## Introduction to Graph Theory

«Poor, sad-eyed stranger!» – this is how American author Mark Twain described the canvasser or the salesman. Why was he so sad and poor? Maybe because he had to solve the big problem of a travelling salesperson – “Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?”. Without the answer, we could not trade today.

A long time ago, this was an inconvenient difficulty not only for the canvasser, but also for the mathematicians. The travelling salesman problem was mathematically formulated in the 1800s by the Irish mathematician W.R. Hamilton and by the British mathematician Thomas Kirkman.

Up to this day, the issue is existent for postmen and travellers all over the world. The most convenient and profitable approach for a solution is graph theory.

Another issue is called the Seven Bridges of Königsberg, named by Leonard Euler in 1736, and it sounds more like a beautiful legend than a problem.

Königsberg in Prussia (now Kaliningrad, Russia), was set on both sides of the Pregel River, and included two large islands, which were connected to the rest of the city by seven bridges. The problem was to devise a walk through the city that would cross each of those bridges once and only once. The residents of the city always challenged visitors and travellers to see whether they could solve the problem. However, no one except Leonard Euler could find a way; not even the locals. Euler proved that the problem had no solution. He was able to find a rule, using which, it was easy to determine whether it is possible to pass through all bridges, without passing twice on either of them. Its negative resolution laid the foundations of graph theory. Euler wrote a paper about the The Seven Bridges of Königsberg and published it in 1736. It was the first paper about graph theory in history and the first page of the history of graph theory.

This is the history. Now, let’s take a look at pop culture. Labyrinths have featured prominently in pop culture for a long time. Starting from the myths of Theseus and Minotaur and ending with the dystopian science fiction action thriller “The Maze Runner”. Or do you remember the fourth movie about Harry Potter and his adventures in the labyrinth? And the “The Shining” by Stephen King? The labyrinth challenges the hero to find the right path to escape. The study of labyrinths is also connected to graph theory. If the protagonists knew the basics of graph theory, they might have found a faster way out in some cases.

What then is the graph theory?

Graph theory in mathematics means the study of graphs. Graphs are one of the prime objects of study in discrete mathematics. In general, a graph is represented as a set of vertices (nodes or points) connected by edges (arcs or line). Graphs are therefore mathematical structures used to model pairwise relations between objects. They are found on road maps, constellations, when constructing schemes and drawings. Graphs underlie many computer programs that make modern communication and technological processes possible. They contribute to the development of thinking, both logical and abstract. For example, maybe you remember this game from your childhood: connect the dots on the piece of paper to make a figure, a dog or a cat – those connections are also graphs.

We all know the saying: “Mathematics is not needed in real life”, but graph theory is actually applicable in real life. In fact, this is how it was discovered.

ITMO University in St. Petersburg, Russia, is a large state university and one of Russia’s National Research Universities. Research priorities of the university are concentrated on information and photonic technologies. The online course „Einführung in die Graphentheorie“ is focused on the study of methods and algorithms of graph theory and their application in practice. The goal of the course is to develop basic knowledge and skills to solve the most important and frequently encountered graph problems.

You will, first of all, gain a readiness to demonstrate a basic knowledge of mathematics (graph theory). And secondly, you will learn to apply effective methods and algorithms for solving typical graph problems in practice. Furthermore, this course will be very useful for all those, who encounter graph theory in university – either on a beginner level or for in-depth studies and research.

The course consists of four chapters and a final exam. The theory will be taught through video lectures. The length of our teaching videos is of the time it would take to teach the same material in a lecture setting. Therefore, each video lecture lasts about 15 minutes. After completion of each video lecture, an online quiz is conducted to verify the that you as a learner understand what has been taught.

The main goal of this course is to gain basic knowledge in graph theory. This knowledge will help you to independently study other sections of graph theory in the future, and to apply it in real life. Do note, however, Graph theory develops very quickly. There are various terms and concepts, the correct use of which is possible only in the context of the problem being solved.

## Learn Agile Management Online à la Hollywood

When we we thought about how one would best learn Agile Management online, we eventually settled on exploring news ways of how to use storytelling in online education. Because learning works best, when it is fun, intuitive and engaging. This is why the course “Agile Management” (which unfortunately for now is only available in German), led by Lean Launchpad Educator and founder of Safari Consulting Stefan Hoch, takes you to a fictional corporate world that might very well feel familiar to you in one way or another.

