analysis is a key area of the assignments and is possible even where the student has not worked with learners directly, provided that the task and learner work is suitable.

An extract from an alternative assignment for ME626

We developed an alternative assignment that afforded students the opportunity for this rich discussion by creating plausible accounts of learners’ work. These accounts were based on our knowledge of likely student and learner interactions on a task, as summarised and analysed in previous assignments. In these accounts we compiled and created dialogue and mathematical working that was messy, contained errors, was sometimes aimless or boring but included evidence of mathematical thinking.

Of course, this alternative has some drawbacks; there is little choice of tasks for students, so there is less flexibility about which module frameworks to write about. We would always recommend working with a real learner where possible. Learner interaction provides the opportunity for a rich discussion about the key ideas from the modules. The authenticity of working with a real learner can prompt students to recognise mathematical thinking that might not exist in an artificial context. The act of choosing and adapting a task with these module frameworks in mind is also pertinent to the assignments; a task that does not allow for generalising, for example, would not be a good choice if the assignment needs to include a discussion of specialising and generalising – something we considered when creating the alternative assignments. Students are also asked to annotate their learners’ work to provide evidence to support the discussion, and so tasks need to allow for learners to produce some work, rather than be wholly verbal.  Working with learners directly (either in person or remotely) is also good practice for those students who want to work in an educational setting.

In total, a handful of students in each module have made use of the alternative assignments. We are certainly glad we were able to provide an alternative for those students; it is the right thing to do to ensure all students have fair access and support for assignments, and it made us look back on past assignments and reflect again on the benefits of working with learners.

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External engagement during the pandemic

Last September I came back from maternity leave, mid pandemic, to reclaim my role as external engagement lead for the school of mathematics and statistics. I had particularly missed this aspect of my job: organising events with our academics, hosting school visits on campus, interacting with members of the public at national science fairs, and generally sharing a love of mathematics. As you can imagine, however, the role I returned to was somewhat different to the one I had left. As a school, we faced the challenge of facilitating our existing external engagement programme within the constraints of the “new normal”.

Working at the Open University, we are no strangers to working with students online, in fact all our mathematics education modules are delivered online with tutorials running through Adobe Connect software. As a university, we are experienced facilitators of online learning and this extends to our public engagement offerings, including our free Open Learn website (http://www.open.ac.uk/about/open-educational-resources/openlearn), our involvement in BBC productions, and sharing live broadcasts of public lectures and workshops through our Youtube and Facebook live channels, allowing us to reach a national (and sometimes international) audience.

Despite this strong online presence, within the school of mathematics and statistics we are proud to offer free face to face outreach activities for school pupils in the areas surrounding our campus. Like many other universities, we feel a strong sense of duty to provide outreach for local school students, encouraging them to take an interest in mathematics beyond compulsory education. Our offerings include our annual Mathematics Christmas lecture, attracting around 400 sixth form pupils and their teachers to our Milton Keynes campus, a six-week mathematics masterclass series for year 9 students every year at Bletchley Park, and hosting the Enigma Maths hub “Welcome to your mathematics A-level” conference, for year 11 students who are about to embark on their sixth form mathematics.

We offer these events to support local pupils in developing their interests in mathematics through exposing them to interesting topics outside of the normal mathematics curriculum. We invite speakers from varied professional backgrounds, from career mathematicians to experts in industry, so the pupils can learn about different career paths and, more importantly, be inspired. These events also give attendees the opportunity to meet pupils from other schools who have similar interests in mathematics, giving them the chance to collaborate and to develop their confidence in exploring and discussing mathematics in an informal setting.

This year, like schools, colleges and traditional universities, we have had to think creatively about how we could offer these face-to-face events through online platforms. We still wanted to offer pupils the same opportunities: to explore interesting topics outside of their curriculum, to hear from experts in different fields, and to work with other pupils to explore and discuss mathematics, whilst adhering to countrywide restrictions.

So how did it work? 

Our outreach events since March 2020

Below you will be able to find links to all our recent events which took place online. We are pleased to be able to offer these recordings as resources to anyone who is interested.

Maths Week Scotland

We offered four interactive workshops as part of Maths Week Scotland, covering topics from Number Theory to Code breaking, Exponential growth to the Butterfly effect.

Ian Short’s session “Navigating by numbers” covered a range of frieze patterns.

