Here’s an example of how this works. Suppose that a student at the beginning of her first Knewton-enhanced course is struggling with a word problem which involves calculating the area of a triangle. Assume we know nothing about the student aside from this fact. (This is an uncommon scenario — students who have made any progress in a Knewton-enhanced course or who have taken previous Knewton-enhanced courses will have already generated proficiency data that can help inform recommendations.) Knewton must determine why the student is getting this exercise wrong, so that we can recommend content that helps her learn the skills and concepts required to solve this problem.

Specifically, we must ask the following question: what is preventing the student from solving the triangle problem? There are several possibilities. It may be the case that she doesn’t know how to calculate the area of a triangle. Perhaps she struggles to read and interpret word problems. Maybe the base and the height of the triangle are given as decimals and she doesn’t know how to multiply decimals. There is even the possibility that she doesn’t know how to multiply integers!

It might also be the case that she can find the area of a hundred triangles with her eyes closed while she taps her head, rubs her belly and hops on one foot, and that she’s simply distracted by a computer game that she’s toggling to and from as her teacher wanders in and out of eyeshot of her computer screen. This last possibility is an important one. However, for the purposes of this example, let’s assume that the student is engaged and that her difficulty stems from the fact that she doesn’t understand one or more of the skills or concepts I described above. As you’ll recall if you’ve read one of our posts on knowledge graphs or checked out our white paper, we refer to these concepts as prerequisites.

Let’s list the prerequisites for the triangle word problem again:

- Multiply integers
- Multiply decimals
- Calculate the area of a triangle
- Read and interpret word problems

Using circles to represent the concepts and arrows to represent the prerequisite relationships, we can draw a diagram:

After we’ve identified the prerequisites, we must then assess the student’s proficiency in these areas so that we can recommend content that helps her learn the necessary concepts to solve the triangle word problem. For example, if she does poorly in assessments on calculating the area of a triangle, we can recommend additional content that helps her master this concept. But where do we start? Should we start by giving her an assessment on prerequisite 3 (Calculate the area of a triangle), or should we start with something more basic, like prerequisite 1 (Multiply integers)?

As you ponder this question, you might notice that the prerequisites in this example are not necessarily independent of one another. For example, it is unlikely that the student can multiply decimals if she cannot multiply integers. In fact, the content in this course that is associated with multiplying decimals expects and assumes that the student is able to multiply integers. In other words, 1 is a prerequisite for 2! Furthermore, prerequisite 1 (Multiply integers) is only important to the triangle problem as it relates to prerequisite 2 (Multiply decimals). In this case, we say that prerequisite 2 subsumes prerequisite 1 because it transmits the knowledge from 1 that is required to solve the triangle problem.

We can adjust our diagram to reflect this as follows:

How does this observation about the relationship between prerequisites 1 and 2 help us determine what the student does and does not know? Let’s imagine that we give the student an assessment on prerequisite 1 (Multiply integers) and she aces it. All we can say is that she’s proficient in prerequisite 1. However, what if we give her an assessment on prerequisite 2 (Multiply decimals) and she aces that? Since we know that the assessments for prerequisite 2 expect and assume that the student is proficient in prerequisite 1, then based on her performance in prerequisite 2, we can be fairly confident that she is proficient in both 1 and 2. Conversely, if she fails prerequisite 1 (Multiply integers), it’s probably safe to say that she is not proficient in prerequisite 2 (Multiply decimals) either. In other words, we can estimate her proficiency in certain concepts without having to directly assess her on them.

There is also the case that the student knows how to multiply integers but does not know how to multiply decimals. We can only determine this by assessing her on both concepts, and therefore, the subsumption relationship does not help us in this scenario. We can, however, use information about how similar students performed in the past to help us identify this scenario. (In a future blog post, we’ll expand on network effects, or how we utilize data about other students’ activity to inform the recommendations we generate for each individual student.)

The example above involves just a few concepts, but for a typical Knewton-enhanced course, we map out the relationships between hundreds of concepts. Knowing what concepts subsume other concepts allows us to eliminate concepts that we think students are already proficient in and more quickly hone in on what they should study to meet their goals. It’s like when the blocks disappear in Tetris!

To summarize: rather than assessing a student on every single prerequisite concept, we can use our understanding of the content — specifically, the relationships that exist between the concepts — to make intelligent inferences about what the student does and does not know. As a result, we can generate recommendations that lead to a more efficient use of the student’s time and energy.

In this post, I’ve described one way that we use our understanding of content to make learning more efficient and effective for our students. In a future blog post, I’ll talk about how we use goals that are defined by students and instructors to help us generate better recommendations.

**For more from the Knewton Adaptive Instruction team, check out Jesse Sternberg’s post on the cross-disciplinary approach of the Knewton knowledge graph and Matt Busick’s post on the power of a knowledge graph. **