# The Proof Hat

In our study group, we read Tommy Dreyfus’ paper Why Johnny Can’t Prove. In the paper, he describes how beginning university mathematics students find it difficult to see the difference between different kinds of mathematical reasoning.

Often, teachers and textbooks use both formal and informal explanations to justify claims. Both are needed, and for an experienced mathematician it is easy to see which one is which. Students, on the other hand, may get confused and cannot form a clear picture on what is meant by a proof.

This is how we came up with the idea of a proof hat. It makes it possible to distinguish between different kinds of justifications. When teacher writes a formal justification for a claim, he/she wears the proof hat. When a topic is discussed more informally, the proof hat is not used.

It seems that the proof hat makes students pay attention to the different levels of justification, since if the hat is missing, they will let the teacher know. And it definitely brings cheerfulness to the classroom.

# Eleven Commandments for Instructors

Instructors working with students. The instructors wear orange vests so that the students can spot them.

At our department we are using the Extreme Apprenticeship method that emphasises students’ active work. Students are assisted in their work by instructors who are usually senior students.

The instructors support students’ own thinking and teach them study skills. They try, at the same time, not to spoil the joy of discovery for the students. This kind of teaching can sometimes be difficult, especially if you are used to thinking that a good teacher is one who can explain everything thoroughly to the students. That is why our instructors take part in training that lasts throughout the semester. The training consists of weekly meetings in which the pedagogy of instruction is discussed.

Together with the instructors, we have put together some guidelines. The list has changed during the years, as we have all learned more about good instruction. This is what the guidelines look like at the moment:

1. Listen. Encourage the student to talk, and listen to what he/she says. Let the student’s needs lead your instruction.
2. Guide individually. Students are different. Some may need help with the basics and require very concrete advice. Others are just asking for a small hint. Try to find out what the student needs, and address that need.
3. Let the student do and discover. The aim is that the student works towards a solution with the support of the instructor. Guide in such a way that the student can have ‘aha’ moments.
4. Be encouraging. Students may be very insecure in what comes to mathematics, and feel that they are not doing well enough. Be encouraging and try to find something good in the student’s work.
5. Be active. Circulate among students on your own initiative and say hi to them. It is easier for the students to ask questions if the instructor has opened the conversation.
6. Divide your attention. Do not let one student take too much of your time. Note also that sometimes it is good to let the student think about the problem on his/her own.
7. Help the student in reading the course material. Reading mathematics is difficult for the students, and they may try to use the instructor as a data bank. The instructor should encourage the students to read the course material and advise them how to do this.
8. The instructor does not need to know everything. He/She can investigate the topic together with the student. This way, the student sees how a more experienced mathematics student works.
9. Teach study skills. The aim of guidance is not only to help the student to solve a task, but to show how mathematicians tackle problems they face.
10. Do not take emotional outbursts personally. React to students’ problems and outbursts with compassion and empathy, but do not let them worry you too much. The reason behind outbursts can, for example, be the insecurity of the student.
11. Encourage co-operation. Students should learn to discuss mathematics. Encourage students to collaborate, especially if many of them are working alone on the same problem.

# The Durham Experience, Part II

In the academic year 2014–2015 I had the privilege to teach a lecture course in Durham University in the United Kingdom. Durham is a rather prestigious university that is very selective in its student intake. In these entries I record some of my thoughts concerning the experience. Although I could only witness the system in Durham, many details can most likely be generalised to other UK universities.

# Lecturing Algebra II

Despite its name, Algebra II is the first course in abstract algebra offered in Durham. It is a second-year course, spanning the whole academic year. Compared to courses taught for example in Helsinki, the course covers roughly the material for a basic one-term algebra course plus a slightly shorter and more advanced course. The course had about 170 students.

I was eager to face the challenge of teaching a full-fledged lecture course in a new country. Having taught algebra before, I knew it to be a difficult topic that demands a lot of abstract thinking and becoming familiar with many new concepts. My colleagues in Durham had also warned me that it would be very difficult to get the students to appreciate how much and what kind of work they would have to do in order to learn the material effectively.

