[LON-CAPA-cvs] cvs: modules /gerd/alt2007 damped.pdf damped.ppt dampedcleaned.pdf dampedcleaned.ppt dampedfit.pdf dampedfit.xls graphing.bib graphing.tex

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  Added files:                 
    /modules/gerd/alt2007	damped.pdf damped.ppt dampedcleaned.pdf 
                         	dampedcleaned.ppt dampedfit.pdf dampedfit.xls 

  Modified files:              
    /modules/gerd/alt2007	graphing.bib graphing.tex 
  Log:
  Next steps in grant proposal
  
  
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Index: modules/gerd/alt2007/graphing.bib
diff -u modules/gerd/alt2007/graphing.bib:1.1 modules/gerd/alt2007/graphing.bib:1.2
--- modules/gerd/alt2007/graphing.bib:1.1	Mon Feb 12 17:48:50 2007
+++ modules/gerd/alt2007/graphing.bib	Mon Feb 19 11:36:21 2007
@@ -381,4 +381,36 @@
    pages="375-382",
    title = {"Why doesn't it start from the origin?": Hearing the Cognitive Voice of Signs}
 }
-   
\ No newline at end of file
+
+@CONFERENCE{kennedy98,
+   author = "David M. Kennedy and Paul Fritze",
+   year = "1998",
+   booktitle = "Proceedings of the 1998 ED-MEDIA/ED-TELECOM Conference on Educational Multimedia and Hypermedia and World Conference on Educational Telecommunications",
+   title = {An Interactive Graphing Tool for Web-Based Courses}
+}
+
+@CONFERENCE{kennedy04,
+   author = "David M. Kennedy and Douglas R. Vogel and Tomson Xu",
+   year = "2004",
+   booktitle = "Beyond the comfort zone: Proceedings of the 21st ASCILITE Conference",
+   pages="493-502",
+   title = {Increasing opportunities for learning: Mobile graphing}
+}
+
+@ARTICLE{elby00,
+   author = "Andrew Elby",
+   year = "2000",
+   journal = "Journal of Mathematical Behavior",
+   volume = "19",
+   pages = "481-502",
+   title = {What students' learning of representations tells us about constructivism}
+}  
+
+@ARTICLE{meltzer05,
+   author = "David E. Meltzer",
+   year = "2005",
+   journal = "Am. J. Phys.",
+   volume = "75",
+   pages = "463-478",
+   title = "College Physics Students' Epistemological Self-Reflection and Its Relationship to Conceptual Learning"
+}
\ No newline at end of file
Index: modules/gerd/alt2007/graphing.tex
diff -u modules/gerd/alt2007/graphing.tex:1.1 modules/gerd/alt2007/graphing.tex:1.2
--- modules/gerd/alt2007/graphing.tex:1.1	Mon Feb 12 17:48:50 2007
+++ modules/gerd/alt2007/graphing.tex	Mon Feb 19 11:36:21 2007
@@ -9,10 +9,10 @@
 	\usepackage{graphicx}
 	
 \textwidth = 16.3cm
-\textheight = 22.8cm
+\textheight = 23cm
 \oddsidemargin = 0.2cm
 \evensidemargin = 0.2cm
-\topmargin = 2cm
+\topmargin = 0.8cm
 \headheight = 0.0in
 \headsep = 0.0in
 
@@ -33,15 +33,22 @@
 \section{Introduction}
 The ability to work with graphs is a basic skill of any scientist. When scientists are discussing concepts and phenomena, they quickly resort to sketches of one variable versus another, sometimes just three lines on a paper: two axes and a graph. But how often do we intentionally teach our students about this important representation format?
 
