Since May 2016, several school talks and workshops on the Arithmetic of Infinity have been run both in the UK and in Italy. These workshops are designed to allow students, working in small groups, to get to grips, in a dialogical and autonomous fashion, with the basics of numerical infinities and infinitesimals and to develop applications to simple problems. If you are interested in running a workshop, please contact Davide Rizza (d.rizza@uea.ac.uk).
Here is what some of the teachers involved in the workshops thought about the activity and the materials:
- Compelling! I love the idea of being able to utilize grossone in familiar ways, as oppose to the notion of ∞. They are thought provoking but achievable and surprisingly rewarding. The exercises have clarified/ help to clarify the concepts;
- The students have been on task and communicating with each other about this throughout the session and have made progress;
- Very accessible, even with no prior knowledge. Questions are well structured with progression in mind;
- Students meet the concepts of infinite series and infinite (indefinite?) integrals in year 12 and often ask about the concept of infinity. The materials give them more insight into this.
Scroll down for more information about the workshops, talks and conferences organised so far.
October 19th, 2019. Workshop at Norwich High School for Girls.
A group of sixth-formers discussed the Hilbert hotel paradox and possible ways of thinking about it and of tackling it. Among others, student examined the possibility of adopting the Arithmetic of Infinity as a way to reflect on the paradox. Interesting observations arose:
'If it [the hotel] is full, it’s full. The new person will have a room but the last person will be left without one.'
'If there’s an infinite number of rooms, they’ll never run out! Can’t be full? BUT. It says it is full so there is no room.'
'This means that Gross-one is the fist number of its own scale, an equivalent to 1 of sorts, and can therefore have numbers such as Gross-one + 1, 2Gross-one following?
A group of sixth-formers discussed the Hilbert hotel paradox and possible ways of thinking about it and of tackling it. Among others, student examined the possibility of adopting the Arithmetic of Infinity as a way to reflect on the paradox. Interesting observations arose:
'If it [the hotel] is full, it’s full. The new person will have a room but the last person will be left without one.'
'If there’s an infinite number of rooms, they’ll never run out! Can’t be full? BUT. It says it is full so there is no room.'
'This means that Gross-one is the fist number of its own scale, an equivalent to 1 of sorts, and can therefore have numbers such as Gross-one + 1, 2Gross-one following?
May 19th, 2018: Workshop at Liceo Scientifico Filolao, Crotone (KR), Italy
Students were asked what thoughts about infinity were stimulated by the workshop.
They said:
'Gross-one enables us to simplify the way we deal with infinity, it allows us to carry out calculations using this concept.'
'A simplification, certainly, because it enables us to make calculations.'
'The practice-based approach; infinity is usually only a theoretical concept, whereas in this manner it was really made more practical'.
Students were asked what thoughts about infinity were stimulated by the workshop.
They said:
'Gross-one enables us to simplify the way we deal with infinity, it allows us to carry out calculations using this concept.'
'A simplification, certainly, because it enables us to make calculations.'
'The practice-based approach; infinity is usually only a theoretical concept, whereas in this manner it was really made more practical'.
May 7th, 2018: Workshop at Liceo Scientifico Pitagora, Rende (CS), Italy
Students were asked whether using gross-one changed their intuitions about infinity.
They said:
'Yes, remarkably so.'
'Yes, has been a new, useful mathematical element that simplified the approach to infinity.'
Students were asked whether using gross-one changed their intuitions about infinity.
They said:
'Yes, remarkably so.'
'Yes, has been a new, useful mathematical element that simplified the approach to infinity.'
April 20th, 2018: Workshop at Thetford Grammar School, Thetford, Norfolk
Students were asked whether using gross-one changed their intuitions about infinity.
They said:
'Gross-one made it easier to explain the reasoning for my thinking about infinity.'
'Doing arithmetic with gross-one, i.e. infinity, means that not all terms including at least one infinite term are equal.'
'I now treat infinity as a number.'
'I understand that now infinity is treat as a number.'
'Infinity can be treated as a number.'
Students were asked whether using gross-one changed their intuitions about infinity.
They said:
'Gross-one made it easier to explain the reasoning for my thinking about infinity.'
'Doing arithmetic with gross-one, i.e. infinity, means that not all terms including at least one infinite term are equal.'
