[ICME 12 Conference]

McMaster University’s new Interdisciplinary Science (iSci) program has been designed to “put into practice many of the innovative concepts linking science education and science research” with the aim to present a “holistic view of science through the interaction between various science disciplines.” As well, this four-year, undergraduate, small-size program “provides students with an opportunity to specialize in a selected discipline including chemistry, chemical biology, biology, mathematics, physics, biochemistry, radiation science, earth science, geography, and neuroscience.” In the first year of the iSci program, within a single course (which carries 80% of the full first-year course load) students study five science subjects (mathematics, physics, chemistry, biology, and earth science). I have been teaching mathematics in the iSci program since its inception. Going beyond anecdotes and personal experiences, I have been asking myself — What is the impact of this interdisciplinary approach on learning mathematics? Do students know and understand mathematics better? In what sense are they different from the students who do not have similar interdisciplinary experiences? These questions (and others) helped me form a research goal — to determine whether (or not) the rich environment within the iSci program enhances learning mathematics, both in terms of content knowledge (calculus of functions of one variable, power series and differential equations) and mathematical skills (formation of a precise mathematical argument, problem-solving, communication of scientific ideas, etc.).

[Canadian Mathematics Society winter meeting]

According to the report BIO2010: Transforming Undergraduate Education for Future Research Biologists, prepared by the National Research Council of the National Academies, "Biological concepts and models are becoming more quantitative, and biological research has become critically dependent on concepts and methods drawn from other scientific disciplines. The connections between the biological sciences and the physical sciences, mathematics, and computer science are rapidly becoming deeper and more extensive." The report continues, claiming that "In contrast to biological research, undergraduate biology education has changed relatively little during the past two decades. The ways in which most future research biologists are educated are geared to the biology of the past, rather than to the biology of the present or future. [...] undergraduate education must be transformed to prepare students effectively for the biology that lies ahead. Life sciences majors must acquire a much stronger foundation in [...] mathematics than they now get."

A number of Canadian universities have taken steps toward creating (or have created) mathematics and/or statistics courses appropriate for future biologists (and life scientists in general). In this session, we plan to discuss issues, challenges and successes surrounding the creation, development and teaching of math and stats courses for life sciences students (MSLS for short). For instance:
o Curriculum for (one-semester and full-year) MSLS courses: what math (calculus, linear algebra, several variables, probability and statistics)? How much math?
o What applications to include? How to best integrate math and applications?
o Institutional opportunities arising from the development of MSLS courses (teaching and collaboration between departments, interdisciplinary approaches)
o Institutional barriers to the creation and teaching MSLS courses
o Challenges in teaching MSLS courses (large classes, assessment, etc.)
o Moving away from the concept of service teaching; making an MSLS course attractive to all students, including math majors
o Resources for teaching MSLS courses, including textbooks and online materials
o Need for departmental resources (instructors, teaching assistants, etc.)

The objectives of this session are:
(1) To provide an opportunity for the faculty in charge of thinking about/ designing/ teaching math and stats for life sciences courses to come together and exchange ideas and experiences.
(2) To create a network of faculty interested in teaching math for life sciences. This could be a good opportunity for faculty from departments across Canada to collaborate, in order to develop good math courses.
(3) Decide on ways to continue this dialogue and further the collaboration between faculty at different universities.

[Ontario Math Education Forum meeting]

Teaching about body mass index, or about strength of a femur, or about forensics figuring out the location of impact from blood splatters or about the spread of pollutants in Lake Ontario in my life sciences math class, I find myself thinking - this stuff is so cool, how can one *not* be interested in it? It's things we must care about: our bodies, our health, our environment, our planet ... However, in the sea of faces in my classroom, at any moment, I can spot a few who are texting or checking their friends? status updates on facebook (and not even trying to hide it as I pass by them). Motivation is a tricky thing. In this presentation, I plan to talk about my attempts at stimulating students' interest in my classroom. What tools do I have at my disposal? Internet, real-life situations with real data, employment statistics of life sciences graduates, medical school acceptance rates. Does it all work?