-Even with the relatively simple concept of position, velocity, and acceleration, students have unexpected difficulties translating between graphical representations and both the mathematical representation and the ``real world.''~\cite{mcdermott,beichner}. These difficulties can sometimes have subtle reasons that lead to cognitive disconnects: for example,  using a graph showing the position of a bouncing ball versus time, Ferrara~\cite{ferrara} found that even the sign, i.e., measuring the distance from the ground versus from the launch points, can make a large difference. Students might misinterpret a graph as a pictorial representation of a situation, mix up what is on the axes, or become confused when approximating the slope of a graph that does not start at the origin~\cite{beichner}. Students need practice both interpreting and generating these graphs, but appropriate formative assessment frequently does not happen in the typically large enrollment introductory courses due to scalability problems: there are simply not enough teaching assistants to give the students appropriate feedback on these complex tasks.
+Even with the relatively simple concept of position, velocity, and acceleration, students have unexpected difficulties translating between graphical representations and both the mathematical representation and the ``real world.''~\cite{mcdermott,beichner,meltzer05}. These difficulties can sometimes have subtle reasons that lead to cognitive disconnects: for example,  using a graph showing the position of a bouncing ball versus time, Ferrara~\cite{ferrara} found that even the sign, i.e., measuring the distance from the ground versus from the launch points, can make a large difference. Students might misinterpret a graph as a pictorial representation of a situation~\cite{elby00}, mix up what is on the axes, or become confused when approximating the slope of a graph that does not start at the origin~\cite{beichner}. Students need practice both interpreting and generating these graphs, but appropriate formative assessment frequently does not happen in the typically large enrollment introductory courses due to scalability problems: there are simply not enough teaching assistants to give the students appropriate feedback on these complex tasks.
 
-Only too often, problems given in physics courses focus on numerical calculations, e.g., ``A car accelerates from rest with $2 m/s^2$ for 10 seconds, what is the distance covered?'' -- students can ``solve'' these problems without any understanding of the underlying concepts~\cite{lin,heuvelen}. More sophisticated problems may ask students to match graphs with physical situations, but do not involve generating the graphs.
-
-Sketching a graph is also different from plotting a function or data, something which students already might have practice in: it is an activity that should be manageable with just a few strokes.
+Only too often, problems given in physics courses focus on numerical calculations, e.g., ``A car accelerates from rest with $2 m/s^2$ for 10 seconds, what is the distance covered?'' -- students can ``solve'' these problems without any understanding of the underlying concepts~\cite{lin,heuvelen}. The sketching of graphs is an example of a more constructivist approach to teaching these concepts. The students need to make a number of decisions:
+\begin{itemize}
+\item Where does the graph start (is the start point known and/or significant)?
+\item Where does the graph finish (is the end point known and/or significant)?
+\item What is the general shape of the curve (e.g., exponential growth or decay, a hysteresis, sinusoidial, asymptotic)?
+\item Are there singularities or significant points along the way?
+\end{itemize}
+ (list expanded from \cite{kennedy04}). Students need to construct the curve, which is quite different from selecting from a series of possible answers in a multiple choice setting, inputting an equation and having the software sketch it~\cite{kennedy04}, or plotting a given function or set of data. Sketching is an activity that should be manageable with just a few strokes to express a general relationship.
 
 Within this project, we will develop an online assessment tool for graph sketching, which can provide randomized scenarios and provide immediate feedback to graph sketches entered online with a mouse or trackpad. We will evaluate usability for both faculty and students, as well as impact on student problem solving strategies and conceptual learning.
 
 The tool will be developed on top of an existing course and learning content management system in order to minimize overhead. However, both the algorithms and the code will be made freely available, so they can be incorporated into other systems.
+\subsection{Learning Goals}
+\subsection{Technology Goals}
 \subsection{Intellectual Merit}
 \subsection{Broader Impact}
 \section{Project Overview}
@@ -57,8 +64,14 @@
 \item disseminate results
 \end{itemize}
 Several of these steps will be carried out iteratively over the duration of the project while refining both tool and content
-\section{Relevant Results from Prior NSF Support}
-\subsection{The LON-CAPA System}
+\section{Relevant Results from Related Projects}
+\subsection{Relevant Results from Prior NSF Support to the PIs}
+\begin{figure}
+\includegraphics[width=3.3in]{figures/coil1}
+\includegraphics[width=3.3in]{figures/coil2}
+\caption{Two versions of the same problem in LON-CAPA. Different students would see different currents as a function of time in coil 1, and need to identify the corresponding induced voltage in coil 2. In this scenario, it is simply $V_2(t)=M\frac{dI_1(t)}{dt}$, with $V_2(t)$ being the induced voltage in coil 2, $I_1(t)$ being the current in the first coil, and and $M$ being the mutual inductance factor between the coils. The second and third part of the question are multiple choice and numerical response (with physical units), respectively.\label{induction}}
+\end{figure}
+
 Gerd Kortemeyer has been Co-PI on the NSF-ITR grant {\it Investigation of a Model for Online Resource Creation and Sharing in Educational Settings} (\#0085921, \$2,055,000, 09/15/00-07/31/06). The project
 developed the cross-institutional learning content management system Learning{\it{}Online} Network with Computer Assisted Personalized Approach (LON-CAPA)\cite{loncapa} and researched methods to assess the educational impact of
 content resources and representations within its shared content pool.
@@ -69,11 +82,7 @@
 
