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Posters – Women in Science Technology Engineering and Mathematics

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Explore stories of Women in STEM through this series of posters. Download one or all of the posters to share in your classroom, community centre or workplace and help us profile amazing ideas and the people who developed them.

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Guidelines for assessment and resource design

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Guidelines for assessment and resource design

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Biographies of diverse mathematicians (QMUL)

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Biographies of diverse mathematicians (QMUL)

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Diversifying the Maths Curriculum at Queen Mary University of London

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A collection of slides that can be added to a lesson designed to raise awareness of lessor known mathematicians. Each mathematician has an activity that can be linked to the curriculum.

We aim to keep adding to these activities so please get in touch with suggestions.

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Significant figures - Top Trumps AMSP February 2025

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This is Storyofmathematics.com (SOM), the go-to website for all things math-related! With the click of a button, you'll be able to access a wide selection of

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Collaboration with IMA to to explore the idea of the identity of ‘mathematician’, using examples from recent history who have been carefully chosen to represent traditionally minoritised and underrepresented identities.

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IMA (Institute of Mathematics and its Applications) in the UK to create a series of posters, a booklet, a worksheet and some bunting to explore this idea of the identity of 'mathematician', using examples from recent history who have been carefully chosen to represent traditionally minoritised and underrepresented identities.

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The Crest of the Peacock: Non-European Roots of Mathematics

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The Crest of the Peacock: Non-European Roots of Mathematics

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The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph | Goodreads

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Increasing participation in Core Maths, AS/A level Mathematics and Further Mathematics

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Designed for GCSE students

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Landing back on Earth with Katherine Johnson

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Biographies Alphabetical Index - MacTutor History of Mathematics

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If you have comments, or spot errors, we are always pleased to hear from you.

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This page contains the original submitted versions of book reviews subsequently published in the British Journal for the History of Mathematics. They are collected here with the authors' permission, as a service to the community.

2019
Vol 34 issue 3
A Heefer review of Ingrid Alexander-Skipnes (ed.) Visual Culture and Mathematics in the Early Modern Period

D Leddy review of Chris Pritchard, A Common Family Weakness for Statistics: Essays on Francis Galton, George Darwin and the Normal Curve of Evolutionary Biology

2020
Vol 35 issue 1
T. Astarte review of Ken Steiglitz, The Discrete Charm of the Machine: Why the World Became Digital

C. Hollings review of David Rowe, A Richer Picture of Mathematics: the Göttingen Tradition and Beyond
Vol 35 issue 2
V. Peckhaus review of Robin Wilson and Amirouche Moktefi (eds.), The Mathematical World of Charles L. Dodgson (Lewis Carroll)

L. Rogers review of June Barrow-Green, Jeremy Gray, and Robin Wilson (eds.), The History of Mathematics: A Source-Based Approach, Vol1

F. Tanswell review of Philip Ording, 99 Variations on a Proof
Vol 35 issue 3
J. Dauben review of Christopher Cullen, Heavenly Numbers: Astronomy and Authority in Early Imperial China

T. Crilly review of Marcel Danesi, Pythagoras' Legacy

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This page contains the original submitted versions of book reviews subsequently published in the British Journal for the History of Mathematics. They are collected here with the authors' permission, as a service to the community. 2019 Vol 34 issue 3 A Heefer review of Ingrid Alexander-Skipnes (ed.)

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ICMS - International Centre for Mathematical Sciences

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ICMS - International Centre for Mathematical Sciences

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Black Heroes of Maths Free Download Poster!

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In celebration of Black History Month in the UK, and for the Black Heroes of Mathematics conference 2022, ICMS has worked with Nira Chamberlain to produce a poster celebrating Black mathematicians. The poster is free to download. There is both a web-suitable file and a high quality print file. We…

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A-Z female and non-binary mathematicians

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A-Z female and non-binary mathematicians

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Posters | Mathematical Institute (ox.ac.uk)

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Famous Mathematicians from around the world (PDF)

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Famous Mathematicians from around the world (PDF)

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From @nathanday314
PDF: https://t.co/e3oIqfPCGp

Editable PPT: https://t.co/skyLbNVvAj

Individual posters: https://t.co/hmPoXfzAK3 https://t.co/iyN95Dbg1d

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Mathematicians of the World - Version 2.pdf

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Pearson - Diversity and Inclusion poster series

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Pearson - Diversity and Inclusion poster series

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These posters show key mathematicians who have inspired us and positively impacted the way that we live today; from Katherine Johnson, the first African-American woman to work as a NASA scientist, to Al-Khwarizmi, the man who is considered to have written the first book on algebra.

Explore our series of posters

Featuring a short description of their mathematical achievements, these posters can be downloaded and used to inspire your students in the classroom or as starter questions to be used in lessons as discussion topics.

We'll be releasing one new poster each month until the end of the year, so please make sure you visit us again to get the whole set!

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Maths: Diversity and inclusion poster series | Pearson UK

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These posters show key mathematicians who have inspired us and positively impacted the way that we live today; from Katherine Johnson, the first African-American woman to work as a NASA scientist, to Al-Khwarizmi, the man who is considered to have written the first book on algebra.

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Includes resources on Islamic algebra

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Diversifying STEM Curriculum Project 2021 - Mathematics

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As part of Black History Month, Dr Nira Chamberlain has produced a presentation which celebrates black mathematicians from history.

"As Black people we have had more than our share of the academically hereditary disaffection all people seem to experience relative to mathematics...Nowadays our promising youth are even more menacingly threatened by exposure to teachers...who have been convinced that Blacks...are abysmally and irrevocably hopeless as far as mathematics is concerned."

Dr Wade Ellis Sr: 1980 (1909-1989)
10 th African-American to gain a PhD in Mathematics

Francis William: 1702 - 1770, Jamaican Mathematician

Mathematics Degree -Cambridge University

Background

Francis Williams was selected to take part in a social experiment devised by the Duke of Montagu who wished to show that black individuals - with the right education - could match the intellectual achievements of whites.

Challenge

A rival once suggested-"The abstruse problems of mathematical institution turned his brain; and he still remains, I believe, an unfortunate example, to show that every African head is not adapted by nature to such profound contemplations."

Achievements

Degree in Mathematics, Latin and Literature from Cambridge University. He returned to Jamaica to set up a school, teaching Mathematics, Latin, Reading and Writing.

Thomas Fuller 1710 - 1790, Slave and Mathematician

Extraordinary powers of calculation

Background

Thomas Fuller , known as the Virginia Calculator, was kidnapped from his native Africa at the age of fourteen and sold to a planter.

Challenge

He was regularly tested. On being asked, " How many seconds a man has lived who is 70 years, 17 days and 12 hours old?" He answered in a minute and a half 2 210 500 800. One of his examiners who was using pen and paper, advise him his answer
was incorrect! The sum was not so great as he had suggested. Upon which Thomas hastily replied: "Stop, you forget all the
leap years! " On adding the amount of the seconds for the leap years, the total in both their sums agreed exactly.

Achievements

Later in his life was he was discovered by antislavery campaigners, who used him as a role model to demonstrate that blacks were not mentally inferior to whites.

Benjamin Banneker: 1731 - 1806, Astronomer and Civil Rights Leader

The sky was not his limit

Background

Benjamin Banneker was a largely self-educated mathematician, astronomer and first civil rights leader.

Challenge

He would exchanged letters with the soon to be Third U.S. President Thomas Jefferson, whom had stated that negroes are inferior to whites when it comes to high level of mathematical thinking.

Achievements

Banneker was the first scientist to study the relativity of time and space, and his revelations on the topic preceded Einstein's
Theory of Relativity by two centuries. He successfully predicted the solar eclipse that occurred on April 14, 1789, contradicting the forecasts of prominent mathematicians and astronomers of the day. Banneker was the first to disclose in his writings that the Star of Sirius is two stars rather than one. His hypothesis was not confirmed until the event of the Hubble Telescope two and a half centuries later at NASA.

Elbert Cox : 1895 - 1969, 1st Black Mathematics PhD

The Pioneer

Background

After fighting for his country in World War I , Elbert Cox, applied for a PhD program. One of his referees wrote " He would have certain difficulties, because of the fact, he is of the coloured race."

Challenge

Universities in England and Germany would not recognise his PhD at the time. However, Japan's Imperial University of San Dei accepted the dissertation.

Achievements

In 1929, became the first Black Mathematician in the world to gain a PhD in Mathematics. From 1929-1961, he taught at
Howard, an Historically Black College and University. Inspiring generations of Black Mathematicians to obtain a PhD

Euphemia Lofton Haynes : 1890 - 1980, 1st Black Female Mathematics PhD

Background

Euphemia Lofton Haynes faced three obstacles in becoming a PhD mathematician. She was female, she was Black and she was in her 50's.

In 1940 the Cambridge Mathematician G.H. Hardy, wrote A Mathematician's Apology , one of the most widely read books about the nature and practice of mathematics. It stated: "No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game."

Achievements

Three years after Hardy's book, Euphemia Lofton Haynes completed her PhD in Mathematics. She was living proof that you are never too old and that mathematics is not just for the few, it is for everybody

Further Reading:

Article by Dr Nira Chamberlain

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The Black Heroes of Mathematics - Maths Careers

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As part of Black History Month, Dr Nira Chamberlain has produced a presentation which celebrates black mathematicians from history. "As Black people we have had more than our share of the academically hereditary disaffection all people seem to experience relative to mathematics...Nowadays our promising youth are even more menacingly threatened by exposure to teachers...who have been convinced that Blacks...are abysmally and irrevocably hopeless as far as mathematics is concerned."

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Let’s Decolonise the History of Mathematical Proofs!

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Let’s Decolonise the History of Mathematical Proofs!

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Professor Agathe Keller 19th October 2022 Introduction Preliminary remarks
A cursory search on the internet, looking for references on the "history of mathematical proofs", brings up videos, blogs or even scholarly papers, that always tell a variation of the same story

- That we owe the idea of mathematical proof to mathematicians of Greek antiquity, especially to Euclid's Elements who himself was inspired by Aristotle's Posterior Analytics.

- That central to the idea of proof is the idea of certitude (and its correlate, rigor);

- Before classical Greece there was no evidence of proof. Furthermore, this unique idea of proof was later developed in Europe. In other words, this idea of proof, constitutes a Greek and European "exception".

- Finally, this idea of proof is at the basis of the proofs practiced in mathematics today.

This standard idea of the history of mathematical proof, we know from the previous speakers, is infinitely more complex than what it might seem at first glance. But what is striking in such histories, is that these texts say nothing about mathematical texts outside of Europe, as if they didn't exist.

And if, by chance, they do evoke texts in other languages than European ones, we encounter for instance the kind of statements given by M. Kline (which came up as the first reference in my search) in 1973[1]:
Perhaps most interesting is the Hindus' and Arabs' self-contradictory concept of mathematics. Both worked freely in arithmetic and algebra and yet did not concern themselves at all with the notion of proof. That the Egyptians and Babylonians were content to accept their few arithmetic and geometric rules on an empirical basis is not surprising; this is a natural basis for almost all human knowledge. But the Hindus and Arabs were aware of the totally new concept of mathematical proof promulgated by the Greeks. (...) Both civilizations were on the whole uncritical, despite the Arabic commentaries on Euclid. Hence they may have been content to take mathematics as they found it...
Such stands are all the more surprising that, as I will show in part in what follows, all sorts of mathematical texts in Arabic, Sanskrit, Chinese, containing proofs and sometimes quite elaborate reasonings, have been known and studied in Europe from the beginning of the 19th century.

In other words, but you probably already know this, the history of mathematical proofs has been and continues to be a very political history.

In what follows, my aim will also be to give a display of the diversity of mathematical reasonings that can be found in Sanskrit mathematical texts of the first millennia of our era, to show that as we " decolonize " the library from which we choose the texts with which we study mathematical reasonings we might also question the focus we have on the notion of proof, that is on the will to find rigorous irrevocable reasonings dealing with the truth of a mathematical assessment.

I think that we are all aware that ideas of what makes a valid reasoning in general and in mathematics have undergone radical changes in the late 20th and beginning of the 21st century as algorithmics and statistical approaches to proofs have gained momentum. Humans in the past too had many different ways of reasoning in mathematics, and many different ways of thinking of what made a valid reasoning, notably in mathematics. This diversity was much greater that we have thought of in the past and is worth studying, this is another point I would like to make in what follows.
State of the Art
Exactly 10 years ago, Karine Chemla edited a book that made quite a sensation in the discipline of the history of ancient mathematics: The History of Mathematical Proof in Ancient Traditions. [2] The studies in this book continue to inspire historians of ancient mathematics today, notably the study of mathematical diagrams and the way they have been edited.

In the introduction, Chemla shows first how values have been attached to the standard notion of mathematical proof and that such a model of proof has been used for all sorts of things that have nothing to do with mathematics; like for instance being a model of rigorous reasoning for philosophers, or the hope that such a feat could help convert savages to Christianism.

She also shows that mathematicians of today read proofs for many reasons, not just to be convinced by the truth of a statement. This might have been also true in the past.

Conversely, as we have seen with our first speaker, when we read mathematical documents of the past, some mathematical texts may adhere to the standard point of view on mathematical proof, but many do not. So that, as we have seen, this way of valuing texts, conveniently selects some texts and discards others. In other words, Kline's point of view, has a long story.

Thus, Jean Baptiste Biot a French mathematician, astronomer and physicist, could write in 1841[3]:
'this peculiar habit of mind, following which the Arabs, as the Chinese and Hindus, limited their scientific writings to the statement of a series of rules, which, once given, ought only to be verified by their applications, without requiring any logical demonstration or connections between them: this gives those Oriental nations a remarkable character of dissimilarity, I would even add of intellectual inferiority, comparatively to the Greeks, with whom any proposition is established by reasoning and generates logically deduced consequences'
The book edited by Chemla has two parts, the first is made of studies focusing on how in the 18th, 19th and 20th century scholars have written on the topic of mathematical proofs. They show that during the first ¾ of the nineteenth century there were many different ideas of proofs that were discussed by mathematicians and historians of mathematics. It was possible to imagine algebraical proofs notably; which implied not to dismiss the history of computation or of algebra as secondary or of no interest for the history of mathematical proof. The adoption of a standard shape of proof took hold in the history and philosophy of mathematics at the end of the 19th century. It led to discarding all sorts of mathematical texts, not only those in Chinese and in Sanskrit but also those in Greek or in Latin which did not follow these norms.