[Topic Session, Canadian Math Education Study Group Meeting]

In words of Michèle Artigue “existing research [in mathematics education] can greatly help us today, if we make its results accessible to a large audience and make the necessary efforts to better link research and practice.” I plan to illustrate how these links between research and practice help me prepare lectures and class activities in teaching infinity within the context of (but not limited to) first-year calculus. Infinity belongs to a collection of those difficult mathematics concepts which are often in conflict with our intuition and understanding of real numbers. Teaching infinity consists of a hard task of stimulating students to reconstruct their existing cognitive models and engage in genuine abstract thinking. As well, I will comment on the ways infinity is covered in university mathematics textbooks. Rather than being general, I will be specific in describing my approach to teaching infinity, in the hope of hearing useful and constructive critiques.

Dans les mots de Michèle Artigue “recherche existante [dans l'éducation de mathématiques] peut considérablement nous aider aujourd'hui, si nous rendons ses résultats accessibles à une grande assistance et faisons les efforts nécessaires d'améliorer la recherche et la pratique en matière de lien.” Je prévois d'illustrer comment ces liens entre la recherche et la pratique m'aident à préparer des conférences et des activités de classe pour l’enseignement au sujet d'infini dans le contexte (mais non limité) du calcul de première année. L'infini appartient à une collection de ces concepts difficiles de mathématiques qui sont souvent en conflit avec notre intuition et compréhension de nombres réels. L'enseignement au sujet d'infini se compose d'une tâche dure de stimuler des étudiants de reconstruire leurs modèles cognitifs existants et de s'engager dans la pensée abstraite véritable. Aussi bien, je présenterai les observations sur les manières que l'infini est couvert en manuels de mathématiques d'université. Plutôt qu'étant général, je serai spécifique en décrivant mon approche à l'infini de enseignement, dans l'espoir d'entendre les critiques utiles et constructives.

[iSci Synthesis Symposium guest lecture]

Certain patterns of behaviour in animals are far from random - as a matter of fact, they can be explained and understood using mathematics. It turns out that animals are lot smarter than we think they are.

[Ontario Math Education Forum meeting]

Math content of the two Newton's books owned by McMaster University (Opticks and Universal Artihmetick). What are they about? What are Newton's thoughts on teaching math? Based on Newton's Opticks, we created a learning object -- to help students to discover and explore the complex mind of Isaac Newton. I will discuss a rationale for its construction, as well as a critique of use of online resources.

[iSci careers event]

Presentation on possible careers and jobs involving mathematics and statistics.

[Last Lecture series]

Learning calculus or algebra, we intuitively feel that there is lot more to math than functions, integrals and vectors. Math has a life on its own, stretching far beyond the rigid boundaries of some theory. What does that life look like? Math challenges and modifies the mind of the person learning it and working with it. How does this happen, and in what sense is a 'math mind' different? Drawing on my own experiences, I will attempt to answer these questions by examining a variety of common situations and contexts. Key words of my lecture include: global warming, advertising industry, sustainable living, bottled water, small and large, definition, critical thinking, terrorism, insurance, chance and risk.

[Dialogue with High School Teachers]

I plan to discuss some of the challenges and benefits associated with teaching math in McMaster’s Integrated Science program, where students in Level I are taught by interdisciplinary ‘teams’ of faculty, in a 24-unit course that involves delivery of content across all science disciplines. The approach uses ‘thematic modules’ (such as ‘Mission to Mars’ or ‘Cancer: The 21st Century Pandemic’) that emphasize the links between discipline areas and their relevance to modern societal issues.

[CLL 'Integrating Practices' Conference]

The aim of my research is to investigate the impact of an interdisciplinary program (in particular, McMaster’s iSci = Interdisciplinary Science Program) on learning of mathematics in the first year of university. To what extent does the rich interdisciplinary learning environment enhance and deepen learning, both in terms of content knowledge and mathematical skills (formation of a precise mathematical argument, communication of scientific ideas, etc.)? A pre-test/ post-test scheme is used to collect the evidence. In the first week of classes in September, students are administered an unannounced 50-minute survey, which gives an initial assessment of their general math knowledge and skills. After students complete the survey, no aspect of it is addressed in lectures. Eight months later, at the end of the school year, students are given the same survey, again unannounced. What is the purpose? None of the survey questions are explicitly discussed in lectures. However, throughout the first-year instruction in iSci students are exposed to a number of activities (such as problem-solving, critical thinking, creating precise scientific arguments, and so on), which can help them answer test questions better than at the start. The purpose of this approach is to determine whether students did learn math in the sense of being able to apply it to situations that were not explicitly addressed in lectures. In my talk I will present the data, discuss interesting and relevant findings, and comment on the implications for other disciplines, within as well as beyond the iSci program.