 The system started in 1992 as a tool to deliver personalized homework to students. ``Personalized'' means that each student sees a different version of the same 
 computer-generated problem: different numbers, choices, graphs, images, simulation parameters, etc\cite{kashyd01}, see Fig.~\ref{induction} for an example.
-\begin{figure}
-\includegraphics[width=3.5in]{figures/coil1}
-\includegraphics[width=3.5in]{figures/coil2}
-\caption{Two versions of the same problem in LON-CAPA. Different students would see different currents as a function of time in coil 1, and need to identify the corresponding induced voltage in coil 2. In this scenario, it is simply $V_2(t)=M\frac{dI_1(t)}{dt}$, with $V_2(t)$ being the induced voltage in coil 2, $I_1(t)$ being the current in the first coil, and and $M$ being the mutual inductance factor between the coils. The second and third part of the question are multiple choice and numerical response (with physical units), respectively.\label{induction}}
-\end{figure}
+
 Over the years, the system was expanded with content management and standard course management features, such as communications, gradebook, etc., similar to those in commercial course management systems, such as 
 BlackBoard, WebCT, or ANGEL. In addition to standard features, the LON-CAPA delivery and course management layer is designed around STEM education, for example: 
 \begin{itemize}
@@ -90,8 +99,10 @@
 LON-CAPA is used by approximately 35,000 students every semester.
 
 LON-CAPA will provide the platform on which the tools will be developed and evaluated. It also provides the initial dissemination platform for problems authored over the course of this project.
+\section{Relevant Results from Other Related Projects}
+In 1997, the Interactive Graphing Tool was developed at the University of Melbourne in support of chemistry instruction~\cite{kennedy98}. Over the years, the project went through a number of technology iterations, and was eventually re-implemented online as the Interactive Graphing Object (IGO) and on PDAs as the mobile Interactive Graphing Object (mIGO)~\cite{kennedy04}.
 
-
+The IGO is more constrained than the proposed tool, in that it models all graphs through B\'ezier curves, and allows the users to then tweak the location and angles of the starting and end points, as well as a number of midpoints. These additional controls have been introduced into the learner interface over the initial freehand graphing as a result of usability concerns~\cite{kennedy04}. The display to the learners includes these coordinates and angles, and the evaluation of the graph is based on agreement of these values and those provided by the educator through the authoring interface.
 \section{Graph Evaluation}
 The LON-CAPA system currently allows to dynamically generate randomized graphs both in the problem text and in the answers. For example, in Fig.~\ref{induction}, different students get different graphs for the current in coil 1 over time, and need to identify the resulting induced voltage in coil 2. 
 
@@ -107,16 +118,16 @@
 
 Fig.~\ref{acccorrect} shows different acceptable solutions. Note that any solution that would show a positive acceleration for a limited time would be correct. Also note that due to the lack of precision in graphical input with a mouse, the left edge of the curve in the right panel actually slightly bends backward -- the software should accept these minor flaws.
 \begin{figure}
-\includegraphics[width=3.5in]{figures/acccorrect1}
-\includegraphics[width=3.5in]{figures/acccorrect2}
+\includegraphics[width=3in]{figures/acccorrect1}
+\includegraphics[width=3in]{figures/acccorrect2}
 \caption{Example of solutions that would be acceptable to the problem described in
 section~\ref{accproblem}.
 \label{acccorrect}}
 \end{figure}
 The panels in Fig.~\ref{accincorrect} are not acceptable. The left panel appears to be the velocity, while the right panel would be position. Using the adaptable hint feature of LON-CAPA, the system should allow to specify the criteria for these anticipated wrong solutions.
 \begin{figure}
-\includegraphics[width=3.5in]{figures/accincorrect1}
-\includegraphics[width=3.5in]{figures/accincorrect2}
+\includegraphics[width=3in]{figures/accincorrect1}
+\includegraphics[width=3in]{figures/accincorrect2}
 \caption{Example of solutions that would not be acceptable to the problem described in
 subsection~\ref{accproblem}.\label{accincorrect}}
 \end{figure}
@@ -129,36 +140,80 @@
 Fig.~\ref{potentialcorrect} shows different acceptable solutions. Note that the scale is irrelevant in this case, but the graphs need to go to infinity and minus infinity at the positions of the charges, respectively, go to zero at infinite distances, and be zero at the point where the potential is flat. 
 
