Published on

March 14, 2022

Category:

Research [R&D]Reading time:

10

Minutes

*Effective skill assessments require a little help of clever data science methods. Here’s a short introduction to how we do it at edyoucated.*

Hey there!

Welcome back for part two of our blog series on skill assessments. This time, we’ll get a bit more technical, but don’t worry, we got your back. If you haven’t read the first part, you maybe want to head over there first. But if you want to start with the details right away, just stick with us here.

In this post we’ll tackle some of the data science behind atomic skill assessments. As a quick reminder, atomic skills are very small bits of knowledge and as such, a learning topic (for example, Microsoft Excel) might contain up to a few hundred of them. In order to let you start with the topic in the best possible way, we somehow need to “locate” you in the topic.

So our challenge is to figure out which of the atomic skills you have **already mastered** and which are **still unknown** to you!

And yes, all of them. Without any further intelligence involved, we’ll have to ask you for every single one. As you can imagine, this can get quite the boring and frustrating task for you as a learner, so we definitely do not want to do this. But at least it can serve as a baseline for our evaluation: Trading one question for one bit of information about you. This is the worst case, and does not create a pleasant user experience, don’t you think?

So let’s start with a simple example to warm up and see how we can improve this. If learners tell us they know how to create pivot tables in Microsoft Excel, how likely do you think it is they know how to enter some numbers in a worksheet as well? Close to 100%, right? Simply because pivot tables are much more advanced than entering content in cells.

So there is an inherent connection between the two atomic skills (pivot tables & content in cells): the mastery of one depends on the mastery of the other. We call the relationship of the atoms in such a situation (where one skill depends on the other) a **prerequisite relation**. In the language of graphs, Atom A is called an ancestor of Atom B, and Atom B is a descendant of Atom A.

How does this help us?

It’s actually quite simple. Let’s say Atom A is a prerequisite for Atom B (depicted above). Then, if you do not know A, you cannot know B either and we do not have to ask for it anymore! Vice versa, if you have already mastered B, then we know that you must have been familiar with A all along. Great! If we leverage that information during the assessment we can save some questions.

But how many can we actually save? To illustrate this, we have prepared a slightly more complicated situation below, with a couple more atoms and prerequisites. As a start, take a few seconds to ask yourself: Using the prerequisites, which atom would you ask for first to figure out as much information as possible?

Make a mental note of your choice and let us walk through this together. Say, we ask you about your mastery of Atom A and you tell us you don’t know it. Jackpot! Since A is a prerequisite for C, D and E, we can directly deduce that you cannot be familiar with those either. For B we’ll still have to ask, but you can see how nicely this cascades:

We’ve gained 4 bits of information with **only one question**!

But what if you tell us you *do* in fact know A? Then we haven’t really gained much, we’ve been trading one question for one bit of information, our worst case. So maybe A isn’t the best atom to ask for as a start.

And we are pretty sure that you’ve figured it out by now. Just verify for yourself that Atom C is actually the best to start with. Why? It gives us 3 bits of information in every case, independent of whether you answer Yes or No. We can either include A and B or exclude D and E from your knowledge state.

For the two remaining atoms of the left subgraph (either A and B or D and E) it’s then a fifty-fifty situation whether we have to ask once or twice more. The same is true for the subgraph containing F and H. And for G we’ll have to ask anyway as it is not connected at all. Following this strategy, we will most likely have to ask 5 questions (or 6 in the worst case) to figure out your knowledge about 8 atoms.

We’ve almost **cut the numbers in half**, that’s quite something, isn’t it?

Mathematically, the strategy we’ve been following can be described as picking the atom with the highest expected information gain in each step, where we assume that your probability of answering with either Yes or No is 50% in both cases. If you want a formula, the expected information gain for any atom could be the following:

Let’s assume that the number (#) of ancestors and descendants count the atom itself. Then Atom D, for example, would give us an expected information gain of 1(D) = 0.5 * 4 + 0.5 * 2 = 3 , which is ... the same as for Atom C?

Okay, you got us!

Interpreting the expected information gain this way, Atom D is actually just as valuable as C. But we’d argue that C still has an edge over D, as it works equally well if you answer with Yes or No. So let’s say this tie in information gain goes to Atom C nevertheless. And this actually hints us at a way we can optimize.

What if we knew some **better a-priori probabilities** for you answering with Yes or No? Then we could shift the probabilities for prerequisites and ancestors in the formula, which might eventually give an edge to Atom D, who knows? There’s plenty more that we optimized on our platform in order to ask as little questions as we can. What exactly? We think at least a little mystery should remain for the moment! 🤫

But wait! Where do the prerequisite relations actually come from?

Let’s solve at least a part of that mystery. The easiest and most accurate, but admittedly time-consuming way is to let experts model the prerequisite relations. We’ve got a whole team for this at edyoucated, working with experts in the respective topics, modeling prerequisites and other relations between atoms to save you time in your assessment process.

But we’ve promised some data science here, so let’s show a simple way to **learn** these relations from data. The following image shows the assessment results of a few of our learners for the topic of Microsoft Excel. As you can see there are quite a number of learners with no previous expertise in the topic (all-grey columns), but also many learners with more refined knowledge states.

This is the kind of data we need in order to learn prerequisites.

As a start, let us quickly re-interpret the idea of a prerequisite. What does it actually mean that A is a prerequisite for B? It means that if you know B, then you definitely know A. Or, if you know B, then the probability of knowing A is 100%. Let’s write this properly and maybe give it a little more freedom:

The conditional probability of knowing A given B is at least 95%, sounds correct, right? Even if you’re not a mathematician, I hope you can acknowledge the beauty of formulas. They just make everything that much more concise! The other way around works as well:

tells us that it is very likely that you do not know B given you do not know A (the little c denotes the complementary event).

And these two conditions actually give us a way to learn the prerequisites from the data above. For each atom, we simply need to find all learners that have marked B as known and, among these, compute the ratio that also marked A as known. This is a nice estimate for the conditional probability above, and we can, of course, compute the other one the same way.

Once we have found two atoms fulfilling both conditions, we found a prerequisite relation!

Neat, isn’t it? Adjusting the threshold (currently 0.95) lets us be a little more or less restrictive and helps us find more or less relations, depending on our goal here (if we set it to 1, then both conditions are actually equivalent, so we only need to check one). We’ve run this for you in Python using networkx and pyvis to give a small glimpse of how this looks like below:

We have been a little more restrictive on the threshold here and added some post-processing of the graph to make the results more visible. As you can see, “Mathematical Operators” are a prerequisite for many other atoms, or the “Excel Charts Introduction” a prerequisite for more specific chart atoms, other atoms remain unconnected.

Sounds about right, don’t you think?

A small side note, if you find a few connections that are “unintuitive”: We need to keep in mind that we are dealing with empirical probabilities derived from sometimes inconsistent learner answers, so everything has to be taken with a little care. In practice the whole thing has a lot more complications to it, but let those be our problem. For today it’s just about the idea.

There’s one last thing to tackle in this post. What if we make some mistakes, that is, model or learn a wrong prerequisite relation? In this case the predictions for your knowledge state might be wrong, too.

Well, of course, we make sure to test all our algorithms so that they make as few mistakes as possible. But we all know how data can be messy, noisy and flawed sometimes, so there will inevitably be some mistakes we make. That’s why we always let you as a learner verify our predictions at the very end. Just to make sure you never miss out on anything you can still learn. Speaking of learning, why don’t you try it out for yourself and learn something new along the way?

And with that, let’s call it a (data science) day! We see you in the next part of our series, where we will tackle a more sophisticated prediction model for our problem, based on conditional probabilities with multiple conditions.

Stay tuned!

And while you’re waiting: It’s often not only about figuring out *what* you should learn but also when!

If you want to read some loosely related work, here you go:

Askar, P., & Altun, A. (2009). CogSkillnet: An ontology-based representation of cognitive skills. Link

Doignon, J.-P., & Falmagne, J.-C. (2012). *Knowledge spaces.* Springer Science & Business Media. Link

Falmagne, J.-C., & Doignon, J.-P. (2011). Knowledge Structures and Learning Spaces. In *Learning Spaces: Interdisciplinary Applied Mathematics* (S. 23–41). Berlin, Heidelberg: Springer Berlin Heidelberg. Link

McGaghie, W. C., Adler, M., & Salzman, D. H. (2015). Mastery learning. *William C. McGaghie Jeffrey H. Barsuk*, 71. Link

Reich, J. R., Brockhausen, P., Lau, T., & Reimer, U. (2002). Ontology-based skills management: goals, opportunities and challenges. *J. UCS, 8*, 506–515. Link

West, M., Herman, G. L., & Zilles, C. (2015). Prairielearn: Mastery-based online problem solving with adaptive scoring and recommendations driven by machine learning. *age, 26*, 1. Link

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