## Where did that substitution come from?

If you want to understand maths, you really have to do it. I recommend going through these examples and using the substitutions given here as hints. Get a blank piece of paper, put your notes away and try to do these examples and see if you get the same answers as in class. If you don’t, write in the comments, and we can see where things may have gone astray.

I’ve been teaching integration by substitution, including by trig substitutions over the last few days, and a frequent question which a newbie substituter will ask is “how did you know to make that substitution?”. It’s a very reasonable question, and one that takes practice to build the correct intuition, but I’ll do my best to give some motivation now as to why we made some of the substitutions we made. We won’t solve the integrals, but we will motivate here why we make particular choices for substitutions.…

## Welcome to Reproducing Kernel Hilbert Space

In a series of posts I hope to introduce Mathemafrica readers to some useful data analysis methods which rely on operations in a little back-water of Hilbert space, namely Reproducing Kernel Hilbert Space (or RKHS).

We’ll start with the “classic” example. Consider the data plotted in figure 1. Each data point has 3 “properties”: an $x_1$ coordinate, an $x_2$ coordinate and a colour (red or blue). Suppose we want to be able to separate all data points into two groups: red points and blue points. Furthermore, we want to be able to do this linearly, i.e. we want to be able to draw a line (or plane or hyperplane) such that all points on one side are blue, all points on the other are red. This is called linear classification.

Figure 1: A scatter of data with three properties: an x_1 coordinate, an x_2 coordinate and a colour.

Suppose for each data point we generate a representation of the data point $\phi(x)=[x_1, x_2, x_1x_2]$.…

## Greetings from Bibliotheca Alexandrina

 I just received the info below on Bibliotheca Alexandrina and its new African Networks and I thought it makes sense to share it on mathemafrica. I really like the idea of historical places connecting to new technologies. Here is a link to a film (in French) about the historical library at Alexandria:   Greetings from the Bibliotheca Alexandrina (BA) in Egypt. We are pleased to announce the launch of “BA African Networks”. Below you will find a description of the networks. You are most welcome to explore our portal and our five networks through the following link. http://afn.bibalex.org/GeneralPortal.aspx The Bibliotheca Alexandrina (BA) follows in the footsteps of the Ancient Library of Alexandria as a meeting point for cultures and civilizations. It aims to rise to the digital challenge in order to develop African innovation through the use of science and technology in networks that extend throughout Africa. The new goal set by the BA is to connect those with common interests and expertise to hasten the development of knowledge and enable immediate sharing of knowledge and contributions.

## Do You Find Mathematics Scary?

A few weeks ago I attended a lecture by Johnathan Lewin, regarding the use of technology when teaching and it was brilliant, and I’m not even talking about his use of technology. The passion that Johnathan speaks with and the passion he has for Mathematics is explosive and practically contagious.

He uses a number of different programmes and applications to assist him in the classroom. He even records his lectures (he captures the audio and a visual of the learning materials and then makes them available to his students). He is in favour of designing the materials in front of the learners in order for them to see how the Mathematics is created rather than to arrive with some neatly prepared sides and show them what Mathematics looks like. He wants them to engage in it at all levels and not just see the perfect final product, if you wish.…

Gallery

## Mathemafrica and the Next Einstein Forum (NEF) #1

I am sitting at the AIMS-Senegal institute in Mbour (about 1.5 hours drive South of Dakar) and together with my team member Sebastian and the AIMS-Senegal staff and students, we are preparing an interactive mathematics exhibition (as part of the IMAGINARY – open mathematics project).. It will be shown as of March 8 at the Next Einstein Forum (NEF), to be held at a huge conference venue just outside Dakar.

There will be many ministers, scientists, politicians, even presidents from (apparently all fifty-four) African countries – and also many international guests – joining the NEF, with the goal to discuss about scientific innovations, collaborations and solutions in Africa!

We have to plan to blog live from the NEF, with insights and views from participants. And of course, we will let you know about our exhibition, about a new competition, we will launch and everything happening around!