The second part of the book edited by Chemla is made of studies of sources in all sorts of languages, notably Akkadian, Sanskrit and Chinese, looking at mathematical proofs as a historical practice, trying to understand and describe how different authors practiced proofs and how they understood them.
What This Presentation Will Be About
So what them can we do from here, is part of what my presentation is about.

In what follows, first I would like to underline that part of the problem in the history of mathematical proofs-which is a wider historiographical problem-is that some discourses have attached modes of reasoning to peoples, nations or religions, in a homogeneous way; and that this fuels identity politics in the history of mathematics.

And second, that we can maybe go beyond the study of mathematical proofs, and document the mathematical reasonings we find in texts, studying together the practices of reasonings they testify of, the kind of texts that are forged and what are the aims of these reasonings. This means going beyond the idea of proof, because past mathematicians have written reasonings whose aims might not have been limited to establishing truth or correctness but maybe also to provide explanations, to show that a mathematical object or a procedure can be interpreted in different mathematical and non-mathematical contexts, for instance.

I will first show how the history of proof is important in the writing of the history of mathematics in South Asia, and underline the dangers of nationalist historiographies this might construe, before giving examples of the diversity of mathematical reasonings that can be found in Sanskrit mathematical sources.

Identity Making in Historiographies of Mathematical Proofs

At the end of the 18th century the Scotish mathematician and philosopher John Playfair (1741-1819) officially called for a search for mathematical texts from South Asia, with an equal interest in geometry, arithmetic and astronomy.[4] Henry Thomas Colebrooke (1765-1837) who is known as the father Indology and who knew Playfair, took an early interest in collecting manuscripts of mathematics and astronomy in Sanskrit during his stay in India first as a member of the East Indian Company and then at the head of the Asiatic Society of Bengal.[5] After spending more than 30 years in India, he composed the most influential and enduring translations of Sanskrit mathematical texts in English of the early 19th century[6].

In particular he worked on Bhāskara II 's (b. 1114) mathematical texts devoted to arithmetic ( Līlavatī) and algebra ( Bījagaṇita). His translation starts with a preliminary dissertation, in which he declares that indeed, Sanskrit mathematical texts contained proofs:[7]
On the subject of demonstrations, it is to be remarked that the Hindu mathematicians proved propositions both algebraically and geometrically : as is particularly noticed by BHÁSCARA himself, towards the close of his Algebra, where he gives both modes of proof of a remarkable method for the solution of indeterminate problems, which involve a factum of two unknown quantities.
Such an affirmation sprang from a prose part of Bhāskara II's Algebra. Sanskrit scholarly texts have a specific shape: treatises are written in sutras- that is aphorisms- often versified, which might be at times very terse. Mathematical sutras are more often then not about algorithms, procedures and not about making statements of mathematical truths. These treatises invite prose commentaries. Bhāskara II composed both treatises and short commentaries of his texts, which were integrated in Colebrooke's translations :[8]
The demonstration follows. It is twofold in every case: one geometrical and the other algebraic. asyopapatiḥ| sā ca dvidhā sarvatra syāt| ekā kṣetragatānyā rāśigatetiti| The algebraic demonstration must be exhibited to those who do not comprehend the geometric one. ye kṣetra-gatām upapattiṃ na buddhyanti teṣām iyaṃ rāśigata darśanīyā
Colebrooke therefore takes a sentence which has a certain meaning in the context in which it is stated, and extends it to all of Bhāskara II's work, and from there to all mathematics in Sanskrit.He is creating a homogeneous 'Hindu' mathematics, with a homogeneous idea of proof, to be able to compare the work of authors writing in Sanskrit with those of authors who wrote in Arabic or in Latin. Note the term used by Bhāskara, upapatti. This will be the standard term understood to mean 'proof' in Sanskrit.

Colebrooke's publication was very influential in Europe and in Asia during all the 19th century. Mathematicians in France and Germany read the proofs his translation contained. Discussions on notions of proof in mathematical education were thus developed. Hermann Hankel (1839-1873) for instance developed the idea that an ideal education should combine together two types of proofs, the analytical (attributed to Greek) and the intuitive (attributed to Indians).[9]

Colebrooke took proofs he found in commentaries and added them as footnotes to his translation, giving the feeling that proofs only belonged to commentaries, and could only be found in bits and pieces. By the beginning of the 20th century, the existence of such proofs seems to have been forgotten; and standard histories overlooked them altogether.

Colebrooke's claim has been taken up at the end of the 20th and the beginning of the 21st century by a certain number of historians of mathematics in South Asia, aiming at having the history of mathematics in Sanskrit enter a global conversation on the history of mathematical proofs.

Thus, M. D. Srinivas writes, after quoting Colebrooke:[10]
Clearly the tradition of exposition of upapatti-s is much older and Bhāskarācārya and the later mathematicians and astronomers are merely following the traditional practice of providing detailed upapatti-s in their commentaries to earlier, or their own, works. The notion of upapatti is significantly different from the notion of 'proof' as understood in the Greek as well as the modern Western traditions of mathematics.
K. Ramasubrahmaniam, also adds, after having quoted Colebrooke also:[11]
The upapatti s of Indian mathematics, unlike the western tradition, are not formulated with reference to a formal axiomatic deductive system. (...) One often finds the statement iyam atra vāsanā, when the commentator is about to begin to explain/demonstrate something. Meaningwise this statement iyam atra vāsanā ≡ atropapattiḥ. Both the forms being equivalent, there is hardly any consideration for choosing one over the other.
For M. D. Srinivas then, the mode of proofs found in Sanskrit text is part of an unchanging, immemorial tradition. While K. Ramasubrahmaniam also adds that all sorts of different Sanskrit vocabulary concerning proofs should be considered as synonymous.

As you can see, they not only take up Colebrooke's idea that proofs are either algebraic or geometric (a part of their argument I did not quote), but also the idea of an existence of an unchanging homogeneous tradition of "non-western " proofs. Behind what looks like a true interest in actor's categories of proofs, rather than imposing on these texts an a-priori definition of proof- something that in what follows I hope we can do a bit together- lies in today's India what is truly a very strong claim; that all Sanskrit mathematical texts written over several centuries in very different contexts all functioned in the same way.

Here then a mode of reasoning is harnessed to a language, if it is not to the geographical space of South Asia, to a religion (Hinduism), or a cast (brahmanism). Such claims then, I think you can see, resonate with what is a wider political use of history of science by the current, nationalist hindu, government in India. It constructs a non-western but homogeneous mathematical identity.

I would like to argue that Sanskrit mathematical texts had very diverse ways of reasoning, that do not always correspond to such a portrait of the mathematical proofs they contain.
2. Diverse Reasonings in Sanskrit Mathematical Texts
In what follows I would like to show that proofs need not be algebraic or geometric in Sanskrit texts, that they need not belong to commentaries, and that some reasonings might be larger or different than proofs in as much that they might not be aiming of establishing correctness or truth.
2.1 A Proof in A Treatise: Brahmagupta on Sines
I would first like to show that contrary to what is usually assumed, at times treatises also contain something that might be about a proof. Proofs are not reserved for commentaries.

-Brahmagupta 7th century gives an important tables of sines, and sets out to establish how he derived it

-A geometrical and a numerical entity

-With several rules for each derivation

For an edition and translation of what is discussed here, see Ikeyama, Setsuro. Brāhmasphuṭasiddhānta (Ch. 21) of Brahmagupta with Commentary of Pṛthūdaka, Critically ed. with Eng. tr. and notes. Reprinted from IJHS [Indian journal of history of science] 38. 1-4 (2003). New Delhi, India: Indian National Comission for the History of Science, Indian National Science Academy, 2003.
2.2 A Commentary with Many Reasonings, Not About Algebra and Geometry
Bhāskara I's commentary on the Āryabhaṭīya presents many types of reasonings, which might not all be proofs. We look at the way he re-reads an algorithm with the help of the Pythagorean theorem and the Rule of Three: this is a typical reasoning found in many astronomical texts in Sanskrit.

Inspired by Keller, Agathe. " Dispelling Mathematical Doubts: Assessing Mathematical Correctness of Algorithms in Bhāskara's Commentary on the Mathematical Chapter of the Āryabhaṭīya ". In The History of Mathematical Proof in Ancient Traditions, édité par Karine Chemla, 487-508. Cambridge University Press, 2012. http://halshs.archives-ouvertes.fr/halshs-00150736.
2.3 A Commentary with Explanations Which Is About Showing How an Algorithm Can Be Read in Many Perspectives.
Prthūdhaka's commentary on Brahmagupta's Corrected astronomical treatise of Brāhma ( Brāhmasphuṭasiddhānta, abreviated as BSS) 628 is called Commentary with explanation ( vāsanābhāṣya)

Vāsanā might be about showing that an algorithm can be understood in many different mathematical and non-mathematical contexts.

Inspired by Keller, Agathe. " Characterizing a Sanskrit Mathematical Commentary: An exploration of Pṛthūdaka's Vāsanābhāṣya on progressions. " In Mathematical Commentaries in the Ancient World, édité par Karine Chemla et Glenn Most, 96-140. Cambridge: Cambridge University Press, 2022.
2.4. Travelling Explanations
All the authors mentioned here are part of a same scholarly world. Older author's works are known and quoted by more recent ones. However, obviously they don't always share the same practice of mathematical reasoning. This does not mean that they didn't share reasonings, but more likely that they had had different aims when reasoning. Part of this shared culture of explanations can be seen in circulating mathematical diagrams that might have served as libraries of mathematical explanations made to be stated at times orally, or to be committed to memory.

Finally, let us take an example from early modern Kerala, where authors developed advanced mathematical propositions. Thus, strikingly, as Śaṅkara Vāriyar's (16th century) comments on Mādhava's (14th-15th century) rule providing approximations of π, he is intent in proving ( sadh-) true ( vāstava- statements). His use of the Rule of Three recalls at times Bhāskara I's usage of this rule, and at others seems to be derived from traditions using proportions, derived from Euclid. This suggests that Sanskrit mathematical texts might be at times (here in the early 17th century) incorporating other traditions of mathematical reasonings, forming new practices of proving and reasoning, that do not seem radically "non-western".

This part is inspired from a work in progress; you can look at the translation and discussion of Śaṅkara Vāriyar's proof in Plofker, Kim. Mathematics in India. Princeton University Press, 2009 and Plofker, Kim. " Mathematics in India ". In The Mathematics of Egypt, Mesopotamia, China, India and Islam. A Sourcebook., edited by Victor J. Katz: 385-514. Princeton, N.J.: Princeton University Press, 2007. Other (quite different) reflections on Śaṅkara Vāriyar's way of proving can be found in Wagner, Roy. " The Kriyākramakarī's integrative approach to mathematical knowledge ". History of Science in South Asia 6 (2018): 84-126.
Conclusion
There is something a bit harrowing in the persistence of this ignorance on traditions of proofs outside of the Euclidean norm which has to do, maybe, in part with what the philosopher Achille Mbembe calls " late europeocentrism ", that is of the persistence of eurocentric and colonial structures of thought in a world where everybody knows however that Europe and the United States have never been the unique place in which worthy intellectual production were composed.[12] The creation of a limited corpus of ancient texts of mathematical proofs, is what is sometimes called a 'colonial library' of proofs since such a corpus was constructed at the height of European imperial expansion.

I have entitled this presentation " Let's decolonize the history of mathematical proofs ", because I believe that the decolonizing of the history of mathematical proofs is possible only through a collective critical effort.That as professional historian of mathematics, but also as teachers, as an educated public such as Gresham we should not let spread a discourse which closes the doors of what mathematics is or is not.

Mathematical reasonings were and still are very diverse, they are not necessarily about truth, they were certainly not homogene across a geographical area, in a same language or by people who had a same religion. Let us remember this when we hear people speaking of Western science, British mathematics or islamic proofs

The good news is that we have resources to write other new histories, that are also more stimulating.

Histories that testify to how just like today, people practicing mathematics had many different ways of reasoning. There were ecologies of reasoning that still need to be investigated.

Most probably these ancient ways of reasoning might not bring new ways of reasoning or provide new proofs for today-although I know practicing mathematicians who find stimulating reading mathematical texts of the past-but beyond this issue, these mathematical texts and their reasonings are part of the legacy of our common humanity, as much as say great epics and beautiful works of art, and if its just for that, they are worth studying and cherishing in themselves.

© Professor Keller 2022
References and Further Reading
Biot, Jean-Baptiste. " Compte-rendu de: Traité des instruments astronomiques des Arabes, traduit par JJ Sédillot ". Journal des savants, 1841, 513‑20; 602‑10; 659‑79.

Chemla, Karine, éd. The History of Mathematical Proof in Ancient Traditions. Cambridge University Press. Cambridge: Chemla, Karine, 2014.

Chemla, Karine, and Glenn W. Most, éd. Mathematical Commentaries in the Ancient World: A Global Perspective. Cambridge: Cambridge University Press, 2022. https://doi.org/10.1017/9781108884488.

Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhàscara. London: J. Murray, 1817.

Ikeyama, Setsuro. Brāhmasphuṭasiddhānta (Ch. 21) of Brahmagupta with Commentary of Pṛthūdaka, Critically ed. with Eng. tr. and notes. Reprinted from IJHS [Indian journal of history of science] 38. 1-4 (2003). New Delhi, India: Indian National Comission for the History of Science, Indian National Science Academy, 2003.

Keller, Agathe. " Making Diagrams Speak, in Bhāskara I's Commentary on the Āryabhaṭīya ". Historia Mathematica 32, no 3 (2005): 275-302.

Mbembe, Achille. " Notes sur l'eurocentrisme tardif - AOC media ". AOC media - Analyse Opinion Critique (blog), 16 mars 2021. https://aoc.media/analyse/2021/03/16/notes-sur-leurocentrisme-tardif/.

Morris, Kline. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972.

Playfair, John. " Questions and Remarks on the Astronomy of the Hindus " Asiatick Researches 4 (1792): 151-55.

Plofker, Kim. Mathematics in India. Princeton University Press, 2009 and

Plofker, Kim. " Mathematics in India ". In The Mathematics of Egypt, Mesopotamia, China, India and Islam. A Sourcebook., édité par Victor J. Katz, 385-514. Princeton, N.J.: Princeton University Press, 2007.

Ramasubramanian, K. 2011. "The Notion of Proof in Indian Science." In Scientific Literature in Sanskrit, edited by Sreeramula Rajeswara Sarma and Gyula Wojtilla, 1:1-39. Papers of the 13th World Sanskrit Conference. Dehli: Motilal Banarsidass.

Rocher, Ludo, Rosane; Rocher. The Making of Western Indology, Henry Thomas Colebrooke and the East India Company. Routledge, 2012.