The Mathematics Education Team (Sue Forsythe, Rebecca Rosenberg and Cathy Smith) delivered a session on “Growing Exponentially”, looking at the famous coins on a chessboard problem, how children grow and of course the exponential grow of the Covid-19

Marc Pradas demonstrated chaos theory using his specially designed double pendulum model as part of his session “Chaos theory and the butterfly effect”.

Katie Chicot and Andrew Potter’s session on Code breaking and cryptography included a range of historic codes from the ancient Greek Polybius square, to a final frequency analysis challenge. Participants were challenged to compete against the clock (and each other) to decipher codes.

Missed these sessions? No problem! One of the best things about our outreach being online at the moment, is that we can record our talks and workshops. You can watch (and play along) with all of our “Maths Week Scotland” sessions here: https://t.co/AaLnbb7Vkl?amp=1

*Please note that the interactivities will not be live but you can still have a go at all the challenges, as the solutions are revealed in each recording.

Ask the experts!

Several of our academics offered interactive talks on topics relating to the current pandemic. You can watch these sessions (and other STEM experts) here: https://www.youtube.com/playlist?list=PLar-D-A84stgtlE7bIHj75vZarXf1T1LK

In Ask the experts: “Tracking how a virus mutates” OU academic Dr Cathy Smith, and guest speaker Dr Nick Goldman, spoke about the way Covid-19 has mutated.

In Ask the experts: “Trusting statistics in the news” with Dr Kaustubh Adhikari, Dr Katie Chicot and Emeritus Professor Kevin McConway, the academic gave expert tips on how to understand statistics in the news, and when to trust what is being published.

Events coming up!

Mathematics masterclasses for year 9 students

We are currently in the middle of our Year 9 masterclass series, with over 100 school students in attendance each week, covering topics from Topology to Islamic Geometry.

British Science Week (5th to 14th March)

The dates of our sessions are being finalised but our planned sessions are:

  • Chaos Theory and the butterfly effect with Dr Marc Pradas, linking to our applied mathematics modules
  • Surprising statistics with Dr Kaustubh Adhikari, linking to our statistics modules
  • Magic mathematics with Charlotte Webb and Rebecca Rosenberg, using ideas from our mathematics education modules

All sessions will be broadcast through the Open University STEM Youtube channel, here: https://www.youtube.com/channel/UCo2SBgH_uGV1563RE5at8Cw

So, what has it been like delivering outreach during a pandemic?

I spoke with colleagues who have been involved in online outreach activities during the pandemic about their experiences. Their responses are summarised below:

Were there any advantages to running events online?

The major advantages for me were fact that we were able to reach a much wider audience, and the fact that we were able to use our MK-based colleagues more easily for events around the country, not just around MK.”

We have been extremely pleased to welcome school students, adults and families who may not be able to attend our face to face sessions. Our year 9 masterclasses, normally run at Bletchley Park and at Bradford University, have attracted over 100 students each week and we have really enjoyed being able to spread our love of mathematics even more widely.

The fact that we could record the sessions means that they are there on YouTube and we can continually reuse them by directed people to them on social media”.

What worked well in online outreach events?

“I had a web camera on my laptop pointing at my face. I also had a movable camera pointing down at my desk. In Zoom and other video conferencing things you can switch between the two cameras at a click of a button. That worked well”.

Many speakers spoke about varying the sessions, for example using short video clips alongside interactivities, rather than just slides in order to keep audiences engaged.

We used Menti.com (a quiz app) which really seemed to work well. I also think it worked well to give the participants a challenge to try at home and to post up on social media. This allowed the learning to continue outside of the session itself”.

Using interactive software meant that participants could interact in real time, sending in answers anonymously. This helped generate a competition element in some of the sessions and graphs of polls were shared in others. The added advantage of anonymity allowed participants to interact in the sessions without fear of getting it wrong.

Most colleagues commented that online sessions needed to be shorter than face to faces ones, since it can be harder to concentrate on a screen for an extended period of time.

Working with another person was also referred to by all the speakers as this allows for a natural “lively” conversation but also any technical support, e.g. checking on the chat box.

“Using break out rooms allowed conversation between participants in smaller groups (often still only in the chat box)”

What were the challenges or limitations of presenting external engagement online?

In most of our sessions, the chat box was utilised far more than participant videos or microphones. As an OU tutor I find this is the case with our students as well, so was not surprised to hear this. Participants appear to feel more comfortable answering through text. Some saw this is a challenge, other speakers were OK with this as a form of interaction.

