RC-Based Modeling of student engagement

Think of your attendance tracker as the charge on two capacitors: Aggregate Tracker (AT) and Consistency Tracker (CT), both with a capacitance of 1 Farad. Since the capacitance is sized at 1 unit, the charge and voltage are equivalent and used interchangeably throughout this document.

The figure below illustrates the overall model. The bottom portion shows the circuit that models your attendance. The battery at the center represents the "class," and you charge your capacitors by "Engaging" with the class. In the circuit, the Engagement switch is closed whenever you attend.

The left capacitor, Aggregate Tracker (AT), located in the green highlighted section, represents your overall attendance. It starts with zero charge at the beginning of the semester. As you attend class and the Engagement switch is closed, your AT charge increases. When you stop attending class, the Engagement switch opens, and there’s no decrease in charge. You can begin accumulating charge again when you return to class. The resistor for this part of the circuit is set to 30. Based on RC-circuit formulas, after attending 30 lectures, you’ll have achieved about 63% (63 units) of the total charge needed. This is above the threshold for the Aggregate Tracker, which is 60 units.

The right capacitor, Consistency Tracker (CT), located in the blue highlighted section, represents your consistency in attendance and works a bit differently. It starts fully charged at 100 units at the beginning of the semester. If you attend every class, the Engagement switch stays closed, and you maintain your 100 units, easily surpassing the threshold of 60. However, if you "disengage" from the class, the Engagement switch opens, and the Disengagement switch closes, causing your CT to rapidly discharge.

Disengaging from class isn't just about missing a single lecture. We understand that life can interfere, and there may be instances throughout the semester where you miss class due to valid reasons. To accommodate this, we allow for four free passes. If you miss a class but return to the next lecture, you lose one free pass but aren’t considered "disengaged" yet. Once all four free passes are used, missing any class will result in being marked as disengaged, and the Disengagement switch will close immediately. Even if you have free passes remaining, missing consecutive classes will trigger the Disengagement switch, quickly providing feedback that you need to re-engage with the class.

The top part of the figure below shows the flowchart that determines when the Engagement and Disengagement switches are open or closed. Together with the circuit, they provide a complete picture of how the model works.

The Consistency Tracker consists of a capacitor to store the charge and two resistors—one for charging and one for discharging. You can think of the battery as the energy source for attending class. When you engage with the course by attending classes, the left switch closes, and the right switch remains open. This allows the CT capacitor to charge, similar to how an RC circuit operates. The figure below illustrates the CT Tracker in "Engagement" mode, where E is 1 and D is 0.

When you disengage from the course by missing consecutive classes or too many sporadic classes, the Engagement switch opens, and the Disengagement switch closes. This causes the CT to discharge rapidly, similar to how an RC circuit operates. The figure below illustrates the CT circuitry in "Disengagement" mode, where E is 0 and D is 1.

As described in the overview, there is a grace period between Engagement and Disengagement Modes, referred to as the "Intermediate" Mode. This mode depends on whether you have any free passes remaining or if you're missing classes consecutively, even if you still have free passes. In Intermediate Mode, both the E and D switches are open, and your CT charge remains unchanged. The figure below shows the state of the Consistency Tracker in this mode when both Eand D are set to zero.

The graph below illustrates the Finite State Machine representing the three states of the Consistency Tracker. As described earlier, your state transitions from Engaged to Intermediate or Disengaged based on your attendance and the status of your free passes.

To get the governing formulas for the CT charge in the "Engagement" mode, using Kirchhoff's Voltage Law (KVL) around the loop on the left, we get:

Note that here, Vi is the battery (class) and Vc is your capacitor's charge (because the size of the capacitor is 1 Farad.) Then, we can do some basic algebra and calculus to obtain the solution for a differential equation modeling charging capacitance:

Note that, as it pertains to our purposes, t is the number of consecutive classes you have attended after a miss or a steak of misses. Thus, to calculate the growing exponential from any given point of your attendance streak, impose initial conditions at the start of the streak (t=0):

Yay! We now have a closed form solution to calculate your CT charge in the Engagement Mode:

Similarly, for the "Disengagement: Mode, we start by writing Kirchhoff's Current Law (KCL) at the node of the capacitor connected to the closed discharge switch:

This is a linear, constant-coefficient, time-invariant, ordinary, homogenous differential equation. So by inspection, we can see the solution takes the form:

Similar to charging, t is the number of consecutive classes you have missed after an attendance or a streak of attendances. And once again, imposing initial conditions, we get:

The "Intermediate" Mode and Free Passes are introduced because we understand that sometimes life can interfere with attendance. You can think of them as a delay in the process that triggers the drop in your CT charge. Each student is granted a total of four free passes throughout the semester. When you use a free pass, the decay curve is shifted to the right by one. This is shown in the below equation. 

In order to provide you with the ability to quickly calculate your grades, there are two options. You can use MATLAB or Python, the code for which is provided below. MATLAB is simple, as you simply only need to install MATLAB and run it. The downside is that the plot generated is not interactive. On the other hand, more advanced users can try setting up a Python environment, installing the dependencies specified by the import statements, and running the script. When run, a dash server will be served on localhost, which can be accessed (usually through http://127.0.0.1:8085) in the browser and you can interactively play with different scenarios to model your grade.

MATLAB:

close all

clear

%% Define Constants

engagement_tau = 12;

disengagement_tau = 6;

lowest_possible_engagement = 0;

highest_possible_engagement = 100;

aggregate_tau = 30;

hold_duration = 1;

num_lectures = 38;

lecture_numbers = 1:num_lectures;

%% Initialize variable data - REPLACE THE 1 CORRESPONDING WITH THE ABOVE LECTURE NUMBER WITH 0 FOR A MISSED LECTURE

lectures_attended = [

   %   1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 16 17 18 19

       1, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1,...

   ...   20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

         1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

   ];

if ~isequal(size(lectures_attended), [1, num_lectures])

   error("The class attendance vector must be a row vector" + ...

       "with %u elements", num_lectures)

end

consistency_result = zeros(1, num_lectures);

aggregate_result = zeros(1, num_lectures);

free_pass_count = 4;

prev_consistency_charge = highest_possible_engagement;

prev_aggregate_charge = 0;

prev_consistency_discharge = highest_possible_engagement;

consistency_en_volts = highest_possible_engagement;

aggregate_en_volts = 0;

attendance_streak = 0;

miss_streak = 0;

%% Calculate grades at each class

for idx = lecture_numbers

   if logical(lectures_attended(idx))

       attendance_streak = attendance_streak + 1;

       consistency_en_volts = highest_possible_engagement + (prev_consistency_charge - highest_possible_engagement) * exp(-attendance_streak / engagement_tau);

       aggregate_en_volts = highest_possible_engagement + (prev_aggregate_charge - highest_possible_engagement) * exp(-attendance_streak / aggregate_tau);

       miss_streak = 0;

       prev_consistency_discharge = consistency_en_volts;

       if idx > 1 && ~(logical(lectures_attended(idx-1)))

           free_pass_count = free_pass_count - 1;

       end

   else

       miss_streak = miss_streak + 1;

       if free_pass_count <= 0

           consistency_en_volts = prev_consistency_discharge * exp(-(miss_streak) / disengagement_tau);

       elseif miss_streak > hold_duration

           consistency_en_volts = prev_consistency_discharge * exp(-(miss_streak - hold_duration) / disengagement_tau);

       end

       attendance_streak = 0;

       prev_consistency_charge = consistency_en_volts;

       prev_aggregate_charge = aggregate_en_volts;

   end

   consistency_result(idx) = consistency_en_volts;

   aggregate_result(idx) = aggregate_en_volts;

end

figure

plot(lecture_numbers, consistency_result, 'b-o')

hold on

plot(lecture_numbers, aggregate_result, 'r-o')

legend('Consistency Attendance Tracker', 'Aggregate Attendance Tracker', 'Location', 'southoutside')

title("Student Engagement Over Time")

xlabel("Time (in terms of completed lectures)")

ylabel("Engagement (En-Volts)")

axis([0 num_lectures lowest_possible_engagement highest_possible_engagement])

xticks(lecture_numbers)

Python: