Content area
This is one of two contributions written to commemorate Professor Uri Leron, who became Associate Editor of what was then the International Journal for Computers and Mathematical Learning, a journal founded by Seymour Papert, and upon which Digital Experiences in Mathematics Education builds on. One contribution addresses Leron’s biographical aspects and commemorative stories authored by former graduate students. This is the second contribution that includes memories from colleagues and former Editorial Board members.
Introduction to Our Memories
This paper brings together four memories that paint a multi-faceted picture of Uri Leron as a teacher, a colleague and a very special human being, with contagious enthusiasm, and remembered by all as a master of language and elegance. Michal Yerushalmy collected memories from Celia Hoyles and Richard Noss, Anna Sfard, Nathalie Sinclair and Orit Zaslavsky; we believe that these memories offer readers a good picture of what Uri stood for the following: mathematics, computers, joy of life, wit, optimism and hopefulness.
Celia Hoyles and Richard Noss’ Memories
When we think of our dear friend and colleague Uri Leron, another name often springs to mind: Eeyore. Not that Uri resembled the lugubrious stuffed donkey of the Winnie-the-Pooh children’s books in any way other than being lovable. It is just that that is how our fellow mathematician, and dear friend, would cheekily and cheerfully describe himself to me and my husband, colleague and co-author of this piece, Richard Noss. But there were certainly no long faces when Uri was around.
Richard’s path and mine crossed that of Uri’s separately. I first got to know Uri at a Psychology of Mathematics Education (PME) Conference in the early 1980s. We happened to be at dinner together one evening when Uri mentioned that he was giving an ‘unscheduled’ talk the next lunchtime and that I should come along. So, I did. There he was with his Apple 2E on a small stage, demonstrating the Logo while talking about its transformational potential for mathematics. Details are lost in time, but I certainly recall being excited and inspired. Uri was a brilliant communicator, and of course he was pointing to a goal I so yearned to achieve in our mathematics classrooms, as well as how the goal might be achieved. I quickly emailed my friend Ros Sutherland, who was looking to move into research: ‘We should try to look into Logo together’. The rest is history. Ros and I wrote a research proposal that successfully raised funds from the Leverhulme Trust for a 3-year Logo Mathematics project.
Meanwhile, Richard was also working with Logo, leading a separate project in the UK for young children (Noss, 1984). His first meeting with Uri came a little later, at a conference in Oxford, as he recalls. Over the years, the three of us became firm friends, meeting not only on many occasions, largely at PME conferences, but also on a more personal basis. We remain humbled to think that we both sometimes found ourselves on an international platform or committee with Uri.
Richard and I visited Uri in Haifa, and he came to London a few times. On every occasion, we would talk long into the night, not only trying to debug programs but also to clear our thinking, especially about the ‘windows’ that we felt were opening through programming—windows that would make geometry more accessible, and more learnable for children struggling with the abstract nature of geometrical knowledge (Noss & Hoyles, 1996). And then there was, of course, group theory and proof: The papers Uri wrote with colleagues about these topics in our view brought some of the greatest insights to our mathematical table (e.g., Leron, 1983).
It is worth pausing to think briefly about his early papers and his claim that programming can assist in bringing the abstract to life by harnessing the potential of programming tools to think about algebraic relationships. He takes as an example Lagrange’s theorem, a theorem about groups—part of a non-trivial story about sets of things each of which have certain properties in common … the gist is that, if F is a subset of G, then the order of F must divide the order of G. The point is that the paper is written from the perspective of the students—many of whom do not see the beauty of the abstract idea of the group concept nor its application.
Uri’s idea is classic Leron—take this most abstract of ideas and turn them into manipulable entities that invite the student to explore them without losing any of the rigour or formalism that is so special about the mathematical way of seeing. Uri became Associate Editor of what was then the International Journal for Computers and Mathematical Learning, a journal founded by Seymour Papert, and we joined him and the other editors not only in retreats in London but also in a very special place, namely Varenna in Italy. We were indeed fortunate to spend so much time together, enjoying Uri’s infectious enthusiasm for mathematics, appreciating his deeply felt aesthetic for what mathematics actually is. And. of course, his sense of humour and excitement for something new.
