Three Mathematical Cultures: What Can We Learn?
On Teaching and Learning Mathematics
by msaul
9M ago
Three Mathematical Cultures: What Can We Learn? Mark Saul July 2023 Everyone needs mathematics.  It is the heavy industry of scientific development, the unseen basis on which the more spectacular advances in science, in technology, and in medicine are often built.  And mathematics is cheap.  We rarely need fancy equipment to pursue our research.  A pad and paper, sometimes a computer, and sometimes even less to start the process. In some sense there is only one mathematics.  Mathematicians everywhere tend to agree on the nature of mathematical truth or whether a partic ..read more
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Finding Pedagogy in Recreational Problem Solving: reflections and lessons learned
On Teaching and Learning Mathematics
by msaul
2y ago
Posted on January 4, 2022 by msaul By Dimitrios Roxanas A few years ago, when I started my tenured job at the University of Sheffield, one of my first initiatives was to start a problem solving seminar for students (undergraduate and graduate) and also academic staff.  I have to admit I was a bit apprehensive when I pitched the idea of setting up these sessions (full disclosure: and for several months after!): I was a new and untested hire, with no obvious relevant credentials (I never competed in mathematical competitions beyond the local level), and a very distinct lack ..read more
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Finding Pedagogy in Recreational Problem Solving: reflections and lessons learned
On Teaching and Learning Mathematics
by msaul
2y ago
By Demitrios Roxanas A few years ago, when I started my tenured job at the University of Sheffield, one of my first initiatives was to start a problem solving seminar for students (undergraduate and graduate) and also academic staff.  I have to admit I was a bit apprehensive when I pitched the idea of setting up these sessions (full disclosure: and for several months after!): I was a new and untested hire, with no obvious relevant credentials (I never competed in mathematical competitions beyond the local level), and a very distinct lack of seniority about me. Despite that, I found a lot ..read more
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Three foundational theorems of elementary school math
On Teaching and Learning Mathematics
by Ben Blum-Smith
2y ago
By Ben Blum-Smith, Contributing Editor This post discusses three very familiar facts from grade-school mathematics. In spite of their familiarity, I believe they tend to go under-appreciated, at every level of math education. In the elementary grades, my experience is that if they do get explicit attention, we generally treat them as tools students should learn to use, rather than as the subject of their own inquiry. Meanwhile, in the later grades—middle school, high school, college, and beyond—we are already used to them, so they tend to be seen as trivialities, not worthy of further reflecti ..read more
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From Teaching Math to Teaching Students Math
On Teaching and Learning Mathematics
by Yvonne Lai
2y ago
by Yvonne Lai (University of Nebraska-Lincoln) I did not want to present. Someone had selected my solution to a geometry problem to present at a Mathfest 1996 session. I wasn’t sure who this person was, but I knew already that I did not appreciate them. I was 16 years old, my father was ecstatic at the honor, and I wished my father had never found out, because then I could have played hooky and no one would be the wiser. The problem that I did not want to present The visual of a solution that I did not want to present That summer, I was a student at the Canada/USA Mathcamp, a residential p ..read more
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Access To Epidemic Modeling
On Teaching and Learning Mathematics
by msaul
2y ago
Kurt Kreith and  Alvin Mendle, University of California, Davis Covid-19 has left teachers seeking topics that are both engaging and lend themselves to online instruction.   As a guiding force for the measures that have reshaped our lives, epidemic modeling stands out as natural.  For teachers at the secondary level and those involved in teacher education, this leads to the question: How can an understanding of epidemic modeling be made accessible to students at large? From the vantage point of evolutionary biology, viewing epidemics as a form of natural selection is a good ..read more
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In Memoriam N. N. Konstantinov
On Teaching and Learning Mathematics
by msaul
2y ago
IN MEMORIAM N. N. KONSTANTINOV by Mark Saul This summer marks the thirtieth year since the end of the Soviet Union.  It also marks the passing of one of the great figures of Russian mathematical culture, Nicholas Nikolayevich Konstantinov.  This note concerns both events, but cannot do justice to either.  Rather, I will here give some personal reminiscences that might contribute to the picture, but not find a place in the historical record.    I leave to other sources the task of a more comprehensive account.  Here’s my story. The year was 1987.  The Col ..read more
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Outcomes-Based Assessment — Structural Change in Calculus
On Teaching and Learning Mathematics
by Carolyn Abbott
2y ago
Traditional Grading Sends the Wrong Message For many years I taught Calculus with a traditional structure, in which the students’ grades were mostly determined by a few high-stakes exams (a final and a couple of midterms).  In my classes, I would tell my students: How important it was to practice regularly;  To carefully review their exams and the solutions; That it’s ok to get things wrong and learn from their mistakes; That the idea that we can improve through practice applies in math just as it would in anything else they want to learn. But the structure of my class was giving t ..read more
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Mathematics as Logic
On Teaching and Learning Mathematics
by msaul
2y ago
Maybe it is obvious, but it is something I’ve come to appreciate only after years of experience: mathematics is logic driven, and teaching and learning mathematics is centered on teaching and learning logic.  I find this to be true philosophically, but also practically, in my teaching.  And even in my own learning. Philosophically, this point of view has deep roots.  Plato’s Academy.  Russell and Whitehead.  Frege, Tarski.  And that’s all I want to say about this area, which is outside my expertise.  I leave it to those who think more deeply about the philoso ..read more
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A Tale of Two Hats (Terrance and Lamar): supporting students in authentic mathematical inquiry
On Teaching and Learning Mathematics
by msaul
3y ago
Terrance Pendleton, Drake University Students who have had me for at least one class are familiar with my alter ego, Lamar. If they were to describe him, they may say that he is the poster child for what not to do in mathematics. They may speak to Lamar’s tendency to arrive at the wrong conclusion by making erroneous assumptions and/or using faulty logic. When a mistake in class is made, it is not unusual for someone to say, “That’s totally a Lamar move to make.” You may wonder how Lamar came to be. Let me take you back to a chilly spring day in March inside a proof-based linear algebra c ..read more
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