Smadja, Ivahn. " Sanskrit versus Greek "Proofs": History of Mathematics at the Crossroads of Philology and Mathematics in Nineteenth-Century Germany ". Revue d'Histoire des Mathématiques 21, no 2 (2015): 217-349.

Srinivas, M. D. 2008. "Epilogue: Proofs in Indian Mathematics." In Gaṇita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva, 1:267-310. Springer; Hindustan Book Agency.

Wagner, Roy. " The Kriyākramakarī's integrative approach to mathematical knowledge ". History of Science in South Asia 6 (2018): 84-126.

[1] Morris, Kline. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972 : p. 198.

[2] Chemla, Karine, éd. The History of Mathematical Proof in Ancient Traditions. Cambridge University Press. Cambridge: Chemla, Karine, 2014.

[3] On y trouve une nouvelle preuve de cette singulière habitude de l'esprit, en vertu de laquelle les Arabes, comme les Chinois et les Hindous, bornaient leurs compositions scientifiques à l'exposition d'une suite de règles, qui, une fois posées, devaient se vérifier par leur applications mêmes, sans besoin de démonstration logique, ni de connexion entre elles: ce qui donne a ces nations orientales un caractère remarquable de dissemblance, et j'ajouterai d'infériorité intellectuelle, comparativement aux Grecs, chez lesquels toute proposition s'établit par raisonnement, et engendre des conséquences logiquement déduites.' Biot, Jean-Baptiste. " Compte-rendu de: Traité des instruments astronomiques des Arabes, traduit par JJ Sédillot ". Journal des savants, 1841, 513‑20; 602‑10; 659‑79. The English translation is given in Charette, François. " The Logical Greek versus the Imaginative Oriental: On the Historiography of "Non-Western" Mathematics during the Period 1820-1920 ". In The History of Mathematical Proof in Ancient Traditions, ed. By Chemla. Cambridge University Press, 2012: 272.

[4] Playfair, John. " Questions and Remarks on the Astronomy of the Hindus " Asiatick Researches 4 (1792): 151-55.

[5] For the relation of Colebrooke to Playfair see Raina, Dhruv. " Contextualizing Playfair and Colebrooke on Proof and Demonstration in the Indian Mathematical Tradition (1780-1820) ". In The History of Mathematical Proof in Ancient Traditions, ed. by Chemla. Cambridge University Press, 2012: 228-59. For a very complete biography of H. T. Colebrooke see Rocher, Ludo, Rosane; Rocher. The Making of Western Indology, Henry Thomas Colebrooke and the East India Company. Routledge, 2012.

[6] Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhàscara. London: J. Murray, 1817.

[7] Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhàscara. London: J. Murray, 1817 ; Dissertation p. xvii

[8] Colebrooke, Henry Thomas. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhàscara. London: J. Murray, 1817 : p. 271 and 272 for the English translation; Hayashi, Takao. " Bījagaṇita of Bhāskara ". SCIAMVS Sources and Commentaries in Exact Sciences 10 (2009): 3-301, for the Sanskrit.

[9] Charette, François. " The Logical Greek versus the Imaginative Oriental: On the Historiography of "Non-Western" Mathematics during the Period 1820-1920 ". In The History of Mathematical Proof in Ancient Traditions, ed. By Chemla. Cambridge University Press, 2012: 272. And also Smadja, Ivahn. " Sanskrit versus Greek "Proofs": History of Mathematics at the Crossroads of Philology and Mathematics in Nineteenth-Century Germany ". Revue d'Histoire des Mathématiques 21, no 2 (2015): 217-349.

[10] Srinivas, M. D. 2008. "Epilogue: Proofs in Indian Mathematics." In Gaṇita-Yukti-Bhāṣā (Rationales in Mathematical Astronomy) of Jyeṣṭhadeva, 1:267-310. Springer; Hindustan Book Agency.

[11] Ramasubramanian, K. 2011. "The Notion of Proof in Indian Science." In Scientific Literature in Sanskrit, edited by Sreeramula Rajeswara Sarma and Gyula Wojtilla, 1:1-39. Papers of the 13th World Sanskrit Conference. Dehli: Motilal Banarsidass.

[12] Mbembe, Achille. " Notes sur l'eurocentrisme tardif - AOC media ". AOC media - Analyse Opinion Critique (blog), 16 mars 2021. https://aoc.media/analyse/2021/03/16/notes-sur-leurocentrisme-tardif/.

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Let's Decolonise the History of Mathematical Proofs!

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Joint lectures with the British Society for the History of Mathematics. What is a "valid mathematical proof"? To inquire into such a hotly debated question we might want to look at how past mathematicians tackled this question.

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BAME - IMA
Interviews with many black mathematicians, including Dr Nira Chamberlain

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"Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country." Howard Eves in Mathematical Circles Squared (Boston 1971) Mathematics is a vital and rewarding career that needs diverse talent from all cultures, races and backgrounds. This page features an exciting range of stories and interviews from BAME (Black and Minority Ethnic) mathematicians.

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Includes postcards, themed literacy tasks, podcast and blog

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At Pearson, we believe everyone should have the same opportunity to achieve their full potential and to be successful in education, and in later life - whatever their background, ability or identity. Which is why we're working hard to ensure that education is inclusive.

We know this is a journey. It's a journey that is complex and far-reaching, and one that won't reach its end overnight. But as a Maths team we're committed to getting there and to working together with schools and young people to drive change, create learning environments that reflect the diversity of the modern world, are accessible, and that provide a choice of pathways for all learning styles and abilities.

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📢 New The Right Angle episode: Diversity and Inclusion in Maths

The Right Angle podcast has maths teachers at the centre of its episodes and aims to help shape a positive future for maths education.

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Virtual event: Diversity in data with Professor Hannah Fry

In November 2021, we held our Diversity in data webinar with leading mathematician Professor Hannh Fry, and explored how we can help all learners feel included and represented.

Through practical advice and Q&A sessions, Hannah Fry helped attendees understand the power of numbers and ethics within research and data collection. She also explored gender inequality in STEM and the ways to remove bias to ensure fair results.

This webinar was designed to promote diversity in data, the maths classroom and beyond. It provided attendees with insights and tools that could be applied to the classroom.

Following the webinar, Hannah Fry shared with us a blog post where she discusses five ways to challenge data bias in order to make maths classrooms more diverse and inclusive.
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Perceptions, diversity and inclusion in maths Stat-Shot

Our Power of Maths Stat-Shot series highlights headline insights into different aspects of maths education based on the views of over 2,000 primary teachers and secondary maths teachers who took part in our Teacher Tapp survey in the 2020/21 academic year.

Here's what teachers told us about perceptions of maths and some of the barriers to students feeling included and engaged in the subject...
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Poster series

If you're looking for our diversity and inclusion poster series, we've moved them to their own page! These posters show key mathematicians who have inspired us and positively impacted the way that we live today.

We'll be releasing one new poster each month until the end of the year, so please make sure you visit us again to get the whole set!
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Comprehension tasks

These comprehension tasks have been designed to allow each year group to engage in one themed literacy task per term.

These tasks should take no more than 15-20 minutes. They can be used as a starter activity or an end of week plenary activity, or can even be embedded within your scheme of learning to promote a particular topic.

We know how important it is to try and develop literacy across the school and hopefully these resources can begin to support you in this task.

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Maths - diversity and inclusion | Pearson UK

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At Pearson, we believe everyone should have the same opportunity to achieve their full potential and to be successful in education, and in later life - whatever their background, ability or identity.

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Where do mathematical symbols come from?

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Where do mathematical symbols come from?

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Where do we get our mathematical symbols from? Why is the set of integers called ℤ ? When was the equals sign first used? How about zero? Good notation tends to catch on quickly, whereas bad notation can obscure beautiful theory.

The lecture explores how the introduction of new notation has paved the way for new leaps in understanding, and considers some mathematical quirks of language, such as what the number 4 in English has in common with the number 11 in Russian.

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where do mathematical symbols come from?

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Where do Mathematical Symbols Come From?

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here do we get our mathematical symbols from? Why is the set of integers called ℤ ? When was the equals sign first used? How about zero? Good notation tends to catch on quickly, whereas bad notation can obscure beautiful theory.

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The mathematical life of Florence Nightingale

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The mathematical life of Florence Nightingale

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The Mathematical Life of Florence Nightingale

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Florence Nightingale is the founder of modern nursing and a pioneer of data science and medical statistics. Her innovative use of statistical diagrams helped people see just how many deaths were being caused by poor hygiene in military hospitals. This lecture will look at the importance and legacy of Nightingale’s work, which led to her becoming, in 1858, the first woman elected to the Royal Statistical Society.

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Alan Turing: Pioneer of Mathematical Biology

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Alan Turing: Pioneer of Mathematical Biology

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Professor Sarah Hart 6th June 2023
Alan Turing is well-known for his work on the Enigma code in World War II, and his theoretical work underpinning computer science. But he is less well-known for his pioneering work on one of the great challenges of biology - how do complex living organisms develop from tiny collections of cells? And in particular, how do the shapes, forms and patterns in animals and plants emerge from seemingly amorphous beginnings? This is the problem of "morphogenesis": the emergence of form. In today's lecture we'll look at Turing's ground-breaking 1952 paper: The Chemical Basis of Morphogenesis, ideas from which can help us understand how patterns like a leopard's spots or a zebra's stripes can occur in nature.
Who was Alan Turing?
At the start of his description of a mathematical model for morphogenesis, Alan Turing begins by saying "This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge."

My brief biographical sketch will also be a simplification and an idealization, and consequently a falsification. We all know something of Turing. He was born on 23 June 1912, so (at the time of writing) it's only a few weeks till his eleventy-first birthday. There's a lovely picture drawn by his mother Sara Turing, of a hockey match in which Alan is supposed to be taking part, but he's become distracted by looking at the daisies. Maybe a sign of his future mathematical interest in how plants and animals grow? His school years were not happy - he didn't fit in, he didn't know how to play the game. Even academically he was criticized - his handwriting was messy, he didn't focus in class. It did however become clear that he was an extremely able mathematician, and he went to Cambridge to read mathematics in 1931. Already by 1935 he'd been elected a fellow of King's College Cambridge for a dissertation on probability theory. In 1936 he published On Computable Numbers, with an application to the Entscheidungsproblem. In this paper, he designed a theoretical computer, which we now call a Turing machine, which had a single tape with 0s and 1s on it, and the computer could, based on the number on the tape and the instruction at that stage of the program, either change the number, do nothing, move one step to the left or one step to the right, and go to a different instruction. The amazing thing was that Turing showed that essentially if any computer could do a thing, then this primitive machine could do it too. He is also known for his much later work on what's now called AI, especially the "Turing Test", which he originally called the Imitation Game in a 1950 paper. A user interacts via a terminal with two other users, one is a machine and one a human; the user must determine which is which. The machine passes the test if the user mistakenly thinks it is the human.

The work for which he is most famous now is his work at Bletchley Park during the Second World War in cracking the German Enigma encryption machines. These machines had "scrambler" dials along with a plugboard that swapped ten pairs of letters, giving 158,962,555,218,000,000,000 possible settings. Brute force search would be impossible - even a machine that could check a million settings every second would take over five million years to try every possibility. Worse, the settings were changed every day so that you only had 24 hours to crack each version. Alan Turing managed to find a mathematical way to cancel out the effect of the plugboard. This reduced the number of combinations that needed checking to 1,054,560. Too high to check by hand, but with several Bombe computers running simultaneously, it could eventually be done each day within a few hours. It's estimated that this breakthrough shortened the war by two years.

What does one do when one has contributed incredibly important mathematical work in two completely different areas, one of which for reasons of national security almost nobody knows about. Naturally, one contributes incredibly important work in a third area. By the end of the 1940s, Turing was a Reader at the University of Manchester, and had some time to think about other interests which had necessarily lain dormant during the War years. In the Autumn of 1951, he submitted a paper on morphogenesis, or how plant and animal forms develop, to the Biological Sciences section of the Philosophical Transactions of the Royal Society, and it appeared in 1952. It's this work we are going to focus on today.

This paper was Turing's only published output on mathematical biology. Why did he not do more work in this area? The tragic answer is that in the same year The Chemical Basis of Morphogenesis was published, he was tried and convicted of "gross indecency" because he had had a relationship with another man; as a condition of parole he was forced to endure 12 months of "organo-therapy": crude hormone "treatment", and on June 7th 1954 he died at home - the verdict at the time was suicide, and sadly it probably was, but we will never know for sure. It was a life tragically cut short. We know he was doing more research in mathematical biology at the time of his death. In his unpublished papers there's even a draft manuscript called Outline of the development of the daisy - so he was still thinking about them, all these years later. In this lecture I'm going to try to give you the gist of what his paper on morphogenesis did and why it was so important. We'll start by looking at some of the other ways that mathematical ideas have historically been linked with the natural world.
Watching the Daisies
Symmetry is prevalent in nature. One reason for this could be that symmetry is an efficiency-saving design solution. If a solution is optimal in one direction, then why not just use it in all directions? The only time symmetry doesn't make sense is if there are constraints that force it to break. Humans, along with other big animals, have bilateral (left-right) mirror symmetry. We have a top and a bottom, because of gravity. We have a back and a front, because we walk and see. But there's no reason why our left and right sides should need to be different, and so they are more or less the same, at least externally. Animals which are simpler and less constrained can have more symmetry. Gravity is irrelevant to the tiny sea creatures called radiolara which float in the deep ocean waters. Of course, we couldn't see these until microscopes were good enough. Some of the first images of single-celled organisms like these were drawn by Ernst Haeckel (1834 - 1919). He took part in the HMS Challenger expedition of 1875 and discovered and drew thousands of microscopic organisms with high degrees of symmetry, and published books such as Morphological Symmetries in the Animal Kingdom. Haeckel was a German naturalist and artist. As well as classifying and discovering many new species, he also coined many new words and phrases, including ecology, phylum, stem cell, and possibly even "First World War" - he seems to have been, in 1914, the first person to use that phrase. Haeckel's drawings of radiolara, which are protozoa (single celled organisms) around 0.1mm in diameter, show startlingly symmetrical geometrical shapes, including what look very like the Platonic solids, the most symmetrical of all polyhedra.

There's symmetry and pattern in the plant world as well - flowers tend to have radial symmetry, but more than that, the study of phyllotaxis - the arrangement of the leaves on a stem, throws up a lot of mathematics. Turing was certainly interested in the spirals in pine cones and of course we already saw how much he liked daisies. There are good reasons why the Fibonacci numbers seem to occur so often in the spiral structures of plants and the petal numbers of flowers - essentially it's because they are linked to the famous golden ratio. I went into this a bit in my Gresham lecture on Harmony and Proportion [link at end of transcript] so I won't say any more here.