Every time we met we looked forward to stimulating conversations, whether it be about mathematical proof or our shared love for Logo, not to forget a lot of eating, drinking and laughing together. In the course of these times, we discovered Uri’s long-standing delight in the Winnie-the-Pooh five books of childhood, something else we had in common, and how he identified with the gloomy donkey Eeyore.1
Finally, we have found what might be his last email to us, sent in January 2022, about a book by Dick Tahta, a mutual friend and another visionary educator. It, too, captures Uri, and his way of talking and sharing.
Hello friends,
Hope you are well in these crazy times.
I am forwarding Dick Tahta’s Memorial book, which I received from his daughter Sophy. She had no idea who I was, but she found my name somewhere among his papers, and wrote to ask if I would be interested, which of course I was. Today, Sophy and I had a long Zoom meeting, mostly sharing memories about Dick. I asked her about the two of you (after all, Celia was the one who said “you must meet Dick Tahta”, and then arranged our first meeting), and she encouraged me to send you the memorial. So here it is in the attachment.
Hugs,
Uri
Which of us could fail to smile as we read this and remember Uri, so lovable, playful, naughty, cheeky, creative and clever.
Farewell Uri, we miss you. And getting lost in the mathematical forest without you will not be much fun, said Pooh, ruefully.
Anna Sfard’s Memories
‘I remain speechless’, declared Uri Leron in the subject line of the email he sent in January 2023 to a list of his mathematically-minded friends. This list, which he created a few years earlier, was his private platform for sharing with others texts that he found interesting and thought-provoking. This time, Uri was more emotional about his topic than usual.
Hello friends,
See below the question I asked the AI program ChatGPT and the answer she gave me. I Just to make it clear: I have been teaching group theory for many years, and I am not sure I could have given a better answer!
The question: What is the intuition behind the quotient group? And the answer: The intuition behind the quotient group is that it allows us to “group” together elements of a group that are related in some way and consider them as a single element. For example, if we have a group of integers under addition, and we want to group together all integers that differ by a multiple of 5, we can form the quotient group of integers modulo 5. This quotient group has only 5 elements: 0, 1, 2, 3 and 4, and the operation is defined such that (x + 5) = x for any x in our quotient group. This means that any element in the quotient group “forgets” any information about its “multiple of 5” component and just remembers the remainder. In this way, we can often simplify the study of a group by studying its quotient groups.
Two days later, another email informed the recipients that Uri’s astonishment would not subside anytime soon. ‘I continue to be amazed’, said the title.
Friends, I find the following exchange with ChatGPT amazing for two reasons: First, it is very precise and correct; second, it is not the sort of text you find in any regular textbook on groups. They define what a normal subgroup is and then proceed to prove some theorems, but they don’t usually pause to ask and answer questions like “why is it important?”. So, the program can’t just copy-paste from some source, and I can’t imagine how it manages to assemble such a text from its huge database.
A: Normal subgroups are important in group theory because they have a number of useful properties that can be used to study and understand the structure of groups. For example, normal subgroups are closed under conjugation, which means that if a is an element of a normal subgroup H of a group G, then the conjugate of a by any element of G is also an element of H. This allows for the study of the group by looking at its normal subgroups and their properties. Additionally, the quotient group (G/H) of a group G by a normal subgroup H is always a group. This means that the study of normal subgroups can provide insight into the structure of the original group. Furthermore, normal subgroups are important in the study of symmetry and in the study of algebraic structures.