[Canadian Math Society Winter Conference]

Report on my research into the impact of an interdisciplinary program (in particular, McMaster’s iSci = Interdisciplinary Science Program) on learning of mathematics. I will present the data, discuss interesting and relevant findings, and comment on the implications for other disciplines, within as well as beyond the iSci program.

Discussion of various aspects of learning math, especially in ther contect of the first year life sciences calculus. Good strategies and things to avoid. Illustrated by specific examples and solutions from past tests.

[ISOMA (Independent Schools of Ontario Mathematics Association) Fall Conference, keynote address]

Yes it is - and it and will remain so for a long time! But what is different? In this talk, I plan to discuss how the new developments and discoveries in mathematics, as well as social and political changes, affect the ways we use mathematics, think about mathematics and teach mathematics.

[ISOMA Fall Conference, workshop]

An exchange of information and experiences on a number of issues that affect the transition from high school to university mathematics. How do we prepare students for a success in their university courses on calculus and/or linear algebra? How do we design good, useful, and motivating university math courses for students who will come to us in fall 2012? Based on actual materials that I use in my calculus courses, I plan to stimulate a mutually beneficial dialogue.

[Big Questions (Science 2B3) guest lecture]

Concept of spatial dimension. Through case studies, we develop ways of thinking about fourth, fifth and higher dimensions. Dimension as an important aspect of space we live in.

[Treehouse talk]

In this talk, I will touch upon some amazing aspects of infinity. Through a sequence of mental experiments we will discover concepts which form the basis of modern mathematical understanding of infinity. Georg Cantor (one of the ‘discoverers’ of infinity) exclaimed 'I see it, but I do not believe it,' as he witnessed his discoveries, unleashed, shaking mathematics to its core and changing it forever. Thinking about infinity is not without danger - Cantor died in a mental institution. Want to experience what drove him to insanity? Come to the talk!

[SHAD Lecture and Workshop]

Mathematics has always been an inexhaustible source of inspiration to artists and architects. Case studies of cubism, and Ndebele African art patterns. Also, Bilbao Guggenheim Museum, Esplanade Performing Arts Centre in Singapore and Sydney Opera House. We will build a model of a simple geodesic dome.

[SHAD Lecture]

In this lecture we will discuss mathematical models that are used in weather forecasting. Analysis of such models will lead us to the concept of chaotic behaviour. We will try to understand why is it not possible to calculate accurately a weather forecast even for short time intervals (say, two-weeks in advance). We will discuss soliton waves, so that we can explain the nature of tsunami waves; also, few facts about rogue (freak) waves.

[Canadian Mathematics Society Conference; Adrien Pouliot Award]

In this talk I plan to share my thoughts, emotions and ideas about two aspects of my life that I’m passionate about: mathematics and teaching mathematics. Going beyond our usual daily frustrations (Why don’t they just get this? They did not do conic sections in high school? What’s so difficult about uniform convergence?), I will suggest strategies that might help us improve things a bit. Drawing on my experiences in designing and teaching within interdisciplinary programs at McMaster, I will argue that communication – at all levels – could be the key to improving the quality of our teaching.

[Ontario Math Education Forum meeting]

Analysis of university students' perceptions of mathematics and mathematicians.

[Big Questions (Science 2B3) guest lecture]

Concept of spatial dimension. Through case studies, we develop ways of thinking about fourth, fifth and higher dimensions. Dimension as an important aspect of space we live in.

[SHAD Lecture]

Infinity has many faces. Sometimes, we perceive it as a number larger than all numbers. For indigenous people of Australia and New Guinea, infinity begins at six. For the Moors - creators of the most beautiful mosaics the world has ever seen - infinity was in a repetition of a single artistic motif. Infinity is eternity, divinity, love, death ... Modern mathematicians have embraced infinity as one embraces a good friend; but a few have contemplated suicide rather than being forced to face it. Georg Cantor, early explorer of infinity, died in a mental institution. What exactly drove him to insanity? Does infinity really exist? If so, what is it? Where can we find it? Is our universe large enough to encompass infinity?