In 1917 the Scottish polymath D'Arcy Wentworth Thompson (1860-1948) published a hugely influential book: On Growth and Form. It was a study of how the shapes taken by living organisms relate to mathematical and physical laws. Among a wealth of fascinating observations, I'll just mention one. It's that the same basic "recipe" for form can result in a huge variety of final shapes by applying simple mathematical spatial transformations like stretching, shearing, or transforming a rectilinear grid into a grid of radial lines and concentric circles. Thompson gives several sketches of these kinds of transformations showing how they link different fish species to the same, or very similar, underlying blueprints.

There are a couple of other places where mathematical models can mimic what happens in nature. The first of these I mentioned in my lecture on Christopher Wren - he came up with an equation that can model a three-dimensional form of a logarithmic spiral (so called because the relationship between the angle moved through and the radial distance from the centre is a logarithmic one: the angle is proportional to the logarithm of the distance). This, he suggested, could give rise to the shape of shells. Meanwhile, and much later, fractals, which are produced by an iterative process where at each stage component lines are replaced by copies of the initial simple design, were shown to be an excellent way to produce designs that look very like ferns and other plants. The full power of these techniques could not be seen until computers became fast and powerful enough to carry out multiple iterations of the calculations required. As we'll see, the same issue affected Turing's work.

The mathematical observations we've talked about so far are great at showing what nature might be doing, and even why. But they do not address how nature might be doing it. That's where Turing comes in.
The Chemical Basis of Morphogenesis
Here is the basic issue that Turing was trying to address - and it's something that was a huge open question in biology at the time. If we think about how animals grow, the starting point is a single cell - the egg. It's essentially a sphere - not a perfect mathematical sphere, of course, there would always be tiny imperfections, but these imperfections are small and random - different eggs are imperfect differently. So how does this homogeneous, symmetrical structure develop asymmetry as it grows? How can an apparently uniform surface like an animal's skin in embryo suddenly, apparently spontaneously, develop pigmentation like spots or stripes? What Turing suggests is one way that this may happen. He shows that phenomena like this could be caused by a system of (as yet unknown) chemical substances, which he calls morphogens to convey, as he says, the idea of "form-producer", that diffuse through the tissue but also react with each other. If they are in equilibrium (the same amounts being created and destroyed), then nothing changes, but if that equilibrium is unstable, then even very small random imperfections, as would, indeed must, naturally occur, can destroy the equilibrium and give rise to the development of inhomogeneous patterns. It's these reaction-diffusion models that are what I'm going to talk you through. (In fact, we now know that at least for the case of a fertilized egg, it's not quite as symmetrical as we thought, proteins present from the mother's body diffuse from one side of the egg but not the other. So this particular mystery has a different solution. But plenty of others do not. And Turing's paper was the first to show that simple physical and chemical laws could create such an effect - no special miracle of life is required.)

I'm going to give you the same introductory example Alan Turing used in his paper, but a bit more talkatively! He's not claiming this precise example exists in nature, but it illustrates the way that something almost homogeneous can lose that symmetry. Imagine we have two adjacent cells, and that they both contain certain concentrations of morphogens 𝑥 and 𝑦. We'll look at the effect of diffusion and reaction on this system. Diffusion is the process of molecules moving from areas of high concentration to areas of low concentration so that eventually the concentration will be uniform (if no other processes interfere). Let's say that in this case, morphogen 𝑥 moves to the area of higher concentration at a rate of 0.5 times the difference in concentration per unit time interval (let's say it's 1 minute, Turing doesn't specify and it doesn't matter for the maths). That is, if Cell 1 has concentration 𝑋1 = 3 and Cell 2 has concentration 𝑋2 = 1, then the rate of transfer from Cell 1 to Cell 2 at that instant will be 0.5(3 − 1) = 1. Again, the units of concentration don't matter as long as we are consistent, maybe it's parts per million. This is a continuous process and so the rate of transfer will be constantly changing based on the concentrations in each cell. We get something like "rate of change of 𝑋1" = 0.5(𝑋2 - 𝑋1). If you know calculus, you can write this as

d𝑋1 = 0.5(𝑋2 − 𝑋1). Meanwhile, morphogen 𝑦 moves to the area of higher concentration much quicker - at d𝑡

a rate of 4.5 times the difference in concentration per unit time. That's the diffusion part. But 𝑥 and 𝑦 are also involved in chemical reactions. We make another assumption: the rates of reaction, and hence the rates of change of the concentrations of our morphogens, depend on the concentrations 𝑋 and 𝑌 of these morphogens that are already present. (Things like this can occur - certain chemicals are more reactive than others, the presence of catalysts can inhibit certain reactions while promoting others - we don't get into the details as this is a toy example, but it's not impossible to imagine.)

In his example, Turing posits that morphogen 𝑥 is produced at the rate 5𝑋 − 6𝑌 + 1 and 𝑦 is produced at the rate 6𝑋 − 7𝑌 + 1, where 𝑋 and 𝑌 are the concentrations in the given cell.

First, suppose each cell has a concentration of 1 in both chemicals. There is no diffusion as the cells have equal concentrations. But also, 5𝑋 − 6𝑌 + 1 = 0 and 6𝑋 − 7𝑌 + 1 = 0 in both cells. So there is no production or destruction via chemical reaction either. We have equilibrium. But now, let's imagine a scenario just slightly different from equilibrium. Suppose 𝑋 = 1.06, 𝑌 = 1.02 for the first cell and 𝑋 = 0.94, 112 𝑌 = 0.98 for the second cell. Then in the first cell 𝑋 is going to be produced at the rate 5 × 1.06 − 26 × 1.02 + 1 = 0.18 and 𝑌 is going to be produced at the rate 6 × 1.06 − 7 × 1.02 + 1 = 0.22, while in the second cell they will be lost at those same rates. There is also diffusion from the first cell to the second, at a rate of 0.06 for X and 0.18 for Y. The net effect is that the amount of X in the second cell decreases at a net rate of 0.12, and the amount of Y decreases at a net rate of 0.04, with the amounts in the first cell increasing by these same rates. The equations can be solved (for the experts, this information can be stated as a system of four simultaneous linear differential equations, for example 𝑋̇ = 0.5(𝑋 − 𝑋 ) + 121 5𝑋 − 6𝑌 + 1 would be the equation governing 𝑋 , so they can be solved with techniques like Laplace 111 transforms). It turns out that whenever the concentrations at any stage have the forms 𝑋1 = 1 + 3𝐾, 𝑋2 = 1 − 3𝐾, 𝑌 = 1 + 𝐾, 𝑌 = 1 − 𝐾 for a given 𝐾, then the solution is 𝑋 = 1 + 3𝑒2𝑡, 𝑋 = 1 − 3𝑒2𝑡, 𝑌 = 1 + 𝑒2𝑡,12 121𝑌 = 1 − 𝑒2𝑡. In other words, the concentrations of both morphogens in the first cell continue to rise, and the 2 drift away from equilibrium occurs at an exponential rate. This would work even if the initial values were crazily close, like 𝑋 = 1.000003, 𝑌 = 1.000001 for the first cell and 𝑋 = 0.999997, 𝑌 = 0.999999. This 11 22 example, while deliberately being rather contrived to keep the mathematics simple, still gives proof of concept for the idea.

In his paper, Turing goes into much greater detail and lengthy mathematical analysis covering more biologically likely scenarios, like rings of cells or tissue, and the two-dimensional case of the surface of an animal's skin. He gives a picture of "dappling" which can be produced by this type of process. Here, at any specific location, the diffusion part of the process means concentrations are affected only by concentrations nearby, so as the foetus grows, diffusion plays less of a role as the distances involved are too far for morphogens to travel. Thus, there would be a point during the growth of the foetus when any patterns of pigmentation (for instance) settle down and become fixed.
The Structure of Radiolara
Turing's paper dealt with the morphogenesis equations in two dimensions. These equations can address surface features like spots and stripes. But by 1953 he was working with his research student Bernard Richards on the three-dimensional version: the shape of an entire organism. In particular, Turing was interested in whether they could replicate mathematically the curious patterns of spines in radiolara recorded by Ernst Haeckel. Radiolara are single-celled marine organisms. They start off essentially spherical, but as they grow, the body can, depending on the species begin to resemble a Platonic solid, with spines emerging from its vertices. Bernard Richards programmed the Ferranti Mark I computer at Manchester to represent the solutions to the 3-dimensional equation, with suitable parameters. This is not as easy as it sounds - nowadays every A level student has a calculator that can draw graphs. At the time, the only output possible was numbers and letters. So what they did was output an array with rows and columns representing the degrees of latitude and longitude, in fixed small increments, and then at each entry, the computer printed the distance of the surface of the radiolara from its centre, at the corresponding latitude and longitude. This allowed Richards (by hand) to draw contour lines on the output sheets, joining places of equal distance from the centre, and that meant he could find the spines. He showed these pictures to Alan Turing in late May 1954, and it was only at this point that Turing explained about radiolara, and asked Richards to have a look at the Challenger expedition report, which was in the University library. He was delighted to discover species of radiolara that precisely matched the shapes the computer had produced, and excited to show these findings to Turing at their next meeting on June 8th. Tragically, this meeting would never take place, because Turing died on June 7th.
Reaction and Diffusion of the theory
So, what happened next? Did others immediately continue Turing's work in mathematical biology? The answer is no - in fact there were almost no developments for the next forty years! The extreme difficulty of the calculations even for the most primitive kinds of organisms, gives us a clue as to one reason for this. It just wasn't possible to run sufficiently complex computer simulations to model the solutions to Turing's equations. The other reason is the huge impact of the discovery of the structure of DNA. The effect of this was that, understandably, DNA immediately became the focus of the research community, which took efforts away from working out how the morphogenesis was actually brought about. But towards the end of the 20th century, there was a resurgence in interest in Turing's ideas. All at once, people were producing incredibly convincing simulations of spots, stripes and other phenomena. I like to think that this is itself an example of an apparently static, homogeneous system suddenly producing interesting patterns apparently out of nowhere. So, let's look next at some work on patterns in animals that is based on Turing's ideas.
Spots and Stripes
Modern mathematical biologists tend to use the term activator-inhibitor for the kind of reaction-diffusion systems that lead to patterns in animals. One morphogen is an "activator". It's actually a catalyst for itself in that the higher the concentration already present, the faster it is produced. The other is an "inhibitor" - it slows down the catalysis of the activator. (This is like the 𝑥 and 𝑦 in Turing's example, because we have that the rate of change of the concentration of 𝑥 is 5𝑋 − 6𝑌 + 1.) If it also happens, as in that example, that the inhibitor (here 𝑦) diffuses faster than the activator, then pockets of net activator production are fenced off by the faster-spreading inhibitor. This is intuitively why we might expect "islands" of the activator to form, in a sea of inhibitor.

Because diffusion is a local phenomenon, we might expect that the size of the region in question is relevant. Patterns of pigmentation are laid down early in the development of an animal foetus, so it's not so much the size of the eventual adult animal that counts, as the size of the foetus at the time the patterns are created. At small sizes we can get stripes, but above a given size stripes are not a stable solution - kind of like collapsing a standing wave, and then we get spots. If we imagine an animal as a big cylinder for its body and smaller cylinders for its tail, then if at the point the pigments are laid down the body can still sustain stripes, then the whole animal can be striped, for example in the zebra or tiger. Then there's a transition point where the body is too big to be striped, but the tail can be, so we can get a spotted body and striped tail (such as the cheetah). Then there's another transition where even the tail is too thick to be striped, and then it too becomes spotted. There are intermediate examples where just the end of the tail is striped (like the jaguar or leopard). But you will never see a striped animal with a spotted tail, and now you know why. It's not just spots and stripes. Models like the ones Turing proposed have been shown to describe everything from Now that computers are sufficiently powerful, all these can be modelling very convincingly using the appropriate equations. There are several online simulators where you can play with different initial conditions and equations, and see pattern-forming in action. I've put a link to one at the end of the transcript.
Other applications
Reaction-diffusion models have also turned out to have applications beyond morphogenesis. I'll finish with a couple of examples. When wind blows over sand, ripples can form, with little ridged peaks and troughs. As the sand grains blow across the surface they will be impeded and thus caught by any little ridges, which then will grow - the higher the ridge the faster it grows, like the activator process. Meanwhile because that ridge is capturing sand, it is inhibiting the growth of any nearby ridges. So we get gaps between the ridges, and hence ripples. Turing patterns have also been found in models of how crime spreads. If we think of crimes like burglaries, it's known that burglars tend to return to the same neighbourhoods that they have been successful in targeting previously. If there are already crimes happening in an area, they tend to increase the crime rate further. This is an activator. The inhibitor is crime-prevention like increased policing. Using these observations mathematical activator-inhibitor models can be set up that show crime hotspots developing just as they do in real life. This can lead us to think about what good solutions look like: one method is to increase the inhibiting effect - more bobbies on the beat and so on. But that's resource- intensive. The other is to address the catalyzing effect of crime and try and break the link between a history of crime in an area and ongoing crimes. This is the "broken window" theory. React swiftly to even minor things in order to dampen the catalyzing effect. I've sometimes wondered if there aren't Turing patterns relating to other things, like popularity. Think about the "hotspots" in art galleries, with massive crowds flocking to a few famous masterpieces while ignoring dozens of equally wonderful paintings. I have a feeling these mathematical models will find ever more applications in future.
Conclusion
I hope I have been able to give you a glimpse of the brilliance of Turing's insights into possible explanations of biological, and other, phenomena. There's some suggestions of further reading below as well as links to websites where you can play with reaction-diffusion systems. This is the last Gresham Geometry lecture of the academic year. From October I'll be giving my fourth and final series of lectures as Gresham Professor, beginning with the mathematics of board games. If you can't wait that long for more mathematics from Gresham, don't panic - on June 13th Robin Wilson, former Gresham Professor of Geometry, will be giving a lecture on the history of graph theory, so make sure to tune in for that.

© Professor Sarah Hart 2023
Further Reading

The chemical basis of morphogenesis, Phil. Trans. R. Soc. London B 237 pp 37-72 (1952) by Alan Turing, is available at https://royalsocietypublishing.org/doi/10.1098/rstb.1952.0012. For an academic paper it's surprisingly clear and easy to read. There is a huge amount of scanned material in the Turing Digital Archive (free to access) at https://turingarchive.kings.cam.ac.uk/

The Art and Science of Ernst Haeckel (Taschen, 2020) is a beautiful book showing many of Haeckel's best drawings, including the radiolara ones. You can find the original Haeckel books online too, as they are out of copyright.

