Education minister under fire after introduction deleted from Ontario’s new Grade 9 math curriculum

Appears that the substantive aspects related to systemic barriers and inclusion remain while the ideologic reference to “non-Eurocentric mathematical knowledges” has been dropped. Much of math has non-European roots (numerals, algebra etc):

Premier Doug Ford’s government has deleted a preamble to Ontario’s new Grade 9 curriculum that said math “has been used to normalize racism and marginalization of non-Eurocentric mathematical knowledges.”

While the updated syllabus remains unchanged, introductory language for teachers was quietly edited earlier this week.

The modernized curriculum was introduced June 9 as the first step of ending the streaming of students so early in high school. That practice that has been tied to poor outcomes for Black and Indigenous youth.

As first reported by the Toronto Sun on Saturday, the Progressive Conservatives initially approved of a curriculum introduction that said “a decolonial, anti-racist approach to mathematics education makes visible its historical roots and social constructions.”

“Mathematics is often positioned as an objective and pure discipline,” said the preamble to the curriculum, which was made public last month.

“The Ontario Grade 9 mathematics curriculum emphasizes the need to recognize and challenge systems of power and privilege, both inside and outside the classroom, in order to eliminate systemic barriers and to serve students belonging to groups that have been historically disadvantaged and underserved in mathematics education.”

But within the past few days, that entire 124-word paragraph entitled “An equitable mathematics curriculum recognizes that mathematics can be subjective” was deleted.

Sources told the Star that “while the section referenced is not in the core curriculum taught to students, we revised it to ensure there is no confusion when it comes to making sure our students are being taught fundamental math concepts.”

“The curriculum did not change. It continues to educate on cultural understandings of math, of the history of these concepts, and attempts to advance that lens throughout the curriculum. What changed was language in the preamble only,” an official said.

In a statement Wednesday, Education Minister Stephen Lecce’s office said the Tories “ended streaming in the Grade 9 math curriculum — a system that disproportionately affected Black, racialized and Indigenous students — along with launching new and specialized supports to ensure these students graduate, enter post-secondary education and get good-paying jobs.”

But the new president of the Ontario Secondary School Teachers’ Federation, which supports destreaming, said the education minister “needs to take responsibility” for the episode.

“It’s time for a mea culpa. If you make a mistake, you have to own up to it,” said Karen Littlewood, who took over the union’s presidency on June 22.

Littlewood said “it seems to be very reactionary” for Lecce to amend the language in the wake of media coverage.

“The preamble really sets the stage for the changes to the curriculum and why it was necessary,” she said.

Despite the editing, the lesson plan still addresses inequities in society.

The revised curriculum emphasizes “there are groups of students (for example, Indigenous students, Black students, students experiencing homelessness, students living in poverty, students with LGBTQ+ identities, and students with special education needs and disabilities) who continue to experience systemic barriers to accessing high-level instruction in and support with learning mathematics.”

“Systemic barriers, such as racism, implicit bias and other forms of discrimination, can result in inequitable academic and life outcomes, such as low confidence in one’s ability to learn mathematics, reduced rates of credit completion, and leaving the secondary school system prior to earning a diploma,” it states.

“Achieving equitable outcomes in mathematics for all students requires educators to be aware of and identify these barriers, as well as the ways in which they can overlap and intersect, which can compound their effect on student well-being, student success, and students’ experiences in the classroom and in the school,” it continues.

“Educators must not only know about these barriers, they must work actively and with urgency to address and remove them.”

Still, the New Democrats expressed concern about the deletion.

“The Grade 9 math program was changed specifically because Ontario had to finally recognize that the existing system treated Black, Indigenous and racialized students inequitably,” NDP MPPs Laura Mae Lindo (Kitchener Centre) and Marit Stiles (Davenport) said in a joint statement.

“It’s pretty clear we need more of an equity and anti-racism lens in schools, not less.”

Source: Education minister under fire after introduction deleted from Ontario’s new Grade 9 math curriculum

How modern mathematics emerged from a lost Islamic library

Of interest, to math and history lovers:

The House of Wisdom sounds a bit like make believe: no trace remains of this ancient library, destroyed in the 13th Century, so we cannot be sure exactly where it was located or what it looked like.

But this prestigious academy was in fact a major intellectual powerhouse in Baghdad during the Islamic Golden Age, and the birthplace of mathematical concepts as transformative as the common zero and our modern-day “Arabic” numerals.

