Introduction: About Uri Leron

Professor Uri Leron was born on Kibbutz Nir David, Israel, on December 22nd, 1938. On December 5th, 2023, at the age of (almost) 85, he was laid to rest back on the kibbutz where he was born. Professor Leron earned his three degrees at the Hebrew University of Jerusalem. In 1964, he received his B.Sc. in Mathematics and Physics; in 1966, an M.Sc. in Mathematics; in 1972, he received his Ph.D. in Mathematics. Although Professor Uri Leron was a mathematician whose research originally focused on ring theory (a branch of abstract algebra), in 1980, based on the realisation that his contribution would be much more significant and unique if he concentrated on mathematics education instead, Uri decided to switch his research focus to mathematics and computer science education.

In 1983, he joined the Technion’s Department of Education in Science and Technology, where he developed his remarkable and influential professional research career in mathematics and computer science education. His work was based on a deep mathematical and scientific understanding that integrated several disciplines, including cognitive and evolutionary psychology, music and educational systems. Indeed, this transition proved to be highly successful and enormously beneficial.

In retrospect, Uri’s decision to transition from mathematical research to educational research significantly influenced the Israeli education system in several areas, with far-reaching impact. Among other professional engagements, Professor Leron published numerous articles in scientific journals, served in several leadership positions at the Technion and on the national level, and pioneered a number of important initiatives related to the integration of technology in education that were innovative in their field. These achievements, separately and as a whole, influenced the Israeli education system both directly, by applying many of his ideas, and indirectly, through the dozens of undergraduate and graduate students he mentored and educated over the years.

In the last 20 years, Professor Leron delved into research in cognitive and evolutionary psychology, in order to deepen the understanding of the source of difficulties students encounter in understanding mathematical and logical formalism. Based on research in cognitive and evolutionary psychology (e.g. Kahneman, 2003b; Stanovich & West, 2000), Uri investigated the conflict between intuitive thinking and analytical thinking, and sought ways in which mathematics educators can bridge these two types of thinking. In particular, he found the dual-process theory very insightful and applicable to the context of analysing these two types of thinking.

The dual-process theoryFootnote 1 (Kahneman, 2003a; Gilovich et al., 2002) proposes an explanation for a cognitive phenomenon, that people consistently make mistakes on simple everyday tasks, even when they are knowledgeable, intelligent people who undoubtedly possess the necessary knowledge and skills to perform those tasks correctly. According to this theory, our cognition operates in parallel in two different modes, System 1 (S1) and System 2 (S2), roughly corresponding to our common-sense notions of intuitive and analytical thinking. S1 processes are characterised as being fast, automatic, effortless, unconscious and inflexible (that is, difficult to change or overcome). In contrast, S2 processes are slow, conscious, effortful and relatively flexible. S2 serves as monitor and critic of the fast, automatic responses of S1, with the ‘authority’ to override them when necessary. In many situations, S1 and S2 work in concert, but there are situations in which S1 produces quick, automatic, non-normative responses, while S2 may or may not intervene in its role as monitor and critic.

Uri loved mathematics, and a large part of his efforts in research and practice were an attempt to instill that love in his students. He applied constructivist approaches even before the term became popular and used them continuously. For example, he pioneered the use of the Logo programming language to teach a variety of mathematics and computer science concepts by creating opportunities for students to discover these concepts within the Logo digital environment. The importance Professor Leron attributed to the accessibility of mathematics in general, and of mathematical proofs in particular, earned him international recognition. Among other career highlights, he was a visiting scientist at MIT, Berkley and Stanford in the USA, and was awarded a residency scholarship at the Rockefeller Center in Bellagio, Italy, in May 2005.

In addition to being a leading researcher, Professor Leron was a dedicated family man. Together with his wife Yael, he had three children and six grandchildren. His cleverness extended beyond his professional life, manifesting in a fine, precise and unique sense of humour. This humour was another lens through which he viewed the world and a central component of his communication. He possessed the rare ability to discern the absurd, the ridiculous or the inflated in any situation, often surprising those around him with a witty statement that never failed to bring laughter.