[SHAD Lecture]

Certain patterns of behaviour in animals are far from random - as a matter of fact, they can be explained and understood using mathematics. It turns out that animals are lot smarter than we think they are.

[SHAD Lecture and Workshop]

Mathematics has always been an inexhaustible source of inspiration to artists and architects. Case studies of cubism, and Ndebele African art patterns. Also, Bilbao Guggenheim Museum, Esplanade Performing Arts Centre in Singapore and Sydney Opera House. We will build a model of a simple geodesic dome.

[Canadian Mathematics Society Conference]

In this session I plan to examine several issues that emerge when we try to create a resource for, or teach, applications in mathematics. Although most examples I will present focus on mathematics for life sciences, my hope is that the discussion will broaden the context. Thinking about creating a resource – how “deep” should an application be? We certainly need to provide motivation and outline the appropriate background, but how far do we go? Would a line or two suffice? A paragraph? A page? How do we address potential complexities of dealing with the mathematics material that’s involved? How do we teach applications, especially in a large class setting? What kind of teaching is required, and what are the most appropriate teaching methods? What do we want students to learn, and what do students learn from applications? There will be plenty of time for discussion. Hopefully, we’ll end up with a small step forward in thinking about some of these issues.  

[Canadian Mathematics Education Study Group Conference]

Miroslav Lovric, Olivier Turcotte, Walter Whiteley

In the international context, Universities in England are closing departments of Chemistry, and Physics – and leaving service teaching to other programs. Under financial pressure, Universities in Canada are also starting to close programs with low enrollment. The key to these decisions seems to be enrollment and graduation rates. Mathematics programs are being judged by recruitment and retention numbers, as well as the quality of the support they provide through service courses. At the least – this impacts whether people are hired into mathematics departments. At the extreme -- some programs within mathematics departments will be closed, or even whole departments might be closed (as has happened in some Ontario Colleges). There are a number of ‘causes’ that have been suggested – some of which are observed on an international scale, some of which are local to the cultures of the region or the institutions, some of which connect to student motivation, and the quality of student experiences. A number of responses have been suggested including:

* collaborations with other disciplines, such as education and interdisciplinary science programs;
* offering challenging mathematics in senior secondary education and undergraduate programs;
* refocusing on the balance between topics and processes in the objectives and courses of post-secondary mathematics programs.

We will draw from the research we can collect, from the data and stories brought by participants from their institutions, and from broader recommendations from groups such as the Mathematical Association of America to address these issues. We are hoping that our group will attract participants with a large spectrum of experiences in mathematics. We will also look at the image of mathematics / preparation for the mathematical sciences in the high schools, as it connects to recruitment. Where the experience of mathematics in the first-year of post-secondary education very different from the high school image - does this contribute to attrition? We will investigate how other factors (such as quality of instruction or availability of adequate resources -e.g. good textbooks and good support) influence students' decisions to stay (or not) in mathematics .Through the three days, we will consider what can be done to recruit and retain students with a strong interest in mathematics either as their major focus, or as part of their broader learning over several disciplines.

[AMCA Lecture]

There is so much to look at and admire in Alhambra Palace (Granada, Spain) - exquisite rooms decorated with stone and wood carvings, finest ornaments and calligraphy; night sky represented in ceilings built of thousands of pieces of wood; gardens, courtyards and fountains; monuments, towers, archways - the list is endless. Quite possibly, an immense wealth of ornamental patterns, friezes, mosaics, star designs, and brickwork motifs tops the list. Among those, mosaics are – mathematically – the most interesting and the most intriguing. Scientists and artists working in the Islamic world pushed geometry to its limits, creating patterns and configurations whose sophistication has not been surpassed. Investigating numerous possibilities, based on experience and long tradition, architects and builders of the Alhambra (14th century) created all possible mosaics – in the sense of the mathematical classification of plane crystallographic groups. Mathematics behind all this is intuitive: we will discuss the concept of symmetry and see how it can be used to analyze mosaics.

Animals and Mathematics

[Arts and Science Lecture Series]

Certain patterns of behaviour in animals are far from random - as a matter of fact, they can be explained and understood using mathematics. It turns out that animals are lot smarter than we think they are.

[Fields Mathematics Education Forum]

Comments on remedial efforts at McMaster University and elsewhere. What does research say about it?

