## The New Chalk: How Machine Learning Can Benefit Higher Education

Machine learning, AI and other algorithmic technologies have long promised to enhance the learning experience for college instructors and students.

But what can these technologies actually deliver? What’s required to implement them effectively? And how can they operate in a way that’s both equitable and transparent?

**Andrew Jones**, a data scientist here at Knewton, joined a panel discussion hosted by EdSurge this week in NYC that sought to answer some of these questions.

The panel, which included a group of educators and education technologists, covered a range of issues, including how machine learning technologies are perceived by students, specific areas where machine learning can make an impact on learning, and the barriers that must be overcome for this technology to be implemented successfully.

When asked to suggest the tough questions that instructors should ask before implementing machine learning technologies in their classroom, Andrew urged instructors to push for greater transparency into how a company’s algorithms work. “Asking what is being optimized for, and why, can give you a sense of [whether a tool] is focused on student outcomes, or whether it is about getting a prediction that’s right more often,” he said.

Toward the end of the session, the focus shifted to instructors themselves — and the role they will play in courses that increasingly feature machine learning technologies, such as virtual assistants

Andrew underscored the central role of the instructor, saying: “I’d rather see machine learning reach the level of chalk.”

You can find a full recap of the event over on EdSurge.

## How does Knewton’s Proficiency Model estimate student knowledge in alta?

Accurately estimating a student’s knowledge is one of the core challenges of adaptive learning.

By understanding what a student knows and doesn’t know, adaptive learning technology is able to deliver a learning experience that will help the student achieve mastery. Understanding the student knowledge state is also essential for delivering accurate, useful analytics to students and instructors.

We refer to our data-driven mathematical model for estimating a student’s knowledge state as Knewton’s **Proficiency Model**. This model lies at the core of our ability to deliver lasting learning experiences to students using **alta**.

How does our Proficiency Model estimate student knowledge? Answering that question begins by looking at its inputs, which include:

**The observed history of a student’s interactions**, including which questions the student answered correctly and incorrectly, the instructional material they studied, and when they performed these activities.**Content properties**, such as the difficulty of a question the student is answering.**The structure of the Knewton Knowledge Graph**, in particular the prerequisite relationships between learning objectives.

The model’s outputs represent the student’s proficiencies in all of the learning objectives in the Knowledge Graph at a given point in time. So what’s in between the inputs and the outputs?

## A basis in Item Response Theory

The foundation for our Proficiency Model is a well-known educational testing theory known as Item Response Theory (IRT).

One important aspect of IRT is that it benefits from *network effects* — that is, we learn more about the content and the students interacting with it as more people use the system. When a student answers a difficult question correctly, the model’s estimated proficiency for that student should be higher than it would be if the student had correctly answered an easy question. But how can we determine each question’s difficulty level? Only by observing how large numbers of diverse students performed when responding to those questions.

With this data in-hand, we are able to better and more efficiently infer student proficiency — or weakness — and deliver content that is targeted and effective.

### Moving beyond the limits of IRT

Because IRT was designed for adaptive testing — a learning environment in which a student’s knowledge remains fixed — it does not meet all of the requirements of adaptive learning, an environment in which the student’s knowledge is continually changing. In a model based on IRT, a student’s older responses make the same impact on the student’s proficiency level as their more recent responses. While this is fine in a testing environment, in which students aren’t typically provided feedback or instruction, it becomes a problem in an adaptive learning environment.

In an adaptive learning environment, we inherently expect that students’ knowledge will change. As a result, we want to give more weight to recent responses than older ones — allowing for the possibility of an “Aha!” moment along the way.

To correct for the limitations of IRT, Knewton has built temporal models that weight a student’s recent responses more heavily than their older ones when determining proficiency, providing a more accurate and dynamic picture of the student’s knowledge state.

### Accounting for relationships between learning objectives

Adaptive learning requires constant, granular assessment on multiple learning objectives embedded in the learning experience. However, traditional IRT also does not account for the relationships between learning objectives. As discussed above, these relationships are an important part of the Knewton Knowledge Graph.