As I reread these emails now, after the flow of Uri’s effervescent notes came to an abrupt stop, I nevertheless feel as if he was still around. His vibrant presence emanates from both what is said and how it is said. For those who were fortunate enough to have met him in person, these breathless communications showcase the quintessential Uri—the special person whose company they appreciated and enjoyed. It is this private Uri—a colleague, a friend—whom I wish to commemorate in this brief piece. To others will be left the task of bidding farewell to the distinguished public figure known for his contribution to the project of turning mathematics education into an academic discipline; to the early enthusiast of digital technologies who once rode Papert’s turtle while spreading the gospel of computers in education; to the interdisciplinary thinker whose idea of structuring mathematical proofs appealed not only to mathematics educators but also to mathematicians and people of computer sciences; and to the teacher who left behind a long list of admiring students.
Yet, it was this public figure whom I met first. Sometime in the mid-1980s, when I was just beginning my doctoral studies, I got a letter (that is, a handwritten note, sent to my university address in a white envelope) from one Uri Leron, whom I knew by name but never met in person. A beginning faculty member at the Department of Teaching Mathematics, Science and Technology (today it would be called STEM) in Technion, he was looking for young people interested in joining his Logo project, possibly as a part of their graduate research. Being already pretty advanced in my own investigations, I declined, but not before we met and found out that we had lots of interests in common. These were the times of great fuss around the seemingly infinite possibilities of revolutionising education due to the recent appearance of personal computers, then heralded as ‘micro’-computers. Given our shared fascination with this new technology and, in particular, with the underlying mathematics, we had many topics to discuss.
Uri was still in the midst of his metamorphosis from research mathematician to researcher in mathematics education, and I was glad to join his informal mathematically-minded Logo group, in which I felt like a native. The conversations that began then continued for years, especially in the mid-1990s, when Uri, at that time the head of the governmental project ‘Tomorrow 98’ aiming at strengthening STEM education, was paying weekly visits to Jerusalem, where my husband and I were his company of choice. With time, the frequency of our contacts diminished, and they became electronic rather than three-dimensional. In the last two decades, during Uri’s retirement, I heard from him almost exclusively through his email list. Still, from time to time, a personal email would arrive with me as the only addressee, carrying a customised recommendation such as this one:
Hi Anna,
Long time, no see. I hope you & family are well.
Lately I have become interested in the recent “argumentative theory” of Sperber and Mercier. I am now reading their 2017 book “The Enigma of Reason”, and their 2013 BBS Target article on the topic. They bring together many ideas and experimental results from psychology (cognitive, social, and evolutionary) to support their “revolutionary” theory. And I am fascinated by it all (even though they are against Dual Process Theory, on which I have read and written so much in the last few years). The reason I am sending you this recommendation is that there seem to be many similarities between their theory and your commognition theory, so I was wondering if you are familiar with this stuff and what you may be thinking about it.
Kol Tuv,
Uri.
All in all, I had sufficient opportunities to get to know Uri in all his private incarnations: as a teacher, a colleague, a researcher, a man of broad knowledge with insatiable thirst for more, an ardent explorer of ever new territories, a music lover who began playing piano only in his retirement, a conscientious citizen deeply concerned about the state of the world at large and of his own country in particular, a devoted family man, a mindful friend who liked being helpful—and the list is still long. I cannot possibly exhaust all that is worth saying about him but suffice to read his emails quoted above to grasp what made him who he was and what endeared him to many people. Probably the most striking thing about these emails is that they emanate uncontainable enthusiasm. In Uri’s texts, whether written or spoken, strong emotional load was a rule. He was a man of passion who enthused about any phenomenon he deemed unusual. His thirst for wonder fuelled a constant quest for novelties. Among the people I know, he was probably the first to notice and appreciate the significance of the AI breakthrough. In the wholehearted devotion with which he threw himself into the investigation of this game-changing invention, almost all his life-long passions—mathematics, computers, and the question of how people think and how they are different in this respect from machines—came together and found their vivid expression.
