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1 parent bcfb996 commit ac98225cd884174adfac2bbe0ff2bc0a9903885b @bvds committed May 10, 2013
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6 LogProcessing/moment-of-learning/moment-of-learning.tex
@@ -215,14 +215,14 @@ \subsection{Method}
the Andes intelligent tutor homework system~\cite{vanlehn_andes_2005}.
231 hours of log data were recorded.
%, covering 85,744 transactions, and 26,204 student steps.
-Each step was assigned to one or more different KC's. The dataset
+Each step was assigned to one or more different KCs. The dataset
contains a total of 2017 distinct student-KC sequences covering a total of
-245 distinct KC's. We will refer to this dataset as student dataset
+245 distinct KCs. We will refer to this dataset as student dataset
$\mathcal{A}$. See Figure~\ref{student-length-histogram} for a
histogram of the number student-KC sequences having a given number of
steps.
-Most KC's are associated with physics
+Most KCs are associated with physics
or relevant math skills while others are associated with
Andes conventions or user-interface actions (such as, notation
for defining a variable). The student-KC sequences with the largest
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37 LogProcessing/moment-of-learning/three-models.tex
@@ -211,10 +211,9 @@ \section{Three models of learning}
These models involve fitting data for
multiple students and multiple KCs and may involve other observables
such as the number of prior successes/failures a student has had for
-a given skill.
-Since we are interested in fitting
+a given skill. However, in this investigation, we are interested in fitting
to the correct/incorrect bit sequence for a single student
-and a single KC, a logistic regression model will take on a
+and a single KC and a logistic regression model takes on a
relatively simple form
%
\begin{equation}
@@ -237,7 +236,8 @@ \section{Three models of learning}
The third model is the ``step model'' which assumes that learning
occurs all at once; this corresponds to the ``eureka learning''
-discussed by \cite{baker_detecting_2011}. It is defined as:
+discussed by \cite{baker_detecting_2011}.
+It is defined as:
%
\begin{equation}
P_\mathrm{step}(j)= \left\{\begin{array}{cc}
@@ -256,6 +256,7 @@ \section{Three models of learning}
model. Thus, this model satisfies criteria \ref{crit:step} and
\ref{crit:perform}.
+
\section{Model selection using AIC}
\label{model-selection}
@@ -326,14 +327,14 @@ \subsection{Method}
the Andes intelligent tutor homework system~\cite{vanlehn_andes_2005}.
231 hours of log data were recorded.
%, covering 85,744 transactions, and 26,204 student steps.
-Each step was assigned to one or more different KC's. The dataset
+Each step was assigned to one or more different KCs. The dataset
contains a total of 2017 distinct student-KC sequences covering a total of
-245 distinct KC's. We will refer to this dataset as student dataset
+245 distinct KCs. We will refer to this dataset as student dataset
$\mathcal{A}$. See Figure~\ref{student-length-histogram} for a
histogram of the number of student-KC sequences having a given number of
steps.
-Most KC's are associated with physics
+Most KCs are associated with physics
or relevant math skills while others are associated with
Andes conventions or user-interface actions (such as, notation
for defining a variable). The student-KC sequences with the largest
@@ -390,7 +391,7 @@ \subsection{Analysis}
Since the goodness of fit criterion, AIC, is valid in the limit of
many steps, we include in this analysis only student-KC sequences that
contain 10 or more steps, reducing the number of student-KC sequences
-to 267, covering 38 distinct KC's. We determine the correctness of
+to 267, covering 38 distinct KCs. We determine the correctness of
each step (Section~\ref{steps}), constructing a bit sequence, {\em
exempli gratia} 001001101, for each student-KC sequence. This bit
sequence is then fit to each of the three models, $P_\mathrm{step}$,
@@ -596,6 +597,26 @@ \subsection{Summary}
better model of student learning, in the usual sense. The better fit does
not predict anything about the nature of learning.
+Our results suggest that the step model may be useful
+for modeling the learning of an individual student.
+However, the step model assumes that learning a skill occurs
+in one step. Is this how people actually learn? Certainly, everyone
+has experienced instances of ``eureka learning'' at some point. However,
+it is unclear how well this describes the acquisition of most skills,
+especially since
+many KCs are implicit and we are not consciously aware that we
+know them~\cite{koedinger_knowledge-learning-instruction_2012}.
+Certainly, if the student performance bit sequence is of the
+form $00\ldots 0 1 1 \ldots 1$, then is seems safe to assume
+that learning occurred all in one step, corresponding to the first
+1 in the sequence. However, it is possible that the transition
+from unmastered to mastery occurs over some number of
+opportunities.
+In a companion paper~\cite{van_de_sande_measuring_2013}, we introduce
+a method (based on AIC)that can correctly describe gradual mastery,
+even though the step model itself assumes all-at-once learning.
+
+
Finally, we see that the scatter plot of Akaike weights for student
data is remarkably similar to the scatter plots for the random model.
This suggests that the student data has a high degree of randomness,
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4 LogProcessing/self-improved-tutor/self-improved-tutor.tex
@@ -465,7 +465,7 @@ \section{Objective function}
The machine learning algorithm finds a function $f$ that acts on
the set of states $\left\{\mathbf{x}_k\right\}$ that minimizes
-the objective function $Z$ summed overs students and KC's.
+the objective function $Z$ summed overs students and KCs.
Since our policies are binary-valued
and many of our features are well ordered (times, counts of transactions,
{\em et cetera}), it is natural to define $f$ in terms of a
@@ -482,7 +482,7 @@ \section{Objective function}
space. All states on one side of the plane are given policy 0
and states on the other side have policy 1.
Numerically, we find $\mathbf{a}$ and $b$ that minimizes $Z$
-summed over students and KC's.
+summed over students and KCs.
Optimum values of $\mathbf{a}$ and $b$ for the
Study 1 log data are shown in Table~\ref{results}.

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