For interactiveness, I find online seriously limiting. I can’t see what the listeners are up to. I would like to get them building models, experimenting. Most don’t want to put their cameras on and share. Really changes the dynamic”.

When running our masterclasses, for example, one of the aims is to connect students with other likeminded students from other schools. In an online environment it is difficult for participants to communicate with each other if not using break out rooms. Break out rooms have safeguarding issues associated with them if working with school students and many of our speakers chose not to use them.

In a face to face session, presenters can adapt based in how the participants respond to questions or if further explanation is needed. It was a challenge for many of our presenters to judge where the group was up to when working on some of the tasks or how they were responding. Though interactive tools gave some indication of how many people were responding and if they got the answers correct, it was impossible to tell whether the tasks were appropriately challenging for the specific audience.

In addition to hearing from the academics who presented these online sessions, we received some feedback from participants who were in the audience.

“I liked how it was interactive. Even if not directly with the speaker we could still join in within the class. We did not have microphones enabled, partly because most students joined as whole classes, but they were able to use the chat box and the speaker used websites that the students could enter their answers into”.

“They (the students) valued the session and interactivities being live”

“Presenting this remotely was a great idea as all our students could benefit rather than be limited to one minibus full. Thank you for sharing.”

“I enjoyed the fact that it was interactive and engaging”

“Start was too slow and too many instructions repeated. I nearly logged off”

Things to take away when planning outreach activities online:

  • Shorter sessions
  • A mix of activities
  • Opportunities for discussion
  • Things to take away and work on later
  • Interactivities – e.g. Menti.com or Desmos
  • Visually interesting activities
  • Using break-out rooms
  • Chat box interaction
  • Start promptly to avoid losing interest from audience who can simply exit with a click

Get in touch

As always, you can get in touch with us on Twitter: @OUMathsStats or you can email our outreach team at: STEM-MS-Outreach@open.ac.uk

We would love to hear from you if you have attended one of our events or have watched them on Youtube.

Please also let us know if you would like to invite some OU academics to take part in your outreach event.

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What is the way forward for Maths exams 2021?

A personal viewpoint from Cathy Smith, Mathematics Education lead at the Open University.

Ofqual and DfE are currently consulting about their proposals for giving students GCSE, AS and A level grades this summer, a change of plan from the problematic awards from 2020. The full consultation document is here  and the way to give your response is here.  These are my thoughts about what they are proposing, distilled into six points.

  1. The purposes of assessment.

The consultation document offers this about the meaning and purpose of grades: “Qualification grades indicate what a person who holds the qualification knows, understands and can do, and to what standard. That is their purpose. People who use qualifications, for example to make selection decisions, need to be able to rely on the grades. For qualification grades to be meaningful, a person who holds a qualification with a higher grade must have shown that their knowledge, understanding or skills are at a higher standard than a person who holds a qualification with a lower grade.”

One trouble with this is that it mixes up the certification and selection purposes of examinations. Grades are meant to certify what someone ‘knows, understands and can do’ so that employers and universities can be reasonably sure that someone with a grade 4 can, for example, increase a quantity by 20%.   This year, students have had unusual and disrupted learning for 10 months.  Most of them will simply not know as much as a pre-2020 student.  Through no fault of their own, this cohort would – overall –  probably get a worse grade distribution if they sat normal GCSEs, and within that the groups  whose education has been most disrupted would have the worst drop.

The government has already conceded that post-2020 result do not mean the same as pre-2020 results. This is why Ofqual is talking about reducing curriculum content or allowing teachers to match tests only to what has been taught. The problem here is that rather than recover students’ knowledge, it risks being narrowed further.

This leaves the selection purpose: a person with a higher grade should have shown that they have ‘a higher standard’ than someone with a lower grade.  The 2020 outcry over grades shows how much people care about children’s futures resting on comparing grades.  To be fair, this has to hold within a school, across schools, and across any different ways of assessing that standard, or opinions of how to interpret it.  It is no small task to achieve this. Existing GCSE and A-level assessments are not perfect, and that is not for want of effort.

  1. Need for assessment

It is worth asking why it is necessary to have this national ranking. Certainly, grades for A-levels, BTECs or other final qualifications allow admissions offices to make quick impersonal decisions where students are applying to university or apprenticeship schemes with limited intake numbers. For GCSE, where all students must now continue in education, most in their current school, there is no longer a strong justification.  16-year olds have years of assessment history behind them that can inform their choice of subjects and courses.  There are many, many countries around the world that manage without the equivalent of GCSE. Although it is not in the scope of the 2021 consultation, there should be a future discussion about the time and money devoted to a wide-scale assessment.