Uri’s enthusiasm was contagious and affected anybody who came his way. Many seemed to appreciate his childish sense of wonder. People were drawn to him by the fact he was ‘child-lined’, as the Polish writer Witold Gombrowicz would have it, and attracting partners to an interesting conversation was what Uri liked best. Good company and an inspiring exchange occupied the highest places on the list of this gregarious person’s priorities. An artful conversationalist himself, he knew how to tell a story, spicing up even the most serious of topics with a witty anecdote. His special knack for metaphors and puns transpired from the titles of his academic papers: ‘Being sloppy about slope’, ‘Direct approach to indirect proofs’, ‘The slippery road from actions on objects to functions and variables’, ‘Turtle goes to school’. The propensity for language games, especially metaphors, was one of those things Uri and I had in common, and this liking found its expression in the title of our only joint paper, written in a transatlantic conversation when I was on a sabbatical in Canada: ‘Just give me a computer and I will move the earth: Programming as a catalyst of a cultural revolution in the mathematics classroom’ (Sfard & Leron, 1996).
For Uri, by the way, writing papers was a social occasion—and one of the most enjoyable he could think of. When he and his co-authors were tolling on a new paper amidst laughter and sounds of music, an uncustomed person walking by the door of his university office might mistake this activity for a birthday party that went somewhat out of hand. Sometimes, I had the impression that for Uri, no experience deserved this name unless it was shared with other people. This is probably why he created his private email list and why, in his retirement, he hosted a small mathematics club in his house.
Uri was a keen observer of human thinking. Although a mathematician to the core, as demonstrated in his first-ever questions to the AI marvel, he had the rare ability—rare at least among the mathematicians I know—to bracket his own understandings while trying to fathom the mathematical thinking of other people. Rather than seeing a student’s unexpected answer as erroneous or incorrect, he would first examine the possibility that this student was playing a game different from his own but still deserving of the name ‘mathematics’. It was probably because of his wish to be more knowledgeable on how people arrive at their mathematical ideas that he inquired ChatGPT about the intuition behind the concept of group. Years ago, when he sent me a photograph showing his first grandchild—a beautiful 3- or 4-year-old examining a flower, his email, lost since then in a computer crash, was full of conjectures on what and how the little person was thinking when facing all these petals, stamens and pistils. If Uri was enthralled by the performances of ChatGPT, it was because he considered them as an unusual, and thus potentially eye-opening example of the phenomenon of thinking, the nature and development of which he tried to fathom all his life. And, indeed, ChatGPT managed to fool him. Contrary to all he believed so far, the machine came up victorious from the Turing test that was expected to prove it dumb.
Long before becoming aware of what AI can do, and along with his careful observations on children, students, and just plain folks with whom he happened to be in contact, Uri, like Hermann Ebbinghaus, arguably the first serious student of the human mind, used his own thinking as an object of careful inquiry. Most of the stories he was telling in both private exchanges and public presentations were running on several levels: in parallel to a tale about the protagonists—be them people, physical things, or mathematical objects—Uri narrated his own thinking about them; occasionally, he would also say a few words about his thinking about his thinking about his heroes! It was not unusual to hear Uri saying things like, ‘Of course, what I am telling you about X is shaped by my thinking about it in a way that, I think, should be carefully examined’. In an imaginary dialogue between a journalist and a mathematician, Uri sent to his list in August 2023, just 4 months before his death, the interviewer’s request, ‘Give me a succinct description of the mathematics learned in school’, was responded with a reflection on how a mathematician, clearly impersonalising the author, decided what was the mathematics worth learning:
I derive my descriptions from my values as a mathematician, from what I like, what I consider beautiful and clean, and what I view as ugly and dirty; and, also, from what I regard as thrifty or wasteful. And I must precede my description with a remark that mathematicians are pedantic and frugal. They want their descriptions to be clean and succinct (for instance, it is of no interest to me what is 34 times 87—this I leave to my phone).
A healthy dose of disarming self-irony protected this multi-level storyteller from accusations of being self-centred and ensured that people would take his meta-remarks for what he intended them to be: his potentially valuable first-person insights into human ways of thinking. In a natural way, Uri’s interest in how he and other people think about thinking went hand in hand with his fascination with the notion of recursion. In my conversation with him, recursion was a recurring theme, which only too often left us moving in, well, recursive loops.