Our Stories

In this section, we present five stories in chronological order of Ph.D. graduation of the storytellers: Rina Zazkis—1989, Orit Hazzan—1995, Meira Levy—2004, Irit Hadar—2004 and Orly Buchbinder—2011. We hope that our reflections on Uri’s impact on our professional development contribute to the rethinking of mathematics and computer science, as well as their learnability and teachability.

A Gift of Mathematics (Rina Zazkis)

My story starts in 1981, when, studying at the University of Haifa towards a major in Mathematics alongside a certification for teaching mathematics in high school, I was enrolled in a ‘Seminar on didactics of Mathematics’ taught by Dr. Leron. (Indeed, back then it was not ‘Professor Leron’ and definitely not ‘Uri’ for me. I believe I dared to say ‘Uri’ only towards the completion of my Ph.D. in 1989.)

This course, and my first interactions with Dr. Leron, opened my eyes to what it means ‘really to understand’ some mathematical idea and how to express this understanding formally. It showed me a few ‘obvious’ things that I had never thought about. Surprisingly, one can successfully complete a degree in Mathematics, and even excel in some courses, without ever engaging with some basics. For example, what was the driving force in introducing integers and so extending the set of natural numbers to a set that includes negative numbers? What was the driving force in introducing rational numbers? These questions can be answered, but only when asked; Dr. Leron asked them to a class of clueless prospective teachers. The answer, which was definitely within our ‘zone of proximal development’, was a desire for closure. That is, the set of natural numbers is closed under addition, but if we want a set to be closed under subtraction, we need the integers— thus, we include negative numbers. And integers are closed under multiplication, but if we want a set to be closed under division, we introduce rational numbers. This was the beginning of the topic of field extensions, such as extending the field of rational numbers \(\mathbb{Q}\) to that of \(\mathbb{Q}\left(\sqrt{2}\right)=\left\{a+b\sqrt{2}|a,b\in \mathbb{Q}\right\}\).

What are field extensions doing in a ‘Seminar on didactics’? Maybe Dr. Leron stated this explicitly, maybe not; I cannot recall more than 40 years later. I believe Dr. Leron considered that expanding teachers’ mathematics was extremely relevant in teacher education. Back then, we did not have distinctions requiring acronyms such as CK and PCK (content knowledge and pedagogical content knowledge: Shulman, 1986), or ‘knowledge for teaching’. What I do recall is that many of my classmates were rather unhappy with the course content. They would rather create lesson plans on multiplying integers or solving quadratics. But I was excited. It was the first time in my student life that I was required to attend to mathematics ‘in depth’, to consider every detail, every phrase and every comma. It was irritating at times, but this undergraduate course marked the beginning of my career.

There is a lot of mathematical content in my courses taught in the Faculty of Education. However, I wrap my intention to expand teachers’ knowledge of mathematics in tasks with a pedagogical flavour, so that my students have to engage with mathematical ideas while attending to pedagogy (e.g. Zazkis & Marmur, 2021). But do not let me digress—this is not about me.

While exploring the idea of graduate school, I noted a poster advertising a graduate program in ‘Mathematics and Education’. Note the ‘and’ that separated (or united? ) the disciplines. ‘Mathematics Education’ was not yet a declared specialisation or profession. The small text at the bottom of this poster named Dr. Leron as the co-ordinator of this program, which made it my obvious choice of pursuit.

That is when Dr. Leron introduced me to Logo as a powerful computer language with an initial focus on turtle graphics, but also as a pedagogy of active learning, where ‘playing turtle’ (Papert, 1980/2020) paves the way towards expressing intuitive mathematical ideas in computer code. First encounters with Logo bring to mind some nostalgic memories:

A computer program defines a square:

To Square

Repeat 4 [Forward 50 Right 90]

End

Then this square is rotated:

To RotateSquare

Repeat 36 [Square Right 10]

End

and the result is … MAGIC. Excitement (see Fig. 1). And the desire to experiment, to try different numbers, different shapes, different designs.