To remedy this shortcoming of IRT, Knewton has developed a novel way to incorporate these relationships in a Bayesian modeling framework, allowing us to incorporate prior beliefs about proficiency on related topics, with evidence provided by the student’s responses. This leads to so-called *proficiency propagation*, or the flow of proficiency throughout the Knowledge Graph.

What does this look like in practice? If, in the Knowledge Graph below, a student is making progress toward the learning objective of “Solve word problems by subtracting two-digit numbers,” our Proficiency Model infers a high proficiency on that learning objective. The model also infers a high proficiency on the related learning objectives (“Subtract two-digit numbers” and “Subtract one-digit numbers”), even without direct evidence. The basic idea: If two learning objectives are related and a student masters one of them, there’s a good chance the student has also mastered the others.

## The effectiveness of Knewton’s Proficiency model

The many facets of the Proficiency Model – IRT-based network effects, temporal effects, and the Knowledge Graph structure – combine to produce a highly accurate picture of a student’s knowledge state. We use this picture to provide content that will increase that student’s level of proficiency. It’s also the basis of the actionable analytics we provide to students and instructors.

How effective is the Proficiency Model in helping students master learning objectives? In his post “Interpreting Knewton’s 2017 Student Mastery Results,” fellow Knerd Andrew D. Jones presents data that shows that Knewton’s Proficiency Model helps students achieve mastery — and that mastery, as determined by the Proficiency Model, makes a positive impact on student’s academic performance.

## Interpreting Knewton’s 2017 Student Mastery Results

*This post was developed with Illya Bomash, Knewton’s Managing Data Scientist.*

Results. Efficacy. Outcomes.

Student success is the ultimate goal of learning technology. Despite this, there exists a startling lack of credible data available to instructors and administrators that speaks to the impact of ed-tech on learning and academic performance.

To provide instructors and administrators with greater transparency into the effectiveness of **alta** and the Knewton adaptive technology that powers it, we analyzed the platform results of students using **alta**. These results represent our effort to validate our measure of mastery (more on that to come) and provide instructors and administrators with much-needed transparency regarding the impact of **alta** on student achievement.

Here, we hope to provide context and explanation that we hope will leave educators and those in the ed-tech community with a clearer picture of how we arrived at the these results — and why they matter.

### Our data set

The findings in this report are drawn from the results of 11,586 students who cumulatively completed more than 130,000 assignments and 17,000 quizzes in **alta** in 2017.

This data set includes all of **alta**’s 2017 spring and summer student interactions. Only cases in which the relevant calculations are impossible have been excluded — such as quiz scores for a course in which the instructor chose not to administer quizzes. So while these results aren’t from randomized, controlled trials, they do paint an accurate portrait of student performance across **alta** users, making use of as much of our student data as possible.

### Why mastery?

Our adaptive technology is based on the premise that if a student masters the concepts tied to the learning objectives of their course, that student will succeed in the course and be prepared to succeed in future courses. It’s also based on the premise that Knewton’s mathematical model of student knowledge states — which we frequently refer to as Knewton’s proficiency model — can determine when a student has reached mastery.

This basis in mastery manifests itself in how students experience **alta**: Every assignment that a student encounters in **alta** is tied to learning objectives that have been selected by the instructor for their course. A student “completes” an **alta** assignment when our proficiency model calculates that a student has mastered all of the learning objectives covered in that assignment.

Our 2017 Mastery Results seek to clarify two things: the frequency with which students achieve mastery in **alta**, and the later performance of students who have (and have not) achieved mastery, as determined by our proficiency model.

### Controlling for students’ initial ability level

In this analysis, we wanted to assess the impact of mastery across the full spectrum of student ability levels. To capture a sense of each student’s initial proficiency, we aggregated the first two questions each student answered across all of the concepts he or she encountered in the course. The percentage of those questions the student answered correctly provides a naive but reasonable estimate of how well the student knew the material entering the course.