  1. Exams or exam-papers

The proposal (read it in full yourself here) is to replace national exams with new ‘smaller’ papers, covering just a few topics, that will be written by the exam boards in the next couple of months. Teachers will not see the questions but will choose a few of these papers for their students to sit, then mark the papers themselves (to provided mark-schemes) and be responsible for how they turn these marks into grades.  (There is no detail about how this will happen.) Maths is actually simpler here, as teachers don’t have to consider how to include practical or oral work in the final grade.  The proposal suggests that teachers might also write their own papers, which will have to be vetted by the exam boards. It also suggests that only papers taken in a particular time period (e.g. one week in June) will count. This in effect throws away the assessment records that teachers have been carefully building up, to be replaced by new ones.  Students who have been sitting a test every fortnight since September will be overjoyed!.

  1. The danger of the new

Perhaps this proposal seems a reasonable compromise. Certainly, given time, teachers could develop skills to make fair judgements between students in their own schools and between other schools. In the late 1980s teachers themselves wrote the exam papers for Mode 3 GCSEs that were accredited by the then-version of Ofqual; this century they moderated GCSE grades and national curriculum levels across schools. But they have not done this for over ten years. Only a few teachers (in some schools) are also examiners. Writing high-stakes exam questions is a skill that examiners have and teachers have not needed.  So is setting grade boundaries. With new, untrialled exam papers, even the exam boards would find it hard to predict how students will perform, let alone when teaching has been disrupted by a pandemic. Inventing a new system simply means we cannot tell in advance how the unfairnesses will appear, not that there won’t be any.

  1. Workload

It is hard to overstate this: teachers and school leaders are being asked to take on the job and the responsibilities of exam designers, markers and checkers, to run the whole appeals process and have the paperwork to defend themselves against any criticism from students, parents or Ofqual itself. This is on top of teaching full classes, running testing and looking  out for children’s wellbeing.

  1. Who loses?

It is hard to see any winners in the situation. Covid is not fair.  Students aren’t getting the education they need, and what they are losing is as much about social skills, about being able to work alongside each other, as it is about what they know.   But when you introduce a new system,  and it goes wrong or it can be exploited or it depends on teachers’ resources and goodwill rather than their actual job, it is always the children who are already disadvantaged that lose out most.

It seems wrong to criticise the proposals without suggesting an alternative, but no one person should be pushing their suggestion.  I would add some different questions to the consultation: What do we think would happen if there were no GCSE grades at all this year? If students were advised by their teachers about which year 12 courses they should take? If all 16-year olds were offered an opportunity to sit GCSE Maths and English for free sometime in the next three years?

Do respond to the consultation –  here.

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Generalising in Mathematics

Generalising in Mathematics by Philip Higgins

 

 

One of the most revealing aspects of ME625 and ‘Developing Thinking in Algebra’ (Mason et al 2012) is that algebra is the search for the general rule and that this generality can be uncovered in most cases, and with some practise it becomes achievable. I say this because I never felt that way in my O’ Level school maths, back in the 1980’s. Most education critique that I currently read says that the curriculum should now be taught differently, with creativity being foremost. Searching out that general rule is a creative challenge. I have learnt all this aged fifty, having graduated in BSc Mathematics and its Learning. The following extract is from my ME625 End-of-Module assignment in which I tried to step up another level and actually teach generalising to a student. It was not easy.

Finding generality through Invariance and Change

My maths learner was my son, twenty years old, and in second year university on a Product Design course. He is educated to A’ Level in Mathematics. It was my difficulties in trying to teach him generalising during earlier work, which he found confusing due to my lack of structure, that prompted me to shake up ‘Question 13.1.5 Invariance’.

Here is the course textbook question:

Task 13.1.5 Invariance

What is changing and what is the same about the three following expressions as a whole?

3 + 5 = 2 x 4          5 + 7 = 2 x 6          3 + 7 = 2 x 5

 

I adapted it and split the whole task into six small parts so that steady progress could be made without feeling overwhelmed by suddenly having to generalise. The ‘Dimensions-of-possible-variation’ task shown at the end allowed him to create his own version.

Part One – finding changes

My learner explores change using spot-the-difference. The image is Paul Morphy, a famous Victorian chess player. My idea is to encourage visual interest.

He stumbled with finding the last three differences. It seems my tiniest of ‘Paul Morphy tweeks’ was challenging to him but this encouraged focus.