[Big Questions (Science 2B3) guest lecture]

Concept of spatial dimension. Through case studies, we develop ways of thinking about fourth, fifth and higher dimensions. Dimension as an important aspect of space we live in.

[Engineering and Science Olympics guest lecture]

Mathematics is not like calculus (finished and unchanged basically since early 1900s). It is a dynamic discipline, that is part of almost any scientific effort, and almost any discipline people are involved with today.

[SHAD Valley lecture]

What does the space that surrounds us look like? Can we determine whether it Is finite or infinite? In this presentation, we will examine studies that will help us understand things a bit better. What is the shortest path between two points in space? How would our mundane three-dimensional existence change if we were suddenly pulled into the fourth dimension? We will develop a reasonable scenario for time travel into the future.

[SHAD Valley lecture]

Case studies of present-day research efforts in mathematics: error correcting codes and computer security, data compression; weather forecasting and understanding disasters (tsunami, freak waves),etc.

 Suggestion For A Theoretical Model For Secondary-Tertiary Transition In Mathematics

[European First Year Experience 4th Annual conference]

Abstract (based on two papers I published recently): The transition ('gap') between secondary and tertiary education in mathematics is a complex phenomenon covering a vast array of problems and issues. Although there is evidence of similar gaps in other disciplines in science and beyond, it seems that the transition in mathematics is by far the most serious and the most problematic. Based on certain anthropological notions, we suggest a model for the secondary-tertiary transition in mathematics. Although we focus on the area of mathematics, and on transition between a secondary institution and university (and, to a lesser degree, college), the model could be applied, with minor modifications, not only to other transitions in mathematics (primary-middle, primary-secondary, etc.) but to other areas as well. This model provides a useful lens through which we can examine the process of transition from secondary to tertiary mathematics, and also offer ways of improving present strategies and suggest new ones.

[May@Mac]

What is integrated science about?

[CMEF - Canadian Math Education Forum]

Panel presentation and discussion. Title says it all.

[CMEF - Canadian Math Education Forum]

Working group on studying the design of textbooks. Mathematics instruction in North Americais strongly textbook-driven. Besides determining material to be taught, textbooks - implicitly or explicitly - suggest teaching strategies. Perhaps contradictory to one’s expectations, textbook-related research is modest, in particular as it relates to the quality of mathematical content and its exposition. I have been looking at textbooks commonly used in Ontario (high school and university), to determine to what extent, and how, they contribute to the creation and strengthening of students’ misconceptions (e.g. sources of systematic errors). Thus, I investigate to what extent textbooks promote (or not) deep, conceptual understanding of the
material they present.

AERA Report

[Ontario Mathematics Education Forum]

Report on the chapters on teacher and teacher education and instructional resources.

[V International Conference on Multimedia and ICT in Education (m-ICTE2009)]

The aim of my presentation is two-fold. I will discuss a multitude of issues - coming from my experience, as well as that of my colleagues - related to teaching mathematics using various information and communication technologies. As well, I will talk about the ways my teaching has changed in the last decade, and critically asses the role that ICT has played (or not) in bringing forward those changes. In a way, this is my attempt at trying to answer the question to which extent ICT can help us teach and learn mathematics better.
I plan to discuss the ways I use internet, WebCT, e-mail and personal response devices (clickers) in teaching, focusing on conceptual issues rather than on specifics of technical problems and situations.
At the moment, I have been involved in creating a curriculum for the Integrated Science Program at my university (as member of the group, my responsibility is mathematics and its integration). I will present my thoughts and ideas on how ICT will help us build a rich and dynamic curriculum that emphasizes research skills, problem-solving and independent learning from the very beginning of a semester.

Discussion on issue of transition from high school to first-year university mathematics. Profile of our first-year mathematics for life sciences course.

[Ontario Mathematics Education Forum, with Mollie O'Neill and Maritza Branker]

Investigation of quality of presentation of mathematics in textbooks K-12 and first-year university

Math, Art and Architecture

[CMS Winter meeting]

I will discuss rationale for and implementation of a new course 'Calculus for Life Sciences' that I created in an attempt to address (among other objectives) certain issues that emerge in transition from secondary to tertiary mathematics. Course design has been, in part, based on my previous research in transition, and on insights that I gained from analyzing high school mathematics background surveys that I have been administering to incoming students for over 5 years. Among the largest challenges that students in transition face are poor algebraic skills, inadequate mathematical reasoning and learning strategies, as well as robustness of their preconceived notions about what a calculus course should be about.