Founded as a private collection for caliph Harun Al-Rashid in the late 8th Century then converted to a public academy some 30 years later, the House of Wisdom appears to have pulled scientists from all over the world towards Baghdad, drawn as they were by the city’s vibrant intellectual curiosity and freedom of expression (Muslim, Jewish and Christian scholars were all allowed to study there).

An archive as formidable in size as the present-day British Library in London or the Bibliothèque Nationale of Paris, the House of Wisdom eventually became an unrivalled centre for the study of humanities and sciences, including mathematics, astronomy, medicine, chemistry, geography, philosophy, literature and the arts – as well as some more dubious subjects such as alchemy and astrology.

To conjure this great monument thus requires a leap of imagination (think the Citadel in Westeros, or the library at Hogwarts), but one thing is certain: the academy ushered in a cultural Renaissance that would entirely alter the course of mathematics.

The House of Wisdom was destroyed in the Mongol Siege of Baghdad in 1258 (according to legend, so many manuscripts were tossed into the River Tigris that its waters turned black from ink), but the discoveries made there introduced a powerful, abstract mathematical language that would later be adopted by the Islamic empire, Europe, and ultimately, the entire world.

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“What should matter to us is not the precise details of where or when the House of Wisdom was created,” says Jim Al-Khalili, a professor of physics at the University of Surrey. “Far more interesting is the history of the scientific ideas themselves, and how they developed as a result of it.”

Tracing the House of Wisdom’s mathematical legacy involves a bit of time travel back to the future, as it were. For hundreds of years until the ebb of the Italian Renaissance, one name was synonymous with mathematics in Europe: Leonardo da Pisa, known posthumously as Fibonacci. Born in Pisa in 1170, the Italian mathematician received his primary instruction in Bugia, a trading enclave located on the Barbary coast of Africa (coastal North Africa). In his early 20s, Fibonacci traveled to the Middle East, captivated by ideas that had come west from India through Persia. When he returned to Italy, Fibonacci published Liber Abbaci, one of the first Western works to describe the Hindu-Arabic numeric system.

When Liber Abbaci first appeared in 1202, Hindu-Arabic numerals were known to only a few intellectuals; European tradesmen and scholars were still clinging to Roman numerals, which made multiplication and division extremely cumbersome (try multiplying MXCI by LVII!). Fibonacci’s book demonstrated numerals’ use in arithmetic operations – techniques which could be applied to practical problems like profit margin, money changing, weight conversion, barter and interest.

“Those who wish to know the art of calculating, its subtleties and ingenuities, must know computing with hand figures,” Fibonacci wrote in the first chapter of his encyclopedic work, referring to the digits that children now learn in school. “With these nine figures and the sign 0, called zephyr, any number whatsoever is written.” Suddenly, mathematics was available to all in a useable form.

Fibonacci’s great genius was not just his creativity as a mathematician, however, but his keen understanding of the advantages known to Muslim scientists for centuries: their calculating formulas, their decimal place system, their algebra. In fact, Liber Abbaci relied almost exclusively on the algorithms of 9th-Century Arab mathematician Al-Khwarizmi. His revolutionary treatise presented, for the first time, a systematic way of solving quadratic equations. Because of his discoveries in the field, Al-Khwarizmi is often referred to as the father of algebra – a word we owe to him, from the Arabic al-jabr, “the restoring of broken parts”—and in 821 he was appointed astronomer and head librarian of the House of Wisdom.

Scholars and translators at the library also took great pains to ensure that their work was accessible to the reading public

Al-Khwarizmi’s treatise introduced the Muslim world to the decimal number system,” explains Al-Khalili. “Others, such as Leonardo da Pisa, helped transmit it across Europe.”

Fibonacci’s transformative influence on modern maths was thus a legacy owed in great part to Al-Khwarizmi. And so two men separated by nearly four centuries were connected by an ancient library: the most celebrated mathematician of the Middle Ages stood on the shoulder of another pioneering thinker, one whose breakthroughs were made at an iconic institution of the Islamic Golden Age.

Perhaps because so little is known about the House of Wisdom, historians are occasionally tempted to exaggerate its scope and purpose, giving it an mythic status somewhat at odds with the scant historical records left to us. “Some argue that the House of Wisdom was nothing like as grand as it became in the eyes of many,” says Al-Khalili. “But its association with men such as Al-Khwarizmi, with his work in mathematics, astronomy and geography, is for me strong evidence that the House of Wisdom was closer to a true academy, not just a repository of translated books.”