Fig. 1
figure 1

Result of RotateSquare

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Yes, I know, nowadays, the magic got lost. No contemporary student will get excited by a square rotated on a computer screen. Colourfully designed fractals are within the click of a button. But engagement with Logo, while seldom mentioned within a current research interest in education on ‘computational thinking’ or ‘students’ computational activity’ (Weintrop et al., 2016), paved the way to this area of study.

My M.A. thesis, completed in 1985, explored models of recursion in computing using Logo and also the connection between computational recursion and mathematical induction (Leron & Zazkis, 1986). The topic appears to have gotten some life recently (Bailey et al., 2024; note that the second author in NOT me! ). My Ph.D. dissertation, completed in 1989, was also based on Logo’s turtle geometry. It was completed at the Technion–Israel Institute of Technology, where Professor Leron continued his career, and I followed him. I explored turtle geometry, focusing on the movement of a turtle on a screen as a group structure, referred to as ‘the group of the turtle’. Then I studied the isomorphism between the group of the turtle and the group of direct isometries on the Euclidean plane (Leron & Zazkis, 1992; Zazkis & Leron, 1991). Prof. Leron’s guidance in my learning of group theory and transformation geometry was invaluable.

However, the main outcome for me was the personal guidance I received in learning and rediscovering mathematics. For example, in considering isometries on a plane, four transformations are usually listed: translation, reflection, rotation and glide reflection. But why do we need a glide reflection, if it is simply a composition of a reflection and a translation? That is the way it is, no questions asked. But Uri (indeed, he gradually transformed for me from ‘doctor’ to ‘professor’ to ‘Uri’) asked me to seek the answer. It brought me back to the main concept that started my undergraduate course on didactics of mathematics: closure, the desire to work with a closed set. That is, by adding a glide reflection to the trio of reflection, translation and rotation, we get a set that is closed under composition of transformations, which is one of the requirements for forming a group. This was an important discovery for me that no textbook cared to mention.

Having completed my doctoral dissertation, I moved to the USA and then to Canada, seeking employment in academia. (For many years, I considered this move to be ‘temporary’, contemplating life and work in Israel … but this was over 30 years ago.) There were many things I had to learn about research that I wish I had learned during my doctoral studies. I was angry with myself for knowing very little about research trends and research communities. I was even occasionally angry with Uri for letting me graduate with very little knowledge of Mathematics Education (it was already a common phrase) as an area of study.

It took me a few years to turn my frustration with initial personal incompetence into a deep appreciation of the enormous gift that I got from Uri. This gift is Mathematics. It is an appreciation of the beauty of mathematics, of which I only had a vague idea when engaged in ‘lemma–theorem–proof–occasional example’ coursework. It is also an appreciation of what it means for me ‘really to understand’ a mathematical idea, as well as a belief in my personal competence in ‘doing mathematics’, at least to a degree that is beneficial for my work as a teacher and a researcher.

EUREC(k)A: Educator Uri’s Lessons on Reflection–Evolution–Cognition–Abstraction (Orit Hazzan)

Uri and I had a long professional acquaintance: I knew him as a graduate student (while working on both my master’s and doctoral theses), a teaching assistant, an R&D co-ordinator of the Information Technology in Technion Teaching (ITTT) initiative that Uri led and then (between 2000 and 2004) as a colleague in the Department of Education in Science and Technology until Uri’s retirement in 2004. Also, after Uri’s retirement, we continued working together at significant junctions of our different intellectual and academic paths. During this long joint voyage, I learned many things from Uri. In my story, I focus on four concepts that are strongly connected to my work with Uri: Reflection, Evolution, Cognition and Abstraction. Appropriately, when the words Educator and Uri come before these concepts, the acronym formed is my Eurec(k)a.

Reflection

On-going reflection on my thinking, feelings and doings is one important skill that I learned from Uri which has guided me during my professional development. In addition to the many other aspects of my life to which I applied this skill, I adopted the reflective practitioner perspective (Schön, 1984) to my work on software engineering education and agile software engineering, which eventually opened new professional horizons and opportunities in my academic career (e.g. Hazzan, 2002, 2008; Hazzan & Tomayko, 2005; Talby et al., 2006). My familiarity with this concept enabled me to foster innovative processes in the software engineering world which, at the beginning of the millennium, underwent a significant change from a rigid process to an agile one, adopting development methods that embraced reflective processes and other human-related habits and conventions.