 \begin{figure}
-\includegraphics[width=3.5in]{figures/correct1}
-\includegraphics[width=3.5in]{figures/correct2}
+\includegraphics[width=3in]{figures/correct1}
+\includegraphics[width=3in]{figures/correct2}
 \caption{Examples of acceptable solutions (blue) for the example problem described in section~\ref{potproblem}. The system provides a graph of a potential versus location along an axis that has two charges placed on it (black). The electric field is given by $\vec E=-\nabla V$.\label{potentialcorrect}}
 \end{figure}
 The graphs in Fig.~\ref{potentialincorrect} are not correct because of a sign error and the assumption that the field is zero where the potential is zero, respectively.
 
 \begin{figure}
-\includegraphics[width=3.5in]{figures/incorrect1}
-\includegraphics[width=3.5in]{figures/incorrect2}
+\includegraphics[width=3in]{figures/incorrect1}
+\includegraphics[width=3in]{figures/incorrect2}
 \caption{Examples of incorrect solutions for the example problem in section~\ref{potproblem}.\label{potentialincorrect}}
 \end{figure}
 Different randomizations of this problem will have other charge distributions or relative strengths, which will result in different system-provided graphs for the potential.
-\subsection{Salient versus spurious features}
-When the system evaluates a student-provided graph, it is important to distinguish salient from spurious graph features. Examples for salient features might include:
+\subsection{Significant versus spurious features}
+When the system evaluates a student-provided graph, it is important to distinguish significant from spurious graph features. Examples for significant features might include:
 \begin{itemize}
 \item linear versus non-linear
 \item asymptotic behavior at infinity or possibly certain points
 \item approximate position of maxima or minima
-\item approximate position of crossing axes
+\item approximate position of axis intercepts
+\item curvature (convex/concave)
 \end{itemize}
 Spurious features might include:
 \begin{itemize}
 \item scale
-\item artifacts from inputting the graph using a mouse
-\item exact position of crossing axes, minima and maxima
+\item artifacts from inputting the graph using a mouse or trackpad
+\item exact position of axis intercepts, minima and maxima
 \end{itemize}
-Which features to which degree are salient depends on the problem posed. For example, for the acceleration problem in section~\ref{accproblem} the exact shape of the graph is irrelevant, as long as the graph starts with zero and ends with zero, and is positive for a finite time in-between. In the potential versus field problem~\ref{potproblem} on the other hand, the asymptotic behavior at certain positions and the position of crossing axes is important. 
+Which features to which degree are significant depends on the problem posed. For example, for the acceleration problem in section~\ref{accproblem} the exact shape of the graph is irrelevant, as long as the graph starts with zero and ends with zero, and is positive for a finite time in-between. In the potential versus field problem~\ref{potproblem} on the other hand, the asymptotic behavior at certain positions and the position of crossing axes is important. 
+\subsection{Rules}
+Rather than constructing the graph using B\'ezier curves and checking if the parameters are within acceptable tolerance~\cite{kennedy04}, we propose to evaluate the graphs using rule sets for the function itself and its derivatives. Figures~\ref{accrule} and \ref{potrule} shows examples of what these rules might look like.
+\begin{figure}\begin{center}
+\begin{tabular}{|l|l|l|l|l|l|}\hline
+{\bf Type}&{\bf From $x$}&{\bf To $x$}&{\bf From $y$}&{\bf To $y$}&{\bf Rules}\\\hline
+Interval&0&&0&&$\displaystyle\frac{df}{dx}=0$\\\hline
+Subsequent Interval&&&&&$f>0$\\\hline
+Subsequent Interval&&&0&&$\displaystyle\frac{df}{dx}=0$\\\hline