We looked at the distribution of this score across all of our students, tagging each student’s history with a label corresponding to where they fell among all users.

**Struggling**: Students whose initial ability fell into the bottom 25% of our population**Average**: Students whose initial ability fell into the middle 50% of our population**Advanced**: Students whose initial ability fell into the top 25% of our population

*Note*: Knewton’s proficiency model neither uses this measure nor tags students with any kind of “ability label.” Our adaptive technology calculates a detailed, individualized portrait of each student’s proficiency levels across a wide range of concepts after each student interaction. But for the sake of this comparative impact analysis, we’ve chosen to use these distinctions as a tool to compare students of similar initial abilities.

## Our findings

### Students of all ability levels achieved mastery with alta at high rates

Analyzing students’ assignment completion revealed that with **alta**, students achieve mastery at high rates. As seen in **Figure 1**, across all students, 87% of the time, students working on an assignment in **alta** achieved mastery. Even among students who struggled to complete a particular assignment, 82% eventually reached mastery.

### Achieving mastery with alta makes a positive impact on students’ academic performance

We know that with **alta**, students are highly likely to achieve mastery. But what is the impact of that mastery? When our model indicates that a student has mastered the material, how well does the student perform on future assignments, quizzes, and tests?

For any given level of initial ability, Knewton’s adaptive learning technology is designed to facilitate reaching mastery effectively for any student willing to put in the time and effort. To validate Knewton’s measure of mastery, we compared the performance of students who mastered prerequisite learning objectives (for adaptive assignments) and target learning objectives (for quizzes) through **alta**with students of similar initial ability who did not master these concepts.

### Mastery improves the quiz scores for students of all ability levels

**Figure 2** shows average Knewton quiz scores for students who did/did not reach mastery of the quiz learning objectives on prior adaptive assignments. Quiz takers who mastered at least ¾ of the quiz learning objectives through previous adaptive work went on to achieve substantially higher quiz scores than similarly-skilled peers mastering ¼ or fewer of the learning objectives.

### Mastery levels the playing field for struggling students

Putting in the work to reach mastery on the relevant adaptive assignments increased initially struggling students’ average quiz scores by 38 percentage points, boosting scores for these students above the scores of otherwise advanced students who skipped the adaptive work.

### Mastery improves future platform performance

Students who master the learning objectives on earlier assignments also tend to perform better on later, more advanced assignments.

### Assignment completion

As **Figure 3** shows, controlling for overall student skill levels, students who mastered ¾ of the learning objectives prerequisite to any given assignment tended to complete the assignment at much higher rates than students who did not. This is the virtuous cycle of mastery: the more students master, the better prepared they are for future learning.

### Work to completion

Mastery of an assignment’s learning objectives also saves students time. When students began an assignment after having mastered most of its prerequisites, they tended to require significantly fewer questions to complete it. For students who mastered at least ¾ of the prerequisites to any given adaptive assignment, completing the assignment took 30-45% fewer questions than for students who did not (see **Figure 3**). Mastery helps students of all abilities learn faster, and struggling students see the biggest gains: for these students, prerequisite mastery leads to an average postrequisite assignment shortening by more than 40%.

### The road ahead

Any self-reported efficacy results will be met with a certain amount of scrutiny. While we’ve attempted to be as transparent as we can be about our data, we understand that some will question the validity of our data or our approach to presenting it.

It’s our hope that, if nothing else, the reporting of our results will inspire others in the ed-tech community to present their own with the same spirit of transparency. In many ways, these results are intended not as a definitive end-point but as the start of a more productive conversation about the impact of technology on learning outcomes.

Lastly, while our 2017 Student Mastery Results are encouraging, we know that they exist in a world that is constantly changing. The challenges in higher education are becoming greater and more complex. The student population is growing increasingly diverse. Our technology and our approach to learning is evolving.

This year, we plan to update these numbers periodically and provide the results of other analyses with the goal of providing greater transparency into the effectiveness of **alta** and deeper insights into how students learn.