Evolution

In the last 30 years, Uri delved into research in cognitive and evolutionary psychology, in order to understand the source of difficulties students encounter in understanding mathematical formalism. Based on research in cognitive and evolutionary psychology, Uri investigated the tension between intuitive thinking and analytical thinking, and sought ways in which mathematics educators can bridge these two types of thinking.

Our research paths have met at this junction over the years. First, during my Ph.D. research, which focused on undergraduate students’ understanding of abstract algebra concepts, we applied the dual-process theory (Kahneman, 2003b; Stanovich & West, 2000). Specifically, we identified that, in many cases, students use Lagrange’s Theorem incorrectly by immediately applying an S1 process, without stopping to think whether the conditions for using the theorem apply—that is, they do not apply any S2 process (Hazzan & Leron, 1996). This finding reflects what students could gain if introduced to the power of reflection on their own mental processes.

We later used the dual-process theory for the analysis of students’ understanding of additional mathematical topics on different mathematical levels, and presented the results in three papers published in Educational Studies in Mathematics (Leron & Hazzan, 1997, 2006, 2009).

Cognition

This familiarity with cognitive psychology was also useful in my research on software engineering and data science education. In software engineering education, Uri and I explored why we resist testing, an important activity in software engineering processes (Hazzan & Leron, 2006). We explained it using confirmation bias, which is exhibited when:

scientists consider only one hypothesis (typically the favored hypothesis) and ignore alternative hypotheses or other potentially relevant hypotheses. This important phenomenon can distort the design of experiments, formulation of theories and interpretation of data. Confirmation bias is very difficult to overcome. (Dunbar & Fugelsand, 2005, p. 709)

Uri and I argued that testing resistance is a special case of confirmation bias. We showed that Test Driven Development (TDD), a practice that incorporates testing into agile development processes, supports developers in overcoming their confirmation bias. The remarkable insight behind TDD is that, by incorporating testing into the development process, testing becomes part and parcel of ‘the hypothesis’ and is no longer conceived as a search for disconfirming evidence.

Inspired by my work with Uri on cognitive biases, later on, when data science started flourishing, my graduate student Dr. Koby Mike and I researched two cognitive biases which may have significant impact if applied in data analysis processes: base-rate neglect (was already known in cognitive psychology) and domain neglect (was identified by us, Mike & Hazzan, 2023).

Abstraction

My Ph.D. research deals with undergraduate students’ understanding of abstract algebra concepts. Abstract algebra was one of Uri’s favourite mathematical fields, and together with Ed Dubinsky, he developed a constructivist pedagogical approach to teaching the subject using the programming language ISETL (Dubinsky & Leron, 1994)—which, in some sense, continued the Logo pedagogical philosophy. The students in my dissertation research, however, did not learn abstract algebra with ISETL, but, rather, in a regular class.

The organising theme I developed in my Ph.D. is called ‘reducing abstraction’ (Hazzan, 1999), according to which, when students cannot make sense of the too abstract abstract algebra concepts, they find mental ways to reduce the level of abstraction to make them more accessible. Later, I applied this framework for the analysis of students’ conceptions of a variety of mathematics topics (e.g. school mathematics) and computer science topics (e.g. abstract data structures, graph theory and computability theory) (see Hazzan, 2003a, b; Hazzan & Zazkis, 2005; Hazzan & Hadar, 2005; Sakhnini & Hazzan, 2008).

Summary

In retrospect, my meta-EURECA is that my professional work with Uri demonstrates the balance between collaboration and individuality in academic work. It was a long academic voyage of 30 years, shared between two colleagues who each developed different careers and research expertise, while continuing to collaborate on joint research that met at junctions of their different professional paths.

