## 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.

## Rethinking Corporate Professional Development: Achieving Quality at Scale

Traditional corporate professional development is stuck between a rock and a hard place. While on-site training is expensive (trainer, venue, travel, accommodation, catering) and bound to a specific time and location, traditional web-based trainings and elearning courses are scalable, but suffer from low didactic quality and thus fail to achieve advanced learning outcomes. Webinars combine advantages and disadvantages of both of these approaches.

What companies need is an effective online learning experience that enables learners to reach learning outcomes on par with the best classroom training – at scale, anytime and anywhere. We believe that asynchronous, active (“lean forward”) learning and social online courses can help to square the circle. That is: achieve advanced learning outcomes at a reasonable price point.

## Where Traditional Corporate Professional Development Falls Short

As I have explained in my post about flexible online learning, classroom learning to this day is considered the gold standard of education. Granted, learning in direct interactions with others can be fun. But as everyone who has ever gone to school will know that is not a given. Even it is though, the practical requirement to be present in a particular location at certain point in time makes it more and more difficult to effectively use classroom training in increasingly global companies. Moreover, due to the fact that all of the learning has to take place in one, usually fairly short, block learners can find themselves overwhelmed by all the input. Corporate classroom training often leaves learners with little to no time for digesting content, entering into discussions with other learners or reflecting on what they have learned. While coming together in a group sure is a powerful way of learning, it reveals certain deficits that flexible online education manages to overcome.

Elearning formats promised a more flexible approach, allowing large numbers of learners to learn outside of the classroom. However, it soon became clear that the WBT format is primarily suited to convey knowledge in the form of content. Learners watch videos or read texts in order to learn certain facts or concepts. In addition they usually answer a few multiple choice questions – but that’s about it. This, however, is not enough to understand a topic in depth, acquire new skills or fundamentally change someone’s mindset. This is why traditional elearning is often one-dimensional and neither encouraging nor challenging. Which is why elearning in many companies is still primarily used in the field of compliance, where the learning objectives are relatively straightforward. For learners to reach more advanced learning outcomes, however, they need to apply their knowledge in different contexts and discuss value judgments with others. Which is why we need a new kind of corporate professional development.

## What Innovative Online Education Has to Offer to Corporate Professional Development

Innovative approaches to effective online learning manage to combine the advantages of both classroom learning and traditional elearning. This approach to online education gives learners the opportunity to work at their own pace while acknowledging the importance of the interaction among peers. Messaging, for example, allows students to exchange and discuss ideas with each other. They can even set up working group chats and thus solve assignments and problems together with their peers. Being part of a community is a key aspect of the learning experience – particularly in the field of greyscale learning.

The Learning Journal provides a room for learners to collectively reflect on different solutions to course assignments and motivates them to engage in active learning. Learners can reward outstanding works of others by giving posts a heart. By following the journals of other users, they can keep track of what is happening inside the community. Many of these features are widely successful aspects of other social networking sites. Now it is time to leverage their potential in the field of online corporate professional development.

Our online courses use a broad range of assignment formats. Learners have to write free text responses, record videos, visualise or develop a concept, answer open-ended discussion questions or participate in an essay competition and post the result of their work in the Learning Journal. The assignments are challenging as well as entertaining, but most importantly: educational. This hands-on approach requires learners to apply what they have learned in many different ways and contexts. Sharing their solutions with others enables them to learn with and from each other.

## A New Frontier for Corporate Professional Development in the Digital Age

We believe that this innovative approach to online education will be able to overcome the shortcomings of both classroom learning and traditional elearning. In so doing it conveniently places itself in the top right corner of the matrix above. On the one hand it achieves advanced learning outcomes by providing high quality content, challenging assignments and a community of peers in a flexible organisational setup. At the same time the courses are asynchronous and require little to no active supervision by an expert. Hence they are much affordable per participant than classroom instruction.

Combining scalability and affordability in this way allows companies to tackle wholly new challenges using digital learning. On the one hand it will help to make digital solutions succeed across a much broader range of subject areas; specifically in fields where it was previously thought to be ineffective such as communications skills. On the other hand it will help corporate professional development to finally deliver on the often-proclaimed aspiration to create a “learning organisation”. Strategic change management initiatives often require the  (re-)qualification of hundreds or even thousands of employees. On-site training programs provide very limited economies of scale and require a lot of time and resources to plan and implement. Exciting alternative solutions that leverage the potential of online learning more fully such as corporate MOOCs (e.g. Deutsche Telekom’s Magenta MOOC) may allow corporate professional development to finally gain the strategic importance HR departments have long claimed, but found hard to deliver on.

The times they are a changin’…

## Video Role-Play Training: How to Teach Communication Skills Online

Our newest feature, video role-play training, is currently undergoing beta-testing and we will begin to roll it out to learners within the next couple of days. Besides answering multiple-choice and written work, learners will be able to record video responses on our platform.