Symmetry, by Hermann Weyl, (Princeton University Press), first published 1952 but still in print, is a classic still worth reading that covers symmetry in nature but far beyond.

I mentioned a couple of my previous Gresham lectures - the Harmony and Proportion one is at https://www.gresham.ac.uk/watch-now/maths-proportion, the Christopher Wren one is at https://www.gresham.ac.uk/watch-now/maths-wren

On Growth and Form, by D'Arcy Wentworth Thompson, is a classic text on why living things have the forms they do. First published in 1917, and extended to a 1116-page second edition in 1942, it has been incredibly influential both to mathematicians and biologists. A trimmed down version (a mere 346 pages!) with helpful editorial commentary was published in paperback by Cambridge University Press in 2014.

Fearful Symmetry: Is God a Geometer, (reprinted by Dover books in 2011), by Ian Stewart and Martin Golubitsky, is an excellent guide to symmetry in nature.

Alan Turing: The Enigma, by Andrew Hodges, is widely regarded as the best biography of Turing. It was reissued in 2014 with a photo of Benedict Cumberbatch on the cover to tie in with the film The Imitation Game, which is a highly entertaining film but does feature Keira Knightley mispronouncing Euler, which the character she plays would never have done. If you watch it, do look out for the Haeckel pictures the set designers have stuck all over the walls of Turing's house in the scenes set in the 1950s.

The Turing Guide, by Jack Copeland and others, OUP 2017. This is a collection of articles and essays on different aspects of Turing's life and work; it includes one by Sir Dermot Turing, Alan Turing's nephew, who kindly gave me permission to use the drawing "watching the daisies grow", and one by Bernard Richards, the student of Turing's who worked with him on reaction-diffusion models.

Breaking the Code is a 1986 play by Hugh Whitemore, about events surrounding his prosecution for homosexuality, and linking with his interest in AI and cryptography. The script is available in book form; in the original run Turing was played by Derek Jacobi, who reprised the role for the BBC adaptation in 1996, which is occasionally available on the BBC iPlayer. There's not much on mathematical biology in it but there is a scene where he talks about pine cones and Fibonacci.

You can play with your own reaction-diffusion models at Jason Webb's free site https://jasonwebb.github.io/reaction-diffusion-playground/app.html

Philip Ball has written several good articles on pattern formation - he gives a bit more detail on Turing patterns in sand, crime hotspots and also ant cemeteries (!) in this one: https://philipball.co.uk/wp-content/uploads/2015/05/Turing_long.pdf

"Watching the Daisies", drawn by Sara Turing, used by kind permission of the Turing family.

I created the "Gresham" reaction-diffusion patterns at https://jasonwebb.github.io/reaction-diffusion-playground/app.html

The puffer fish photo is by Chiswick Chap, CC BY-SA 3.0; the Zebra stripes are a close-up of a

photo by Rainbirder, CC BY-SA 2.0; the cheetah is by flowcomm, CC BY 2.0, the leopard is by Davidvraju, CC BY-SA 4.0, the tiger is by Charles J. Sharp , CC BY-SA 4.0, all via Wikimedia Commons.

To the best of my knowledge, every other image in this talk is either made by me and © Sarah Hart, or in the public domain (either because out of copyright or because its creator has waived copyright), or use is allowed under "fair use" rules.

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Alan Turing: Pioneer of Mathematical Biology

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Alan Turing is well-known for his work on the Enigma code in World War II, and his theoretical work underpinning computer science. But he is less well-known for his pioneering work on one of the great challenges of biology - how do complex living organisms develop from tiny collections of cells?

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Pearson - Diversity and inclusion in maths with Hannah Fry podcast

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Pearson - Diversity and inclusion in maths with Hannah Fry podcast

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From the Right Angle podcast The Right Angle Podcast | Pearson UK

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Diversity and Inclusion in Maths with Hannah Fry | Pearson UK

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Listen to The Right Angle podcast episode where Professor Hannah Fry shares valuable insights, tips and tools that teachers can use in their Maths classrooms >

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From the Right Angle podcast The Right Angle Podcast | Pearson UK

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Ep 08: 'Girls do STEM too!' with Dr Anne-Marie Imafidon

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Listen to this episode from The Right Angle on Spotify. Listen to host, Nicola Woodford-Smith talk chat with Dr Anne-Marie Imafidon about the huge success of the Stemettes to being the recent arithmetician on Countdown, as well as how teachers can encourage young people to explore more STEM inspired roles.

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1. Speak up and challenge assumptions

A lot of harm can be done by someone who is not malicious but simply isn't thinking about the impact of their words or the data that is presented. And a lot of good is done by someone simply speaking up to point it out.

Data bias makes assumptions about the population and can omit a group of people by design. This can lead to racial, gender and exclusion of other groups of individuals such as learners with special educational needs and disabilities (SEND). It is essential that children and young people are not given the message that 'you don't belong' and we must not be afraid to challenge where intentional biases appear.

Instead of reinforcing assumptions, we can spread positivity and representation through data and promote diversity through numbers and Maths education.

2. Pay careful attention to stereotypes in the classroom

Flip the stereotypes and ensure there is gender, racial and SEND representation in the portrayal of careers, home life and achievements. Use visual and verbal representation to reinforce the message. Our classrooms must represent a version of the world that we want.

Stereotypes are particularly problematic within STEM subjects. During an experiment in the 1960-70s, less than one percent of students drew a female when they were asked to draw a scientist. However, this is changing, and girls are driving the shift. By 2016, around one third of drawings were female.

Change is happening more prominently in younger students and so it is important that we pay special attention to stereotypes and provide students of all ages with diverse role models. These don't have to be the leading mathematicians and scientists. Evidence suggests that role models that make the biggest difference to students are just one step ahead, so employ the help of diverse students one or two school years ahead, to inspire in the classroom.

3. Debunk the myth of innate ability

Boys are NOT better at Maths than girls and yet girls struggle with maths anxiety and less choose maths as a subject. Scientifically, there are very little differences in the male and female brain and very little difference in maths performance. Gender differences in confidence, imposter syndrome, maths anxiety and choices of math-intensive career choices do exist, but evidence suggests that socialisation and social influencers create these differences in the brain.

A growth mindset is essential in STEM. The false idea that some people just inherently 'get it' is so prevalent and will invariably tip towards a gender imbalance.

Effort must be rewarded and resilience praised. We must remind our learners that any difficulty and struggle only reflect on the complexity of the subject, NOT on the ability of the person. Constant reassurance and a reward system are needed to ensure that everyone is equally equipped to succeed in Maths.

4. Accept that you have biases so that you can act against them

The fact we are working to stop data biases being ingrained in the young people of today means we must recognise it is inevitably ingrained within ourselves as teachers and educators.

Take time to pause and recognise what you personally struggle to overcome within yourself, so you're better prepared to question your own decisions and actions.

5. Recognise the scope of the problem

Try as much as you can to mitigate against data bias and point out issues when they arise but recognise that this is not something that can be easily solved.

Data bias is global, nuanced, pervasive, and persistent. Minor changes can easily feel meaningless and hollow when held up against the world at large but they might mean the world to the students in your classroom.

It may be chipping away at an iceberg, but when every chip is a child who feels like they belong where they previously didn't, it's worth it.

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1. Speak up and challenge assumptions A lot of harm can be done by someone who is not malicious but simply isn't thinking about the impact of their words or the data that is presented. And a lot of good is done by someone simply speaking up to point it out.

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Katherine Johnson loved to count. "I counted everything. I counted the steps to the road, the steps up to church, the number of dishes and silverware I washed ... anything that could be counted, I did." And so it began for this young girl from West Virginia in the US . Born in 1918 in White Sulphur Springs, W.Va., Johnson's love for mathematics was inherent, an inclination she had from birth. At a young age, she was ready and anxious to go to school. She can vividly remember watching her older siblings go to school and wishing so much that she could go with them. The opportunity to attend school finally did come. Johnson so excelled that she began her studies in the second grade, then moved into advanced classes. By age 10, Johnson was in high school.

In school, one teacher stood out to Johnson. Miss Turner taught geometry, and Johnson couldn't wait to take her class. The teacher was a great encourager to the students and a strong mentor to many of them. Johnson did so well in her classes that she graduated early from high school, and at age 15 she entered West Virginia State College. She had two years before having to declare a major, so Johnson wavered between English, French and mathematics. One of her professors at West Virginia State College helped Johnson with her choice. She told Johnson, "If you don't show up for my class, I will come and find you." And so it was, through part threat and part joke, Johnson steered her way into what was already her first love: mathematics.

At West Virginia State College, Johnson became immersed in academia and the mathematics program. She loved being surrounded by smart people, she said, and knew all of the professors and students on campus. One of her professors, the renowned Dr. William W. Schiefflin Claytor, recognized the bright and inquisitive mind that Johnson had. "You'd make a great research mathematician, " he told her. Then professor Claytor did something else. He told Johnson that he would help her become one. Johnson said, "Many professors tell you that you'd be good at this or that, but they don't always help you with that career path. Professor Claytor made sure I was prepared to be a research mathematician." He saw that Johnson took all of the mathematics classes listed in the catalog that were needed to pursue her life's passion, and even went so far as to create a class in analytic geometry of space just for her. At age18, Johnson graduated summa cum laude with Bachelor of Science degrees in mathematics and French. Johnson recalls of her professor, "Claytor was a young professor himself, and he would walk into the room, put his hand in his pocket, and take some chalk out, and continue yesterday's lesson. But sometimes I could see that others in the class did not understand what he was teaching. So I would ask questions to help them. He'd tell me that I should know the answer, and I finally had to tell him that I did know the answer, but the other students did not. I could tell."

Johnson ended up teaching after college; at that time, teaching was the only option for her in her community. She left teaching to marry and start her family. When her husband fell ill in 1952, she began to teach again. And then one day, at a family function in the 1950s, a relative mentioned to Johnson that the National Advisory Committee for Aeronautics, the predecessor to NASA, was hiring. They were specifically looking for African-American females to work as "computers" in what was then their Guidance and Navigation Department. In the 1950s, pools of women at NACA did calculations that the engineers needed worked or verified. Johnson immediately applied for the job, but the agency already had filled its quota for the year. By the time the next year rolled around, Johnson had applied again and found herself with two contracts on her table. One was a contract to teach, and one was to work for NACA. Remembering what professor Claytor had always told her about becoming a research mathematician, she took the job at NACA.

Johnson began working for NACA in 1953. Her work in the agency was a day-to-day progression. She started as one of the women who worked on problems assigned from the engineers in what was then the Guidance and Control Branch. As Johnson worked on the problems, she would ask questions. She didn't want to just do the work - she wanted to know the "hows" and the "whys" and then the "why nots." None of the other women had ever asked questions before, but by asking questions, Johnson began to stand out. She was told that women didn't participate in the briefings or attend meetings; she asked if there were a law against it. The answer, of course, was no, and so Johnson began to attend briefings. NACA was just beginning its work on space. Space itself may be perceived as a series of plane surfaces, and as Johnson became known for her training in geometry, she began to work with the team more and more. Eventually, she became known as a leader, and the men increasingly relied on her. She remembers quite clearly her experience at the time. "The women did what they were told to do," she explained. "They didn't ask questions or take the task any further. I asked questions; I wanted to know why. They got used to me asking questions and being the only woman there." It was this inquisitive nature that made her a valuable resource to the team and the only woman at the time to ever be pulled from the computing pool to work on other programs.

In 1957, Katherine provided some of the maths for the 1958 document Notes on Space Technology, a compendium of a series of 1958 lectures given by engineers in the Flight Research Division and the Pilotless Aircraft Research Division (PARD). Engineers from those groups formed the core of the Space Task Group, the NACA's first official foray into space travel, and Katherine, who had worked with many of them since coming to Langley, "came along with the program" as the NACA became NASA later that year.

In 1960, she and engineer Ted Skopinski coauthored Determination of Azimuth Angle at Burnout for Placing a Satellite Over a Selected Earth Position, a report laying out the equations describing an orbital spaceflight in which the landing position of the spacecraft is specified. It was the first time a woman in the Flight Research Division had received credit as an author of a research report.

Then in 1962, President John F. Kennedy charged the country to send a man to the moon. Johnson became part of the team, and she began to work on calculating the trajectory for America's first space trip with Alan Shepherd's 1961 mission, an early step toward a moon landing. She went on to do the calculations for the first actual moon landing in 1969.

Johnson worked at the agency until 1986, when she retired after 33 years of service. During her tenure at NASA, Johnson received many prestigious awards. Among them were the NASA Lunar Orbiter Award and three NASA Special Achievement Awards. She was named Mathematician of the Year in 1997 by the National Technical Association. In addition to these NASA awards, Johnson has been honoured with an honorary Doctor of Law degree from the State University of New York and honorary Doctor of Science degrees from Capitol College in Maryland and Old Dominion University in Virginia. At the degree ceremony in New York, Johnson discovered at the last minute that not only was she an honouree, but she was also the keynote speaker. With her esteemed career to draw upon, Johnson rose to the occasion and spoke with the audience. She referred to it as a "chat" with the graduates. A 45-year-old relative of a graduate told Johnson after the ceremony that, because of her "chat," she was returning to school the following fall to complete her degree.

Today, Johnson is enjoying her retirement. She likes to travel, play bridge, watch sports and spend time with her family. She has participated in many panels and conferences, including the NASA Trailblazers and Legends STEM Conference in Cape Canaveral, Fla., in 2010. She often speaks to students about her own extraordinary career and encourages all of them to pursue STEM careers. Johnson tells them, "We will always have STEM with us. Some things will drop out of the public eye and will go away, but there will always be science, engineering and technology. And there will always, always be mathematics. Everything is physics and math." Johnson has served on different panels and has heard from students who told her they decided to go into a STEM career after listening to her talk. She keeps in touch with NASA employees and will call on them, when needed, to help drive home the importance of STEM to audiences. At one recent event in North Carolina, Johnson mentioned to the students that she could get an astronaut to talk to them. The school hooked up her phone to the PA system, and they were able to have a conversation with astronaut Leland Melvin. In 2011, Johnson was honoured at the dedication of the Katherine G. Johnson Science Technology Institute at Alpha Academy in Fayetteville, N.C.

When asked if she still counts things, Johnson said, "Oh, yes. And things have to be parallel. I see a picture right now that's not parallel, so I'm going to go straighten it. Things must be in order." Geometry is still an important part of who she is, a fact evident as she began to talk about her father. She recalled him as "the tallest, straightest man in the area." She loved her father a great deal, and he was a big influence in her life, teaching her many things. One lesson she has carried everywhere, taught to her own children, and let be a guide throughout her education, career and every other aspect of her life.

