What the outstanding thinkers of the twentieth century Simone and André Weil can teach today's mathematics teachers

Like most mathematiciansI hear confessions from complete strangers: the inevitable “I've always been bad at math.” I suppress the reply: “You are forgiven, my child.”

Why does failing math feel like a sin? Why are so many individuals traumatized by their math classes? Is there some extent in learning math?

Sometimes agreeing, sometimes disagreeing, Andre And Simone Weil were the type of siblings who would argue about such questions. André rose to fame as a mathematician; Simone was a formidable philosopher and mystic. André focused on applying algebra and geometry to deep questions on the structure of integers, while Simone was concerned with how the world could be soul-destroying.

Both grappled with the query of how best to show mathematics. Their findings and contradictions point to the basic role that mathematics and arithmetic education play in human life and culture.

André Weil’s rigorous mathematics

Unlike the outstanding French mathematicians of previous generations, André, who was born in 1906 and died in 1998, spent little time philosophising. For him, mathematics was a living subject with a protracted and wealthy history, but, as he noted, he saw “no must defend yourself (It).”

When coping with people, André was a ruthless critic. Although admired by some colleagues, he was feared by and sometimes contemptuous towards his students. He was a co-founder of the Bourbaki mathematics collective, which used abstraction and logical rigor to Restructuring mathematics from scratch.

However, Nicolas Bourbaki’s commitment to ranging from first principles didn’t fully capture his idea of ​​worthwhile mathematics. André was coordinated on how mathematics must be taught in another way to different goal groups.

In a tone that was softened by Bourbaki’s spirit, he defined rigor as “(not) proving everything, but … trying to assume as little as possible at every stage.”

In other words, absolute rigor has its place, but teachers should be willing to contemplate their audience. He believed that teachers must motivate their students by providing them with meaningful problems and provocative examples. For advanced students, excitement comes from encountering the unknown; for beginners, it comes from solving questions of, as he put it, “theoretical or practical importance.” He insisted that mathematics “must be a source of intellectual excitement.”

André's own mental excitement got here from applying knowledge from one area of ​​mathematics to other areas. Letter to his sisterAndré described his work as a seek for a metaphorical “Rosetta Stone“ of analogies between advanced versions of three fundamental mathematical objects: numbers, polynomials, and geometric spaces.

Andre described mathematics in romantic terms. At first, the connection between the various parts of mathematics is that of passionate lovers, exchanging “furtive caresses” and having “inexplicable quarrels”. But when the analogies finally give solution to a single unified theory, the affair cools: “Gone is the analogy: gone are the two theories, their conflicts and their delicious mutual reflections… alas, everything is only one theory whose majestic beauty can no longer excite us.”

Although this theory, which mixes numbers, polynomials and geometry, is dispassionate, it goes to the center of mathematics and André pursued it intensely. In the words of a colleague, André was in search of the “true meaning of each fundamental mathematical phenomenon.” Unlike his sister, he found this true meaning within the careful definitions, precisely formulated theorems, and rigorous proofs of probably the most advanced mathematics of his time. Romantic language merely described the sentiments of the mathematician confronted with mathematics; it didn’t point to a deeper meaning.

Simone Weil and the philosophy of mathematics

Simone, however, who was born three years after André and died 55 years before him, used philosophy and religion to explore the worth of mathematics for non-geniuses, alongside her work on politics, war, science and suffering.

There is something maddening about all her writings – indeed, her whole life – in her polished essays, but in addition in her private letters and diaries, she often makes extreme claims or enigmatic comments. Such claims might concern the motivations of scientists, the psychological state of a sufferer, the character of labor, an evaluation of trade unions, or an interpretation of Greek philosophers and mathematicians. She isn’t a scientific thinker, but circles around complexes of ideas and themes. When I read her writings, I’m often baffled. I begin to argue along with her, offering counterexamples and qualifications, but I find yourself agreeing along with her in essence. Simone was known for her single-minded pursuit of her ideals.

Despite the discomfort their views cause, they’re price engaging with. her childhood was largely completely satisfiedall their lives felt silly compared along with her brother. She channeled her feelings of inadequacy into an exploration of how one can live a meaningful life within the face of oppression and suffering. Over the course of her life, she developed a Interpretation of beauty and suffering intertwined with the geometry.

In addition to her lifelong mathematical discussions with André, her views were influenced by one in every of her first jobs as a teacher. In a letter to a colleague, she described her students struggling because they “viewed the various sciences as bodies of fixed knowledge.” Like André, Simone saw the flexibility to motivate students as the important thing to good teaching. She taught mathematics as a subject embedded in culture and emphasized overarching historical themes. Even the scholars “completely ignorant of science” followed her lectures with “passionate interest.”

For Simone, nonetheless, the major purpose of mathematics education was to develop the virtue of attention. Mathematics confronts us with our mistakes, and the Consideration of those inadequacies brings out the flexibility to deal with one thing and block out the whole lot else. As a math teacher, I often see students grit their teeth and frown, which only causes headaches and resentment. However, based on Simone, true attention comes from joy and desire. We hold our knowledge loosely and wait with detached thoughts for the sunshine to dawn on us.

For Simone, the “first duty” of teachers is to assist students to develop the flexibility to grasp God through their studies, which she sees as a combination of Plato’s description of the the most effective with Christian ideas of a self-denying God. A real understanding of God leads to love for many who suffer.

Simone might even discover the lingering anxiety and frustration felt by many former math students as a consequence of a scarcity of attention from their teachers.

Authors examine Weil’s legacy

More recently, others have grappled with Weil’s legacy.

Sylvie Weil, André’s daughter, was born shortly before Simone’s death. Her Family experience was that she was mistaken for her aunt, was ignored or humiliated by her father, and received neither recognition nor appreciation from the people round her.

Also writer Karen Olsson uses Simone and André to explore their very own conflicted relationship with mathematics. Their desperate quest to grasp André's mathematics eerily mirrors Simone's desire to grasp André's work and Sylvie's desire to be seen as an individual in her own right and never to be in Simone's shadow. Olsson studied under exceptional mathematics teachers and students, but on a regular basis felt misplaced, overwhelmed, and intimidated by her fellow students. Most painfully, while writing her book concerning the Weil siblings, Olsson asks a mathematician who had studied along with her for assist in understanding some facets of André's mathematics. She was ignored. Both Sylvie Weil and Karen Olsson reside witnesses to Simone's remark that every of us is crying out to be seen.

Christopher Jackson, however, testifies to how mathematics can do justice to Simone's vision. Jackson is in a federal prison, but finds a brand new life through mathematics. His correspondence with the mathematician Francis Su forms the backbone of Su's book “Mathematics for human flourishing”, which uses as its leitmotif Simone’s remark that “every being silently cries out to be interpreted differently”. Su identifies facets of mathematics that promote human flourishing, equivalent to beauty, truth, freedom, and love. In their very own ways, Simone and André would probably agree.