The course uses the so-called hero’s journey format. The learner follows the hero of the story, who after mishandling a corporate innovation project gets one last chance to save his career from ruin. He is tasked with – finally – getting a product innovation off the ground and confronted with a host of challenges. In every chapter of the course, you will learn theoretical and methodological principles which you will then apply to your own course project at a later stage. Movie scenes, case studies and assignments make the theory come to life and help the learner to transfer what they have learned to their professional context.

### The Making Of “Agile Management”

We decided that when conceptualising and filming effective learning videos that revolve around the theme of “agility”, the approach should be just as innovative as the topic itself. That is why we decided to explore the hero’s journey concept of storytelling in online education. In order to create the best possible learning experience for this topic and the target audience of this couse, our film team did not spare any efforts:

The nine day journey began in a forest in the outskirts of Berlin. Our team was accompanied by the expert Stefan Hoch and the lead protagonist of the course story. It’s December – minus two degrees Celsius – and they were equipped with Moon Boots, gloves, hats and winter jackets. Despite the cold, some parts of the story had to be filmed outside. Alongside cameras, lights and many pages of script our team brought bags full of groceries and a self-appointed team cook in order to prepare for a couple of intense days of shooting.

### From Brandenburg, via Mainz and Berlin to Hollywood

Five days of isolation in the woods combined with commitment and full concentration eventually resulted in lots of video material; including some that would later become bonus content. But that was not all. In order to tell a well-rounded story that work for those who seek to learn Agile Management online, additional material had to be filmed in Mainz at the Safari Consulting headquarters, and at a coworking office space in Berlin. Here, five experienced actors came into play. During the filming, the actors themselves learned first hand what agile management is all about. They enjoyed working with the various methods and learning about the topics covered in the course. So that at the end of the nine days of production, not just the actors but also the production team left the set with a thorough understanding of Agile Management.

The videos that have been created in the process are probably different from other learning videos you may know. The whole concept is more reminiscent of a movie or a TV series and the whole learning process unfolds very casually; almost as a side-effect.

### How Storytelling in Online Education Helps You Learn Agile Management Online

Most traditional learning videos show little change of scenery. Instead what you see is an instructor standing in front of a wall or a green screen, which always looks the same. Such a monotonous setup does not provide any “visual landmarks” for the learners. These, however, would greatly help them to better remember the concepts being taught. The video material of “Agile Management” on the other hand, offers learners frequent changes of scenery. This aids memory and inspires conversations at the water cooler: “Remember the scene in which Stefan Hoch sits by the lake and talks about freeing yourself from outdated thinking patterns?” In other words: these scenes do not just offer cognitive, but also emotional messages that allow us to connect and remember. Therefore they offer a learning experience that is distinctly different from seeing someone lecture in front of the same white wall for hours on end.

Another important aspect that differentiates the learning videos of “Agile Management” is that they use multiple characters, alongside the expert Stefan Hoch. Through the presence of more than one character in many of the videos, the learner can follow a storyline. Of course someone could also explain the situation as a case study. But if you learn Agile Management online surely it is much more exciting to see the drama unfold as it happens, rather than listening to the monologue of an expert. Just like in a movie, the key learnings are embedded in conversations. This makes it easier for learners to listen and comprehend the content within the given context of the story.

### Making Theory Come to Life: Because Seeing is Believing

Stefan Hoch could simply state the fact that a lot of times, employees’ lack of motivation and buy-in are responsible for the failure of a project. Of course stating a fact is a reliable way of conveying information. This, however, far from ensures that the importance of that point is understood by the learner. Nor does it ensure that it sticks. So instead, the participants of our “Agile Management” course witness the story of the frustrated protagonist Marc, who struggles with the lack of commitment among his colleagues – the dialogues, the action itself, the tone of voice, the facial expressions as well as the subtext of the conversations all leave a lasting impression.

Oftentimes the instructor does not simply state certain facts or solutions right away. Instead the learners who learn Agile Management online have to identify the key problems themselves, by keeping up with the storyline. In one of the videos, the learners get to witness the failure of Marc’s project. Subsequently they have to recapitulate why and how they think the project failed, according to what they saw in the video.

In addition learners dive right into the story by working on the very assignments that Marc has to solve, which helps them retain the key learnings. Similarly, when Marc’s mentor Stefan sends him some notes to recap what they discussed in the video, learners receive these very notes so they can slip into Marc’s role. By having to solve the same problems as the story’s protagonist, the learners begin to identify with the hero of the story. Being emotionally involved makes it easier for them to see the relevance of what is being taught.