While I was planning my course, it soon became clear to me that to encourage the students to work in an effective manner for their own learning, I should implement methods that in the University of Helsinki go under the name “Extreme Apprenticeship Model”. This model has been applied to teaching certain mathematics undergraduate courses in the Department of Mathematics and Statistics since 2011. After its introduction, the model has enjoyed great success. In particular, it has been successful in engaging the students, teaching them good, professional-style studying habits, as well as making them more committed to their studies in the department.

It was, however, impossible for me to run a full-fledged Extreme Apprenticeship course since I had much less control over the teaching resources than is usual in Finland. Therefore I needed to adapt my teaching to a new environment. In such cases I find it useful to consider the motivations behind the practices in order to decide what is essential and what is not. Doing so, I found that I wanted to focus on

1. Improving the students’ working habits.
2. Enabling strong bi-directional feedback that would allow the students to know what is expected from them and the teaching to be guided by the students’ needs.
3. Teaching abstract thinking and communicating these thoughts in speaking and writing.

Considering these points led me to implement roughly the following set of methods:

• a form of flipped learning, in which the students would work individually for their own learning, and the lectures are used to solidify the students’ knowledge, build a general picture, weed out misconceptions, and discuss effective study methods
• course notes designed for the students to be able to read and understand to a great extent by themselves
• detailed descriptions of learning objectives and self-evaluation exercises based on them
• weekly meetings with homework markers to ensure they would know what I require of the students, and that I would find out how the students are doing
• the possibility of resubmitting certain homework questions in order to employ the markers’ comments if the first solution was not good enough
• activating the students during the lectures with small group tasks, questions, and polling exercises
• activating conceptual thinking through tasks that would require explaining, connecting and evaluating.

In short, the course ran in the following way. First, the students were given typed, well-prepared notes. They familiarised themselves with new concepts through simple homework problems. They would hand in the problems to be marked. Once the markers had had a few days to look at the scripts, we would meet together to discuss what the students had come up with. Then the markers would go back to finish the marking, while I would discuss the most common mistakes and important misconceptions in the lectures. I would use online voting systems in the lectures to help gauge the students’ learning pace and to promote meaningful discussions. The lectures would also progress to deeper theory and examples, and the next week’s homework would – in addition to preparatory material – also contain harder problems on topics that were discussed during the week.

Running Algebra II. The arrows between boxes of different colour indicate the flow of information or feedback. The double arrows ML and SL symbolise bi-directional feedback between markers and lecturer and between students and lecturer.

There were two types of questions in the homework. Most of the questions were marked based on honest attempt whether the answer was right or wrong. For these questions, the markers were not required to comment anything if they did not have time for it. The students could later compare their solutions to the model solutions. On the other hand, a few questions were more substantial, usually requiring a written proof. These questions were marked based on the quality of the answer – not only of the mathematical correctness, but also the quality of writing. It the solution was not adequate in the first week, the student was encouraged to resubmit the question the following week and would receive the marks after the resubmission.

In applying the above methods, I faced several challenges. As already mentioned in my previous post, the lecture system in Durham is very rigid. My lecturing hours (twice 50 minutes a week plus a fortnightly “problems class”), as well as tutorials (50 minutes per a 15 person group fortnightly) were all prescribed to me. Also the teaching assistants giving the tutorials and marking the homework (mostly separate people, to my surprise) were decided for me. One major difference with Extreme Apprenticeship method in Helsinki was that I could not provide the students with a drop-in workshop with instructors that would always be present. Another challenge was that the other teachers in Durham that I got into contact with had very little experience about any of the methods I was going to apply, so I could not count on their advice as much as I was used to in Helsinki.

The biggest obstacle, however, was that the students were in general unaccustomed to anything else than the traditional teaching style where the lecturer writes on the blackboard and students take notes, which they then later refer back to when doing their homework. This limited background, of which I only learned gradually when I was already teaching the course, created a lot of mental opposition towards my teaching right from the start of the course. Of course, I was prepared to explain any new methods I would be using, so that the students would be aware of what was expected from them and how they would benefit from the diverse exercises, but in Durham I felt I was required to explain myself much more than usual. There was a constant need to reassure the students of the fact that they were indeed learning the necessary skills even when they did not recognise the “unconventional” methods we were using.

Despite initial misgivings, the students gradually learned to work in the new way, some quicker, some at a slower pace. However, student feedback questionnaires half-way through and at the end of the course show that the students were not in general happy with the lectures, saying that these were useless and not teaching them anything. Being inefficient is a well-known problem of mass lectures, and the amount of negative feedback on this aspect seems only to indicate that the students have not previously recognised this inefficiency. On the other hand, the prepared course notes received a lot of positive feedback. It seems to be very rare in Durham that the students would be given such complete sets of notes.

During the course, I observed the students closely in order to spot any learning difficulties resulting from the change in teaching style. Some individuals indeed claimed that they were facing serious difficulties, but on the whole I could see no significant indication that anything would be going wrong. Even if the students complained, they were still clearly learning. Their results in the final exam were as good as would be expected from such a course in general.

While one might not experience any obvious adverse effects from the different kind of teaching, it would be good to know whether my efforts actually made a difference in the end. From the teacher’s point of view, I certainly felt that I was spending my time much more meaningfully than I would have using a more traditional style, and that kept me highly motivated throughout the year. Many students also changed their minds about my methods and gave me praise in private emails and conversations. One student said that I had taught him to write proofs, and claimed that this had helped him in the exams also on other courses. Some physics students thanked me for giving examples of physical applications and for teaching not only the content but also the method of studying mathematics. Another student, who had originally despised my lecturing style, thanked me in the end, saying that during the course he had realised the value of his own work and that this was the most important thing he had learned from the course.

The last mentioned comments show that the methods I employed were about so much more than straightforward delivery of definitions and theorems. The students were taught to read and write mathematics on their own and construct their personal understanding instead of simply memorising results and methods. Although this kind of instruction may be unusual to see in the Durham maths community, I trust that my example shows that, if found valuable, it is fully possible to implement.

# Extreme Apprenticeship

By Johanna Rämö.

Since 2011 we have been using the Extreme Apprenticeship method in teaching first year mathematics in the University of Helsinki. The core idea of Extreme Apprenticeship is to support students in becoming experts in their field by having them participate in activities that resemble those of professionals. The main method of teaching is one-on-one instruction, and the students are encouraged to work collaboratively. During the past few years, Extreme Apprenticeship has changed the way the teaching staff and students in our department view learning and teaching.

Extreme Apprenticeship promotes active engagement of the students. Each week, the students start studying a new topic by solving problems given by the teacher. They get as much help as they need from the teaching assistants.

The teaching assistants do not give answers but guide the students towards solution. They help the students in reading course literature and gaining studying skills. The teaching assistants are undergraduate or graduate students who undergo a training that lasts throughout the semester. Training is important, as teaching in a new way can be very challenging.

The physical learning environment we have created in the middle of our department is an important part of the Extreme Apprenticeship method. It encourages the students to collaborate with each other and interact with the teaching assistants.

The students hand in their solutions so that the teaching assistants can read them and give feedback. Emphasis is on learning how to write mathematics, which is very difficult for most students. If the solution is not good enough, the student has an opportunity to improve it. The feedback is two-directional: by reading the students’ answers, the teachers get to know what kind of problems the students have and can react accordingly when planning the tasks and lectures.

After the students have worked on the course assignments, there are lectures. The students have already familiarised themselves with the topic of the lectures by doing the tasks. This is a simple and effective way of making the students prepare for the lectures. The lectures are not for delivering content or going through details. Instead, it is possible to discuss the meaning and consequences of definitions and to address misconceptions. The students do not need to just sit and listen, but they get to work in pairs, discuss with each other and vote on questions posed by the lecturer.

After the lectures, students are given more challenging tasks concerning the topics that have been discussed in the lectures. At the same time, studying a new topic starts with relatively easy tasks.

Compared to traditional teaching, Extreme Apprenticeship has increased student engagement and effort. It has enabled moving from rote learning towards conceptual understanding. Even though the students have to work hard, the feedback from them has been overwhelmingly positive. All in all, the method has had a considerable impact on the atmosphere of our department. The corridors and classrooms are filled with students who solve problems together and talk about maths with excitement and enthusiasm.

