Starter!

• Picture a mathematician.
• Without a calculator, list or compute the first 10 square numbers.
• Without a calculator, list or compute some cubes.
• Where was The Elements written?
• Which is larger, Europe or South America or Greenland?
• What groups of people have privilege in Scotland or in your classroom?
• What implicit biases do students enter/leave your classroom with?
• Hold onto these images, numbers, ideas for later!

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From LGBT+ History Month to Anti-Racist Education: ��Telling more of the story of Mathematics

Daniel Wolf-Root (He/Him)

Hutchesons’ Grammar School

wolfrootd@hutchesons.org

SMC Conference

16 May, 2026

Starter!

• Picture a mathematician.
• Without a calculator, list or compute the first 10 square numbers.
• Without a calculator, list or compute some cubes.
• Where was The Elements written?
• Which is larger, Europe or South America or Greenland?
• What groups of people have privilege in Scotland or in your classroom?
• What implicit biases do students enter/leave your classroom with?
• Hold onto these images, numbers, ideas for later!

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Anti-Racism, LGBTI-Inclusive Education, LGBTQ+History Month, Black History Month, … �in Maths Class?!

• Education Scotland’s Anti-Racist Principles
• Scotland’s LGBTI-Inclusive Education
• In Mathematics?
• How can we teach all this?
• Meaningful and connected to our curriculum?
• Not tokenistic?
• Solution: Tell more of the story of Mathematics throughout the year!

Education Scotland’s Anti-Racist Principles

https://education.gov.scot/resources/anti-racist-education/principles-for-an-anti-racist-curriculum/

Education Scotland and Anti-Racism

Breaking the mould: Principles for an anti-racist curriculum | Resources | Education Scotland

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For students:

Will be critical thinking global citizens that challenge discrimination and prejudice through an understanding and awareness of the behaviours, practices and processes that create injustice in the world.

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For educators:

Will foster an anti-racist culture where racism can be discussed openly, honestly and with humility, and with a willingness to take risks and make mistakes while remaining accountable for their actions.

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Education Scotland and Anti-Racism

Breaking the mould: Principles for an anti-racist curriculum | Resources | Education Scotland

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For students:

Will be critical thinking global citizens that challenge discrimination and prejudice through an understanding and awareness of the behaviours, practices and processes that create injustice in the world.

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For educators:

Will foster an anti-racist culture where racism can be discussed openly, honestly and with humility, and with a willingness to take risks and make mistakes while remaining accountable for their actions.

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Three Maths topics for teaching Anti-Racist Principles

• The Mathematics of Mapping
• Conditional statements and their converses
• Euclid’s Elements

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Scotland’s LGBT-Inclusive Education

https://www.gov.scot/publications/guidance-lgbt-inclusive-education/pages/1/

Seven LGBT-Inclusive Learning Themes

• Understanding the Equality Act, UNCRC, Human Rights
• Identifying prejudice, discrimination, bullying
• Recognising and challenging gender stereotypes
• Diverse families, including parents and siblings
• Celebrating diversity and difference
• History of LGBT equality movements
• LGBT past and present figures and role models

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https://www.gov.scot/publications/guidance-lgbt-inclusive-education/pages/3/

Seven LGBT-Inclusive Learning Themes

• Understanding the Equality Act, UNCRC, Human Rights
• Identifying prejudice, discrimination, bullying
• Recognising and challenging gender stereotypes
• Diverse families, including parents and siblings
• Celebrating diversity and difference
• History of LGBT equality movements
• LGBT past and present figures and role models

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https://www.gov.scot/publications/guidance-lgbt-inclusive-education/pages/3/

Picture a mathematician

Another reason to tell more of the story of Mathematics

Dan Reynolds, Unwrapped; Google search 29/09/21; R3 Heinemann textbook

Identity has always mattered in Mathematics teaching

• Pythagoras
• Pascal
• The “Greek” mathematicians

What is the expected value of a roll of a fair die?

Pascal’s Wager

Pascal’s Wager

Pascal’s Wager

Pascal’s Wager

Pascal’s Wager

Pascal’s Wager

Devlin, K. (2008) The Unfinished Game: Pascal, Fermat and the Seventeenth-Century letter that made the world modern. Basic Books.

