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Date: 2007-03-31 13:54:56
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Modified files:
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Index: modules/gerd/alt2007/graphing.tex
diff -u modules/gerd/alt2007/graphing.tex:1.7 modules/gerd/alt2007/graphing.tex:1.8
--- modules/gerd/alt2007/graphing.tex:1.7 Thu Mar 29 15:38:50 2007
+++ modules/gerd/alt2007/graphing.tex Sat Mar 31 09:54:54 2007
@@ -51,25 +51,22 @@
{\Large\sc Online Assessment of Back-of-the-Envelope Graph Sketches in Introductory \
Physics} \end{center}
\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 curve. \
But how often do we intentionally teach our students about this important \
representation format? +The ability to work with graphs is a basic skill of any \
scientist. Sketching is a skill that allows one to express general relationships with \
just a few strokes. 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 curve. 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,meltzer05,clement81}. 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 lar! ge 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,clement81}. \
These difficulties can sometimes have subtle origins 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}. \
-Only too often, instead the 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}.
-Going beyond these types, there may be problems that require selecting from a series \
of possible graphical 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. It was found however that these traditional representation-translation \
problem types do not lead to significantly increased conceptual or less procedural \
solution strategies~\cite{kortemeyer05ana}, i.e., they do not lead students to \
construct any new knowledge in a manner different from numerical or other \
multiple-choice problems. +Students need practice in both generating and \
interpreting graphs, but appropriate formative assessment frequently is lacking 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. Too often, instead the problems given in physics \
courses focus on numerical calculations, e.g., ``A car accelerates from rest at $2 \
m/s^2$ for 10 seconds. What is the distance covered?'' Students can ``solve'' these \
problems without any understanding of the underlying concepts~\cite{lin,heuvelen}. \
+Going beyond these types, problems may require selecting from a series of possible \
graphical 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. It \
has been found, however, that such traditional representation-translation problem \
types do not lead to significantly increased conceptual (or less procedural) solution \
strategies~\cite{kortemeyer05ana}, i.e., they do not lead students to construct any \
new knowledge in a manner different from numerical or other multiple-choice problems. \
-The sketching of graphs is an example of a more constructivist approach to teaching \
physics concepts, as well as representation-translation and visualization skills. The \
students need to make a number of decisions: +The sketching of graphs is an example \
of a more constructivist approach to teaching conceptual and visualization skills, \
requiring students 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 Where does the graph start and finish (are the start and end points 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, not \
reproduce it or select it from a set of prefabricated solutions. + (list expanded \
from \cite{kennedy04}). In short, students need to {\it construct} the curve, not \
reproduce it or select it from a set of prefabricated solutions.
-Sketching is a skill that allows one to express general relationships with just a \
few strokes. In this project we will develop an online assessment tool for graph \
sketching, which will provide randomized scenarios and 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. \
+The main objective of this project is to develop an online assessment tool for graph \
sketching, which will provide randomized scenarios and 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. To minimize overhead, the tool will be developed on top of an \
existing course and learning content management system. However, both the algorithms \
and the code will be made freely available, so they can be incorporated into other \
systems. \subsection{Learning Goals}
A sketched graph can be thought of as a pictorial description of a phenomenon. It \
emphasizes the major features of the phenomenon while ignoring details that may be \
distracting or better represented in other ways. Sketched graphs are regularly used \
by scientists and engineers in informal discourse, as means for describing and \
supporting claims. In the Benchmarks for Science Literacy~\cite{aaas93}, Project \
2061 makes an explicit recommendation regarding the importance of using graphs in \
making arguments (12D/H7). Graphs are best used to present data, describe ideas, and \
support claims that involve trends, comparisons, and general behavior. The ability \
to sketch a graph and use it in an argument involves not only an understanding of the \
holistic characteristics of graphs, such as general shape, number and general \
location of intersections with axes, and asymptotes, but also the general behavior of \
the phenomenon being graphed. As such, opportunities to sketch graphs and! receive \
formative feedback on them should foster students' conceptual understanding of the \
phenomena being graphed.
@@ -101,7 +98,7 @@
\item make algorithms and tools available open-source and freely\vspace*{-2mm}
\item disseminate results\vspace*{-2mm}
\end{itemize}
-Several of these steps will be carried out iteratively over the duration of the \
project while refining both tool and content +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 Related Projects}
\subsection{Relevant Results from Prior NSF Support to the PIs}
\subsubsection{LON-CAPA}\label{loncapa}
@@ -118,11 +115,11 @@
LON-CAPA is open-source (GNU General Public License) freeware, there are no \
licensing costs associated. Both aspects are important for the success of this \
project: the open-source nature of the system allows researchers to modify and adapt \
the system in order to address research needs, and the freeware character allows \
easier dissemination of results, in particular adaptation and implementation at other \
universities. +The system started in 1992 as a tool to deliver personalized homework \
to students, ``personalized'' meaning 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).
-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.
-
-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 +Over the years, LON-CAPA has been \
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}
\item support for mathematical typesetting throughout (\LaTeX\ inside of XML) \
(formulas are rendered on-the-fly, @@ -133,21 +130,19 @@
\item and full support of physical units.
\end{itemize}
-LON-CAPA developed into a content sharing network of more than 40 institutions of \
higher education including community colleges and four-year institutions, as well as \
50 middle and high schools~\cite{loncapainst}. In addition, LON-CAPA houses \
commercial textbook content from seven major publishing companies, and a commercial \
service company was established around the product at the end of 2004. +LON-CAPA has \
developed into a content sharing network of more than 40 institutions of higher \
education including community colleges and four-year institutions, as well as 50 \
middle and high schools~\cite{loncapainst}, and serves approximately 35,000 students \
every semester. +In addition, LON-CAPA houses commercial textbook content from seven \
major publishing companies, and a commercial service company was established around \
the product at the end of 2004. The shared content pool currently contains over \
250,000 learning resources~\cite{loncapashared}, including more than 80,000 \
randomizing homework problems. Disciplines include astronomy, biology, business, \
chemistry, civil engineering, computer science, family and child ecology, geology, \
human food and nutrition, human medicine, mathematics, medical technology, physics, \
and psychology.