Uri lived in a country whose history, briefer than his own, was punctuated with wars and whose existence could never be taken for granted. Against all odds, he remained optimistic. On 11 May 2020, at the pick of the COVID scare, when social distancing was the name of the game, Uri sent an email, this time addressed mainly to his female colleagues. It contained a link to a video showing a group of people on Zoom, singing and dancing to the tunes of Gloria Gaynor all-time hit ‘I will survive’.2 As it became customary in those days, singers made their successive solo contributions from their kitchens, living rooms, and bedrooms. Uri’s sparse text explained that they were ‘women who excelled in the mathematics of computer science’. ‘Watch the credits at the end of the clip’, Uri urged, ‘and you will surely be able to recognise at least some of them’. I returned to this highly exhilarating, less-than-3-min-long piece when I first heard about Uri’s death, and I urge you, the reader, to do the same now. I hope it will do to you what it did to me: make me smile it will bring together all those things that Uri stood for: mathematics, computers, joy of life, wit, optimism and hopefulness.
Nathalie Sinclair’s Memories
My first encounter with Uri was when he was an associate editor for IJCML. As a Ph.D. student at Queen’s University in Canada, I had submitted a paper and he was the associate editor in charge of handling the review process. I had read his work, of course, and was amazed to find that he was a real person! Not only did he patiently guided me through the process of improving my paper—he was both demanding and supportive. I learned that writing a journal article could be a joint endeavour, even though the work of reviewers and editors often remains invisible.
After the paper had been published, I found out that he had been invited to a conference at Brock University. This fact came up in our email correspondence. He was not planning to attend. I informed him that Brock was quite close to Niagara Falls and that I was sure it could be arranged to have him visit this wonder of the world. After some back-and-forth, he finally decided he would make the trip. I learned that as an academic, sometimes it is the venue that matters, and not just the conference!
Through a back-and-forth email exchange that I had with him, I learned how knowledgeable he was in evolutionary psychology, which I had been reading about for one of my courses. Indeed, he went on to publish several articles on the topic. In every email, he would mention one or two or three books that he was reading. Humphrey’s The mind made flesh, Deacon’s The symbolic species, Bjorklund and Pelligrini’s ‘The Origins of Human Nature’, and on and on. Every time an email arrived, I was back at the bookstore. I think I finally stopped trying to keep up when he wrote, in one email, ‘I have recently acquired 100 new books and am reading simultaneously 10 (this is really a Darwinian process of competition between the books and survival of the fittest), and I must slow down for a while’. This is typical tongue-and-cheek Uri. But I learned that reading outside education—and reading a lot—was valuable, interesting and good conversational fodder!
Uri used to comment, in these email exchanges, about how much he appreciated the writing of a particular author, or how helpful it was when people could synthesise whole areas of research. He remarked often on beautiful writing, on the difference between beautiful writing and interesting writing, and on the importance of reading things with which you do not agree. He also stated that, ‘no matter how hard you try, people will misunderstand what you write’. From these comments, I learned that writing was not just ‘sharing results’, but something that was worth working on, trying to improve and, indeed, that care and attention in writing were worthwhile.
Over the course of the subsequent years, our email correspondence waned. One is often too busy to engage in wandering, exploratory, ‘unproductive’ email exchanges. I am so thankful that Uri was at a stage in his career where he could take the time to read and write with a new researcher who had so much to learn. I wish he had told me that this could be some of the most productive thinking I could do.
Orit Zaslavsky’s Memories
Uri has been part of my professional life for over 40 years. Our relationship started when I was a doctoral student at the Technion and took a course that he taught, then we became colleagues, and throughout the years we developed a special kind of friendship, together with Yael—his extraordinary wife. I would visit them occasionally in their home and always enjoyed their warmth, wisdom and genuine concern about me and my family.