Scholars and translators at the library also took great pains to ensure that their work was accessible to the reading public. “The House of Wisdom is fundamentally important, as it’s through translations there – Arabic scholars who translated Greek ideas into the vernacular – that we formed the bedrock of our mathematical understanding” says June Barrow-Green, professor of history of mathematics at the Open University in the UK. The palace library was as much a window into numerical ideas from the past as it was a site of scientific innovation.

Long before our current decimal system, the binary number system that programs our computers, before Roman numerals, before the system used by ancient Mesopotamians, humans were using early tally systems to record calculations. While we might find each of these imponderable or antiquated, differing numerical representations can actually teach us something valuable about structure, relationships, and the historical and cultural contexts from which they emerged.

They reinforce the idea of place value and abstraction, helping us to better understand how numbers work. They show that “the Western way wasn’t the only way”, says Barrow-Green. “There is a real value in understanding different numbers systems.”

When an ancient trader wanted to write “two sheep”, for example, she could inscribe in clay a picture of two sheep. But this would be impractical if she wanted to write “20 sheep.” Sign-value notation is a system in which numeric symbols add together signify a value; in this case, drawing two sheep to represent the actual quantity.

A global shift away from Roman numerals underscores a creeping innumeracy in other aspects of life

A vestige of sign-value notation, Roman numerals somehow persisted despite the introduction of Al-Khwarizmi’s system, which relied on the position of digits to represent quantities. Like the towering monuments on which they were inscribed, Roman numerals outlived the empire that gave birth to them – whether by accident, sentiment or purpose, none can say for sure.

This year marks the 850th anniversary of Fibonacci’s birth. It could also be the moment which threatens to undo the journeywork of Roman numerals. In the UK, traditional time-pieces have been replaced with easier-to-read digital clocks in school classrooms, for fear students can no longer tell analogue time properly. In some regions of the world, governments have dropped them from road signs and official documents, while Hollywood has moved away from using Roman numerals in sequel titles. The Superbowl famously ditched them for its 50th game, worried it was confusing fans.

But a global shift away from Roman numerals underscores a creeping innumeracyin other aspects of life. Perhaps more importantly, the disappearance of Roman numerals reveals the politics that govern any wider discussion about mathematics.

The library was home to many groundbreaking texts, such as this book of "ingenious inventions", published in 850 (Credit: Photo12/Universal Images Group/Getty Images)

The library was home to many groundbreaking texts, such as this book of “ingenious inventions”, published in 850 (Credit: Photo12/Universal Images Group/Getty Images)

“The question of whose stories we tell, whose culture we privilege, and which forms of knowledge we immortalise into formal learning are inevitably influenced by our Western colonial heritage” says Lucy Rycroft-Smith, editor and developer at Cambridge Mathematics. A former maths teacher, Rycroft-Smith is now a leading voice in mathematics education, and studies differences across global curricula. While Wales, Scotland and Ireland do not include Roman numerals in their learning objectives, and the US has no standard requirements, England explicitly states that students must be able to read Roman numerals up to 100.

Many of us will find nothing special about the figure MMXX (that’s 2020, if you’re unaware). We may dimly recognise Fibonacci for the famous pattern named after him: a recursive sequence that starts with 1 and is thereafter the sum of the two previous numbers.

The Fibonacci sequence is certainly remarkable, showing up with astonishing frequency in the natural world – in seashells and plant tendrils, in the spirals of sunflower heads, in pine cones, animal horns and the arrangement of leaf buds on a stem, as well as the digital realm (in computer science and sequencing). His patterns often make their way into popular culture, too: in literature, film and visual arts; as a refrain in song lyrics or orchestral scores; even in architecture.

But Leonardo da Pisa’s most enduring mathematical contribution is something rarely taught in schools. That story begins in a palace library nearly a thousand years ago, at a time when most of Western Christendom lay in intellectual darkness. It is a tale that should dismantle our Eurocentric view of mathematics, shine a spotlight on the Islamic world’s scientific achievements and argue for the continued importance of numerical treasures from long ago.

Source: How modern mathematics emerged from a lost Islamic library

Nobel laureate’s discovery revealed the patterns behind breathtaking works of Islamic art

Interesting commentary by Sheema Khan:

This year’s Nobel Prize in physics was awarded to Roger Penrose, Andrea Ghez and Reinhard Genzel for their research on black holes. Dr. Penrose, a mathematician, proved the existence of black holes from Einstein’s theory of relativity; Dr. Ghez and Dr. Genzel spent decades gathering evidence of a black hole in our own galaxy.