 

Part Two – visualising the nth term

In part two my idea was to familiarise my learner with the general number ‘ n ‘ but without the trauma of having to extract it from a ‘sequence of numbers’ or a ‘story question’. It is a simple approach.

Question

 

The ‘n&n’ task is undoubtably easy and sure enough it was for him but notice his use of ‘n-2’ and ‘n-4’. I prompted him for an alternative expression and he found ‘ n/2 ‘.

 

Part Three – moving from iconic to symbolic

For part three I asked my learner to explore change in a numerical context, so more symbolic than visual.

Question – Invariance & Change

What Stays the Same (invariance) and What Changes (change) for the three following expressions as a whole?

3 + 5 = 2 x 4, 
5 + 7 = 2 x 6,
3 + 7 = 2 x 5

On writing the headings ‘Invariance’ and ‘Change’ he was seeing the language patterns and using them. His articulation was valid. He noticed operators ( x and +), and values (3+5=8, and 4 is half of 8) but he did not see the 4 as being in the centre of 3+5. Learners see different things.

 

Part Four – using the nth term towards a general formula

Only now in part four did my learner need to generalise. The idea here is that he has settled into ideas of ‘change’ and ‘invariance’ without having had an algebraic equation launched at him.

Question – invariance & change

Think back to the “ n & n’s challenge “, earlier. We are going to use this idea now.

The aim is to find a general expression for our three equations above.

  1. Take your first equation.
  2. Let n=3.
  3. Now follow through the remaining numbers, term by term, for each of those terms that you found earlier that change, and in so doing, develop a general expression.

Those terms that did not change, the ‘invariant’ terms, will just stay the same.

  1. Repeat, to obtain a general expression for the remaining two equations.

Test it – exploring the range of possible change for “ n “

We suspect that ‘ n ‘ is an integer, since we made it so when we first set it at 3, but now ask if your general formula works for other types of number.

Question

  • Does it work for other whole numbers?
  • Does it work for fractions?
  • Does it work for square roots?

My learner’s generalising,

 

And his testing by specialising,

 

Part Five – can I create my own maths question?

School maths often ends there and the student thinks ‘so what?’ but now I encouraged my learner to push on and to create his own unique bit of maths using the theme of ‘Dimensions-of-possible-variation’.

Question – explore which bits of your general expression you can change

We want to create a whole new maths problem, which has the spirit of the existing, but which you alone have made.

  • Looking now at your general expression, which elements can you change?
  • Now change them.
  • Make sure both sides of the equation still equate to each other
  • Now insert some numbers to get three different expressions, much as the original question above did to start with.

This is your new maths problem. You could present these expressions to somebody and ask them “What Stays the Same and What Changes for these expressions as a whole?”

We reached a key learning moment; could he vary the generalisation? I did explain at this point what was being asked for (to adapt an element of your choosing in this general equation) but not what to vary. He hesitated, thought, and nervously said;
“The coefficient?”
I asked for more.
“n?” he said.
I told him to go ahead and our ‘n’ now morphed into 3n2

 

All that remained was to insert new numbers for ‘n’ and recreate the original question in his own vision. With care he substituted the numbers 2, and ⅓, and ¼ . Interestingly, despite squaring of fractions he did all of this without recourse to a calculator, which suggested competence.

Substituting for 2,

 

Substituting for ⅓, and ¼

 

This was his unique version of the original question. He had used ‘Imagination’ and he had ‘Got a Sense of’ the themes of Invariance and Change.

His rewritten version of Task 13.1.5

 

Part Six – The Conclusion – is generalising relevant to me?

This part of the task involved no mathematical work and no question to do. It was given as a simple analogy. The idea was to appeal my learner’s Product Design skills and to explain and inform him as to what we mean by ‘developing thinking’.

Mathematical thinking, as shown in the above process, can be summarised:

  • You are given a particular set of numbers in an equation.
  • You generalise that equation, so it works for any number ‘ n ‘.
  • You classify it as a simple linear relationship and explore its range.
  • You manipulate it by changing those dimensions which can vary.
  • You gain control over the structure of the problem.
  • This control enables you to create a new and extended problem. This new problem is your brand.

Here is a simple analogy:

“You dedicate time to studying some particular timber materials; Ash, Birch, Ebony, Iroko and others. You generalise them into ‘Hardwoods’. You classify the hardwoods in terms of texture. You manipulate that texture into your proposed design of a piece of furniture. This gives you control over the look of the finished product. This control and finish gain you kudos. This kudos informs your brand.”