## What are Knewton’s Knowledge Graphs?

Imagine you are tutoring a new student who is working on a homework assignment about **solving word problems that require two-digit subtraction**. The first question on the assignment is:

*“Jessica wants to buy a book that costs $15 and she has $32 in her wallet. After she buys the book, how much money will she have left?”*

Your student looks at the problem and looks back at you, terrified. He has no idea what to do. Clearly he hasn’t yet mastered the ability to solve word problems with two-digit subtraction. But what exactly is the problem? Is he struggling with the word problem itself, or is he missing crucial prerequisite knowledge as well?

Answering this question correctly requires a student to be able to perform the following steps:

**Translate a word problem about subtraction into a mathematical expression**: Question text → 32–15 = ?**Solve the two-digit subtraction problem**: 32–15**Solve one-digit subtraction problems (while performing the two-digit subtraction problem)**: 12–5, 2–1

From the above breakdown, we can see that this word problem requires a student to know three separate skills, or learning objectives. These learning objectives also happen to build on each other: a student can’t solve a two-digit subtraction problem without first knowing how to solve one-digit subtraction problems, and a student can’t solve word problems that require two-digit subtraction without knowing how to perform two-digit subtraction. We can represent the relationship between these learning objectives in the following way:

In the picture above, we are representing each learning objective as an oval. An arrow pointing from oval A to oval B indicates that a student must know A in order to be successful in B — we describe this relationship by saying that “A is a prerequisite to B”. This simple framework forms the foundation of **Knewton’s Knowledge Graphs**, a key tool that helps Knewton provide adaptive assignments and analytics in **alta**.

## Extending the Example

The tiny Knowledge Graph above is a very simple example of how relationships between learning objectives can be represented. In many cases, a given learning objective can have multiple direct prerequisites. A simple example of a learning objective with two prerequisites can be found by looking at the learning objective requiring students to **divide fractions**. To divide fraction X by fraction Y, you multiply X times the reciprocal of Y. This means that in order to be able to divide fractions, you must already be able to (1) **multiply fractions**and (2) **find reciprocals of fractions**.

By connecting each learning objective to its prerequisites, we can create a large Knowledge Graph full of connections between our content. Learning objectives can be connected to each other even if they appear in different assignments, chapters, or courses. Below, we show an example section of a large Knowledge Graph, where different colored learning objectives can be thought of as different assignments, chapters, or course levels.

The notion of representing content relationships in a graph structure is not new, and Knewton’s Knowledge Graphs build on previous work (Novak 1990; McAleese 1999; Doignon and Falmagne 1999; Hwang 2003). Knewton’s Knowledge Graphs are **created by trained subject matter experts**, who identify the learning objectives in a course, the relationships between these learning objectives, and the pieces of content that teach and assess each learning objective. This **direct alignment of each piece of content to learning objectives** allows Knewton to precisely diagnose which learning objectives a student has mastered, in addition to providing targeted instructional content when a student is struggling.

## Powering Just-In-Time Remediation

Knewton’s Knowledge Graphs allow us to generate adaptive recommendations based on pedagogical criteria, such as those reviewed by Graesser et al. (2012), including frontier learning, building on prerequisites, and providing remediation. For example, let’s go back to our struggling student. It is possible that our student may not have the prerequisite knowledge necessary to succeed in solving word problems with two-digit subtraction. If he struggles when solving these word problems in an **alta** adaptive assignment, Knewton’s recommendation engine can diagnose his prerequisite knowledge by using the information contained in the Knowledge Graph and provide just-in-time remediation in the form of **targeted instructional content and formative assessment on the prerequisite learning objective(s) that he is struggling with**. As the student masters the prerequisites, Knewton can move him forward towards his ultimate goal of learning how to solve word problems with two-digit subtraction.

Since Knewton’s Knowledge Graphs are an abstract representation of relationships between learning objectives, prerequisite relationships can also be defined between learning objectives existing in different books, subject areas, or school years, enabling **cross-domain adaptivity**. For example, Knewton can recommend remediation on a math prerequisite that’s required for mastering a chemistry learning objective. As we’ll see in a future blog post, Knewton’s Knowledge Graphs are also a **key input in the Knewton predictive analytics engine**, enabling Knewton to estimate student mastery on each learning objective in a graph.

