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\begin{document}

\newpage


\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\ROCKS IN THE HEAD}




\markboth{Modeling Perception of Mass of Rocks}{Modeling Perception of Mass of Rocks}







\author{Brian Winkel\\
Director SIMIODE\\
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
 We describe an experiment and offer data from a previously conducted experiment on the perception of the individual mass of a collection of rocks in comparison to a 100 g brass mass. We lead students to use the logistic differential equation as a reasonable model, estimate the parameters, compare the model with data, and ask them to develop a criterion for determining which group of perceivers is best.
\end{abstract}





\section*{SCENARIO DESCRIPTION}


Sensitivity is a subject that invites notions of change, e.g., how sensitive is a system to a slight change in a variable? How discerning are we to change around us?  If we could change the wattage to a light bulb some 5 feet from us from 100 to 102 watts, would we notice the change?  How about from 100 watts to 500 watts?

We offer a description of a project which measures the ability of individuals to sense the discernible difference between the masses of small rocks.  Basically, we  lay out a number of small rocks. It is best if these rock samples are from the same ``geological stock," i.e.  have the same density and color, and that they be smooth, for sharp edges have the effect of appearing heavier as the edge cuts into the bare hand holding the rock used to assess its relative mass, thus distorting one's perception (mass discernment) with another (pressure). 

This experiment will require a number of volunteers to each conduct an experiment. The experiment consists of comparing the mass of  a number of small rocks to a 100 g mass and simply saying, for each rock, if it has more mass  or less mass than the 100 g mass and reporting  observations on a score sheet, one sheet per individual. We shall summarize the data for our rock collection. Suppose we have $n$ rocks and compute the percent of our population which says rock $i$, $i = 1, 2, \ldots, n$ has more mass than the 100 g mass. A discerning population should have a low percentage report that low mass rocks, say 20 g, have more mass than 100 g mass.  Similarly, a high percentage might be expected to say a 180 g rock has more mass more than a 100 g mass.  

\subsection*{Equipment needed}

\begin{itemize}
\item 5-10 small rocks or stones (use stones of the same density or a ``geological stock,'' try to get them
smooth without jagged edges, and make them in the 50-150 g range);
\item 100 g mass from an old science lab scale set (use this as a standard mass or in lieu of the lab piece you can use a rock of known mass as the standard with which you compare the other rocks);
\item shoebox or small container to hold the rocks and the standard mass;
\item pencils, attached by string to the table or box, and forms (described below) which are used to anonymously collect data;
\item  table on which to set out this equipment (table should be in a public area where students, faculty, and staff can see it, but in a protected area where it would not be subject to vandalism); and 
\item ballot box used to collect respondents' forms.
 \end{itemize}


\subsection*{The experiment}
The goal of this project is to determine how good people are at distinguishing between different masses. Accordingly, set out some 5-10 rocks in the 50-150g
range and a standard mass of 100g. Label the rocks with a marker in random order, 1, 2, 3, \ldots  , 9, say. Do not label them in increasing mass order! Identify the standard mass and its actual mass for the respondents. We offer data from an experiment we conducted over 35 years ago with 9 rocks.

Ask participants to indicate whether or not the sample rocks are of greater mass than the standard mass. Provide a slip of paper in which to ask them some information you might want for later references, e.g. their major, their position (student-undergraduate or graduate), faculty and discipline, administrative, staff, etc. Do not ask for names unless you wish to have a contest for best guesses.

The real information is respondent opinion on whether the mass of the rocks, numbers 1, 2, 3, \ldots , 9, are more than the standard mass. Provide a form such as the following:


\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|}\hline
Sample rock number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8& 9 \\ \hline
Check if mass $>$ 100 g &\qquad \qquad &\qquad \qquad &\qquad \qquad &\qquad \qquad &\qquad \qquad &\qquad \qquad &\qquad \qquad &\qquad \qquad &\qquad \qquad \\ \hline
\end{tabular}\end{center}

Set the collection table out for whatever time is reasonable to get sample data. One week will be sufficient, especially if there is a sign, with words like `Rock-A-Rama', `Let's Rock', `Science Experiment', or `Which Major is More Perceptive', and some catchy graphics.

Once the data are collected, compute the percentage of respondents who say that a given rock has mass greater than the 100 g standard mass. Plot this percentage against the actual mass of the given rock for each rock.

\subsection*{Statement of the problem and first actions}


We will attempt to fit a curve (or curves) through the response data, perhaps attempting to differentiate groups with a parameter or feature in the graphs we obtain for different groups. We might even wish to rank the groups based on this criteria. For example, can geology majors differentiate rock mass better than political science majors?