This will be especially useful in the context of courses with a focus on greyscale learning in the broad field of communication skills (leadership, sales etc.). Because becoming great requires practice. The ability to record, replay and record again until the learner is satisfied is one aspect that makes video responses such a powerful tool for effective online learning. In addition, sharing video responses in the Learning Journal enables learners to comment on each other’s work, complement each other or point out things that need improvement.

## Moving Beyond Multiple Choice Using Video Role-Play Training

Video role-play training assignments thus allow us to teach things online that are often thought of as being hard to teach in a digital format. Namely courses on real-life communication skills where there is no dichotomous black or white. Where there are many right and many wrong answers and many different shades of grey in between.

Examples of such courses are those dealing with topics such as sales, leadership or customer care. Consider one example: a course on leadership. This is exactly one of the “soft” or “human” topics that, as it is often argued, can only be taught well in a face-to-face environment. In a traditional elearning format it often is either all theory or a fairly simple “multiple-choice game”. In web-based training one would, for example, watch a video of two people fighting in the hallway and would then have to respond to a couple of questions that offer a few options to choose from. The user has to choose a course of action – that usually results in stating the obvious.

However, knowing what is the right thing to do and actually doing it, are of course two fundamentally different things. The way we envision online education is that after watching the video, students have to use the camera of their device to formulate and record an original response to the characters in the conflict situation in form of a video comment answering the question: “What would you say now? 30 seconds. GO!”

Assignments such as video role-play training allow students to respond to a complex problem in an infinite number of ways and require them to move from multiple choice to infinite choice. Aggregating the user-generated content in the Learning Journal and letting all learners give each other detailed feedback on the basis of sophisticated grading rubrics can take this one step further.

## Life Comes in Many Shades of Grey – Online Learning Should Reflect That

In short: with this feature we want to teach learners complex topics such as leadership by moving beyond simple multiple choice formats. We try to encourage them to bring their context and experience to the table. An old person will respond differently than a young person. A woman differently than a man. Yet all of these different answers may well be correct in their own unique way – or not. And that’s for the learners to discuss and work out together. This approach to effective online learning allows them to be creative and to think outside of the box. It also shows learners that a lot of times, there can be many different ways to solve a problem. This truly embodies what we mean by greyscale learning. We believe that video role-play training can help learners to broaden their horizons and to see the bigger picture of complex topics.

## Social Learning Online is Learning in a Social Network

We envision learning on iversity as learning in a social network. Therefore, we created a social experience for our users that enables them to interact with each other in ways that they are already familiar with from social networks.

The messaging feature, for example, allows learners to privately communicate with others on the iversity platform through a convenient messenger system – one on one as well as in groups. Learners can message each other regardless of whether or not they attend the same courses. All they need to do to start a conversation, is to search the user directory.

When searching for another user on our platform, learners can identify other learners who are members of the same organisation. This is a useful feature to help facilitate communication among people belonging to the same company. However, an organisation is not visible to non-members. That way only members of the same company can recognise each other as such. This is how we avoid harassment of our users, for example by headhunters or competition. Another way to ensure the learner privacy in a social network is the possibility to block other users in order to prevent spamming and other annoyances.

The messaging feature also allows for group conversations. Here, learners have the possibility to add new members at any time or leave the conversation if they wish to do so. This allows users to turn to their peers when facing a problem and solve it together as a group. The feature makes it easy to exchange ideas and discuss assignments with more than one person; while keeping the discussion among a select group of peers, instead of the entire course community.  Learners also have the option to name group conversations. This makes it easier for them to distinguish between multiple group chats – because convenience is key.

While the discussions feature is course public and intended solely for exchanging thoughts about the course content, the messaging feature can of course be used for personal chit chat among peers inside our social network.

## Community Managers – Facilitators of Learning in a Social Network

A course member can be appointed community manager by the course admin and thus gains access to special messaging functions. He or she can send announcements – email messages that are sent to either all course participants or specific subsets of the group – e.g. in order to draw attention to specific posts that are particularly relevant, helpful or controversial. They can also contribute content of their own in order to provide inspiration or feedback. Through announcements, learners of a course are further encouraged to engage with the course content and to think outside the box. Thus, they help with community management and tutoring. They can provide learners with assistance regarding the substance of the course and help to ensure their successful progress throughout the course, fostering effective online learning in a social network.

## Recent Activities in a Course

On the dashboard page, learners can see a short preview of recent activities. Like in the picture below, they can see who joined the course, as well as who posted a comment or an entry to the Learning Journal. Much like the newsfeed feature in other social networks such as Facebook or Twitter, the recent activity overview allows users to see at one glance what is happening in a course. This enables them to find recent contributions, active discussions and to connect with other learners – even when they have been absent from the platform for a couple of days. Seeing other users’ activity is a key motivating factor. Instead of learning in isolation you can see what assignment your colleague has been working on yesterday. This can create a healthy form of competition. It also helps to create a sense of belonging where learners feel that they are part of an active community of peers.