-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. +LON-CAPA will provide the platform on which the \
tools will be developed and evaluated, and will be the initial dissemination platform \
for problems authored over the course of this project. \subsubsection{Investigating \
and Questioning our World through Science and Technology}\label{iqwst} David Fortus \
is a co-PI on the NSF grant {\it Investigating and Questioning our World through \
Science and Technology} (ESI-0439352, \$712,034 for MSU subcontract, \
10/01/04-09/30/10). The project is developing a coordinated and comprehensive middle \
school science curriculum that emphasizes several scientific practices, one with is \
DGOA - data gathering, organization, and analysis.
-\section{Relevant Results from Other Related Projects}
+\subsection{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.
-
-The activity of identifying the correct graph however is very different from \
sketching the right graph. A system that can evaluate student input of graphs needs \
far more sophisticated algorithms than are needed for problems like in \
Fig.~\ref{induction}. +The LON-CAPA system currently allows dynamically generated \
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. +The \
activity of identifying the correct graph however is very different from sketching \
the right graph. A problem that can evaluate student input of graphs requires far \
more sophisticated algorithms than does a problem such as that in \
Fig.~\ref{induction}.
\subsection{Example Problems}
\subsubsection{Open-Ended Problems}
@@ -156,7 +151,7 @@
\begin{quote}
Draw a graph of acceleration versus time for a car that first stands at a red light, \
drives off when the light turns green, and then coasts with a constant velocity. Take \
``forward'' to be positive. \end{quote}
-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. +Fig.~\ref{acccorrect} shows possible 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=3in]{figures/acccorrect1}
\includegraphics[width=3in]{figures/acccorrect2}
@@ -164,7 +159,7 @@
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. +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 \
specifying the criteria for these anticipated wrong solutions. \begin{figure}
\includegraphics[width=3in]{figures/accincorrect1}
\includegraphics[width=3in]{figures/accincorrect2}
@@ -177,7 +172,7 @@
\begin{quote}
The graph shows the electric potential along an axis connecting two charges. Draw \
the component of the electric field along this axis. \end{quote}
-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. \
+Fig.~\ref{potentialcorrect} shows possible acceptable solutions. Note that the scale \
is irrelevant in this case, but the electric field must diverge at the positions of \
the charges, go to zero at infinite distances, and tend to zero where the potential \
plateaus.
\begin{figure}
\includegraphics[width=3in]{figures/correct1}
@@ -336,7 +331,8 @@
\item[David Fortus] is a senior scientist at the Weizmann Institute of Science in \
Israel and an assistant professor of science education at Michigan State University. \
He is a co-PI on the IQWST project (section~\ref{iqwst}), has won research awards \
from the National Association for Research on Science Teaching (NARST) and from the \
American Psychological Association (APA), and has worked as a project manager in the \
hi-tech industry. \item[Stuart Raeburn] is currently a postdoctoral staff member in \
the Division of Science and Mathematics Education (DSME) at Michigan State \
University, where he is a member of the LON-CAPA development team. In the past \
decade at Michigan State University (prior to joining DSME) he was a visiting faculty \
in the Department of Geological Sciences, where he taught a variety of geology \
courses, from introductory to graduate level, and he was also a developer and \
administrator for a number of course management systems in the centrally-supported \
Faculty Facility for Creative Computing.
-\item[Sarah J. Swierenga] is the director of the MSU Usability \& Accessibility \
Center (UAC, section~\ref{uac}) and Professor by Courtesy in the Department of \
Telecommunication, Information Studies, and Media. She is responsible for developing \
and disseminating innovations in theory building, research methodologies, and \
technologies to enhance usability and accessibility in Web and information technology \
contexts. She has worked with users in commercial, military, and academic \
environments. Swierenga received a Ph.D. in human factors psychology with a \
concentration in human-computer interaction from the University of South Dakota, and \
a B.A. in psychology from Calvin College. She is also a Certified Professional \
Ergonomist (C.P.E.). +\item[Sarah Swierenga] is the director of the MSU Usability \& \
Accessibility Center (UAC, section~\ref{uac}) and Professor by Courtesy in the \
Department of Telecommunication, Information Studies, and Media. She is responsible \
for developing and disseminating innovations in theory building, research \
methodologies, and technologies to enhance usability and accessibility in Web and \
information technology contexts. She has worked with users in commercial, military, \
and academic environments. Swierenga received a Ph.D. in human factors psychology \
with a concentration in human-computer interaction from the University of South \
Dakota, and a B.A. in psychology from Calvin College. She is also a Certified \
Professional Ergonomist (C.P.E.). +\item[Alan Denton] is an associate professor of \
Physics at North Dakota State University. He facilitated NDSU's implementation of \
LON-CAPA, which now is the local course-management, homework-assignment, and testing \
system for all introductory Physics courses (serving 600 students per semester). \
Over the past 10 years, he has taught a spectrum of physics courses from \
undergraduate to advanced graduate, using LON-CAPA from introductory mechanics to \
senior-level quantum mechanics. His keen interest in physics pedagogy was sparked by \
attending an APS/AAPT-sponsored Physics and Astronomy New Faculty workshop. His \
research interests are in theoretical and computational condensed matter physics, \
with emphasis on modeling of colloids, polymers, and other soft materials. \
\end{description} \pagebreak
\bibliography{graphing}
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