 

My reflections on the above task

I tried to conduct this question in the spirit of Jo Boaler’s piece on Sarah Flannery, European Young Scientist of the Year (Boaler, 2013). She writes (using her italics),

“The first thing I realized about learning mathematics was that there is a hell of a difference between, on the one hand, listening to maths being talked about by somebody else and thinking that you are understanding, and, on the other, thinking about maths and understanding it yourself and talking about it to someone else.”

I conversed with my learner on Task 13.1.5, explaining why we were doing the steps and what it achieved. He proceeded smoothly, using Invariance and Change in both spot-the-difference and in his generalising, but it seems my approach backfired somewhat. He was simply listening to ‘maths being talked about by me’. My conversing had failed in the chance to let him talk and reveal his own understanding. I then questioned why he had done so well and he said,
“You explained quite a lot, which made it easy.”
I immediately sensed didactic tension and realised that I could not now undo all of that talk. Did he just display behaviours that I had asked of him? Was he generalising or did he just blindly convert numbers into n’s? I sense a lot of the latter, though he did work actively on Dimensions-of-possible-variation, creating his own set of three equations, and likewise the work on Invariance and Change was all his own. Fortunately, the ‘Heads & Tails’ task came next, so I then felt the urge to do just the opposite and maintain a steady silence whilst he experienced the mathematical struggle, intervening only where needed.

Finding generality through a recreational maths problem

Posamentier’s problem on Heads and Tails

You are seated at a table in a dark room. On the table, there are twelve pennies, five of which are heads up and seven of which are tails up. Now mix the coins and separate them into two piles of five and seven coins, respectively. Because you are in a dark room, you will not know if the coins you are touching were heads up or tails up. Then flip over the coins in the five-coin pile. When the lights are turned on there will be an equal number of heads in each of the two piles. How can this be possible?”

In ‘Heads & Tails’ I chose a question beyond the course material, and also as a way to challenge myself in applying the course techniques. I found the author’s original question (Posamentier, 2017) a little awkward (why sit in a dark room?) and he also revealed the solution, so in the spirit of the ME625 course textbook I developed it into a more eye-catching ‘Story Problem’, which related to my learner as a design student, and I kept the pattern hidden so I could register his surprise when he spotted the outcome of the puzzle, since if we could both take delight in his reaction then it acted as a motivator.

 

 

“You take a break from your studies, lean back in your chair, think about pizza, deadlines, assignments, software, formulae, design work, and you commit to acquiring a new set of watercolour markers for £11.99. You stare vacantly at the twelve pound-coins on your desk. The shops are shut. It’s late. Only the pizza place is trading.

You lean forward and playfully arrange five of the coins to form a group showing heads and the seven remaining coins to form a group showing tails. It looks concise, deliberate, and organised, but suddenly there’s a power cut. As the lights whine down you sit in the dark, waiting for something to reboot.

Nothing reboots!

The lights in the pizza parlour across the way have gone out too! Maybe it’s a temporary street-thing.

You think about reaching for your phone-torch but before you do you playfully shuffle the coins around in the pitch dark on your desk, and once again, by touch alone, separate them into a group of five, and a group of seven.

You flip over all the coins in the group of five.

The lights soon reboot, much to your delight, and you look down at the group of five and the group of seven coins, expecting something rather random.

Weirdly you notice something about the heads in each group!

How can that possibly be???”

Can you explain it;
i) in a sentence, that a friend or colleague could follow?
ii) in a diagram?
iii) using a table (ICT, not kitchen)?
iv) using a generalisation?

I had further adapted the task by asking the four questions shown above, three to help articulate, and a final challenging task of generalisation, with each step improving his understanding toward the next. The third question set up the problem on an Excel spreadsheet and my idea was to explore what awareness he had about the structure of the problem, which by this point should have been established. Would he see generalising or is it ‘just a spreadsheet’? At the final step he must generalise, but ‘finding x’ feels like ‘school algebra’. Would he still have absences in his grasp of it (you must first understand what ‘x’ is), despite progress on the earlier work?

The following photos show how the puzzle was played out after my learner had initially read the question and before he started on the written element.