Rämö, J., Oinonen, L., & Vikberg, T. (2015, February). Extreme Appreticeship – Emphasising conceptual understanding in undergraduate mathematics. Paper presented at the 9th Congress of European research of mathematics education, Prague, Czech Republic.

Rämö, J. & Vikberg, T. (2014). Extreme Apprenticeship – Engaging undergraduate students on a mathematics course. In Proceedings of the Frontiers in Mathematics and Science Education Research Conference 1-3 May 2014, Famagusta, North Cyprus (pp. 26-33).

Hautala, T., Romu, T., Rämö, J. & Vikberg, T. (2012). Extreme apprenticeship method in teaching university-level mathematics. In Proceedings of the 12th International Congress on Mathematical Education, ICME.

# The Durham Experience, Part I

In the academic year 2014–2015 I had the privilege to teach a lecture course in Durham University in the United Kingdom. Durham is a rather prestigious university that is very selective in its student intake. In these entries I record some of my thoughts concerning the experience. Although I could only witness the system in Durham, many details can most likely be generalised to other UK universities.

Compared with Finland, the English universities hold a much tighter grip of both students and the teaching staff. For someone used to making all the decisions themselves, from planning your own schedule as a student to choosing the most suitable date for your exam as a lecturer, the inflexibility of the UK system may come as a shock. On the other hand, there is also a feeling of security and comfort when things are planned for you in advance.

Studying an undergraduate degree in mathematics in England takes typically 3–4 years and costs up to 9000 pounds in tuition fees per year (Scotland has no tuition fees). As a maths student, you will not be studying any other subject, although courses in some subjects such as theoretical physics can in manyuniversities be incorporated in a maths curriculum. You will be able to choose between a few optional courses, but mostly the schedule will be decided for you. You will simply receive a timetable in the beginning of each year telling you what lectures you will be taking and when.

In Durham, a typical maths course (they are called “modules”) takes a full academic year, consisting of lectures, tutorials and homework assignments. There are three terms: autumn, spring and summer (these are dubbed “Michaelmas”, “Epiphany” and “Easter”, respectively). To give a sense of how the three terms are organised, this year the autumn term lasted from 6 October until 12 December, spring term from 12 January until 13 March and summer term from 20 April until 19 June. Spring term ends at Easter, after which there is a five-week holiday before the start of the summer term. The summer term is aimed at revision: there are 2–3 weeks of revision classes, and then the exams begin.

All exams are held in the summer term, in either May or June, regardless of when the course itself took place. On a typical course, the final grade depends entirely on the exam. Homework or mid-term exams, if any, will not contribute. If a student fails in the exam, they can retake it in August, but in the resit it’s not possible to score more points than the minimum for a pass. No further resits are allowed.

After the exams, the students wait in anguish until all the scripts have been graded before they receive their final grades. They are not allowed to view their own scripts, at least not in the maths department, nor even know which questions scored them points and which did not. This is in sharp contrast to the practice for example in the maths department in Helsinki, where the exam, its solutions and some notes on the grading are published, together with question-by-question statistics, and where a student can at any time request to see their script in presence of the lecturer.

Let us switch to the teacher’s point of view. As for the students, the timetable is completely decided in advance, and the teacher cannot affect the amount or length of lectures (typically two 50-minute lectures a week in Durham) or tutorials (50 minutes per a 15 person group fortnightly). The content of the course is also prescribed, but in practice you are allowed to take some liberties.

The final exam, as well as the resit exam, need to be written in January, which can be quite difficult as half of the course is yet to be lectured. (Even more so for those teachers who are teaching a half-year course starting in January!) After the exam is ready, together with solutions and a grading scheme, it will be checked by a colleague, and subsequently by an external checker from another university. This is to ensure the exam is of a suitable level of difficulty and free of errors. After the exam, there is a fervent period of grading, which has to be done in a week or so for a typical exam. The grading is also checked by a colleague for omissions or mistakes in adding the points.