How would the conclusion of Pascal’s Wager be different if Pascal had a different religious background?

Who else discovered Pascal’s Triangle?

Images from Wikipedia Commons and Creative Commons

Yang Hui’s Triangle

China

13^th Century

Meru Prastaara

India

Al-Karaji’s Triangle

Arabic

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Euclid’s Elements was written in Alexandria

• Do you think differently about “Western Civilisation” and “The Greeks” if you know that the most famous mathematics text was produced in Africa?
• Do you think differently about contributions to mathematics if you know that Euclid and Hypatia and Eratosthenes worked in modern day Egypt?

Teaching Euclid’s Elements to understand the world—literally!

Accessible visual version of The Elements

https://www.c82.net/euclid/

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Oliver Byrne’s 1847 visual edition of the first six books

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(Byrne, 1847/2013)

A map of the World

Wikipedia Commons

Maps: Airline Routes

www.flightroutes.com

The Parallel Postulate

• In Euclid:

If the sum of the interior angles α and β is less than two right angles, the two straight lines, produced indefinitely, meet on that side.

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(Byrne, 1847/2013)

The Parallel Postulate

• In Euclid:

If the sum of the interior angles α and β is less than two right angles, the two straight lines, produced indefinitely, meet on that side.

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Playfair's Equivalent:

Given a line and a point not on it, one line parallel to the given line can be drawn through the point.

A Proof of the Triangle Sum Theorem

(Byrne, 1847/2013)

Does this proof work on the Sphere?

Wikipedia Commons

The Parallel Postulate Fails on a Sphere and this implies the Triangle Sum Theorem is False on a Sphere!

https://sites.pitt.edu/~jdnorton

The angle sum of a Spherical Triangle

The Gauss-Bonnet Theorem:

The area of a triangle on a sphere is proportional to the amount by which its angle sum is exceeds 180 degrees (defect). 

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Corollary: All triangles on a sphere have angle sum greater than 180 degrees.

There are no ideal maps

Why no map preserves both geodesics and angles

Rouleaux Triangle, Wikipedia Commons​

A nice application of Trigonometry

A nice application of Trigonometry

• Which region is bigger?
• How distorted are regions on a map?

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A nice application of Trigonometry

(Maor, 1998)

A nice application of Trigonometry:

(https://gis.stackexchange.com/questions/110730/mercator-scale-factor-is-changed-along-the-meridians-as-a-function-of-latitude)

Which is bigger?

Did you know that South America is larger than Europe?

geospatialworld.net

Did you know that Greenland is smaller than Europe, South America, North America, Africa, …?

Conclusion about Mapping

• Maps of the Earth cannot simultaneously preserve geodesics and angles.
• The maps students see and learn from are thus distorted representations of their world and can lead to implicit bias about relative size and centrality.
• This distortion can be quantified using trigonometry!
• The mathematics we already teach our students can help them understand their world!
• If we engage more with mathematics and its reasoning and less with preparing students for exercises on exams, then we can increase understanding and undo or address implicit biases.

References for Euclid and Mapping

Byrne, O. (1847/2013) The First Six Books of the Elements of Euclid. Facsimile Edition. Taschen.(Available online

 https://www.c82.net/euclid/)

Euclid, Heath, T.L. (trans). (1952). The Thirteen Books of Euclid’s Elements. Cambridge University Press.

Feeman, T.G. (2002) Portraits of the Earth: A Mathematician Looks at Maps. AMS.

Kitagawa, K. and Revell, T. (2023) The Secret Lives of Numbers: A Global History of Mathematics and Its Unsung Trailblazers. Viking.

Maor, E. (1998). Trigonometric Delights. Princeton University Press.

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Further Reading: Pascal’s Wager

Devlin, K. (2008) The Unfinished Game: Pascal, Fermat and the Seventeenth-Century letter that made the world modern. Basic Books.

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Ward, Sophie. (2020) Love and Other Thought Experiments. Corsair.

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Queer Mathematics*

• Taxicab Numbers
• The Partition Problem
• Category Theory

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*by which I mean mathematics(!) discovered by queer-identifying people or those we now think of as queer through 21^st century eyes.