 

This was the starting position, where heads were indicated by the white circles,

 

Now he shuffled the coins. We did not sit in the dark. I improvised a canopy to allow the coins to be hidden,

 

 

The coins are further shuffled prior to being flipped over, all still hidden under the canopy,

 

Joy was evident in this task. He whispered to himself upon reading the question, “So true …”. Idle time is universal. As the task unfolded for the very first time he said with sheer delight;
No Way!! No way does that happen every time!”
He did it again and again. Same result!

Question (iii) – using a table

In approaching the questions my learner did not explicitly use the theme of ‘Do / Talk’ to himself or draft sentences nor even replay the pieces. Instead he sat quietly and thought and eventually drew the table shown below. He had unexpectedly opted to answer question three first, despite my spreadsheet-in-waiting. His table lacked good labelling to identify the flip of the coins but you can see that ‘3 and 2’ becomes ‘2 and 3’. It favours the iconic with its visual cues like columns and place-holders for his numbers. He had the number ‘2‘ at the foot of each column, which was the correct answer for that shuffle.

 


Question (i) – using a sentence

My learner moved onto question one. Articulating through a sentence is tricky. I could not make full sense of his and there was a struggle to clarify the ratio found in the table. Again, there was no reference to the flip of the coins, so it lacked detail.

 


Question (ii) – using a diagram

Next he tackled question two but he was confused about a diagram. I had to first assist him on how to apply this diagram when moving from the enactive to the iconic. He followed my lead and then took a random selection of five circles, and then drew the flipped version, which allowed him to articulate the equal numbers of heads. You can see that his ‘Original Pile’ has two dotted circles and his ‘Taken Slot Inverted’ has the same.

 

Question (iii) – using my speadsheet

I now had to reintroduce question three with my pre-prepared spreadsheet. His task was to fill in the correct formulae in the cells. The Excel sheet is the key learning moment because inputting the cells is generalising and he recognised the need for the ‘Step-3 left cell’ to be a random number of heads (shown here as 32), and he succeeded with the formulae in Step-3 and Step-4 boxes with ease as they are simple subtractions or else unaltered. We did commence with 5 heads and 7 tails, but he soon experimented with 45 heads and tails.

 

Question (iv) – using a generalisation

Question four proved the trickiest for my learner despite him now having some grasp of the situation from the previous three.

His generalising version began with two variables, x and y, but see how he ended up with y = x, which got a bit confusing and he then scribbled out the y variable. When the coins are flipped he wrote down ‘ 7-x ‘ at the foot of the page and then scribbled out the ‘ 7 – ‘. He was almost there.

His first attempt,

 

In his following page the algebra still seemed difficult to grasp for him. Here he has 1 and 6, so no matching heads. There was no sense of the theme ‘Do / Talk’ to himself first in an effort to grasp the unknown before attempting to solve. In the bottom of his page I had by now prompted him to “think of what x is”. See how he succeeded but only by clinging on to Specialising at the same time, here treating x as 1, (despite having two heads). It shows more confidence is needed.

His further attempt,

 

My Reflection on the above task

The ‘Heads & Tails’ task embraced one of Posamentier’s ‘Effective Techniques’ (Posamentier, 2016),

Technique 4Entice the Class with a ‘Ghee-whiz’ Amazing Mathematical Result: One natural way to stimulate interest in mathematics among students is through the curiosity that nestles within all of us. Such curiosity can be awakened through new ideas, paradoxes, uncertainties, or complex results. Here the teacher’s talents come into play to find illustrations of easily understood situations that lead to unexpected results and leave the students intrigued (ghee-whiz), resulting in a motivation to pursue the topic further.”

Poems pare back word into some visceral cry and mathematical puzzles do likewise. This task offered that ghee-whiz result. My learner’s reaction, described above, resonated with this. It was sheer excitement at such a simple trick.

My learner Imagined-and-Expressed the process behind this trick in the form of a table. It was unexpected. I imagined he would sit for some time flipping and un-flipping coins (Do / Undo) to understand, but he thought it all through quietly in his mind. His table was accurate but see how his generalising fell apart in places. Where does the need for a second variable ‘y’ come from? It looked like a residue from linear equations (y=mx). No pause occurred to first grasp ‘x’ and follow its path from ‘5-x’ in one group, to ‘x’ in the other. It was hurried, evidenced by much scribbling. It was the last question-part so perhaps he was disengaging. Should I have intervened and talked more? It seems that conversing itself requires mastery. I asked him why, having found a general number in the spreadsheet, did he struggle with generalisation. Surely the spreadsheet simply ‘gave the game away’?
“It didn’t,” he said, “that was just filling in boxes, and besides those equations are all hidden anyway.”
So oddly, despite his setting up of those very generalised equations he could not easily connect that to generalising. Software, as powerful as it is, is not a panacea. It conceals. My view is that this is a problem of technique and method. He has not yet automated the concept of filling in Excel cells as a process of generality. More Do / Talk was required by him, both at the Excel and the generalising question. If I intervened at all it could have been to point this out. I was hoping to witness (maybe naively) the ‘Manipulate – Get a Sense of – Articulate’ spiral of increasing sophistication with each part. Instead it was curiously mixed.