+\end{tabular}\end{center}
+\caption{Example for a possible rule set for the acceleration problem~\ref{accproblem}\label{accrule}}
+\end{figure}
+\begin{figure}\begin{center}
+\begin{tabular}{|p{0.8in}|l|l|l|l|l|}\hline
+{\bf Type}&{\bf From $x$}&{\bf To $x$}&{\bf From $y$}&{\bf To $y$}&{\bf Rules}\\\hline
+Interval&0&\$pos1&&$-\infty$&$f<0$; $\displaystyle\frac{df}{dx}<0$; $\displaystyle\frac{d^2f}{dx^2}<0$\\\hline
+Interval&\$pos1&\$pos2&$\infty$&$\infty$&$f>0$; $\displaystyle\frac{d^2f}{dx^2}>0$\\\hline
+Interval&\$pos2&\$zerofield&$-\infty$&0&$\displaystyle\frac{df}{dx}>0$; $\displaystyle\frac{d^2f}{dx^2}<0$\\\hline
+Interval&\$zerofield&&$0$&&$f\ge0$; $\displaystyle\frac{df}{dx}>0$; $\displaystyle\frac{d^2f}{dx^2}<0$\\\hline
+Subsequent Interval&&&&&$f>0$; $\displaystyle\frac{df}{dx}<0$; $\displaystyle\frac{d^2f}{dx^2}\le0$\\\hline
+Optional Subsequent Interval&&&&&$f>0$; $\displaystyle\frac{df}{dx}<0$; $\displaystyle\frac{d^2f}{dx^2}\ge0$\\\hline
+\end{tabular}\end{center}
+\caption{Example for a possible rule set for the potential problem~\ref{potproblem}. The variables \$pos1 and \$pos2 denote the positions of the charges and are determined by the randomization of the problem. The variable \$zerofield denotes the axis intercept and is calculated by the problem.\label{potrule}}
+\end{figure}
+Instead of applying tolerances to parameters, the system now needs to allow for degrees of ``fuzziness'' in the application of the rules: sketches are not plots, and students who correctly sketch the significant features of the graph need to receive credit. To ensure this, student input from the sketching client needs to be processed server-side and appropriate fuzzy algorithms need to be developed to apply the rules. Figure~\ref{processing} shows a possible sequence of processing steps. The server receives raw data of the student sketch, in this example, the current in an RLC-circuit (blue) and the enveloping exponential decay functions (red and green). In the next step, several of these artifacts are removed by applying a smoothing algorithm to the data. In a subsequent step, the data is fit by a function. As it turns out, in this freehand drawing, while being a correct sketch, the frequency increases slightly with time, so if in the last step, the differential equation itself is used to verify the function, sufficient fuzziness needs to be applied to accept the sketch.
+\begin{figure}
+\begin{tabular}{p{1.8in}l}
+\vspace*{-1.6in}
+
+Raw data received from client. The sketch was made freehand with a trackpad on a laptop computer in less than a minute and clearly shows the artifacts from having to move the finger multiple times across the pad.&\includegraphics[width=2.6in]{figures/damped}\\\hline
+\vspace*{-1in}
+
+Removal of artifacts, smoothing.&\includegraphics[width=2.6in]{figures/dampedcleaned}\\\hline
+\vspace*{-1in}
+
+Fitting the data to a function.&\includegraphics[width=2.6in]{figures/dampedfit}\\\hline
+Applying rules. The parameters $c_1$ and $c_2$ in the differential equation need to be fit to the graph, since the problem itself does not specify their values.&{\small \begin{tabular}{|l|l|l|l|l|p{1.5in}|}\hline
+{\bf Type}&{\bf From $x$}&{\bf To $x$}&{\bf From $y$}&{\bf To $y$}&{\bf Rules}\\\hline
+Interval&0&&\$i0&&$\displaystyle f+c_1\frac{df}{dx}+c_2\frac{d^2f}{dx^2}=0$;\newline $c_1>0$; $c_2>0$\\\hline\end{tabular}}
+\\
+\end{tabular}
+\caption{Server-side processing of sketches\label{processing}}
+\end{figure}
 \section{Tool Development}
 \section{Usability Testing}
 \subsection{Authoring Interface}
+The ease of authoring is crucial for the widespread adoption of the tool, and has been one of the limiting factors to the dissemination of the original Interactive Graphing Tool.~\cite{kennedy04}.
 \subsection{Learner Interface}
 \section{Initial Content Development}
 \section{Analysis of Problem Solving Behavior}