### In Summary: If You Want to Learn Agile Management Online, Storytelling is Key

Pioneering a new way to use storytelling in online education resulted in an online course that is both challenging and rigorous, but also engaging and entertaining. Because they can work on their own project in the course of the various assignments, every individual learner can embark on their own unique learning journey. This journey, however, will not only lead to broad knowledge of what the term “agility” means, but because of the way it is being taught it truly conveys both a “new way of thinking” (as Marc states in the course trailer) as well as practical skills that allow the learners to apply this thinking in their everyday work.

In short, Agile Management is a course that provides learners with a thorough grounding in the necessary theory, while keeping them on the edge of their seats!

If all of this made you curious, sign up for a free preview chapter of the agile management online course now.

## Video Production: A Hands-On Tutorial for Online Learning Content

Video production can be done in many ways – from Hollywood movies to selfie videos. In the following we have collected some of the basics for you.

### Camera

For professional quality video production, you need a Full-HD Camera with a resolution of 1920 x 1080 pixels. Also make sure to have enough memory sticks and spare batteries.
Recommended cameras: Canon EOS, Nikon D, Sony HDR.

### Video Format

• High Quality HD videos
• 16:9 Format
• H.264 codec or similar
• we recommend 1920 x 1080 (1080p or 1080i), or at least 1280 x 720 p
• supported file format: mp4, m4v, flv, avi, mpg, mov, etc.

### Sound

In learning video production, capturing quality audio is key. Arguably, this is the single most important technical aspect of creating a good learning video. Here are some guidelines.

• Using an external microphone is imperative!
• Before recording, check to make sure the volume is correctly adjusted.
• Make sample recordings and have someone else give you feedback about the sound quality.
• If you are recording outside, you should use a microphone that you can attach to your jacket.

Recommended settings for recording: 48kHz and 24 bit

### Lighting

We advise using the standard 3 point lighting in video production. If you prefer to use natural light, keep in mind that the sun moves and your picture therefore changes.

3 point lighting: back light, key light, fill light

For more details see:

Three Point Lighting Explained

### General Tips for Presenting

You should be aware that your voice and body language do not seem the same online as in your classroom.

• Prepare your message – Write a script. What is it that you really want to say for this particular video? Narrow it down and be sure you know what is most important. Memorise and practice your script (see scripting). You do not have to write out every single word you want to say. If you are more comfortable working with bullet points, feel free to work like this, but try to remember your first and last sentence of a scene, so that a cut feels natural.
• Relax and smile – An inhale and exhale followed with a soft smile works wonders (NOT one that’s ear to ear). Be sure you’re breathing regularly in the course of taping. Do not hold your breath. Also, blink on occasion. It’s a natural lubricant for your eyes and will help you stay “bright eyed” – you will look less “robotic”.
• Body language – A little movement is fine; too much movement makes you appear nervous. Use your hands, because hands can enhance decisive statements and support nuance in your speaking dynamics, but don’t overuse them. If you’re standing during your taping, watch the typical rocking back and forth. Try to stand comfortably with your feet about six inches apart, your weight equally distributed and your shoulders relaxed. Don’t stand on the balls of your feet, as you run the risk of losing your balance. When sitting, lean just slightly forward, shoulders down, and then sit as tall as you can but without stiffness.
• Look directly into the camera – Imagine a person where the camera is and imagine a conversation. Tilt your head slightly and position the camera a little above your eye level.
• For tablet capture – Imagine you are sitting next to your student, explaining the material.
• Take some time – Don’t start talking as soon as you push the recording button: give it 2-3 seconds. This will make the post-production easier.
• It’s a process – Record yourself in front of the camera multiple times, take a look at the content you have produced, and try to identify what works and what doesn’t.

Based on:

Wendy Scharfmann

edX101 – How to Make an edX Course

### Commandments of Quality Video Production

There are a few commandments of good filmmaking. Here is a very basic list:

• Know what you are going to shoot! – Have a plan (storyboard) of what you are going to say and show.
• The object moves, not the camera! – Try to hold the camera still, as handheld filming usually looks unprofessional. The best way to steady the camera is by using a tripod.
• Be thrifty with the zoom! – Only use zoom where necessary, otherwise it looks nervous. Zooming can be useful to point out something really important or show details. Remember to try out the zoom of each scene in order not to over- or under-zoom: if you zoom, be on point!
• Each scene needs some extra meat! – Remember to give each scene three extra seconds before you start the actual scene and after you finish it. In doing so, you make sure to have better cuts in post-production.
• Every scene needs light! – Never shoot against the light. If you shoot indoors, take care of good lighting.
• Know who’s talking! – When shooting, ensure absolute silence during the recording. Noises from people not in the picture can be especially annoying.