The Teacher
My first encounter with Uri was in the early 1980 s in a graduate course he gave on LOGO (Turtle Geometry). In those days, he was considered a pioneer in promoting LOGO in Israel. His course was a mind opening experience for me, both mathematically and pedagogically, and his enthusiasm was catching. Uri introduced to us the concept of microworld, through which a learner can develop a deep understanding of the focal content (Geometry), with minimal intervention of the teacher. At that time, teacher-centred learning environments were the most common. Uri created a student-centred learning environment within which we would experience a different kind of learning and offered us numerous opportunities to reflect on this environment, both as learners and as teachers. He emphasised the potential of LOGO in establishing new classroom norms, where the main source of feedback a learner receives is the turtle’s precise response to the instructions (i.e. commands) that the learner enters. Even though I have not incorporated programming in my work, this shift in my thinking accompanied me for the rest of my career, as teacher and researcher.
The Colleague
After a decade, I joined the Technion as a faculty member and became a colleague of Uri. When he served as Chair of our department, I learned of his integrity and commitment to equity. He would not favour anyone, particularly not his close friends. For example, when I requested departmental funding for the participation of one of my graduate students in an international conference, Uri insisted that he first formulate a set of formal criteria for graduate students to be entitled to such support. Only after setting the rules did Uri approve my request and made the criteria public, encouraging other graduate students that met these conditions to apply.
Uri was a challenging and stimulating colleague. He took interest in several aspects of my work—specifically, those that involved subtle mathematical thinking. The first led to our joint paper (together with Hagit Sela, then my graduate student), entitled ‘Being sloppy about slope’ (Zaslavsky et al., 2002).3 I was inspired by the increasing use of graphic software in the 90s, where students were encouraged to zoom in and out as they were investigating the ‘behaviour’ of various functions, with the message that ‘changing the scale does not change the behaviour of the function’. However, observing a representation of a given function in a homogeneous co-ordinate system, where the units on both axes are the same, and its representation in a non-homogeneous co-ordinate system, where the units are different, was thought-provoking.
Contrary to the above message, students and teachers began noticing that some properties remain unchanged (e.g. points of intersection with the axes), but some do change (e.g. angles). These encounters led to a broader question regarding the notion of slope of a function: Is it a visual/geometric entity or an analytic one? After collecting data from students and prospective teachers, we decided to interview a few mathematicians to learn what experts think of the notion of slope. Uri was one of the mathematicians whom we interviewed. Through this interview, he became intrigued by the question of ‘what happens to slope under a non-homogeneous change of scale?’ and asked to join us in this endeavour. Working on this question with Uri was an empowering experience. As expected, he was particularly interested in reconciling (mathematically) the two ‘conflicting’ perspectives on slope, which he did to perfection.
My most significant collaboration with Uri came later. Both of us were working on the interplay between proof and examples. In June 2005, I took part in organising a 3-day international workshop together with John Mason and Anne Watson at the University of Oxford on: Exemplification in mathematics education. We invited Uri together with a few other mathematics education scholars who were interested in discussing and exploring this theme. Each participant was asked to prepare a 1-h session for the entire group. Uri chose to discuss and illustrate the notion of generic proving for the following theorem: Every permutation has a unique decomposition as a product of disjoint cycles. Some of the participants, including myself, argued that what Uri presented was an illustration of a procedure that results in such a decomposition, but it lacked insight about the reason why this will always be possible. Our reaction led Uri to rethink his choice of theorem to illustrate the idea of generic proving.
Two years later, in October 2007 I took part in organising another 3-day international workshop together with Maria Alessandra Mariotti at the University of Siena, Italy, on: The role of examples in argumentation and proof and in concept definition and formation. Most of the participants in the second workshop took part also in the first one, including Uri. By then, Uri presented the idea of generic proving for one of his favourite theorems, the Lagrange’s theorem: If H is a subgroup of a finite group G, then the order H divides the order of G. He used a rather simple example, but this time through his example, Uri managed to convey the main idea of the proof. His generic proof indeed demonstrated the general process which serves to carry out the proof in general, yet some parts of the full proof needed more attention and unpacking (Leron & Zaslavsky, 2013).