For a Finn, all this railroading seems a bit excessive. Not being able to retake the exam more than once, to retake it for a better grade or to manage one’s timetable oneself seems to make the students’ life rather stressful. There is a tendency to work less in the autumn and then attempt to absorb most of the content in the revision period. The stress and worry of the students is also reflected in the course feedback: if the students feel the teacher is not preparing them well enough for the exam, their criticism often worded in strikingly harsh language.

However, the students are not left to struggle on their own, but actually rather conscientiously taken care of. Each student is assigned an “academic advisor”, a permanent staff member with whom the student has regular meetings throughout the year. Each student’s homework activity and tutorial participation is monitored, and if there is cause for concern, the student is contacted for an explanation. This manages to ensure that the students keep doing their homework even though the homework does not count towards their final grade.

The staff enjoys similar fostering. Every new lecturer gets a “mentor” who can be approached if case of unforeseen problems, practical or otherwise. The mentor will also observe the teaching of their mentee to see that everything is going well. Furthermore, every teacher is asked to observe at least one lecture or tutorial of another randomly selected teacher. I find this system quite appealing, as following someone else’s class very often gives me new ideas and revelations. I also found the checking of other people’s exams very useful. One can quickly become blind to one’s own mistakes, and it is easy to misjudge the skills of the students, especially in a new environment. My own exam was much improved after hearing the comments of my checker.

Another detail in the Durham system I was happy about was the three-week revision period and the holiday preceding it. During that time I felt that the pieces were finally coming together in the students’ minds. I have had a similar experience during my own time as a student, when for some reason I have not taken the exam at the first possible instance, but rather deferred it a couple of weeks. It made me much more confident in the actual situation, and I’m sure the same is true with many of my Durham students. If only a similar revision period could be arranged without having to delay all exams until the end of the year!

In conclusion, the system of undergraduate studies in the UK seems quite strange from a Finnish point of view, and I cannot say I appreciate the feeling of lack of freedom. I’m afraid the majority of students will have difficulties growing to be independent thinkers when they have so little to say regarding their own studies. Also the teachers have to strive really hard if they want to experiment and develop their own teaching into a new direction. However, there is definitely something to learn about in the way students and lecturers are tended and their progress constantly followed. To the students such instruction is certainly very helpful, as they are often not yet as independent as we might think them to be.

# Passion, Love, Math

Written by Anssi Mirka.

# “To play without passion is inexcusable.” –L. V. Beethoven

## Cultivating my Mathematical Garden

I love math. It lights my day, opens my mind, brings me to bliss, gives meaning to my life like nothing else I have experienced. It challenges me like a good sport. It surprises me like a new twist in the plot of a detective story. It allows me to express myself like a poet distilling the essence of reality. I cannot get enough of it.

Our romance did not come easily, however. Definitely it was not love at first sight. Initially I treated this babe only as a tool for physics (my first love), or as a fun way to challenge myself and get cheap validation from being able to solve problems. Then I started breaking up with physics – mostly I had issues with the way it was taught – and found I enjoyed spending time more with math courses than physics courses.

That’s, of course, when I began falling for math, but at first I did it for all the wrong reasons: I was bored with physics. I felt I was better at math. The credits came easier. I could juggle more courses. I was good at puzzles. It made me feel clever… such embarrassingly immature reasons. None of them provide the healthy foundations of a strong relationship. I defined myself by the exercises solved, by the courses accomplished, by the grades gained. And the better I did, the deeper I sank into my illusion of knowledge. I thought I loved math, but in reality I was graving for acceptance. I was at the mercy of these superficial indicators, and speeding so fast I had no time to question their justification.

But the true tragedy here was this: no one—be it fellow student, lecturer, lecture material, no one—warned me about this knowledge-illusion, or clearly encouraged to look for a ‘’deeper understanding’’. If anything, I had absorbed the attitude, that a respectable mathematician should steer clear off such vague musings. In fact, I think I did not even realize there existed a higher level of mastery to aspire for. Now, I know some people trust that these profound aspects are somehow ‘’implied’’ in the standard stuff and the meanings transpire to the ‘’talented’’ ones. I don’t care to argue about the elitism of such unfinished defense, but I deeply regret to say that most of the really important insights simply do not ‘’transpire’’ even to the best of students. That is a fact. We’ll come back to this fatal fallacy after I finish my little story.