## Sources

Doignon, J. P. and Falmagne, J. C. (1999). Knowledge Spaces. Springer.

Graesser, A. C., Conley, M. W., and Olney, A. (2012). Intelligent Tutoring Systems. APA Handbook of Educational Psychology. Washington, DC: American Psychological Association.

Hwang, G.-J. (2003). A Conceptual Map Model for Developing Intelligent Tutoring Systems. Computers & Education, 40(3):217–235.

McAleese, R. (1999). Concept Mapping: A Critical Review. Innovations in Education and Training International, 36(4):351–360

Novak, J. D. (1990). Concept Mapping: A Useful Tool for Science Education. Journal of Research in Science Teaching, 27(10):937–949

## What does knowing something tell us about a related concept?

At Knewton, we’ve built an adaptive learning platform that powers digital education around the world based on cutting-edge algorithms that leverage the diverse datasets we receive. One of the core data-driven models that powers everything we do is our *Proficiency Model*, which we use to infer each student’s knowledge state. We do this by combining a “knowledge graph”, time-tested psychometric models, and additional pedagogically motivated modeling. We’ll show you how the relationships in the knowledge graph get realized in Knewton’s Proficiency Model and answer the question: **“What does knowing something tell us about knowing a related concept?” **This has important pedagogical consequences, as well as an enormous impact on how our recommendations get served (and how confident we can be in their accuracy!).

**The Knowledge Graph**

One of the core components of Knewton adaptivity is the *knowledge graph*. In general, a graph is composed of nodes and edges. In our case, the nodes represent independent *concepts*, and the edges represent *prerequisite*relationships between concepts. An edge between concepts A and B (A → B) can be read as *Concept A is prerequisite to concept B*. This means that the student generally must know concept A before being able to understand concept B. Consider the example portion of a knowledge graph below:

In math-speak this is a directed acyclic graph (DAG). We already covered what the “graph” part means. The “directed” part just means that the edges are directed, so that “A prerequisite to B” does **not** mean “B prerequisite to A” (we instead say “B *postrequisite* to A”). This is in contrast to undirected edges in social networks where, for example, “A is friends with B” *does* imply “B is friends with A”. The “acyclic” part of DAG means there are no cycles. A simple cycle would involve A → B → C → A. This would imply that you need to know A to know B, B to know C, and then C to know A! This is a horrible catch-22. You can never break the cycle and learn these concepts! Disallowing cycles in the graph allows us to represent a course, without contradictions, as starting with more basic concepts, and leading to more advanced concepts as the student progresses (this progression is top-to-bottom in the graph above).

Another crucial aspect of the knowledge graph is the content: i.e. the assessing questions and the instructional material. Each concept has a number of such content pieces attached, though we don’t show them in the picture above. You can think of them as living inside the node.

**How do we know what you know?**

Of course, we can never know exactly what you know– that’d be creepy! Instead we *estimate* the student knowledge state using a mathematical model called the** Proficiency Model**. This takes, as inputs, the observed history of a student’s interactions, the graph structure, and properties of the content (question difficulty, etc.) and outputs the student’s proficiency in all the concepts in the graph at a given point in time. This is summarized below:

Abstractly, *proficiency* on a concept refers to the ability for a student to perform tasks (such as answer questions correctly) related to that concept. Thus, we can use the estimated values of the proficiencies to *predict* whether the student answers future questions correctly or not. Comparing our predictions to reality provides valuable feedback that allows us to constantly update and improve our model and assumptions.

The foundation for our Proficiency Model is a well-tested educational testing theory known as Item Response Theory (IRT). One important aspect of IRT is that it accounts for *network effects*— we learn more about the content and the students as more people use the system, leading to better and better student outcomes. IRT also serves as a foundation for our Proficiency Model on which we can build additional features.

