Just out of curiosity

I get it when fiction-writer Dean Wesley Smith says that we have a 'creative voice' and a 'critical voice'. Smith always writes with his creative voice, never with the critical one, not even when rewriting the text, simply because he never rewrites. He says the best work is done with the creative voice (and actually, it is the way to enjoy the process the most). It is easy to distinguish between the two voices: the critical voice is always finding problems, and difficulties, it is negative, complaining, judging; the creative voice says 'let's play!' . I think this is the best description I have ever encountered about creative voice: 'let's play!'. No judgment, no evaluation, no worries about reaching the objective. That is why the creative voice is so important: because it does not create barriers, it allows anything to happen ; it does not stop action by analysing it, evaluating it, judging its value, measuring its importance.  The creative voic...

It was 2004 when I started mathematics at University.  In 2004 studies were not divided between Bachelor and Master. It was just one degree and it lasted 5 years. It is my understanding that that was pretty much the case in every European country, and things changed with the "Bologna plan". So, back in the day, in Spain, to become a mathematician, one had to study for 5 years, from which 3 years and a half were filled with compulsory subjects. Now, of course, things are very different in the study plans.  But some things do not change: the transition from high school to university was tough. Very tough. And we had to pass all the subjects of the first year (a total of 8) to be allowed to go into the second year (and you had to pay for every course that you registered to, and you could not de-register - you were given a fail -, and every time you had to repeat that subject, it became more expensive). Exams were written, at the end of each semester, and they had a fixed date. S...

Let's apply what we learned about logic to some particular scenarios: Definition: we say that a sequence \((x_n)_{n\in \mathbb{N}}\), \(x_n\in \mathbb{R}\) converges to \(x\in \mathbb{R}\) when for all \(\varepsilon>0\) there exists a \(N\in \mathbb{N}\) such that for all \(n\geq N\) it holds that \(|x_n-x|<\varepsilon\) Definition: we say that a function \(f: \mathbb{R}\to \mathbb{R}\) is continuous at \(x\in \mathbb{R}\) when for all \(\varepsilon>0\) there exists a \(\delta>0\) such that if \(0<|y-x|<\delta \), then \(|f(y)-f(x)|<\varepsilon\).  Quiz: Write the negation of these definitions, i.e., what does it mean that a sequence \((x_n)_{n\in \mathbb{N}}\) DOES NOT converge to \(x\in \mathbb{R}\), and what does it mean that a function \(f: \mathbb{R}\to \mathbb{R}\) is NOT continuous at \(x\in \mathbb{R}\).  Cognitive bias We have many cognitive biases, which include biases on how we think about things. For example, there is a well-kno...

If you want to learn maths, you need to learn how to write maths and how to explain maths. The reason is simple: the process of communicating is a process of assimilation. If you cannot explain something clearly, then something is not clear in your thinking, so practice explaining things to other fellow students and let others explain things to you (and reflect on how they communicate). So far, so good. But there is an aspect that is often ignored: writing maths. And this is crucial. I did not learn to write maths properly until very late (and, actually, to write them really properly, until my post-doc). The reason? Two-fold: I did not give it so much importance (I was thinking that the important thing is to have the right answer), and nobody taught me (nor I knew how to go around learning it). Writing and thinking are inseparable. Fix that in your mind. Writing is the most powerful tool that you have to empower your thinking. Mathematical thinking is an iterative process of refining i...

After today's coaching session, I decided that I will do various blog entries on these topics: - It is normal that it is tough to study mathematics (and especially struggling to do proofs). What is different now from mathematics in high-school. - Writing mathematics, doing proofs, the way of communicating maths, and how important it is to write properly to refine your thinking. The entry is now here . - Make sure you understand the exercises (point 1 of Polya's problem-solving). Does it sound like ridiculous advice? Well, it tends to be a common mistake. - How was it for me when I started mathematics at University (and the main reason why I am doing this coaching). There is quite a lot to say about all these points.

These reflections came from a discussion about how long should be the blocks of time dedicated to solving problems. First, I said: - Allocate time to activities rather than to specific goals . For example, decide "today from 5-7 p.m. I will work on the problem set" rather than "today from 5-7 p.m I will solve 3 problems". The first goal is under your control, the second is not. If you do not fulfill a goal, you will feel bad about yourself. But then again, how long is a reasonable block of time?  That is a question without a right answer, in my opinion. But this question came from a student that is working and studying at the same time, so he does not have long blocks of time.  This reminded me of an old belief of mine: Belief : I need a long block of time to be able to work on my math homework. Do you believe that? Do you have 15 minutes. Can you do some homework with that time? What about 10 minutes? What about 5 minutes? I thought for a long time that I needed a...

Today was the first coaching session (with 3 interested people out of around 25). One student shared that during the theoretical class, he was copying everything written on the blackboard and had no time to follow what the teacher was saying. I did this too. Many people do this too. But unless you have no other resource (like no lecture notes or no books), this is a very ineffective way of making the most of the theoretical class. You will learn little, understand not much, and be unmotivated. And then you need to spend a lot of time at home assimilating your transcribed notes. If you have lecture notes the best is to prepare the theoretical class by: (1) Revising the content of previous classes; (2) Reading in the lecture notes what is coming in the next class. Try to understand as much as possible, and make note of your questions or any unclear points. Then in class do not copy the blackboard. Copy only when new material, information, or examples are given that are not on the lecture...