Month: September 2020

Alice in Wonderland enthusiasts recently celebrated the story’s anniversary with creative events like playing with puzzles and time — and future Alice exhibits are in the works. The original 1865 children’s book Alice’s Adventures in Wonderland, sprung from a mathematician’s imagination, continues to inspire exploration and fun.

But is a connection between math and creativity captured in schools? Much discussion across the western world from both experts and the public has emphasized the need to revitalize high school mathematics: critics say the experience is boring or not meaningful to most students. Experts concerned with the public interest and decision-making say students need skills in critical thinking, creativity, communication and collaboration.

Mathematicians, philosophers and educators are also concerned with the excitement and energy of creative expression, with invention, with wonder and even with what might be called the romance of learning.

Mathematics has all the attributes of the paragraph above, and so it seems to me that what’s missing from high school math is mathematics itself.

I am now working with colleagues at Queen’s University and the University of Ottawa to develop RabbitMath, a senior level high-school math curriculum designed to enable students to work together creatively with a high level of personal engagement. My preparation for this has been 40 years of working with teachers in high-school classrooms.

In partnership with grades 11 and 12 math teachers, we will be piloting this curriculum over the next few years.

Mathematical novels

When students study literature, drama or the creative arts in high school, the curriculum centres on what can be called sophisticated works of art, created in response to life’s struggles and triumphs.

But currently in school mathematics, this is rarely the case: students are not connected to the larger imaginative projects through which professional mathematicians confront the world’s problems or explore the world’s mysteries.

The author, Peter Taylor, right, at a Lisgar Collegiate Institute Grade 11 math classroom in Ottawa, 2018. (Ann Arden), Author provided

Mathematician Jo Boaler from the Stanford Graduate School of Education says that a “wide gulf between real mathematics and school mathematics is at the heart of the math problems we face in school education.”

Of the subject of mathematics, Boaler notes that:

“Students will typically say it is a subject of calculations, procedures, or rules. But when we ask mathematicians what math is, they will say it is the study of patterns that is an aesthetic, creative, and beautiful subject. Why are these descriptions so different?”

She points out the same gulf isn’t seen if people ask students and English-literature professors what literature is about.

In the process of constructing the RabbitMath curriculum, problems or activities are included when team members find them engaging and a challenge to their intellect and imagination. Following the analogy with literature, we call the models we are working with mathematical novels.

For example, one project invites students to work with ocean tides. It would hard to find a dramatic cycle as majestic as the effect of that sublime distant moon on the powerful tidal action in the Bay of Fundy.

Student engagement

In the 1970s, the extraordinary mathematician and computer scientist at Massachusetts Institute of Technology, Seymour Papert, noticed that in art class, students, just as mature artists, are involved in personally meaningful work. Papert’s objective was to be able to say the same of a mathematics student.

I had a parallel experience in 2013 when I was the internal reviewer for the Drama program at Queen’s. I marvelled at students’ creative passion as they prepared to stage a performance. And they weren’t all actors: they were singers, musicians, writers, composers, directors and technicians.

In Papert’s curriculum model, students with diverse abilities and interests work together on projects, whereby they collaborate on problems, strategies and outcomes.

As a pioneering computer scientist, Papert understood that students could directly access the processes of design and construction through digital technology. Papert used his computer system LOGO for this technical interface. LOGO was limited in its scope, but Papert’s idea was way ahead of its time.

Students in the RabbitMath classroom will work together using the programming language Python to construct diagrams and animations to better understand their experiments with springs and tires, mirrors and music. They will produce videos that can explain to their classmates the workings of a sophisticated structure.

Today, technology, the internet, computer algebra systems and mathematical programming provide possibilities for immediate engagement in processes of design and construction — exactly what Papert wanted. The platform for RabbitMath is the Jupyter Notebook, a direct descendant of LOGO.

Technical skill

For too many years, real progress in school mathematics education has been hamstrung by a ridiculous confrontation between so-called “traditional” and “discovery” math. The former is concerned with technical facility and the latter is about skills of inquiry and investigation.

There is no conflict between the two; in fact they support each other rather well. Every sophisticated human endeavour, from conducting a symphony orchestra to putting a satellite into orbit, understands the complementary nature of technical facility and creative investigation.

Stanford University Graduate School of Education mathematician Keith Devlin advises parents to ensure their child has mastery of what he calls number sense, “fluidity and flexibility with numbers, a sense of what numbers mean, and an ability to use mental mathematics to negotiate the world and make comparisons.” But for students embarking on careers in science, technology or engineering, that is not enough, he says. They need a deep understanding of both those procedures and the concepts they rely on — the capacity to analyze and work with complex systems.

A high-school math class is a rich ecosystem of differing abilities, capacities, objectives and temperaments.