The Social Aspects of Learning Computer Science Basics (Meira Levy)

I met Uri during my maternity leave from Rafael (where I worked as a software engineer) between 1999 and 2000. I came to his office and checked the option of starting another master’s degree in Education. Uri asked me, ‘You have a master’s degree already, why not start doctoral studies?’ I was a bit surprised and rather afraid, but he assured me that this was the right path and sent me to meet his student who has just finished her Ph.D. studies for further consultation. I remember him telling me that we do not develop sophisticated systems here, as at Rafael, and that we sometimes even play with computer science ideas to make them vivid and realistic to young students. That moment introduced me to the human aspects of my profession, which I had missed during my education and professional development.

Following my own experience with my children, who did not want to choose computer science as a major subject in high school, I knew that I was interested in teaching students in junior high school the principles of computer science in a way that was appealing and motivating. I developed a computer science curriculum, based on the Logo programming language, for 8th-grade students, combining distance, on a WEB social platform, and frontal learning activities, allowing students to advance at their own pace.

The theoretical frameworks on which I based the curriculum’s development were the Scaffolded Knowledge Integration (Linn, 1995), which deals with learning in technology-based environments designed to support knowledge integration and to bridge the naive world concepts of the students, and the agile approach in software engineering, which emphasises software development in short iterations and on-going integration. The model of the mechanism that evolved during the learning experience distinguishes between three process types:

  • External processes which enhanced class communication and established a learning community

  • Internal processes that strengthened students’ emotional expressions, preferences and motivation, while developing meta-cognitive ability of translating spoken language to formal programming language

  • Control processes for dealing with the short learning units and learning follow-up

The study showed the important aspects of learning in a shared social space, where project-based learning of problems that are relevant to students’ interest (e.g. game development) occur, and where knowledge is managed and shared. Such learning can foster formal understanding of programming which is often too abstract and unintuitive.

I took the insights gained in this study to the research of knowledge management in organisations, where I realised the importance of the field and identified its most crucial aspects (Levy et al., 2010a, b, 2019). In particular, these include social networks (Levy et al., 2021) and the human barriers and cultural aspects of applying knowledge management within them (Levy et al., 2010a, b).

Uri was the best mentor I could ask for. He encouraged me and gave me freedom and support to pursue my idea, as if it was an innovative venture. Although the platform I used was based on his Logo expertise, he did not advise me on how to build the curriculum and allowed my imagination and creativity to flourish. This is a lesson I apply whenever I guide or teach someone. When a question arises, I try to reply with my own guiding questions instead of a direct answer (although not in the case of an informative question). Uri’s mentoring taught me to foster deep understanding and interest in the topic and to provide opportunities for self-development and independence. He always searched for independent expressions and insights I had gained in my research and, once they had occurred, he praised them—thus instilling in me self-confidence in my research and my ability to pursue new research directions. But he also helped me to develop self-criticism and attention to every sentence I write that should be backed up with research data.

The most important contributions from Uri to my professional development was a curiosity regarding the human aspects of technology and a desire to do research. After my Ph.D. studies, I had to return to Rafael, where I tried to apply the most important insights from my research on knowledge management of collaborative work. After realising that I would not be able to pursue this direction, I moved to the academic world, starting as a post-doc in Deutsche Telekom, at Ben-Gurion University of the Negev, as a researcher in a knowledge management project related to the cellular phone. Later, I arrived at Shenkar College of Engineering, Design and Art where I realised the importance of design thinking in technological development. Since then, it has become my main research topic, focusing on challenges in requirements engineering, health, business processes, education, sustainability and more. In addition, I pursue the design thinking approach with many collaborators. Specifically, as a research fellow at the University of Haifa, with Prof. Irit Hadar (whom I first met at Uri’s office), I have established the Design Thinking Research Hub for Socio-Technical Innovation (https://www.designthinkinghub.org/) where I pursue multidisciplinary research (Levy et al., 2023).

Uri followed my professional development and was interested in my studies that foster creativity and innovation. I believe that the study I conducted with Uri set the groundwork for my research development. I continued to meet Uri in seminars he arranged and at social meetings, in which I continued to learn from him and enjoyed his special open attitude, intellect, knowledge and humour.