Katherine Johnson celebrated her 100th birthday on Aug. 26, 2018.

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Katherine Johnson loved to count. "I counted everything. I counted the steps to the road, the steps up to church, the number of dishes and silverware I

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In 2018, I watched the excellent Marvel film Black Panther, which has taken over a billion dollars at the box office worldwide! The film had a number of themes, including the question 'what if an African country, named Wakanda, lead the world in technology?' The film offered a cinematic picture of this, with an important emphasis on STEM subjects.

Now, one of the most interesting characters in the film is Shuri. Wikipedia describes her character like this:
Shuri, the princess of Wakanda, designs new technology for the country. She is 'an innovative spirit with an innovative mind' who 'wants to take Wakanda to a new place'. Shuri is a good role model for young black girls as well as being one of the smartest persons in the world.
One of the technologies that Shuri designed was Black Panther's suit. The suit is special because it can distribute the kinetic energy from an impact. The idea is that the kinetic energy will not be focused on one area, but move to another part of the suit where it can be absorbed. Okay, nice Hollywood science fiction stuff... or is it? Watching this scene took me back to my postgraduate days, when I was doing an MSc in Industrial Mathematical Modelling at Loughborough University. Here, I did a dissertation titled 'Impact on an adhesive joint'.

The mathematical model I had to create was to show how a material behaved in aggressive impact circumstances. My placement was at a start-up firm run by three PhD students in the material science department. They had this crazy idea that if you created a design in a composite material, it would be able to take an impact by distributing the kinetic energy away from the impact zone to places where it can be absorbed. Sound familiar? But, materials under impact react like this simulation:

Materials under high impact tend to buckle and fracture. The future of Black Panther's suit is not looking good. Nevertheless, one of the key skills of a mathematical modeller is to listen to the subject-matter experts, and turn their knowledge into a mathematical form. Then, by solving the mathematical form, we will see if we have created a viable solution that can be applied to a real world problem.

Material scientists are very familiar with a stress-strain graph like this:

The three PhD students argued that under impact conditions, if we could minimise the stress and strain in the material it would remain in its elastic region. In the elastic region, once the stress has been removed the material will return to its original size and shape and Black Panther's suit would survive. The material scientists wanted to know if there was a mathematical way to test out their idea; that there is a material design that can deflect an impact. By using the wave equation with an impact forcing term, it was argued that the kinetic energy is distributed away from the impact zone fast enough such that the material remains in the elastic region.

The figure below shows the schematic from my original MSc dissertation. An impact force strikes the material, and creates a wave. Some of the wave is transmitted through the interface between the two parts of the material, and some of it is reflected.

The impact force was represented by a triangular pulse forcing term of duration $a$. That is, the strength of the impact increases up to a maximum value then fades back to zero, with total duration $a$.

Now another interesting thing is that, in the film, Black Panther's suit, which was made up of nanomites, started to disintegrate when it was hit by a sufficient force, like the horn of a rhino or a point-blank grenade launcher. The sharp energy forced the nanomites apart. Okay, not so Hollywood, but in my dissertation research it was identified that the interface is where the composite material was most likely to fail.

However, if the interface remained in place, the adhesive layer serves to absorb the energy by bouncing the reflected wave between its boundaries. So theoretically, a Black Panther suit might actually work. Well done Shuri!

This was one of my first mathematical models, and the dissertation did receive a distinction. At the time I thought the idea of an impact-absorbing composite material was crazy, but after watching Black Panther it motivated me to have another look at my MSc thesis. Though I don't think I can invent the Black Panther suit, I hope that this film inspires children all over the world to greater and better things in STEM.

#WAKANDAFOREVER
This post was adapted from an original article on my blog.

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The mathematics of Black Panther - Chalkdust

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In 2018, I watched the excellent Marvel film Black Panther, which has taken over a billion dollars at the box office worldwide! The film had a number of themes, including the question 'what if an African country, named Wakanda, lead the world in technology?'

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How 'Black Panther' influences STEM diversity beyond Wakanda

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How 'Black Panther' influences STEM diversity beyond Wakanda

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by Zomona Brown | Feb 21, 2018 | Diversity, STEM diversity | no comments.

Wakanda, the homeland of Black Panther/ T'Challa may be a fictional place, but it's "power" is very real.

Yes, to some 'Black Panther' is just another superhero movie. But it is way more than that. The preliminary notes to its importance are the obvious:

While these are amazing feats, there is so much more that scratches the surface - Black people creating, innovating and advancing STEM (Science, Technology, Engineering and Math), a sight unseen on the big screen before 'Hidden Figures'. But once again, 'Black Panther' is different.

While 'Hidden Figures' is documented in a real Jim Crow world, the "fictional" Wakanda in 'Black Panther' is a wealthy, uncolonized, technically and scientifically advanced world filled with kings and queens. This sets the depiction of greatness as opposed to the visual presence of Africa that we see in the media today. The representation of a successful black world is something that elderly, baby boomers, Gen Y & Z and millennials have not seen...until now.

Within all the themes that lie in 'Black Panther', one remains constant: Black women in technology. Shuri (Letitia Wright), T'Challa/Black Panther's sister, spearheads the all the innovation in the Wakanda world. From creating Black Panther's suit to defeating villains with virtual cars, Shuri is the only person that that can provide T'Challa the tools he needs to be the Black Panther superhero that he is. Think of her as the Q in Bond movies; in case you are one of the few people in the US who has not seen the movie yet.

This representation matters. As hard as it is to find a woman or man of color acting as a member of the STEM field in the media, it's even harder to find in real life. According to the United States Department of Commerce, 7 out of 10 STEM positions in the US are held by white, non-Latinx. Black, Native American, and Latinx (any race) workers are half as likely to hold STEM positions compared to the overall workforce, while Asian workers are nearly three times as likely.

STEM diversity is an issue that TCGi CEO Avis Yates Rivers is extremely passionate about. Not only has she received a White House Champion of Chang e in STEM awarded by President Barack Obama, her book, ' Necessary Inclusion: Embracing the Changing Faces of Technology ', addresses the need for more women and people of color in the STEM field and proposes solutions.

(Serena Williams Surprises Black Girls Code Participants at 'Black Panther' screening)

There are many STEM based organizations that are currently involved with minority communities. We support everyone to get involved, as STEM is the way of the future! Here are some organizations below:

NCWIT Black Girls Code Black Girls Rock! Technolochicas
Hidden Genius Project

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HOW 'BLACK PANTHER' INFLUENCES STEM DIVERSITY BEYOND WAKANDA - Technology Concepts Group International

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Wakanda, the homeland of Black Panther/T'Challa may be a fictional place, but it's "power" is very real. Yes, to some 'Black Panther' is just another superhero movie. But it is way more than that. The preliminary notes to its importance are the obvious: First Major Superhero of Color Predominantly Black Cast Highest Budget for a [...]

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As part of Black Mathematician Month, we spoke to the Bristol University professor about access schemes and the importance of mentors.

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Florence Adepoju – Beauty – Cosmetic Scientist | Equals TV - Full Edit

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When we talk about LGBTQ+ mathematicians, there might not be many people that come to mind. There's Alan Turing, of course, and there are theories that maybe Leonardo Da Vinci was gay or that perhaps Florence Nightingale was a lesbian, but no one really knows for sure.

Mathematics, and STEM in general, have typically been thought of as heterosexual, male-dominated fields and there hasn't always been a visible community for gender and sexual minorities. But this is changing! With organisations such as Spectra, 500 Queer Scientists, LGBTQ+ STEM, and Pride in STEM, we are not only seeing a more visible and active community, but the next generation of academics in STEM has a supportive network of role models and mentors.

To celebrate LGBTQ+ History Month, we've put together a list of some modern day LGBTQ+ mathematicians we think you should know about.

Ron Buckmire
Originally from Grenada but now living and working in California, Ron Buckmire was named LGBT Educator of the Year in 2011 by Out to Innovate. He is the Associate Dean for Curricular Affairs and Professor of Mathematics at Occidental College in Los Angeles. His research interests lie in applied mathematics, computational aerodynamics and numerical analysis. Ron is passionate about widening participation for historically excluded groups, particularly the LGBTQ+ community and racial and ethnic minorities. Coming out in the late 80s, he became involved in LGBTQ+ activism in the early 90s and was one of the founding members of Spectra, the association for LGBTQ+ Mathematicians.

Autumn Kent
Autumn Kent specialises in topology and geometry and is currently a Professor of Mathematics at the University of Wisconsin. She has published over 20 papers and received a number of awards throughout her career, including a Simons Fellowship in 2019. A pansexual trans woman, Autumn is active in the LGBTQ+ community and in 2019 she organised the LG&TBQ+ conference to foster collaboration between LGBTQ+ mathematicians working in geometry, topology, and dynamical systems.

Emily Riehl
Emily Riehl is an Associate Professor of Mathematics at John Hopkins University in Baltimore, specialising in category theory and homotopy theory. In 2020, she was awarded JHU President's Frontier Award, a $250,000 award that support individuals "who are breaking new ground and poised to become leaders in their field". A graduate from Harvard, Cambridge and the University of Chicago, Emily has held teaching positions in the US, Europe and Australia. As one of the founding board members of Spectra, Emily is an active member of the LGBTQ+ community in mathematics.

Anthony Bonato
Anthony is a Professor in Mathematics at Ryerson University in Toronto. His research is in Graph Theory and Network Science and he has authored over 130 publications. He also writes a great blog called The Intrepid Mathematician, which has over 280,000 views and covers everything from Maths to pop culture to queer issues. Anthony is the chair for LGBTQ+ Math Day at Ryserson and sits on the EDI board for Banff International Research Station.

Tyler Kelly
Tyler is a reader in geometry at the University of Birmingham and is the recipient of a UKRI Future Leaders Fellowship. Their research focuses on algebraic geometry, specialising in its interactions in mathematical physics and mirror symmetry. Tyler works hard to promote the inclusion of the LGBTQ+ community in Mathematics and STEM and is a member the LGBTQ+ STEM's Steering Committee. Tyler is currently working onscheduled for July 2022, which will be thefirst conference to celebrate and promote research advances of LGBTQ+ mathematicians specialising in algebraic geometry, arithmetic geometry, commutative algebra, and number theory.

Juliette Bruce
Juliette is an early career mathematician whose research interests lie in algebraic geometry, commutative algebra, and arithmetic geometry. After getting her PhD from University of Wisconsin, Madison, Juliette moved to UC, Berkley to work as a NSF Postdoctoral Fellow. She is passionate about inclusivity, diversity and justice in the mathematics community, and has organised and spoken at a number of conferences and seminars dedicated to celebrating the contributions of gender and sexual minorities to the field of mathematics. She is currently working on Spec

Below we have included a list of useful resources for LGBTQ+ mathematicians and their allies.
http://lgbtmath.org/ https://500queerscientists.com/ https://lgbtstem.wordpress.com/ https://prideinstem.org/ https://www.noglstp.org/outtoinnovate/
And some general LGBTQ+ resources for Queen Mary staff and students.
LGBTQ+ Society QMOUT, QMUL's Lesbian, Gay, Bisexual and Transgender (LGBTQ+) staff network group LGBTQA+ History Month 2022

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6 inspirational LGBTQ+ mathematicians we think you should know about

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When we talk about LGBTQ+ mathematicians, there might not be many people that come to mind. There's Alan Turing, of course, and there are theories that maybe Leonardo Da Vinci was gay or that perhaps Florence Nightingale was a lesbian, but no one really knows for sure.

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Diversity in the curriculum toolkit - DiverseED

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Diversity in the curriculum toolkit - DiverseED

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Diversity in the Curriculum Toolkit | Diverse Educators

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'Diversity in the curriculum' usually applies to the concept of a curriculum that is inclusive of the experiences of those with protected characteristics

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Resource - AMSP

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Resource - AMSP

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Resource - AMSP

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Increasing participation in Core Maths, AS/A level Mathematics and Further Mathematics

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Resource - AMSP

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Resource - AMSP

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Perhaps the oldest mathematical artifact in existence, the Ishango Bone (above), was unearthed in 1950 in the then Belgian colony of the Congo (now the Democratic Republic of Congo). It was discovered by the Belgian anthropologist Jean de Heinzelin de Braucourt (1920-1998) and named after the region in which it was found. The bone, probably a fibula of a baboon, large cat, or other large mammal, has been dated to the Upper Paleolithic Period of human history, approximately 20,000-25,000 years ago. It is 10 cm long and bears an articulated, organized series of notches readily identifying it, to many observers, as a tally stick. However, its original purpose remains a subject of debate. The Ishango Bone is now housed at the Museum of Natural Sciences in Brussels, Belgium, with whose cooperation the image above was obtained.

For more information and images, see:

Museum of Natural Sciences, Royal Belgian Institute of Natural Sciences: Ishango Bone (2007-2009) or Ishango Bone (2001)

Smithsonian National Museum of Natural History: Ishango Bone

Alison S. Brooks, David M. Helgren, et al, "Dating and Context of Three Middle Stone Age Sites with Bone Points in the Upper Semliki Valley, Zaire," Science, New Series, vol. 268, no. 5210 (Apr. 28, 1995), pp. 548-553.
Index to Mathematical Treasures

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Perhaps the oldest mathematical artifact in existence, the Ishango Bone (above), was unearthed in 1950 in the then Belgian colony of the Congo (now the Democratic Republic of Congo). It was discovered by the Belgian anthropologist Jean de Heinzelin de Braucourt (1920-1998) and named after the region in which it was found.

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10 Questions To Begin Decolonizing Mathematics In Your Classroom

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10 Questions To Begin Decolonizing Mathematics In Your Classroom

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10 Questions To Begin Decolonizing Mathematics In Your Classroom

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In order to rehumanize mathematics, which has been an emergent theme recently in mathematics, we must first start to tell the correct…

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Gender parity and queer awareness needed in mathematics

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Gender parity and queer awareness needed in mathematics

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Gender parity and queer awareness needed in mathematics

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Mathematics departments in Canada have a poor record on equity, diversity and inclusivity says a gay mathematics professor. Here he speaks about the hopeful changes he sees coming.

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March is Women's History Month, a time to honor, celebrate, and acknowledge generations of women of all backgrounds, their struggle to achieve equality in society, and their incredible contributions in fields including civil rights, athletics, art, politics, mathematics, and every other facet of life imaginable.

Women's History Month is a special opportunity for students to learn about famous women mathematicians throughout history and their contributions to the field of study-ranging from discovering fundamental theorems to rocket science and putting astronauts into orbit.