'I now treat infinity as a number.'
'I understand that now infinity is treat as a number.'
'Infinity can be treated as a number.'
November 16th: Workshop at Liceo Scientifico di S. Giovanni in Fiore (CS), Italy
Students were asked about their impressions of the concept of gross-one.
They said:
'It is odd but very intuitive, in the sense that we are able to understand a new concept of infinity. Earlier, we thought that an infinity minus a quantity was always an infinity. Now we can put things in better focus: an infinity is smaller than another infinity or greater than another infinity.'
'The geometric series of 1/2 that tends to 1 at infinity without reaching it: gross-one can get very close to this quantity and allows us to grasp it better.'
'It asked how many times a lamp could be turned on an off in one minute [Thomson's Lamp] and it was impossible to do this with traditional arithmetic, because you sum half plus half of this half [...] and you never get there. [...] Gross-one solved the paradox, I think.'
Students were asked about their impressions of the concept of gross-one.
They said:
'It is odd but very intuitive, in the sense that we are able to understand a new concept of infinity. Earlier, we thought that an infinity minus a quantity was always an infinity. Now we can put things in better focus: an infinity is smaller than another infinity or greater than another infinity.'
'The geometric series of 1/2 that tends to 1 at infinity without reaching it: gross-one can get very close to this quantity and allows us to grasp it better.'
'It asked how many times a lamp could be turned on an off in one minute [Thomson's Lamp] and it was impossible to do this with traditional arithmetic, because you sum half plus half of this half [...] and you never get there. [...] Gross-one solved the paradox, I think.'
July 5th, 2016: Teaching the Arithmetic of Infinity Conference at the University of East Anglia
This was a one day-event, primarily aimed at teachers, but students were also involved. It was devoted to discussing the prospects for introducing the Arithmetic of Infinity in A-Level Mathematics teaching or earlier. Both teachers and students tried their hand at selected exercises from the booklet First Steps in the Arithmetic of Infinity, available in pdf from this website, under Resources.
This was a one day-event, primarily aimed at teachers, but students were also involved. It was devoted to discussing the prospects for introducing the Arithmetic of Infinity in A-Level Mathematics teaching or earlier. Both teachers and students tried their hand at selected exercises from the booklet First Steps in the Arithmetic of Infinity, available in pdf from this website, under Resources.
May 27th, 2016: Pilot Workshop at Norwich High School for Girls, Norfolk
Students offered several insightful remarks about the pilot workshop, in which they were asked to learn how to carry out basic computations in the Arithmetic of Infinity, how to assign numerical measures to certain infinite collections of numbers and how to apply this knowledge to geometric series and Thomson's lamp.
Here are somme comments from the students involved:'It was pretty cool the idea that you could do stuff with it like count with it and do arithmetic with it, and use calculations with it. So I thought it was quite cool.'
'It made sort of a lot of sense that that sort of thing would exist and the conclusions you could draw from it also.'
'Even this year or last year, like not being able to use infinity even though it was like necessary seems a bit weird. So I suppose like it was kind of necessary to have something that you could use as an infinite number.'
'I kind of want to know how gross-one can be compared to other infinities, so like you can have bigger infinities than others and where is gross-one on the list of sizes of infinities. That’s a big question.'
'We do bring it up sometimes in Maths lessons- actually quite a bit.'
Students offered several insightful remarks about the pilot workshop, in which they were asked to learn how to carry out basic computations in the Arithmetic of Infinity, how to assign numerical measures to certain infinite collections of numbers and how to apply this knowledge to geometric series and Thomson's lamp.
Here are somme comments from the students involved:'It was pretty cool the idea that you could do stuff with it like count with it and do arithmetic with it, and use calculations with it. So I thought it was quite cool.'
'It made sort of a lot of sense that that sort of thing would exist and the conclusions you could draw from it also.'
'Even this year or last year, like not being able to use infinity even though it was like necessary seems a bit weird. So I suppose like it was kind of necessary to have something that you could use as an infinite number.'
'I kind of want to know how gross-one can be compared to other infinities, so like you can have bigger infinities than others and where is gross-one on the list of sizes of infinities. That’s a big question.'
'We do bring it up sometimes in Maths lessons- actually quite a bit.'