When Intuition and Logic Clash: A Tale About Experienced Software Engineers (Irit Hadar)

It was the year 2000. The tech industry was in dire need of software engineers. Courses offered to graduates of science and engineering, for ‘converting’ them to software engineers, appeared like mushrooms after the rain. I came very prepared to my first meeting with Uri, with a grand research proposal (or so I thought) for developing scalable, customisable and efficient pedagogic programs for creating the ‘perfect’ software engineer. After patiently listening to my enthusiastic pitch, Uri smiled and said: ‘I am charmed by your naivety; it reminds me of my six-year-old grandson’.

We spent the next three hours unraveling this naivety, during which I started to realise the level of depth one can achieve when really thinking of something, no matter how trivial it may first appear. Over our next meetings, brainstorming about possible research directions, I shared with Uri an opportunity I had been offered: to participate in workshops on object oriented (OO) modelling for advanced and experienced software engineers. Uri found this opportunity very interesting: ‘Look for difficulties demonstrated by strong populations’, he suggested, ‘it’s much more interesting than finding what they do well, or where weaker populations struggle’. Following this idea, I started studying how experienced software engineers design software, with the aim of identifying difficulties they face. It was my expectation to find them struggle or make mistakes in the most complicated, advanced tasks. To my utter surprise, I found them making simple mistakes, e.g. confusing inheritance direction between classes or defining inappropriate objects, when performing OO design. These were the kind of mistakes we do not expect to see even from novices, following their very first OO programming course. Interesting indeed!

Searching for explanations, Uri introduced me to the dual-process theory, explaining how our minds operate in two different modes: the automatic, experience-driven S1, also referred to as intuitive thinking, and the rule-based S2, also referred to as logical thinking. As we learned in this research, there are some contradictions between the OO paradigm and our intuitive thinking, creating pitfalls that are hard to avoid even for experienced software engineering practitioners (Hadar, 2013; Hadar & Leron, 2008).

The OO programming paradigm was created partly to deal with the ever-increasing complexity of software systems. The idea was to exploit the human mind’s natural capabilities for thinking about the world in terms of objects and classes, thus recruiting our intuitive powers for building formal software systems. It has commonly been assumed that the intuitive and formal systems of objects and classes are similar and that fluency in the former helps one deal efficiently with the latter.

However, our study, as well as several previous ones, showed that OO programming, analysis and design are quite difficult to learn and practice. Our research was the first to offer a theory-based explanation of the cognitive mechanisms from which some of the observed difficulties stem. Our empirical findings, collected using tools based on the qualitative research approach and analysed through the lens of the dual-process theory, suggest that a tension between intuitive and logical thinking modes may lead to simple mistakes software engineers exhibit when practicing OO analysis and design. This finding helps explain some of the previously documented, and some newly identified, difficulties in learning and practicing the OO paradigm, as well as guide the design of future quantitative experiments to understand how prevalent these phenomena are.

Following the completion of my Ph.D., Uri remained my mentor and friend for all these years, giving good advice and teaching me something new every time we met. After my graduation, I received a position as a faculty member at the Department of Information Systems, University of Haifa (in which I currently serve as a department head). Highly motivated by the results of my doctorate research, I have continued studying cognitive aspects of software engineering.

In 2020, my colleague, Prof. Pnina Soffer, and I invited Prof. Leron to collaborate with us on a research project aimed at supporting the cognitive processes of data miners, using an AI system as their extended cognition. The research builds on post-cognitivism theories and principles. In particular, we examined the cognitive processes taking place during process mining (data mining performed on business process logs) and analysed them based on the Prediction Error Minimization (PEM) principle. According to this principle, people make hypotheses in their minds, next, they observe the world to test them, and then they refine their hypotheses to reduce the error as reflected by the gap between the hypothesis and observation. It is an iterative process ending when one perceives their prediction and the real world as similar enough.