Taxicab Numbers: G. H. Hardy and Srinivasa Ramanujan (1919)

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Image sources: Trinity College Library (Hardy), Encyclopedia Britannica (Ramanujan)

The Partition Problem

• Hardy and Ramanujan (1917)
• Amanda Folsom and Ken Ono (2011)

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Image Sources: Amherst College (Folsom), University of Virginia (Ono)

Category Theory: Emily Riehl and Eugenia Cheng

Image source: The women taking math to the next dimension. Lauren J. Young, 2017. Available at The Women Taking Math To The Next Dimension

Emmy Noether and Category Theory

Image source: Talitha Williams. Power in Numbers: The Rebel Women of Mathematics.

What is Category Theory?

• The mathematics of structures and their relations.
• “The mathematics of mathematics” (Cheng)

What are the factors of 30?

A Category Theory Diagram for the factors of 30

Cheng, E. (2020) X + Y: A Mathematician's Manifesto For Rethinking Gender. Profile Books.

Category Theory, Privilege and Intersectionality

Cheng, E. (2020) X + Y: A Mathematician's Manifesto For Rethinking Gender. Profile Books.

A Category Theory, and Privilege and Intersectionality

Cheng, E. (2020) X + Y: A Mathematician's Manifesto For Rethinking Gender. Profile Books.

Another example: Are the shapes along the bottom row analogous?

Shape

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Polygons

Circles

Triangles

Quadrilaterals

Kites

Equilateral Quadrilaterals (Rhombi!)

Square

Is the experience of people along the bottom row analogous?

Cheng (2018).

people

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oppressed people

straight white men

women

minorities

gay people

visible minorities

Black people

Is the experience of those along the bottom row analogous?

Shape

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Polygons

Circles

Triangles

Quadrilaterals

Kites

Equilateral Quadrilaterals (Rhombi!)

Square

Emily Riehl’s diagrams

Image Source: Johns Hopkins Magazine: The Mathematical Mind of Emily Riehl

Spectra: The Association for LGBTQ+ Mathematicians

Emily Riehl and Amanda Folsom are members of Spectra

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QED: Queer, Equality and Diversity Network

https://sites.google.com/view/qednetwork/home

Further Reading: Category Theory

Taxicab Numbers: G. H. Hardy and Srinivasa Ramanujan (1919)

• Time to get those squares and cubes out!

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Image sources: Trinity College Library (Hardy), Encyclopedia Britannica (Ramanujan)

1729 is the smallest number that can be expressed as the sum of 2 cubes in 2 different ways!

1729

1729

What is the smallest number that can be expressed as the sum of two squares in two ways?

What is the smallest number that can be expressed as the sum of two squares in two ways?

That’s cheating!

I meant the smallest number that can be expressed as the sum of two distinct squares in two ways!

That’s cheating!

More on Taxicab Numbers

G. H. Hardy, from A Mathematician’s Apology

• “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because thay are made with ideas.”

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• “Very little of mathematics is useful practically, and that little is comparatively dull.”

Hardy, G.H. (1940). A Mathematician’s Apology. Cambridge.

The Partition Problem

• Hardy and Ramanujan (1917)
• Amanda Folsom and Ken Ono (2011)

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Image Sources: Amherst College (Folsom), University of Virginia (Ono)

The Partition Problem

In how many ways can you express a natural number (1, 2, 3, …, n, …) as the sum of other natural numbers?

The Partition Problem

In how many ways can you express a natural number (1, 2, 3, …, n, …) as the sum of other natural numbers?

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For example:

3 = 3, 2+1, 1+1+1

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There are 3 partitions of the number 3.

The Partition Problem

In how many ways can you express a natural number (1, 2, 3, …, n, …) as the sum of other natural numbers?

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For example:

3 = 3, 2+1, 1+1+1

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There are 3 partitions of the number 3.

How many partitions are there of the numbers 1, 2, 4, 5, 6, … n?

The partition problem: Is there a pattern?

1 = 1 p(1)=1

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2 = 2, 2 = 1+1 p(2)=2

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3 = 3, 3 = 2+1, 3 = 1+1+1 p(3) = 3

p(4)

4 = 4

= 3+1

= 2+2

= 1+1+2

= 1+1+1+1.