Why is it important for students to learn about female mathematicians? Because the History of Mathematics is overly focused on the contributions of a small group of male mathematicians like Issac Newton, Albert Einstein, and Euclid, while often ignoring the diverse contributions of women and minorities

If you are looking to promote gender equality and celebrate Women's History Month with your kids this month and beyond, sharing the contributions of famous female mathematicians is a great way to inspire students, teach them about the history of mathematics, and remind them that math is accessible to everyone.

In Febraury, we celebrated Black History Month by sharing 11 Famous African American Mathematicians You Should Know About. And now that the calendar has turned to March, we are sharing 11 Famous Women Mathematicians and Their Amazing Contributions in honor of Women's History Month.

Some ideas for celebrating these famous women is by focusing on a different individual each day, assigning individuals for a famous woman mathematician research assignment, or by sharing this article on your classroom's bulletin board.

Note that you can certainly share this list of famous female mathematicians with your kids during any time of the school year. However, March, being Women's History Month, is an extra special opportunity to provide your students with a deeper understanding of these women and their contributions within the context of history and the fight for gender equality.

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11 Famous Women Mathematicians and Their Incredible Contributions! - Mashup Math

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Celebrate Women's History by learning about these famous women mathematicians and scientists and their amazing contributions. A full biography of each famous female mathematician is included!

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11 Famous African American Mathematicians

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11 Famous African American Mathematicians

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February is Black History Month, a time to celebrate and honor the generations of African American men and women and their struggle to achieve citizenship and equal rights in American society.

Black History Month provides a special opportunity for students to explore and learn more about famous African American mathematicians and their contributions to our culture and modern institutions.

When it comes to STEM classes, one of the best ways you can celebrate Black History Month in your classroom is by sharing the contributions of famous African American Mathematicians that you and your kids may not be familiar with (but definitely should!).

Why? Because most lists of famous mathematicians in history focus heavily on men like Isaac Newton, while the stories of so many other mathematicians from various walks of life go untold.

Some ideas for celebrating these famous African American Mathematicians include focusing on a different individual each day, assigning individuals for a famous African American Mathematician research assignment, or by sharing this article on your classroom's bulletin board.

And while you can share this list with your students and admire the accomplishments of these incredible mathematicians any time of the year, the month of February provides a focused opportunity to provide your students with deeper historical context and make connections between the past, the present, and the future.

11 Famous African American Mathematicians

1.) Benjamin Banneker (1731-1806)

Banneker, a mostly self-educated man in mathematics and astronomy, is best known for building America's first clock-a wooden device that struck hourly. Using his rich understanding of trigonometry and astronomy, he accurately predicted a solar eclipse in 1789.

Banneker was also a passionate civil rights advocate. In 1791, he famously penned a letter to then secretary of state Thomas Jefferson asking for his help in improving conditions for Africans living in the United States, which highly impressed Jefferson and convinced many that blacks were intellectually equal to whites.

He died at age 74 on October 9, 1806.

2.) Fern Hunt (1948-Present)

Fern Hunt is best known for her work in applied mathematics and mathematical biology. Throughout her great career, she has been involved with biomathematics, patterns in genetic variation, and chaos theory.

She currently works as an educator and presenter with the aim of encouraging women and minority students to pursue graduate degrees in mathematics and other STEM fields.

Hunt is a strong support of student choice, following one's passion, and surrounding yourself with a strong support system, which are all factors that she credits to her own personal success in mathematics.

3.) Mark Dean (1957-Present)

Dean is a famous computer scientist and engineer, credited with assisting in the development of several breakthrough computer technologies for IBM.

He invented the first gigahertz computer chip and co-invented the Industry Standard Architecture System for computing, which allows for common plug-ins such as personal printers and modems.

Dean was inducted into the National Inventor's Hall of Fame in 1997

4.) Elbert Frank Cox (1895-1969)

In 1925, Elbert Frank Cox was the first black man to earn a Ph.D in mathematics not only in the United States, but in the entire world.

After receiving this honor, Cox became a professor of mathematics at Howard University in Washington D.C. and eventually became chairman of the Mathematics Department in 1957. He would hold this role with great esteem until his retirement in 1965.

In 1975, the Howard University Mathematics Department established the Elbert F. Cox Scholarship Fund for undergraduate math majors to encourage young black students to pursue degrees in mathematics.

5.) Katherine Johnson (1918-2020)

Katherine Johnson was the main character of the critically acclaimed film Hidden Figures. Her contributions in the field of orbital mechanics, alongside fellow female African American mathematicians Dorothy Vaughan and Mary Jackson, were critical to the United States' success in putting astronaut John Glenn into orbit in 1962.

During her 35-year career at NASA, she held a reputation for being a master of complex manual calculations and helped pioneer the use of computer programming for performing complex calculations.

She was awarded the Presidential Medal of Freedom by President Barack Obama in 2015.

Video: Katherine Johnson: The Girl Who Loved to Count by NASA

6.) Valerie Thomas (1943-Present)

Valerie Thomas is a scientist and inventor best known for inventing the Illusion Transmitter in 1980. This technology was the first of its kind in the field of 3D-imagery and is the basis for modern 3D-televisions, video games, and movies.

From 1964 -1995, Thomas worked for NASA, developing real-time computer data systems and managing the team that developed the Landsat technology that supported the first satellite to transmit images from outer space.

Today, she continues to serve as a mentor for youth through the Science Mathematics Aerospace Research and Technology and National Technical Association.

"I always look forward to getting my Mashup Math newsletter email every week. I love the free activities!" -Christina R., 5th Grade Math Teacher, Dallas, TX Do YOU want free math resources, lesson activities, and puzzles and games for grades 1-8 in your inbox every week? Join our mailing list and start getting tons of free stuff!

7.) Lonnie Johnson (1949-Present)

Lonnie Johnson is a famous inventor, mathematician, and engineer who holds over 120 patents. He served as an engineer for the U.S. Air Force, where he worked on developing the stealth bomber, and later for NASA's Jet Propulsion program.

He is best known for inventing the Super Soaker water gun while working at the U.S. Air Force, a product which has grossed nearly $1 Billion since 1990. He also invented a "toy projectile gun," which eventually became the Nerf Gun.

8.) John Urschel (1991-Present)

John Urschel excelled in both mathematics and playing football at Penn State University and earned bachelor's and master's degrees in mathematics. In 2014, Urschel was drafted from Penn State into the NFL by the Baltimore Ravens, where he played his entire NFL career before retiring at age 26 to further pursue mathematics.

He is currently working towards his Ph.D. in mathematics from MIT where some of his mathematical fields of interest include graph theory, numerical linear algebra, and machine learning.

9.) Euphemia Lofton Haynes (1890-1980)

Euphemia Lofton Haynes is famous for being the first African American woman to earn a Ph.D. in mathematics, which she achieved from the Catholic University of America in 1943.

After earning her Ph.D., she embarked on a 47-year career of advocacy for students of color, improving schools, and overcoming the harmful effects of racial segregation. During this time, she was also a math professor at the District of Columbia Teachers College, where she managed the Division of Mathematics and Business Education department.

Haynes died at the age of 89 on July 25, 1980.

10.) Annie Easley (1933-2011)
Annie Easley is famous NASA computer and rocket scientist and mathematician who contributed to several space programs, inspired others through her participation in numerous outreach programs, and broke down barriers for both women and African Americans in STEM. Most notably, she was a leading member of the team that developed the breakthrough Centaur rocket, which opened the door for the launch of many of NASA's most important missions.

Easley was known for being a "human computer" and always fought to do her best in the face of adversity. In a 2001 interview she shared that "I just have my own attitude. I'm out here to get the job done, and I knew I had the ability to do it, and that's where my focus was."

11.) Mae Carol Jemison (1956-Present)

Mae Carol Jemison is a mathematician, engineer, physician, and astronaut. In 1992, she became the first African American woman to visit space when she went into orbit aboard NASA's Space Shuttle Endeavour.

Jemison also spent time as a general practice physician and in the Peace Corps before joining NASA's astronaut program.

She continues to be involved in several initiatives and recently served on the council for an initiative called " Science Matters" which aims to encourage young children to understand and pursue careers in STEM.

Famous Black Mathematicians Video and Free PDF

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11 Famous African American Mathematicians You Should Know About - Mashup Math

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Mathematics in Africa
Mathematics in the Americas
Mathematics in China
Mathematics in India
Mathematics in Japan
Mathematics in the Middle East
Mathematics in Oceania

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Decolonising the Curriculum through the History of Mathematics

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Decolonising the Curriculum through the History of Mathematics

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Decolonising the Curriculum through the History of Mathematics

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Decolonising the Curriculum through the History of Mathematics

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As part of Black History Month, June Barrow-Green and Brigitte Stenhouse gave a presentation in which they explored how historical sources can be used to decolonise the mathematics curriculum.

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Celebrating Women Role Models in Science, Technology, and Math

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Celebrating Women Role Models in Science, Technology, and Math

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Free Posters Celebrating Women Role Models in Science, Technology, and Math | A Mighty Girl

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Fair living wages including gender wage gap

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Created by Frances Harper

(adapted from CPM task - "The Big Race")

Algebra lesson - appropriate for 7th-8th grade or Algebra I

Mathematics topics: Connections between graphs, tables, and equations by identifying slope and y-intercept in each representation; Point of intersection

Social justice issues: Recognize that the current minimum wage does not allow an individual to afford the fair market price of a moderate 2 bedroom rental unit, assuming that housing should require about 30% of income; also highlights the gender wage gap

Download lesson plan

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linear equations and graphs + fair living wages

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Students work as a group to figure out the hourly wage necessary for a family in Chicago to afford housing using linear equations and graphs. They look at real data about hourly wages and the cost for renting each month. The goal is to use mathematics to decide whether or not you think six families...

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How modern mathematics emerged from a lost Islamic library

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How modern mathematics emerged from a lost Islamic library

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How modern mathematics emerged from a lost Islamic library

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Centuries ago, a prestigious Islamic library brought Arabic numerals to the world. Though the library long since disappeared, its mathematical revolution changed our world.

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Vicky Neale: 7 things you need to know about prime numbers

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Vicky Neale: 7 things you need to know about prime numbers

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Vicky N

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Millennium Mathematics Project - maths.org

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'7 Things You Need to Know About Prime Numbers' - Dr Vicky Neale

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Lucy Rycroft-Smith and Darren Macey unpick the legacy of some of the most ubiquitous names in statistics

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Statisticians have the wholly unreasonable reputation of being the reprobates of the mathematical world: the drummers of the band, the viola players of the orchestra, the Chalkdusts of the scientific magazines. Much of this contempt seems to stem from statistics not being seen as a 'real' discipline within mathematics; statistics is so often the butt of mathematical jokes that it has spawned an entire joke genre of the 'lies, damned lies and statistics' type (for example, 'A statistician is a person who draws a mathematically precise line from an unwarranted assumption to a foregone conclusion', or 'Did you know that 87.166253% of all statistics claim a precision of results that is not justified by the method employed?'). A power struggle as to whether statistics is a field of mathematics, an entirely separate field, or a science (or something else) has been ongoing for decades and shows no sign of abating. It may come as little surprise, dear readers, that many of of us with a love of mathematics seem to also have a love for forensic debate about boundaries, classification, and domains.

But there is another, far more urgent power struggle that bridges the worlds of statistics and mathematics, too: the question of how we respond to the emergence of unpleasant historical details surrounding the lives of those we have previously considered heroes of our discipline. This is not a hypothetical question. Books and research papers from the last few decades have increasingly recognised that the founding fathers of statistics built the field on a dark and sinister foundation. The fact is: statistics as we know it is so intertwined with eugenics as to be inseparable. It could be said, in fact, that every time we speak the names of the men who gave their names to the statistical tests we might see as so innocuous, we are breathing life back into the memory of racists, obsessed with measuring 'difference from the norm', who built a 'scientific' reputation by metaphorically climbing on top of the broken bodies and reputations of entire peoples that they considered less than human.

Don't believe us?

In Germany, an investigation into the workings of the organisation now called the Max Planck Society, before 1948 known as the Kaiser Wilhelm society, suggested scientists and academics there played a key role in legitimising Hitler's ideology of racial purity. Not only did they contribute to the theoretical basis for Nazi politics, but in some cases they helped to set up concentration camps, gas chambers and conducted unimaginably cruel experiments on incarcerated humans. But this is Nazi Germany, you might say; nothing so evil could happen on British soil. (But you would be wrong.)

Angela Saini writes in their bestseller Superior: "The truth: that it is perfectly possible for prominent scientists to be racists, to murder, to abuse both people and knowledge-doesn't sit easily with the way we like to think about scientific research. We imagine that it's above politics, that it is a noble, rational, and objective endeavour, untainted by feelings or prejudice...[but] science sits at the mercy of the personal political beliefs of those carrying it out."

Some of the scientists most influential to the thinking inherent in the Holocaust worked at UCL in Gower Street, London, and give their names to statistical measures we continue to teach in classrooms and lecture halls up and down the UK.

The genesis of eugenics

Francis Galton was the cousin of Charles Darwin. He coined the term 'eugenics' ('good breeding') to describe the process of "improving stock, which is by no means confined to questions of judicious mating, but which, especially in the case of man, takes cognisance of all influences that tend in however remote a degree to give the more suitable races or strains of blood a better chance of prevailing speedily over the less suitable than they otherwise would have had." As you can see, the word 'improving' is used here in a way that depends on a hierarchical framework of 'suitability' of 'races'-in other words, white supremacy.

The US National Human Genome Research Institute defines eugenics as "the scientifically inaccurate theory that humans can be improved through selective breeding of populations. Eugenicists believed in a prejudiced and incorrect understanding of Mendelian genetics that claimed abstract human qualities (eg intelligence and social behaviours) were inherited in a simple fashion." While the scientific rebuttal alone should be sufficient to discredit eugenics as a discipline, there is a further ethical issue here: the underlying model of white supremacy. White supremacy is the idea that whiteness constitutes superiority and confers rightful power; but this concept of 'whiteness' is not just about skin colour. "Racism is based on the concept of whiteness-a powerful fiction enforced by power and violence. Whiteness is a constantly shifting boundary separating those who are entitled to have certain privileges from those whose exploitation and vulnerability to violence is justified by their not being white." (Kivel, Uprooting Racism, 2011.)

This means that various peoples throughout history have been marginalised and oppressed as part of white supremacy, even those who might in other contexts be considered 'white'. This is because race is not a scientific concept-it is a social construct, and remarkably hard to pin down. It is, as Jenée Desmond-Harris says, neither "permanent, scientific, objective, logical, consistent, [nor] able to stand up to scrutiny." So why does race loom so large in our society and our histories? Desmond-Harris also argues, along with many others, that the concept of race itself was invented by scientists to justify slavery.