One thing that basic IRT does not include is any notion of temporality. Thus older responses count the same as newer responses. This is fine in a testing environment, where “older” responses mean “generated 20 minutes ago”, but isn’t great in a learning environment. In a learning environment, we (obviously) expect that students will be learning, so we don’t want to overly penalize them for older work when in fact they may have had an “Aha!” moment. To remedy this, we’ve built temporal models into IRT that make more recent responses count more towards your proficiency estimate than older responses on a concept*.

Another thing that basic IRT does not account for is instructional effects. Consider the following example. Alice got 2 questions wrong, watched an informative video on the subject, and then got one question right. Under basic IRT we’d infer that her proficiency was the same as Bob who got the same 2 question wrong, did **not** watch the video, and then got one question correct. This doesn’t seem accurate. We should take Alice’s instructional interaction into account when inferring her knowledge state and deciding what’s best for her to work on next. We have extended IRT to take into account instructional effects.

Finally, basic IRT does not account for multiple concepts, nor their interrelationships in the knowledge graph. This will be the main focus of the rest of this post.

**Proficiency propagation**

The titular question of this post: “What does knowing something tell us about knowing a related concept?” is answered through *Proficiency Propagation*. This refers to how proficiency flows (propagates) to different concepts in the knowledge graph.

To motivate why proficiency propagation is important, let’s consider two different scenarios.

**Propagating mastery**

**Propagating mastery**

First, consider the example shown below, where the only activity we’ve observed from Alice is that she performed well (a ✔ indicates a correct response) on several more advanced concepts.

We can’t know everything Alice has ever done in this course– she may have done a lot of work offline and answered tons of “*Add whole numbers*” questions correctly. Since we don’t have access to this information, we have to make our best inference. Note that all three concepts Alice excelled at are reliant upon “*Add whole numbers*” as a prerequisite. Let’s revisit the definition of the prerequisite relationship. We say “A is prerequisite to B” (A → B) if A must be mastered in order to understand B. In other words:

Concept B is mastered ⇒ Concept A is mastered

In our case, there are three different “concept B’s” that Alice has clearly mastered. Thus, by definition of the prerequisite relationship Alice almost certainly has mastered “*Add whole numbers*” (it’s the concept A). So let’s paint that green, indicating likely mastery.

By similar reasoning, if Alice has mastered “*Add whole numbers*”, then she has likely mastered its prerequisite “*Understand the definition of whole numbers and their ordering*”. However, we might be slightly less certain about this inference, since it is more indirect and relies on a chain of reasoning. So let’s paint that slightly less bright green:

What about the remaining two concepts? First consider “Multiply whole numbers”. Alice has mastered its prerequisite, which is promising. But she may have never received any instruction on multiplication, and may have never even heard of such a thing! On the other hand, she may be a prolific multiplier, having done lots of work on it in an offline setting. In this case, we don’t have the definition of “prerequisite” working in our favor giving us a clean inference. But certainly if we had to guess we’d say Alice is more likely to have mastered “Multiply whole numbers” than someone else who we have no info on. Thus, we give Alice a small benefit of the doubt proficiency increase from the baseline. Similar considerations apply to the last, most advanced concept:

Let’s summarize the lessons we’ve learned:

- Mastery (i.e. correct responses) propagates strongly ‘backwards’ to prerequisites.
- As we get further from direct evidence in the prerequisite chain, there is more uncertainty. Thus we infer slightly less mastery.
- Mastery propagates weakly ‘forwards’ to postrequisites.

*Propagating struggling*

Now let’s consider Bob, who has struggled on “Add whole numbers”, getting 3 incorrect:

Recall our deconstruction of the prerequisite relationship A → B:

Concept B is mastered ⇒ Concept A is mastered

Unfortunately, this doesn’t directly help us here, because Bob hasn’t mastered any concepts as far as we know. However, the contrapositive is exactly what we need:

Concept A is **not **mastered ⇒ Concept B is **not** mastered

Let’s take “struggling on” to be equivalent to “not mastered” for our purposes to get:

Struggling on Concept A ⇒ Struggling on Concept B

Thus, we now know that struggling-ness propagates strongly down to the postrequisites of “Add whole numbers”!