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Thus p(4) = 5.

Partitions of n for n=1 to n=6

Image Source: Wikipedia

p(1) = 1

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p(2) = 2

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p(3) = 3

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p(4) = 5

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p(5) = 7

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p(6) = 11

p(n) = ?

Partitions

p(7) = ?

Partitions

p(7) = 15!

See Online Encyclopedia of Integer Sequences A000041: A000041 - OEIS

Is there a formula for p(n)?

How many ways are there to partition 2026?

Is there a formula for p(n)?

Albers, D.J. et. al. (2015) The G. H. Hardy Reader. MAA/Cambridge (pp 295-299).

How good is the asymptotic formula?

York, A. (2020) Improving the Accuracy of the Hardy-Ramanujan Asymptotic Partition Formula, available at Improving the Accuracy of the Hardy-Ramanujan Asymptotic Partition Formula

But what about an explicit, finite formula?

• Folsom and Ono (2011)

Amanda Folsom, Zachary Kent and Ken Ono (2011). l-adic properties of the partition function. American Institute of Mathematics.

But what about an explicit, finite formula?

• Folsom and Ono (2011)

There is such a formula, but it’s complicated:

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Amanda Folsom, Zachary Kent and Ken Ono (2011). l-adic properties of the partition function. American Institute of Mathematics.

But it contains a new insight about partition numbers

Amanda Folsom, Zachary Kent and Ken Ono (2011). l-adic properties of the partition function. American Institute of Mathematics.

This means there is a fractal-like structure to the Partition Numbers (in terms of divisibility properties)!

Conclusion about Partition Function

• Hardy and Ramanujan found asymptotic and infinite formulas for the nth partition number.

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Conclusion about Partition Function

• Hardy and Ramanujan found asymptotic and infinite formulas for the nth partition number.
• Folsom and Ono found a finite formula for the nth partition number that uncovers fractal-like properties for divisibility of partition numbers using modular arithmetic.

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Further Reading: Partition and Taxicab Numbers

Online articles:

Online Encyclopedia of Integer Sequences A000041: A000041 – OEIS

Mathematics' Nearly Century-Old Partitions Enigma Spawns Fractals Solution | Scientific American

York, A. (2020) Improving the Accuracy of the Hardy-Ramanujan Asymptotic Partition Formula, available at Improving the Accuracy of the Hardy-Ramanujan Asymptotic Partition Formula

Amanda Folsom, Zachary Kent and Ken Ono (2011). l-adic properties of the partition function. American Institute of Mathematics.

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Books:

Hardy, G.H. (1940). A Mathematician’s Apology. Cambridge.

Albers, D.J. et. al. (2015) The G. H. Hardy Reader. MAA/Cambridge.

Hardy, G.H. and Wright, E.M. (1938/1990) An Introduction to the Theory of Numbers. Fifth Edition. Oxford.

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So: What does a mathematician look like?

• Picture a mathematician now.
• Where was The Elements written?
• Which is larger, Europe or South America or Greenland?
• What implicit biases do students enter/leave your classroom with?
• How can we change those biases through learning Mathematics?

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Hardy, Ramanujan, Folsom, Ono, Riehl, Cheng, Noether

Adding people into the story doesn’t subtract anyone

Pair up mathematicians, connect discoveries and continuities, make the story of mathematics more reflective of the world of mathematics and mathematicians.

Thank you and Acknowledgements

Thank you!

Thanks also:

To Ms. Mélina Valdelièvre, who asked

To Mrs. Lucy Anderson, who invited

To Dr. Heather Cochrane and Mr. Jack MacLeod, who taught me about Alexandria

To Mr. Anuj Choudhary, for helpful feedback

wolfrootd@hutchesons.org

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Curriculum References

Scottish Government LGBT Inclusive Education Guidance: LGBT inclusive education: guidance - gov.scot

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Education Scotland Principles for an Anti-Racist Curriculum: Principles for an anti-racist curriculum | Anti-racist education | Resources | Education Scotland

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Further reading

Photo credit: Sophie MacDonald