Since the first encounter in Oxford, Uri and I would often meet to discuss the notion of generic proving and to refine the criteria for an illustration to qualify as a generic example of generic proving. These discussions led to our article (Leron & Zaslavsky, 2013). We also kept searching for a more convincing generic example of generic proving. Finally, we came up with the following theorem (Leron & Zaslavsky, 2009): A graph has a Eulerian cycle if and only if it is connected and all vertices have even degree. The graph in Fig. 1 is an example of a graph that meets the conditions of the theorem, while Fig. 2 is an example of generic proving.
[See PDF for image]
Fig. 1
An example of a graph that meets the conditions of the theorem
[See PDF for image]
Fig. 2
A generic example of generic proving
In addition to illustrating the procedure of finding a Eulerian cycle, this example captures the main ideas of the proof. It clearly shows why this method will work for any graph satisfying the conditions of the theorem:
For each ‘free’ vertex we can always find a partial Eulerian circle starting there, because of the evenness of the vertices;
Because of the connectedness of the graph, once we complete a partial Eulerian circle, if it is not a full circle, then there must exist a ‘free’ vertex that is on the partial circle.
Finally, we found an example of generic proving with which both of us were content.
Going back to our 2013 article, there were colleagues of ours that criticised this article in a draft paper of theirs, which they sent us for comments. Here is part of our response to them by email (Uri’s fingerprints are all over it …):
As Orit has mentioned, we have met and discussed our response to your paper. In what follows we first discuss some difficulties we see in your reaction to our paper, and at the end we make a practical suggestion on ways to proceed.
The main difficulty we have with your manuscript is that the two papers (ours and yours) deal with different issues and address different audiences, and thus cannot properly be viewed as being in conflict. Specifically, our paper is a quasi-philosophical essay trying to explicate the view of experts (such as research mathematicians or members of the Advanced Mathematical Thinking community of PME) on the nature and educational value of generic examples and generic proofs. Your paper looks at the same topics from the student’s point of view, which is an entirely different issue.
In general, we welcome an academic debate and criticism, but we believe that the criticised views should be presented precisely and fairly before being criticised (so that the criticised party would endorse the presentation of their position). We believe that this has not been the case with your paper, that is, we don’t feel that our position has been presented accurately before being criticised. We can give more examples, but instead we’d like to end with a positive practical suggestion.
In our view, as we have attempted to explain, your discussion of generic examples and generic proofs is not in conflict with our paper; rather, it addresses a different situation and a different audience. Our suggestion is that we all embrace John Mason’s dictum, that instead of starting a response by “no, but …”, let us try “yes, and …” instead. Specifically, we suggest that you try to express your ideas on generic proofs as complementing or extending ours (perhaps more towards classroom practice? ), or even just referencing them, rather than as conflicting with them.
We hope that you find these comments useful.
Sincerely,
Uri and Orit.
At a certain point, the journal editor invited us to write a reaction to our colleagues’ paper that would be published as a short communication. Instead, Uri suggested responding in writing to the journal editor privately. The title he chose for his response letter was: ‘Much fuss about fussiness’—a title typical of Uri. Below are some excerpts from his letter:
Our paper deals with two main topics: The practical pedagogical issues surrounding the use of generic proofs, and the theoretical issues involved. The context for the discussion of these two issues is critical. Our pedagogical discussion is aimed at the audience of teachers, researchers and curriculum planners. All our statements about what is or is not acceptable mathematically relate, explicitly or implicitly, to the norms of the professional mathematical community. … Clearly, if someone wants to discuss these issues in other contexts (middle-school children, for example), the norms should be adjusted accordingly.