Index: modules/gerd/alt2007/damped.ppt
+++ modules/gerd/alt2007/damped.ppt
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Index: modules/gerd/alt2007/dampedfit.xls
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				</dict>
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</plist>
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<!DOCTYPE plist PUBLIC "-//Apple Computer//DTD PLIST 1.0//EN" "http://www.apple.com/DTDs/PropertyList-1.0.dtd">
<plist version="1.0">
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</plist>
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<plist version="1.0">
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			<array>
				<dict>
					<key>com.apple.print.PaperInfo.PMPaperName</key>
					<string>na-letter</string>
					<key>com.apple.print.ticket.client</key>
					<string>com.apple.print.pm.PostScript</string>
					<key>com.apple.print.ticket.modDate</key>
					<date>2003-07-01T17:49:36Z</date>
					<key>com.apple.print.ticket.stateFlag</key>
					<integer>1</integer>
				</dict>
			</array>
		</dict>
		<key>com.apple.print.PaperInfo.PMUnadjustedPageRect</key>
		<dict>
			<key>com.apple.print.ticket.creator</key>
			<string>com.apple.print.pm.PostScript</string>
			<key>com.apple.print.ticket.itemArray</key>
			<array>
				<dict>
					<key>com.apple.print.PaperInfo.PMUnadjustedPageRect</key>
					<array>
						<real>0.0</real>
						<real>0.0</real>
						<real>734</real>
						<real>576</real>
					</array>
					<key>com.apple.print.ticket.client</key>
					<string>com.apple.printingmanager</string>
					<key>com.apple.print.ticket.modDate</key>
					<date>2007-02-18T19:36:48Z</date>
					<key>com.apple.print.ticket.stateFlag</key>
					<integer>0</integer>
				</dict>
			</array>
		</dict>
		<key>com.apple.print.PaperInfo.PMUnadjustedPaperRect</key>
		<dict>
			<key>com.apple.print.ticket.creator</key>
			<string>com.apple.print.pm.PostScript</string>
			<key>com.apple.print.ticket.itemArray</key>
			<array>
				<dict>
					<key>com.apple.print.PaperInfo.PMUnadjustedPaperRect</key>
					<array>
						<real>-18</real>
						<real>-18</real>
						<real>774</real>
						<real>594</real>
					</array>
					<key>com.apple.print.ticket.client</key>
					<string>com.apple.printingmanager</string>
					<key>com.apple.print.ticket.modDate</key>
					<date>2007-02-18T19:36:48Z</date>
					<key>com.apple.print.ticket.stateFlag</key>
					<integer>0</integer>
				</dict>
			</array>
		</dict>
		<key>com.apple.print.PaperInfo.ppd.PMPaperName</key>
		<dict>
			<key>com.apple.print.ticket.creator</key>
			<string>com.apple.print.pm.PostScript</string>
			<key>com.apple.print.ticket.itemArray</key>
			<array>
				<dict>
					<key>com.apple.print.PaperInfo.ppd.PMPaperName</key>
					<string>US Letter</string>
					<key>com.apple.print.ticket.client</key>
					<string>com.apple.print.pm.PostScript</string>
					<key>com.apple.print.ticket.modDate</key>
					<date>2003-07-01T17:49:36Z</date>
					<key>com.apple.print.ticket.stateFlag</key>
					<integer>1</integer>
				</dict>
			</array>
		</dict>
		<key>com.apple.print.ticket.APIVersion</key>
		<string>00.20</string>
		<key>com.apple.print.ticket.privateLock</key>
		<false/>
		<key>com.apple.print.ticket.type</key>
		<string>com.apple.print.PaperInfoTicket</string>
	</dict>
	<key>com.apple.print.ticket.APIVersion</key>
	<string>00.20</string>
	<key>com.apple.print.ticket.privateLock</key>
	<false/>
	<key>com.apple.print.ticket.type</key>
	<string>com.apple.print.PageFormatTicket</string>
</dict>
</plist>
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