So ... what topics in calculus have been kept, and what is out? Do we still do proofs by contradiction, are there any theorems left in the course? Why are students now studying discrete-time dynamical systems and stability? Relevant applications, presented in appropriate context replaced related-rates and other artificial problems that populate calculus textbooks. We study allometric models (blood circulation time vs. body mass, scaling of bones), population and growth models, radioactive decay and dissolution of drugs, growth of cancer, to mention a few.

It is too early to provide any kind of answer to the ultimate question - does it work? However, I will share my initial experiences and reactions, as well as my students' reactions and critiques that I will collect during the semester.

[Psych 3TT3 workshop]

Teaching assistants as teachers.

[Ontario Math Education Forum]

Experience with the new Math/Calculus for Life Science course that we started at McMaster University.

[Science 2B3 (Big Questions) guest lecture]

Concept of spatial dimension. Through case studies, we develop ways of thinking about fourth, fifth and higher dimensions. Dimension as an important aspect of space we live in.

[SHAD Valley program]

What does the space that surrounds us look like? Can we deternine whether it Is finite or infinite? In this presentation, we will examine studies that will help us understand things a bit better. What is the shortest path between two points in space? How would our mundane three-dimensional existence change if we were suddenly pulled into the fourth dimension? We will develop a reasonable scenario for time travel into the future.

[SHAD Valley program]

Case studies of present-day research afforts in mathematics: error correcting codes and computer security, data compression and transmission of large amounts of data; weather forecasting and understanding disasters (tsunami, freak waves)

[Second Canada-France Congress] .

Mathematics instruction, in many North American secondary and tertiary institutions, is strongly textbook-driven. Textbooks determine not just what is taught, but also suggest strategies that are to be used in teaching. Moreover, math textbook represents a message from the larger mathematical community about what student should learn.

Perhaps contradictory to one's expectations, textbook-related research is far from developed. There have been various attempts at evaluating textbooks, or studies exploring the relationships between textbooks and curriculum or between textbooks and learners; or studies that compared textbooks in different countries. However, the amount of research related to the quality of mathematical content and its exposition appears to be quite modest. Very few mathematics education researchers have taken a really close look at what is in the textbooks, with the focus on how the material is presented and what kind of learning may be implied.

By examining a variety of case studies, I will illustrate several findings of my research (collaboration with Ann Kajander). We took a closer look at textbooks commonly used in Ontario (grade 12 and first year university), to determine to what extent, and how, mathematics textbooks potentially contribute to the creation and strengthening of students' misconceptions (e.g. sources of systematic errors). This way, we investigate to what extent textbooks might promote (or not) deep, conceptual understanding of the material that they present. I will attempt to convince the audience that this is an area of research that requires genuine collaboration between mathematicians and mathematics educators - that, in the end, will benefit both groups.

[Ontario Math Education Forum] ...

Discussion of a variety of issues that affect students' decision to pursue (or not) mathematics in high school and in university.

[OCMA (Ontario Colleges Mathematics Association) Conference] ...

What is the place of mathematics in modern society? The range of applicationsof mathematics has never been as broad as it is today. Paradoxically perhaps, the number of people who are interested, or say they are good at math, or decide to study math has been dwindling ... How important is it to know mathematics? Examination of the role and politics of mathematics and an analysis of issues in teaching and learning mathematics will lead us to fundamental questions, including the one which, quite possibly, cannot be completely resolved: what is mathematics?

[Math and Society; Thematic Fridays series of lectures]

Interesting facts, stories on historic development of teaching and learning mathematics in various cultures.

[lecture by Miroslav Lovric]

Infinity has many faces. Sometimes, we perceive it as a number larger than all numbers. For indigenous people of Australia and New Guinea, infinity begins at six. For the Moors - creators of the most beautiful mosaics the world has ever seen - infinity was in a repetition of a single artistic motif. Infinity is eternity, divinity, love, death ... Modern mathematicians have embraced infinity as one embraces a good friend; but a few have contemplated suicide rather than being forced to face it. Georg Cantor, early explorer of infinity, died in a mental institution. What exactly drove him to insanity? Does infinity really exist? If so, what is it? Where can we find it? Is our universe large enough to encompass infinity?