So I was speeding and about to crash, hopefully before it was too late. How did I save myself? Books. Self-study. Learning to love the math, not the scores. I was lucky, and it was not easy. This is how I re-emerged: I remember being, once again, frustrated about the course material, you know the lecture notes and the scarcity of motivation, so I sought for a good book in Amazon.com. And damn… those bibles of knowledge, those gems of wisdom, those thick, beautiful, respectable monographs. Many of them had stellar peer reviews: ’’A must-have for any aspiring analyst…’’, ‘’The best math book ever written…’’ etc. Immediately I felt I had been seriously missing out and there is a new level of skill to be reached (although at this point my idea of ‘’skill’’ was saddeningly superficial). I was thinking: ‘’Why have I been wasting my time with some last-minute put-together lecture notes when there are such dime pieces of art to be found!’’ So, I dived into literature head on—the best decision I ever made.

At first the reality hit in hard. It was impossible to keep up juggling multiple courses while studying from the fat books instead of the slim lecture notes. Not because the books were more difficult. On the contrary, as the reviews promised, they were awesome compared to most notes. But most notes are compiled together from two or three books and have conventions of their own, and that makes it two to three times more difficult to complete the course self-studying even the best book around, as compared to just cramming trough the course with the provided minimal material.  As a side point, if you, like me, think we should encourage students to read literature, this is something to think about.

So, I had to give up chasing credits and courses. I could no longer draw my pathetic validation from superficial pursuits. It was either the ‘’dirty high’’ by performance or the superior leaning by books; and the right choice was painfully clear. Credit-hunting, multiple courses, and lecture notes went out the god damn door—one of the toughest decisions I ever made.

This all implied I had to build my value system anew, on the most solid ground possible: genuine love for math itself. I could afford no illusions anymore: credits were for fakers and the only thing that matters, the only thing, is math itself: how well I understand it and how beautiful I find it. There was no one to observe my process, so I had to be brutally honest about my skills. At first this was hard because of the uncertainties, self-doubts, and lack of feedback. So hard, in fact, that I really feel sorry for anyone who has to go through it alone… unless you’re a hermit, in which case it’ll be a blast. Most people, though, are not. Then again most never even learn about this option.

And so I finally learned to truly love math for its own sake. As time goes by our relationship is only becoming more interesting. Alas, it has nothing to do with being able to solve more advanced problems or collecting more theorems under my belt. No, the reason I love math more by the day is I can appreciate its beauty more purely and feel its meaning ever more strongly. Like a piano player, the more refined you become, the better you can cherish the nuances—your enjoyment has nothing, no-thing, to do with some dots on a piece of paper.

## Victims of Purity

That is how I transformed from an outcome-oriented mathematician to almost purely aesthetics-oriented mathematician. I am not saying everyone should go crazy with this ‘’meta-math’’ hype, but for me it was (and is) the only way. Unfortunately, awesome as it is, it certainly hasn’t made my academic survival any easier; indeed, it would be much more efficient, in the short run, to ditch the ‘’deepeties’’ and acquire the ‘’just do it’’-attitude. I cannot do it, however. If you can, fine, but keep in mind that there at least exist other levels you should eventually aim for, if you want to excel that is. But, most importantly, never forget that there are other individuals who simply cannot survive without deeper meaning to their studies. They are precious innovators enriching our scientific idea-pool. For their sake, learn to generate motivation and provide inspiration beyond the details, the puzzles, the credits.

The reason why I am telling my story is to convince you that we must actively encourage students to seek for the beauty and beware of the temptation to ‘’just perform and pass tests’’. Do NOT make the mistake of counting on students’ ability to avoid these pitfalls on their own. It is a terrible miscalculation. The reality is that even most successful students often don’t have a clue about the true depth of the mathematical theories.  This problem is so devious because it goes largely under the radar. Traditional exams and grading systems do very little to measure this dimension of learning. And the crazy thing is, that you can easily perform perfectly on those canonical examinations without cultivating any deep understanding.

