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We develop a mathematical framework, based on natural language processing models, for tracking and characterizing the acquisition of conceptual knowledge. Our approach embeds each concept in a high-dimensional representation space, where nearby coordinates reflect similar or related concepts. We test our approach using behavioral data from participants who answered small sets of multiple-choice quiz questions, interleaved between watching two course videos from the Khan Academy platform. We apply our framework to the videos' transcripts and the text of the quiz questions to quantify the content of each moment of video and each quiz question. We use these embeddings, along with participants' quiz responses, to track how the learners' knowledge changed after watching each video. Our findings show how a small set of quiz questions may be used to obtain rich and meaningful high-resolution insights into what each learner knows, and how their knowledge changes over time as they learn. 

Data visualizations can reveal trends and patterns that are not otherwise obvious from the raw data or summary statistics. While visualizing low-dimensional data is relatively straightforward (for example, plotting the change in a variable over time as (x,y) coordinates on a graph), it is not always obvious how to visualize high-dimensional datasets in a similarly intuitive way. Here we present HypeTools, a Python toolbox for visualizing and manipulating large, high-dimensional datasets. Our primary approach is to use dimensionality reduction techniques (Pearson, 1901; Tipping & Bishop, 1999) to embed high-dimensional datasets in a lower-dimensional space, and plot the data using a simple (yet powerful) API with many options for data manipulation [e.g. hyperalignment (Haxby et al., 2011), clustering, normalizing, etc.] and plot styling. The toolbox is designed around the notion of data trajectories and point clouds. Just as the position of an object moving through space can be visualized as a 3D trajectory, HyperTools uses dimensionality reduction algorithms to create similar 2D and 3D trajectories for time series of high-dimensional observations. The trajectories may be plotted as interactive static plots or visualized as animations. These same dimensionality reduction and alignment algorithms can also reveal structure in static datasets (e.g. collections of observations or attributes). We present several examples showcasing how using our toolbox to explore data through trajectories and low-dimensional embeddings can reveal deep insights into datasets across a wide variety of domains.