Working on this topic with Uri, in 2021, we won together a research grant from the Israel Science Foundation for our research proposal, titled ‘An Extended Cognition Approach for Supporting the Process of Process Mining’. In 2023, we published our first results in our paper, titled PEM4PPM: A Cognitive Perspective on the Process of Process Mining (Sorokina et al., 2023), presenting our analysis of the process of process mining (PPM) through the lens of the PEM principle. We used the think-aloud protocol, in which participants are asked to verbalise their thoughts, to elicit the cognitive processes of both novices and experts when performing process mining tasks. One of our main findings was that the participants who explicitly formed a prediction (i.e. hypothesis) about the process they were investigating demonstrated better performance in terms of providing accurate answers to their tasks compared to their peers (of similar level of expertise) who did not.

Intellectual Generosity (Orly Buchbinder)

My first encounter with Uri was when I was doing my undergraduate studies in mathematics education at the Technion–Israel Institute of Technology. For me, he was ‘Uri’ from day one. As a recent repatriate from Moscow, Russia, leaving behind and never looking back at the crashing Soviet Union, I wholeheartedly embraced the liberal mentality of the Israeli culture. This included calling my professors by their first names and a loose regard for authority. But Uri did not have to resort to authority to get students’ attention—every word he spoke, his whole demeanour, the warmth and excitement about knowledge in general and mathematics in particular were mesmerising. The course on Logo was an eye-opening experience for me, both mathematically and pedagogically. I had never experienced anything like it. Somehow, he made it seem that it was us, the students were the ones who wanted to learn and do more, rather than him telling us things or giving us tasks—Uri just asked really good and intriguing questions. Years later, when I was looking for a post-doctoral position, Uri told me jokingly that instead of writing a letter of recommendation for me, he wrote a single sentence, saying to my potential employers that I tend to ask really good and insightful questions. I felt honoured, as I knew that in Uri’s world, the ability to ask ‘good’ questions was held in high regard.

A much closer and more involved experience with Uri also occurred during the last year of my undergraduate studies. I do not recall the details, but I was missing one course credit in order to graduate and there was nothing in the course offering that I could take.

Uri stepped in to save the day and I did an independent study with him. This was another profoundly transformative experience for me. He gave me an abstract algebra book in English (the language I was far from fluent with, at the time) and asked me to read a few chapters containing the proofs of why three construction problems of antiquity—trisecting an angle, squaring a circle and doubling a cube—were, in fact, impossible to perform using only a compass and a straight edge. As a prospective mathematics teacher, I had never been asked to read and comprehend a mathematical text. This was a profound experience for me. Uri and I met weekly to discuss the proofs I read, and my understanding of them, and how these proofs gradually illuminated the main topic. By the time the course was over, I felt a deep sense of accomplishment and excitement about abstract algebra that was unparalleled by any other mathematical experience I had ever had.

The point I am trying to make through these detailed episodes is to convey the type of educator and mentor Uri was, his unique and personal way of connecting with others about mathematics, and his ability to spark joy and excitement about mathematics learning.

My master’s thesis was completed in 2005, under the supervision of Professor Orit Zaslavsky, on the topic of students’ understanding of the roles of counterexamples in disproving mathematical statements. Uri was on my dissertation committee and his influence on my work was profound. By that time, he had developed a deep interest in cognitive psychology, in particular the dual-process theory. Inspired by Uri, in the paper describing the key findings of my research, Orit and I used the dual-process theory to explain how students approach proving or disproving mathematical statements and how they treat counterexamples (Buchbinder & Zaslavsky, 2007). Specifically, I observed that students make an initial, immediate and intuitive guess about whether the statement is true or false (a type of S1 thinking), and then seek to validate their guess using a more deliberate, analytic approach (S2). Encountering a counterexample along the way may or may not lead to a correct conclusion. However, if students changed their mind once (from true to false or from false to true), their confidence in their new answer increased dramatically, even if the answer was not correct or not completely correct.