What about “*Understand the definition of whole numbers and their ordering*”? Similarly to the flipped situation of propagating mastery to postrequisites, we cannot make any strong pedagogical inferences just from the prerequisite relationship. However, we can still assert that it is more likely that Bob is struggling on it given we’ve seen him struggle on “Add whole numbers” than if we hadn’t seen him struggle on that concept:

Let’s summarize what we’ve learned about propagation of struggling-ness:

- Struggling (i.e. incorrect responses) propagates strongly forwards to postrequisites.
- As we get further from direct evidence in the postrequisite chain, there is more uncertainty. Thus we infer slightly less struggling.**
- Struggling propagates weakly backwards to prerequisites.

Notice these rules are just the mirror-opposites of the ones for propagating mastery! And all of this comes simply from the definition of “prerequisite-ness”, and some pedagogical reasoning.

**Details details**

While we now have a nice picture of how we want proficiency propagation to behave, that doesn’t count much unless we can rigorously define a mathematical model capturing this behavior, and code up an algorithm to efficiently compute proficiencies in real time for all possible cases. As they say, the devil is in the details. To give a flavor of what’s involved, here are some of the technical details our mathematical model and algorithm must obey:

- Convexity: This essentially means that the proficiencies are efficiently and reliably computable.
- Strong propagation of mastery up to prerequisites, and of struggling-ness down to postrequisites, with a slight decay in propagation strength at each ‘hop’ in the graph.
- Weak propagation of mastery down to postrequisites, and of struggling-ness up to prerequisites, with a large decay in propagation strength at each ‘hop’ in the graph.
- The above two points imply
*asymmetric propagation*: The impact of a response on neighboring proficiencies is asymmetric, always being stronger in one direction in the graph than the other. - All of this proficiency propagation stuff must also play nicely with the aforementioned IRT model and the extensions to include temporality and instructional effects.

Coming up with a well-defined mathematical model encoding asymmetric strong propagation is a challenging and fun problem. Come work at Knewton if you want to learn more details! !

**Putting it all together**

So what good exactly does having this fancy proficiency model do us? At the end of the day, students care about being served a good educational experience (and ultimately, progressing forward through their schooling), and in Knewton-land that inevitably means getting served good recommendations. Certainly, having a pedagogically-sound and accurate proficiency model does not automatically lead to good recommendations. But having a bad proficiency model almost certainly will lead to bad recommendations. A good proficiency model is necessary, but not sufficient for good recommendations.

Our recommendations rely on models built “on-top” of the Proficiency Model, and answer questions such as:

- What are useful concepts to work on next?
- Has the student mastered the goal material?
- How much instructional gain will this material yield for the student?
- How much will this piece of material improve our understanding of the student’s knowledge state and therefore what she should focus on next?

All of these questions can only be answered when equipped with an accurate understanding of the student’s knowledge state. As an example, consider Alice again. If we had a bare-bones proficiency model that did not propagate her mastery to “Add whole numbers”, we might consider this a valid concept to recommend material from. This could lead to a frustrating experience, and the feeling that Knewton was broken: “Why am I being recommended this basic stuff that I clearly already know?!”

At the end of the day, it’s user experience stories like this that motivate much of the complex data analysis and mathematical modeling we do at Knewton. And it’s what motivates us to keep pushing the limit on how we can best improve student learning outcomes.

**There are other temporal effects that kick-in if you’ve seen the same question more than once recently*.

*** There is a whole other layer of complexity in our Proficiency Model that we’ve glossed over. We actually estimate a student’s proficiency and a measure of our confidence in that estimate. These are the proficiency mean and variance, and can be combined to obtain confidence intervals, for example. For the purposes of this blog post, we are only considering the propagation of proficiency means.*

*This post was written by Michael Binger, a data scientist at Knewton.*