Prepare yourself for the NEF at:

gg2016.nef.org/

iameinstein.org

Please find a picture from our first technical setup yesterday at the AIMS Institute.…

## How Many Languages Do You Speak?

I’m not sure how, but it’s been a month since my last post. It feels like it was just the other day that I was working on its first draft… Since my first blog dealt with the language of Mathematics, I thought I might continue the language theme for now as it is something that really interests me.

Let me start by asking you this: How often do you take being a First Language English speaker for granted? (Has this thought ever even crossed your mind?) Have you ever traveled to a foreign country and needed to communicate and found it difficult? Were you frustrated by this? What happens when you don’t have a very good grasp of a particular Language, would you want to speak it? Or read it? Or perhaps worse still, write it?

Well, I think this is the challenge that a number of learners face and they are often left feeling frustrated and misunderstood in their classrooms, particularly in South Africa, where we have 11 official languages.…

By | February 22nd, 2016|English, Fun|4 Comments

## You’re (probably) a Bayesian – whether you like it or not!

Statisticians have long been separated into two camps as to how they philosophically interpret their trade. These schools of thought are usually called Frequentists and Bayesians.

Frequentists believe that a probability, $p\in[0~ 1]$, associated with a specific possible outcome of an observable occurrence or process, is simply telling you that, could you observe this occurrence (or process) infinitely many times, the fraction of such observations that would yield that specific outcome is $p$ . Using the age-old coin toss example: tossing the coin is the occurrence or process and recording a Heads or Tails are the two observation. The number 0.5 $\left(P(\text{Tails})=0.5=P(\text{Heads})\right)$ tells a Frequentist that, in the pursuit of infinitely many coin tosses, the ratio of Heads recorded to the number of tosses performed asymptotically approaches 0.5. And that’s all! The value should not be interpreted as the most likely outcome for the next observation or sample taken from the process (though I’ve always wondered how a Frequentist would gamble…).…

Gallery

## The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics

Edit: I had a great deal of help for the poster from Andreas Matt, Antonia Mey and Adam Weston.

Tomorrow I will head to Port Elizabeth to the Elephant Delta 2015 conference: The Tenth Southern Hemisphere Conference on the Teaching and Learning of Undergraduate Mathematics and Statistics – quite a mouthful of a title! It looks like it’s going to be an incredibly full week with a huge amount of new information, new people and new ideas. I am going to attempt to blog as much as I can from the conference. The program is spectacularly full with parallel sessions running through the day. The program can be found here.

I am already having a tough time deciding which talks to attend, so if you have a look at the program and see something that you would really like me to write about, then please leave a comment and I will do my best to get there and write up what I learn.…

## AIMS-Senegal – picture story of a maths communication adventure

Dear all,

this is a picture story, so be prepared to see many pictures! And it is an adventure too, since we (I will explain later who is “we”) tried something quite new: an interactive highly technical mathematical exhibition in Senegal, including a road show with high school students and Master maths students, talks, conferences, workshops and discussions, and games! And: all in three days!

Let’s start:

Day 0:

The day before the opening of the exhibition and the start of the roadshow. And my first day in Senegal. I arrive around 1:30 am in the morning and spend the night in Dakar. At 11:00 I am picked up by a car from AIMS-Senegal, who is organising the event together with the Next Einstein Initiative and IMAGINARY. In the car, I meet two other international participants: Niane Kode from Senegal, who works at AIMS-Cameroon, and Marcos Cherinda from Mocambique, an ethno-matematician, who also attended the AIMS-IMAGINARY workshop in 2014.…

## A little medical statistics

Originally written by John Webb

tw: fictionalised statistics of disease rates.

———

Today there are many tests that are widely used to detect life-threatening diseases early. How effective are they? Should they be believed?

At a routine checkup, your doctor tells you that there is a simple and inexpensive blood test that can detect a rare but particularly
nasty form of cancer. You agree to have the test done, and the doctor takes a blood sample and sends it off to the pathology laboratory.

Two days later the doctor calls to tell you that the test has come up positive. The good news is that the cancer can be cured since it
has been caught at an early stage. The bad news is that the treatment, though effective, is very expensive and has a number of unpleasant side-effects.

Before agreeing to treatment you need to do a little bit of basic arithmetic.…