The educator’s goal must be to enable a diverse mix of students to work together in a math class as creatively and intensely as students in the drama program, or to bring the same personal passion as they might to writing fiction.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Peter Taylor

Abstract algebra class gave me the kick in the rear I needed to focus on the relationships between notes

During my senior year of college, I decided I wanted to expand my musical horizons, so I joined the early music ensemble. I had entered college with a viola scholarship, so I had played in the orchestra throughout my time there as well as doing chamber music and working on solo viola pieces, but I had always enjoyed early music and wanted to try something new. In the ensemble, I played Baroque violin, and a lot of my technique as a modern violist translated well. I held the instrument and bow a little differently, but on the whole, I was able to pick it up quickly. Reading the music was another story.

Many people are exposed to treble and bass clefs in music classes. I was very young when my mom started to teach me how to read music, but I still remember the feeling of accomplishment I had when I mastered the idea that the position of a spot on an array of lines and spaces corresponded to a particular key on a piano or a particular pitch I could sing. The treble clef is based on the G above middle C. The bass clef uses the F below middle C.

A C major scale written across bass (bottom) and treble (top) clefs. The two dots on the bass clef indicate the F below middle C, and the treble clef circles the G above middle C. Credit: Martin Marte-Singer Wikimedia (CC BY-SA 4.0)

Many instruments’ ranges fit comfortably onto either the treble or bass clef, perhaps adjusted by an octave when necessary. Piano music, of course, uses both clefs. But the viola is a little too low to use treble clef all the time and a little too high to use bass clef all the time. When I started to play viola, I learned to read alto clef, which has middle C smack dab in the middle of the staff, and eventually I was the music-reading equivalent of trilingual.

A C major scale written in alto clef, starting on middle C. Credit: Hyacinth Wikimedia

My multilingualism had its limits, though. I could read all three clefs, but if I wanted to play music originally written for cello an octave higher or music originally written for flute or violin an octave lower—tasks that would have been trivial on a piano—I would struggle to read bass clef up an octave or treble clef down an octave, respectively. Tenor clef, which is like alto clef but with the C one line higher, flummoxed me entirely. I wasn’t as fluent as I wanted to be.

Early music ensemble pushed my limits. Music from the Baroque era and before was not always notated using the small number of clefs we tend to use now. I was reading music in French violin clef (ooh-la-la, this one looks like treble clef but has the G on the bottom line instead of the line above the bottom), soprano clef (a C clef like alto and tenor clefs with the C on the bottom line), and other currently unusual clefs. It was overwhelming. I made a lot of mistakes in rehearsal, despite the many note names I had to write in my music.

The same semester I started playing with the early music ensemble, I took an abstract algebra class. Abstract algebra looks at structures of sets of numbers and symmetries. It encourages people to see connections between sometimes very different mathematical objects and transformations and to view the relationships between objects as fundamental to understanding those objects.

At some point in the semester, a switch flipped in my brain, and my early music clef struggles virtually disappeared. At the beginning of a piece, I would look at the clef to get my bearings, and I could see the rest of the notes as representing relationships between one pitch and the next. I read intervals, not pitches. I was not perfect, but I felt like almost overnight I had unlocked a new music-reading level.

I have always felt like my journey into more abstract algebra and my new clef fluency were related, but I have struggled to put that connection into words. I feel like the structural and relational aspects of abstract algebra helped me to see clefs as descriptions of relationships between notes rather than as absolute pitches, but I can’t point to a particular theorem or insight in abstract algebra that would apply explicitly.

Last year, I learned about the Yoneda lemma, an important theorem in the mathematical field of category theory. (According to our My Favorite Theorem guest Emily Riehl, it’s every category theorist’s favorite theorem.) I am no category theorist, but I found Tai-Danae Bradley’s description of the Yoneda lemma helpful, particularly the big idea she shared in this post on the Yoneda perspective. She writes that the punchline of the Yoneda lemma, or at least two of its corollaries, is “mathematical objects are completely determined by their relationships to other objects.”

It has taken me a while to make the connection explicitly, but I think the “Yoneda perspective” describes the mental shift I made in early music ensemble. It’s not the exact notes that matter when you’re reading music written in an unfamiliar clef but the relationships between them. Since having this shift in perspective, it’s been easier for me to transpose music into different keys and read treble and bass clefs in whatever octaves I need to.

Some organists and pianists can transpose music seemingly effortlessly to accommodate the needs of their church choirs or musical theater performers, and I think it’s because they’ve already shifted to the Yoneda perspective, even if that’s not how they would describe it. They didn’t necessarily get there via advanced mathematics classes, but for me, I think abstract algebra class gave me the kick in the rear I needed not to be tied to any one set of exact pitches but to focus on the relationships between them. I won’t claim that studying abstract algebra or category theory will improve your music-reading skills—making music, not studying a math book, is usually the best way to get better at making music—but pondering these connections enriches my experience of both math and music, and I hope it can do the same for you.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Evelyn Lamb