Specifically, our critics fuss about “fussiness”, which is the term we chose simply to describe what is required of normative proofs. It all centres on our claim that a real proof needs a certain amount of “fussiness”. Admittedly the word “fussiness” is vague, so a certain amount of good will on the part of the reader is needed to extract the meaning intended by the authors. In retrospect, we might have been clearer if we replaced “certain amount of fussiness” by a “certain level of rigour and generality”. We maintain that had they understood “the fussiness of a real proof” to stand for “the proof norms of the professional mathematical community”, as was clearly our intention, all the presumed conflicts would disappear. Instead, it would turn out that we are simply discussing proof standards in different contexts.
I am grateful to Uri for this amazing journey we took together, aimed at unpacking ideas of generic proving. His wisdom and thoughtfulness made every meeting productive and enjoyable, and his command of English made writing with him a unique experience.
The Friend
As I mentioned earlier, parallel to our professional collaborations, I developed a close friendship with Uri and Yael. In addition to visiting them at their home and seeking their advice on personal matters, I spent some time with them in New York, while I was living there. At one time, Uri came to New York by himself and was eager to meet some of my friends. I introduced him to close friends of mine (a poet and a photographer), with whom he immediately connected and kept in touch after he left. Uri was usually eager to play his flute at social events, including at my retirement party at the Technion. The music he chose and the way he played was always very moving.
Occasionally, Uri would send me a message inquiring how I was doing. During COVID, we had some exchanges, through which he realised that I was completely isolated for over 2 months in my NY apartment. When he heard that I was back in Israel, he wrote to me (translated from Hebrew):
Dear Orit,
I was sorry to hear of your difficult time isolated in NY and am happy to hear that you are back in Tel-Aviv. If you come to Haifa, how about visiting us, so we can exchange experiences and impressions, and Celebrate Life?
Uri was great at celebrating life!! That is how I will always remember him.
Coda
Finally, we hope the readers enjoy and treasure our memories of Uri Leron and can join us in celebrating his profound and passionate influence on mathematics education. His innovative approach to teaching through programming, his enthusiasm for mathematics and life and his unique blend of intellect, humour and kindness will continue to inspire and guide colleagues and students.
Author Contribution
Celia Hoyles, Richard Noss, Anna Sfard, Nathalie Sinclair and Orit Zaslavsky contributed to the memories presented; Michal Yerushalmy co-ordinated the project.
Funding
Open access funding provided by University of Haifa.
Data Availability
No datasets were generated or analysed during the current study.
Declarations
Competing Interests
The authors declare no competing interests.
Eeyore is a fictional donkey in the Winnie-the-Pooh books by A. Milne, generally characterised as pessimistic and gloomy. But everybody loved him.
2https://www.youtube.com/watch?v=4Wl-3kadvgw.
3Uri was the one to suggest this title. He was a master of language and elegance.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Leron, U., & Zaslavsky, O. (2009). Generic proving: Reflections on scope and method. In F.-L. Lin, F.-J. Hsieh, G. Hanna & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and proving in mathematics education (vol. 2, pp. 53–58). ICME.
Leron, U., & Zaslavsky, O. (2013). Generic proving: Reflections on scope and method. For the Learning of Mathematics, 33(3), 24–30. [This article was included in M. Pitici (Ed.) (2014). The best writing on mathematics (pp. 198–215). Princeton University Press.].
Leron, U. Structuring mathematical proofs. The American Mathematical Monthly; 1983; 90,
Noss, R. Children learning Logo programming: Interim report No. 2 of the Chiltern Logo Project; 1984; Advisory Unit for Computer-Based Education:
Noss, R; Hoyles, C. Windows on mathematical meanings: Learning cultures and computers; 1996; Kluwer Academic: [DOI: https://dx.doi.org/10.1007/978-94-009-1696-8]
Sfard, A; Leron, U. Just give me a computer and I will move the earth: Programming as a catalyst of a cultural revolution in the mathematics classroom. International Journal of Computers for Mathematical Learning; 1996; 1,
Zaslavsky, O; Sela, H; Leron, U. Being sloppy about slope: The effect of changing the scale. Educational Studies in Mathematics; 2002; 49,
© The Author(s) 2024. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.