Intersections with racism

Galton's father made money from the slave trade, which he inherited and which allowed him to work as an explorer and a scientist. Galton travelled to Africa, describing the people who lived there as 'savage races'. In our office, we have a Galton board: a lovely object, a dynamic normal graph, showing as you rotate it hundreds of bouncing, jostling ball bearings which eventually settle into a beautiful bell curve (see opposite). But both the words 'Galton' and 'bell curve' (the name of a 1994 book by Herrnstein and Murray explicitly connecting race and intelligence) don't feel right in our mouths. They taste horrible. Would we be better off ignorant, not worrying about these issues and just enjoying the mathematical ideas here? Perhaps, as individuals. But, as Aris Winger, assistant professor of mathematics at Georgia Gwinnett College, says, 'How white people interrogate their own racial identity is a requirement for any meaningful racial, professional or personal progress in maths. The future of the discipline is tied to whiteness, how we understand whiteness, and white supremacy.' Because we, as white people, have the luxury of not having to deal with racism in our professional lives-no one has ever assumed we are not mathematicians because of the colour of our skin. So we also have a painting in our office-the one you see displayed here-which helps remind us of some of these issues.

Karl Pearson is often described as the father of statistics; working as recently as the 1900s, in many ways he founded the discipline. He was the first professor of national eugenics at UCL. He was racist and antisemitic, writing that Jewish immigrants were a 'parasitic race' and that those who were not white were 'inferior'.

His namesake, the Pearson product-moment correlation coefficient (or PMCC) is something we have both taught, and used in our research-it is the first port of call for those investigating linear correlation, or asking 'might these two numbery things have a relationship?' This concept has many, many names-but the 'Pearson' part seems remarkably sticky. It's almost as if someone is doing a PR job behind the scenes. It is remarkable how insistent people can be on proper credit and value for scientific work when said work involves white men, and how these principles seem to vanish without trace when minoritised people are involved. It's almost as if we have a skewed (stats joke) idea of what constitutes authority and status based on kyriarchy (the social system that keeps all intersecting oppressions in place).

Implications and complexities

In a world where toppling statues of slavers has become international news, it is worth considering what a scientist must do in order that we stop hallowing them in curricula, in assessment, in the simple act of speaking their name. Is a mathematical breakthrough ever neutral? Are there figures in our discipline for whom their ideologies are so distasteful that presenting their work to students, or in our research without proper contextualisation is in fact unethical, particularly in a subject like statistics where the two seem so very intertwined?

It is impossible, of course, and indeed undesirable to remove mathematicians' ideas from the wider picture; but at what point does it become reasonable to strike their names from their most famous methods? Remove their faces from our textbooks? Or at least include historical notes alongside their mathematical legacies telling the dark stories of their distasteful ideologies which, in some cases, allowed their work to continue? If Galton's work was created on the backs of murdered and dehumanised slaves, should all those using it be aware of this fact? If Pearson's legacy was built on the idea that humans can be used as data without regard for their humanity, should that history lesson not stand next to his statistical methods, as a warning for all future statistics students not to make the same mistakes?

Indeed, these are not just hypothetical questions: in 2020, UCL announced that it was renaming buildings which had been named after both Pearson and Galton because of their connection with eugenics, and they have said this is one of many measures they plan to take in this vein. Of course the venerable institution with which we are associated, Cambridge University, does not escape critique; for example, there have been calls for the removal of the stained glass window in Gonville \& Caius College which commemorates statistician Ronald Fisher, who was similarly enamoured of eugenics. In 2020, Daniela Witten, professor of statistics at the University of Washington, ran a successful campaign for the Committee of Presidents of Statistical Societies to remove Fisher's name from the Fisher award and lectureship (now called the COPSS distinguished achievement award and lectureship), saying "If Quaker Oats can retire Aunt Jemima, then statistics can retire the Fisher lecture. I hope that this tiny little gesture of a name change can be just the beginning of a much larger conversation in our field about how we can work to create a statistical community that is committed to our shared goals of diversity, equity, and inclusion."

The felling of one stone monument to the slaver Edward Colston signalled for many the beginning of the important work of questioning which other figures' more metaphorical pedestalling might similarly be challenged. The Colston statue was toppled by protesters in June 2020 in the context of longstanding campaigns by Bristol residents to remove references to Colston throughout the city. They advocated to remove the statue, or even just to have a small plaque added, pointing out that the 84,000 African lives he helped sell into slavery were worth considering as a counterpoint to his much celebrated 'philanthropy'. However, the veneration of history's white men has a certain inertia, and multiple attempts to draft a plaque that satisfied everybody failed spectacularly, with one local councillor claiming that vandalising it would be an appropriate course of action should it ever appear. The plaque was never mounted.

Possible futures

But what does this mean for mathematics, and in particular, statistics? Cases like that of Colston's statue and others such as Cecil Rhodes' statue outside Oriel College, Oxford, seem to demonstrate that long dead men can still count on the support of powerful institutions, individuals, and even government ministers whenever people have the temerity to threaten to peel a layer or two of varnish from their legacies and expose the rotten cores beneath. As mathematicians, maths communicators, and maths educators seeking to do the same within our discipline, perhaps we should expect the same forces to swing into action to defend the 'good' names of Galton, Pearson, Fisher and their ilk. How then do we develop the collective power of our community to call for change? What should our demands even be? And how do we counter the arguments that it was ever thus; that these people were born in different times; and that we should not seek to 'vandalise' our history by tit-for-tat retaliatory silences, overwrites, and heavy-handed edits?

We do not have the solution. But we are mathematicians (and statisticians, and we do not differentiate between the two): to us this is familiar ground. We humbly suggest the same approach we take to our beloved discipline of mathematics when we are stuck: listen, learn, think, spend time with the problem; most of all, seek out greater expertise.

One of the places we often turn to when navigating these issues is the excellent podcast Mathematically Uncensored. In the seemingly complicated and sometimes fraught discussion about inclusive practices in mathematics education, Aris Winger and Pamela E Harris, associate professor of mathematics at Williams College in Massachusetts and featured in Chalkdust issue 10, challenge us with a very simple question: what does it actually cost us to be inclusive? If we are only prepared to take action that is comfortable, that does not materially cost us in some way, then are we really making any meaningful challenge to the status quo? Ultimately, no matter how horrified we might be at reading some of these biographies, if we cannot even 'give up' somehow the names of some long dead men, or take action to prioritise the inclusion of those whose contributions have been historically marginalised or completely erased-do we really care at all?

The artwork below is a painting by Lucy Rycroft-Smith. The name of the piece is Galton Under Fire (2021), so named in reference to the frustration the artist feels at the apparently fireproof reputation of some mathematicians, Francis Galton included. It poses the question: how unethical do someone's actions have to be before we choose to take action against them as a mathematical community, and what should that action be?

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Statisticians have the wholly unreasonable reputation of being the reprobates of the mathematical world: the drummers of the band, the viola players of the orchestra, the Chalkdusts of the scientific magazines.

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Books that Lucy recommends on or around mathematics, with a particular focus on authors of colour and female authors

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Books that I recommend on or around mathematics, with a particular focus on authors of colour and female authors.

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Decolonizing Educational Design for School Mathematics

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Decolonizing Educational Design for School Mathematics

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COLIN FOSTER, LEONARDO BARICHELLO, BUSTANG BUSTANG, ROVINCER NAJJUMA, İPEK SARALAR-ARAS

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Foster et al. Decolonizing educational design for school mathematics

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Japanese temple geometry

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JAPANESE TEMPLE GEOMETRY

TSUKIOKA YOSHITOSHI

(1839-1892)

GOSHO GOROZŌ BATTLING A SHADOW

Japanese Temple Geometry Board

Japanese temple geometry refers to the practice of carving verbal geometrical problems, theorems, and shapes into wooden tablets and presenting them as an offering at a shrine or temple. A Japanese math teacher, Hidetoshi Fukagawa, discovered the tablets in a library book while looking for interesting ways to teach his students. His interest in such a little known topic led to his quest throughout Japan to find and study the tablets. Around 900 of these tablets have survived the years and exist today.

In Japanese, this practice was called "sangaku" which translates as "mathematical tablet. " The setting for the sangaku craze was the Edo period. During the Edo period Japan was ruled by a military dictatorship called the Tokugawa Shogunate (1603-1868). With very few exceptions, the Tokugawa government's strict isolationist polices cut Japan off from contact with other countries between 1638 and 1854. During its isolation, Japan experienced a cultural renaissance. One of the many arts that flourished during the period was a unique type of Japanese mathematics that developed separately from western mathematics . Sangaku, therefore, are examples of a distinctly Japanese style of geometry.

Sangaku tablets were presented as offerings at Shinto Shrines and Buddhist temples, the two dominant religions of Japan. Buddhist temples were especially known as being centers of learning. But most of the tablets discovered came from Shinto shrines . This is most likely because the sangaku tradition followed in the steps of a practice common to Shinto of presenting decorated wooden tablets as offerings at religious sites . This practice served as a way for worshippers with few resources to make offerings to the Shinto gods. For example, if a worshipper wanted to offer a horse to the gods, he might choose to draw a picture of a horse on a piece of wood as an acceptable substitute .

The sangaku were placed under the roofs of these religious sites. Sometimes they were even fixed to the ceiling of a Shrine. The reason they were placed in the rafters or on the ceiling is unknown. Similarly, while the sangaku may resemble older ritual offerings on wooden tablets, their exact purpose remains a topic of speculation. Rothman claims the sangaku were either paying homage to Shinto spirits or they were a display of mathematical ability, challenging other worshippers to solve the math problems.

Each tablet was inscribed with a language called Kanbun. Kanbun is an old form of Japanese that consists of Chinese characters and grammar, accompanied by annotations to allow a Japanese reader to understand it . It was used for intellectual documents of the Edo period. The mathematical designs of the tablets varied, but many contained 3-dimensional objects and some had mathematical proofs. Most contained plane geometry, but some tablets displayed a type of mathematical calculation akin to algebra .

Sangaku were created by people from many classes and occupations, from samurai to merchants to farmers . The level of mathematical sophistication displayed on a tablet often corresponded to the education level of the person behind it. The designs on many tablets were done with the intent of attracting the mind of a non-mathematician; from their facedown placement on the ceiling, the tablets' colorful designs were meant to inspire the intellect and issue challenges to other aspiring mathematicians.

It is speculated that Japan's elite samurai class were primarily responsible for the creation and continuation of the sangaku. Following the ascent of the strict Tokugawa regime in 1603, the samurai were no longer needed in the military, and many took government posts that afforded them quite a bit more leisure time. Mathematics seems to have been a common pursuit of the samurai, and it was taught in the schools the samurai operated. While the practice of creating sangaku was widespread and supported by such elites, it would not last once Western mathematics arrived to Japan. Much like the role of the samurai class in Japanese society, the creation and popularity of sangaku faded away after the fall of the Tokugawa shogunate in 1868.
Michael Gott
First posted 11/14/14

PRIMARY RESOURCES

Temple Geometry Boards

See below for enlargements.

SECONDARY SOURCES
Hidetoshi Fukagawa and Tony Rothman, Sacred Mathematics: Japanese Temple Geometry (Princeton: Princeton University Press, 2008). Hidetoshi Fukagawa and Kazunori Horibe, "Sangaku--Japanese Mathematics and Art Dennis Normile, "'Amateur' Proofs Blend Religion and Scholarship in Ancient Japan," Science, New Series, 307/5716 (March 18, 2005): 1715-1716. Tony Rothman and Hidetoshi Fukagawa "Japanese Temple Geometry," Scientific American (May 1998): 85-91. Jill Vincent & Claire Vincent, "Japanese Temple Geometry," Australian Senior Mathematics Journal, 18/1 (2004): 8-20. in the 18th,19th and 20th Centuries," Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014): 111-118.

The Rabbit Hole

CURATED SEARCH

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JAPANESE TEMPLE GEOMETRY TSUKIOKA YOSHITOSHI (1839-1892) GOSHO GOROZŌ BATTLING A SHADOW Japanese Temple Geometry Board Japanese temple geometry refers to the practice of carving verbal geometrical problems, theorems, and shapes into wooden tablets and presenting them as an offering at a shrine or temple.

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Anne-Marie Imafidon passed her computing A-Level at the age of 11 and by 16, was accepted to the University of Oxford to study Maths and Computer Science.

She's used to the 'child prodigy' label that's followed her throughout her career, but that doesn't mean she's had an easy ride.

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In conversation in front of an audience at the 2023 Cheltenham Science Festival, she tells Jim Al-Khalili about her quest for equality and diversity across the scientific community - and explains why she thinks everyone has the potential to be a 'child prodigy', given the right opportunity...

Produced by Lucy Taylor.

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BBC Radio 4 - The Life Scientific, Anne-Marie Imafidon on fighting for diversity and equality in science

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Jim Al-Khalili speaks to Anne-Marie Imafidon about championing girls in STEM.

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Thanks to Ada Lovelace Day sponsor F Hinds for their help developing this poster.
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Thanks to ARM for providing design support for this poster.
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Thanks to Stack Overflow for data and design assistance.
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Feedback

If you have any feedback on these posters, or the rest of the education pack, or if you would like to provide suggestions for improvements, please contact Suw Charman-Anderson at suw@findingada.com.

About this pack

This free education pack comprises of:

All resources have been produced by Ada Lovelace Day, and are available to download for free from their website, findingada.com. These files will be continually updated so please do check the website for the latest versions.

Acknowledgements

We are very grateful to our sponsors ARM, and to Professor Averil Macdonald, the WISE Campaign, the Science Council, Practical Action, AGCAS and Prospects for their support and assistance in the preparation of this education pack.

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We have five posters for you to download and print, or buy from our web shop. All downloads are 5100 x 7000 pixels in size, but most printers will allow you to select your paper size and scale the image down in order to print correctly.

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Resource - AMSP

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Resource - AMSP

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Draft guidelines for assessment and resource design in mathematics education

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Draft guidelines for assessment and resource design in mathematics education

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https://www.cambridgemaths.org/Images/651814-draft-guidelines-for-assessment-and-resource-design-in-mathematics-education.pdf

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Teaching maths for social justice PADLET

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Teaching maths for social justice PADLET

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Dr Nira Chamberlin

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Posters – LGBT Inclusive Education | National Platform

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Access posters providing educational role models or awareness-raising opportunities that can be used within your classroom or across your school community.