How big a problem is this? Sure, some of the students simply enjoy solving puzzles and they are fine with the current system. Then there are those who are very persistent and goal driven; they might go to ‘’perform-mode’’ and become speed blind (this was me—the mindless racehorse).  Usually these individuals do admirably on the traditional scales, at least up to the point when the reality kicks in.  Meanwhile, however, I think most students are just withering away, coping with the curriculum. And the saddest news is this: especially the rare gems of individuals who express, at the same time, both passionate curiosity and critical thinking—the very characteristics of a scientist—are going to have it depressingly, crushingly hard. They usually—more often than not, I’m afraid—lose interest, even though they might score high on tests. This presents a huge, MASSIVE, problem for mathematics: some of our very best students become casualties of education: we’re starving our offspring, and we don’t even realize it.

## Plant the Seeds

Love is never easy. First you seek for it, or at least be open to it, and then you work for it. Above all, you must learn how to share it. That is why I also love teaching math. Just like I could not enjoy life without being able to love, I cannot enjoy math without being able to share it, teach it.

And we need to teach the students to love math for its own sake. If we only teach technical skills we drive away the most passionate individuals, and those that survive are doomed to remain disabled to communicate math with radiating love. I could not care much less about the technical skillset, neither my own nor my students. That is not the essence of mathematics, not the true form of understanding. Yes, one must know how to play an instrument a bit before delivering a musical masterpiece; or know how to read before opening a book on poems, but if you want to inspire greatness, you must provide the experience of joy. You must plant the seed of knowledge: wonder.

It is, therefore, not enough to present the proofs and details to the students and hope that they somehow know how to dig into the deeper message; that they automatically draw passion out of the deltas and epsilons. My experience with students of all skill levels tells me it just won’t work like that: the development of passion and intuition requires, initially at least, a clear view from the right vantage point and a good injection of passionate interpretation. Each student, of course, is different and ideally one should leave room for an individual discovery, but if you only hand out the details you’re leaving it to the chance—no, you’re fooling yourself and letting your students down. I’ve said it before, I say it again: Every passing day promising students are unnecessarily losing their interest and bitterly giving up math. And for no better reason that no-one has ever showed them how things could be, lifted off the veil of obscuring details, inspired them with fulfilling joy. Finally, again and again it seems to be the case that the more passionate you already are, the more difficult it becomes to survive here. The situation is maddeningly messed up. Even if one manages to develop a hunger for the divine food that is the beauty of mathematics, one receives no help in gathering it; the very first step in the path to passion is loaded with hardships. Eventually, then, one either withers away or, maybe even worse, forgets the hunger and settles for coping with unnourishing food.

# Maths bazaar – Creating new kinds of learning spaces

By Johanna Rämö.

These pictures are taken in the main corridor of the Department of Mathematics and Statistics in the University of Helsinki. They show how learning and teaching does not have to happen in a classroom or lecture hall.

The main corridor of our department is filled with tables, so that the students can work there. Everything is close: student common room, school office, classrooms.

The corridor has become a huge drop-in class where students can spend as much time as they want. The teaching assistants, who are either senior students or members of the teaching staff, provide help 8-10 hours per day. They walk around the tables wearing colourful vests, so that the students can easily approach them.

The teaching assistants are not supposed to give answers but lead the students subtly towards a solution and help them improve their studying skills. As this kind of teaching is new to many of the teaching assistants, training is provided for them.

The tables are arranged into groups to encourage student collaboration. For the same reason the tables have been turned into whiteboards. This way it is also easier for the teachers to talk with the students about the problems they are tackling with.

The walls are covered with blackboards for the students to share their thoughts with each other. Also the researchers and professors use the blackboards in sketching their ideas.

Some time ago a student suggested that we bought gym balls for the students to sit on. They are more comfortable and ergonomic than normal chairs. We thought that it was a very good idea, and bought the balls. Here you can see the head of our department testing them.

The corridor is a real maths bazaar: it is full of students working together and having enthusiastic conversations about mathematics. The more tables we bring to the corridor, the more students come to study there. They hang around there on their spare time too, playing games and chatting with their friends. All this has had a huge impact on the atmosphere of our department.

Pictures: Veikko Somerpuro