Conclusion

Both tasks had the course textbook’s key ideas integral in their solutions and for me and my learner it showed that one can take ‘Developing Thinking in Algebra’ and apply it to the outside world, to problems randomly found in curiosity books. In other words, I could answer Root Questions using these general techniques.

Two things emerged from our solving:

  1. The level of verbal interaction in a lesson, and especially the learner talking of their maths understanding back to you, is extremely sensitive to how independently your learner will solve the problem.
  2. This conversation has to be tailored to your learner’s ability. This means that you have to know your learner (difficult in a large classroom).

It informed my approach to algebraic thinking, and serves my goal to seek control and power over the teaching process, and this is not dissimilar to the learner’s own quest for control and power over their maths problem.

Both draw inspiration from Kate Clanchy, on the poet Melissa Lee-Houghton, (Clanchy, 2019):

She says, frankly and simply, that she suffers from depression and poetry is not the cure for it, but that poetry can give her a way of understanding and formulating herself, both as she writes it, and as she reads herself back afterwards. It gives her some distance and control.
The kids are mesmerized by this, and so am I. Control. Not turnaround but control. This word has somehow never occurred to me before, in all my anxious considerations of poetry and therapy, but it seems the right one.”

She adds,

And if they dig deep, and find effective images, and make a good poem out of the truth of their lives, then that is not just control, but power. It’s different from being happy; it isn’t a cure for anything, but it is profoundly worth having. And actually, I don’t need anyone to tell me that; I know that from my own experience. I know it for myself.”

Care must be taken with such comparisons and the sense of control and power is by far an idiosyncratic thing but the spirit is the same; I want to understand, and dig-deep, and find effective methods, seek profundity, so as to be adept at producing the maths resource that converses, motivates and invokes the art of generalisation. Furthermore, what learner would refuse a chance at acquiring control and power; control in knowing how to start, where to start, preferred strategy, direction of travel and when to reverse, and of the power and delight when the problem yields because you fought for it, all of which alleviates (not cures) your maths fear.

Interestingly, none of it is for turnaround of the student, and the course textbook itself states, arguably, “I cannot change others”. The end desire is that the learner will change of their own volition, much as I have done in the course of my own algebraic thinking.

Seeking such personal power and control is rooted in human nature. It is enlightening to see ‘Developing Thinking in Algebra’ tie mathematical power to the humanist subjects of English and History, and by the prompting of learners to “develop their powers to imagine, and to express what they imagine to others”. We can rejoice that this notion is still vibrant thirty-three years after Davis & Hersh (Davis & Hersh, 1986) embraced it when the omnipresent I.C.T. was still cutting its coding teeth,

Metaphor and analogy exist in mathematics and physics as well as in poetry and in religion. Rhetoric exists in mathematics (despite claims to the contrary) as it exists in politics. Aesthetic judgement exists in mathematics as it exists in the graphical or performing arts.”

Those authors wholly reclaim mathematics as being a human institution, and if it ever ceases to be so then they argue that we must “let it decay”.

 

References

Boaler, Jo. 2013. The Elephant in the Classroom. Souvenir Press.

Clanchy, Kate. 2019. Some Kids I Taught and What They Taught Me. Picador.

Davis, Philip. Hersh, Reuben. 1986. Descartes Dream. Penguin Group.
The authors conclude on page 305, “If a synthesis cannot be achieved, if it comes to a showdown between man and mathematical science, then man would be best advised to stop the process. Let it fall into decay…”

Mason, John. Graham, Alan. Johnston-Wilder, Sue. 2012. Developing Thinking in Algebra. The Open University in association with Sage Publications Ltd.

Posamentier, Alfred S. Krulik, Stephen. 2016. Effective Techniques to Motivate Mathematics Instruction. Routledge.

Posamentier, Alfred S. 2017. The Joy of Mathematics. Prometheus Books.

 

 

 

 

 

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