Uri’s fascination with cognitive psychology and the dual-process theory also affected my thinking about my Ph.D. project, which was conducted under the joint supervision of Orit Zaslavsky and Uri Leron. In my dissertation, I examined student understanding of proofs and refutations of universal (‘for all’) and existential (‘there exist’) mathematical statements. While conducting task-based interviews with students, I continued to observe the patterns of immediate insight versus elaborated, laborious search for proof or disproof that are so characteristic of S1 and S2 thinking.

Eventually, I did not use the dual-process theory as an analytic framework, as it did not lend itself naturally to my study design. Nevertheless, Uri provided incredibly supportive and insightful advice along the way. Our meetings were rather infrequent, but I left each one of them with new insights and clarity. Uri had an incredible ability to capture the essence of a subject from a bird’s-eye view and to provide structural direction without imposing his perspective. I try imitating his techniques in my work with graduate advisees and other mentees.

One of the things that stuck with me from my encounters with Uri is his often-repeated statement: ‘I am the most pedantic person that I know’. The lesson that I took from this to my work as a researcher and educator is that when you become ‘the most pedantic person’ about your own work, and you hold yourself to the highest standard, then you are neither surprised nor concerned about others’ critiques, because you have (hopefully) anticipated and even addressed them in advance. I have yet to become ‘the most pedantic person that I know’. It is a work in progress, but with Uri as a role model, at least I know that it is a goal that is possible to achieve.

On a final note, perhaps the most important lesson I learned from Uri is the value of intellectual generosity. Uri was passionate about sharing his passion with the people around him. He actively sought ways to give to others—of his knowledge, experience, new ideas and of his time. During the time of Logo, he found creative and innovative ways to share this with his colleagues, graduate students and undergraduates—including myself. When he developed an interest in cognitive psychology, he organised colloquia and gave numerous talks on the subject. He organised a reading group of faculty and graduate students to read and discuss articles on dual-process theory, and once the group dissipated after many of its members graduated, Uri continued sending the latest articles via email. Uri’s intellectual generosity is what I will miss the most about him.

Conclusion

Our stories of learning, supervision and professional collaboration are spread over four decades. They differ in style and in focus, reflecting not only upon the unique experience of each storyteller, but also upon the evolution of Uri Leron’s career. What do these stories have in common, other than commemorating the same teacher, supervisor, mentor and colleague?

One tenet of abstraction is ignoring the detail. As such, when particulars are ignored, there is a main thread that runs through every story, as well as through the opus of works of Uri Leron: it is the interplay between intuition and formalism (Fig. 2).

Fig. 2
figure 2

Intuition and formalism in different knowledge domains

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This interplay originates with a mathematician–teacher who brings a formalisation of knowledge and experience in terms of mathematical concepts and systems. The interplay continues with a mathematician–educator, and with the pedagogy related to Logo and the Logo turtle, where the imaginary turtle embodies a learner’s personal intuition awaiting a command in a formal language.

This interplay is also present in the ‘reducing abstraction’ theoretical framework, which reflects Uri’s work as researcher–theoriser. We can consider ‘abstraction’ as a level of formalism and reducing it as a learner’s attempt to interpret ideas in a more intuitive way, which is sometimes helpful but is often incorrect. Scaffolding for knowledge construction can be interpreted as an attempt to make the formal more intuitive, especially when it is applied in the design of technology-based learning environments for learning computer science. The bridging between students’ naive concepts and their formalisation emphasises the human aspect of computing and features Uri Leron as a computer science educator. The System 1–System 2 distinction is the awakening of a mathematician–psychologist, and it demonstrates a more explicit differentiation between what is intuitive and where the immediate intuition has to be pushed aside and temporarily ignored in order to allow for an advancement of thought within a discipline.

Furthermore, in exploring existential and universal statements, the formal mathematical conventions often appear in discord with immediately available intuitive reactions. For example, when one claims that ‘swans are white’, we tend to agree, intuitively disregarding the implied universality of the statement and the acknowledged existence of black swans. Here, again, we see a mathematician–teacher, a core that was not lost by welcoming a researcher, an educator, a theoriser, a computer scientist and a psychologist. Our stories are only a small sample of Uri’s stamp on the next generations of teachers and researchers.