Dr. Penrose also discovered “non-periodic tiling” in 1974, known as Penrose tiling. Think of your kitchen floor: It is completely covered by a repeating pattern of tiles. One simple arrangement is a set of identical square tiles, placed side by side. You can do the same with a set of triangles or a set of hexagons. However, if you try it with a set of identical pentagons, a problem arises. The pentagons will not fit snugly next to each other – in contrast to squares, triangles or hexagons. Dr. Penrose was able to formulate a tiling formation, in which a number of basic tile shapes are used to fully cover a flat surface, such that the resulting tiling pattern does not actually repeat. However, if you were to take a floor covered with Penrose tiling, you could rotate it in multiples of 72 degrees, clockwise or counterclockwise, and obtain the same pattern – an example of five-fold symmetry. The foyer of Texas A&M University’s Mitchell Institute is covered with Penrose tiling.

The tiling stood as a unique mathematical breakthrough – until 2005, when Harvard graduate student Peter Lu discovered variations of the same non-periodic tiling patterns on a 17th-century madrassah in Uzbekistan. With his keen mathematical eye, Dr. Lu was able to distinguish between this unique non-periodic tiling, and the equally breathtaking periodic tiling patterns found in Islamic architecture and artwork throughout history. In the latter, simple circles and squares were transformed into stars and overlapping lattices to form intricate symmetric patterns. The 13th century Alhambra Palace in Granada, Spain, provides many beautiful examples of these geometric lattices.

Upon further investigation, Dr. Lu and Paul Steinhardt of Princeton University discovered further examples of non-periodic tiling dating from the 10th to the 15th century, in varied locations such as Iraq, Iran, Turkey, Afghanistan, India and Uzbekistan. They were astounded to find a near-perfect example of Penrose tiling on the façade of the 15th century Darb-I Imam shrine in Iran, created five centuries before Dr. Penrose’s discovery. They also found that a set of five basic tile shapes, called “girih” tiles, were used by craftsmen to create these exquisite patterns. While it is not known exactly how artisans created these patterns on site, the 15th-century Topkapi scroll (housed in the Topkapi Palace in Istanbul) provides a template of 114 different patterns of girih tiles. The patterns crafted by the artisans are not the actual tiles, but outlines thereof, thus giving the impression of an intricate latticework (or “girih,” which means “knot”). These Islamic non-periodic tiling patterns also have five-fold symmetry.

What is unknown is how these Muslim artisans and mathematicians discovered girih tiles and their alignment. These unique patterns are also found in naturally occurring quasicrystals, a form of matter with atom patterns that don’t repeat, like normal crystals.

What is known is that the “Golden Age” of Islam flourished from about the 8th century to the 14th, during which time Muslim scientists made advancements in the fields of algebra, geometry, calculus, chemistry, biology, medicine and astronomy. It began with the Abbasid caliphate, which built Baghdad from scratch as its capital, located strategically along many trade routes. The caliphs put a premium on the pursuit of knowledge. They established the House of Wisdom in Baghdad where scholars of different faiths collaborated. They also undertook a massive effort to translate Greek scholarship into Arabic, which was disseminated widely. Scholars built on this information to forge new advances. The Istanbul Museum of the History of Science and Technology in Islam provides a comprehensive look of that era.

What stands out is an era where faith, science and reason worked in harmony. Unlike the Western approach, where science and faith are deemed irreconcilable, the history of Islam is replete with the opposite.

The first verses revealed in the Koran included the command “Read,” reflections of our humble origins (“created from a clot”) and a reminder that God teaches individuals “what they knew not.” Islam’s holy book contains exhortations to study the natural world as a means to know God and a means of worship. Scientists of other faiths, such as Isaac Newton, Gregor Mendel and Thomas Bayes, have charted a similar path.

Whether it is the intricate pattern on a leaf, the sonar system of bats, or the fabric of the universe – all reflect the signs of a Creator within the Islamic paradigm. The key is that knowledge should lead to humility.

Which brings us back to the girih tiles. However they were discovered, it is no surprise that art and mathematics combined to adorn Islamic houses of worship, given that in Islam, the pursuit of knowledge is in harmony with spirituality.

Source: https://www.theglobeandmail.com/opinion/article-nobel-laureates-discovery-revealed-the-patterns-behind-breathtaking/