Once the data are collected, compute the percentage of respondents who say that a given rock has mass greater than the 100 g standard mass. Then plot this percentage against the actual mass of the given rock. If all respondents had perfect knowledge, e.g., a scale, the plot of the returned data would look like  Figure~\ref{fig:1-19-ModelingTwoPerfectRockResponse}. 

\begin{figure}[h]
\centering
\includegraphics[width=10.5cm]{1-19-ModelingTwoPerfectRockResponse.eps}

%\vspace*{-1.7 true cm}

\parbox{12.2 true cm}{\caption{\label{fig:1-19-ModelingTwoPerfectRockResponse} Plot of responses for a group that got it 100\% right all the time for whatever rock masses come there way. }}
\end{figure} 
 
\vspace*{-.7 true cm}
\begin{itemize}\item [1)] Think about it:  can you explain the dot at the point $(100,50)$ in Figure~\ref{fig:1-19-ModelingTwoPerfectRockResponse}?
 \end{itemize} 
 


\begin{figure}[!h]

\noindent If all respondents guessed then the plot of the returned data could look something like  Figure~\ref{fig:1-19-ModelingTwoLoserRockResponse}.

\vspace*{.3 true cm}
\centering
\includegraphics[width=10.5cm]{1-19-ModelingTwoLoserRockResponse.eps}

%\vspace*{-1.7 true cm}
 
\parbox{12.2 true cm}{\caption{\label{fig:1-19-ModelingTwoLoserRockResponse} Plot of responses for a group that just guesses all the time. }}
\end{figure}



In reality, neither perfect (Figure~\ref{fig:1-19-ModelingTwoPerfectRockResponse}) nor no knowledge (Figure~\ref{fig:1-19-ModelingTwoLoserRockResponse}) exists in a population. Rather, as shown in   Figure~\ref{fig:1-19-ModelingTwoRockParameterEstimationDataPlot},  we see a plot of responses for a reasonable (not perfect) population.  Notice the sigmoidal shape going from low values for percentage (saying low mass rocks have more mass than 100 g) while there will be a sharp rise around 100 g rocks and then a leveling off as we approach higher and higher mass rocks to about 100\% of the respondents saying  HUGE mass rocks have more mass than 100 g.




A possible model for this might be the sigmoidal shaped solution to the logistic equation (\ref{eq:RevisedRockLogistic}) where $M(s)$ is the percentage of respondents who say a rock of mass $s$ has more mass than 100 g mass. Notice that $M(0) \sim 0$.

\begin{equation}
\displaystyle{M'(s) = r M(s)(100-M(s)), \qquad M(0) = M_0}\, . \label{eq:RevisedRockLogistic}
\end{equation}


\begin{itemize}\item [2)]  Explain why $M(0) \sim 0$ is probably true. 
 \item [3)] Explain why the differential equation (\ref{eq:RevisedRockLogistic}) is reasonable. Consideration of several issues might  help you in your explanation.

\begin{itemize}
\item [a)] What happens around the mass of 100 g?
 \item [b)] What happens as the mass gets larger and larger?  smaller and smaller?
\item [c)] How big can the percent who guess a large rock's mass correctly actually go?  
 \end{itemize}
 \end{itemize}


In (\ref{eq:RevisedRockLogisticSolved}) we see the solution to differential equation (\ref{eq:RevisedRockLogistic}). You should be able to solve this by hand with separation of variables and partial fraction techniques or Mathematica's {\tt DSolve} command.

\begin{itemize}\item [4)] Solve (\ref{eq:RevisedRockLogistic}) now and compare your answer to (\ref{eq:RevisedRockLogisticSolved}).
 \end{itemize}  

\begin{equation}\label{eq:RevisedRockLogisticSolved}
M(s) = \frac{100 M_0 e^{100 r s}}{M_0 e^{100 r
   s}-M_0+100}
\end{equation}


\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|} \hline
Rock Mass ($s$ in g)& \% saying Rock Mass $>$ 100 g mass ($M(s)$)\\ \hline
 10 & 2 \\ \hline
 40 & 10 \\ \hline
 70 & 21 \\ \hline
 90 & 45 \\ \hline
 100 & 50 \\ \hline
 120 & 65 \\ \hline
 125 & 75 \\ \hline
 150 & 90 \\ \hline
 175 & 98\\ \hline
\end{tabular}

\parbox{13.5 true cm}{\caption{\label{table:AlbionData} Summary of data collected from students at Albion College over 40 years ago. }}

\end{center}

\vspace*{-.6 true cm}
\end{table}


Table~\ref{table:AlbionData} shows some data students  collected over 40 years ago from  students at Albion College (a small liberal arts college in Michigan). If you cannot collect your own data  then you can use this data. Incidentally, the students determined that the geology students performed the worst!  




Further, we offer a plot of the data from Table~\ref{table:AlbionData} in Figure~\ref{fig:1-19-ModelingTwoRockParameterEstimationDataPlot} where the sigmoidal shape of the logistic equation can be seen. Figure~\ref{fig:1-19-ModelingTwoRockParameterEstimationOptDataPlot} renders a plot of how good your model might just be in predicting the performance of these students.



\begin{figure}[h]
\centering
\includegraphics[width=11cm]{1-19-ModelingTwoRockParameterEstimationDataPlot.eps}

%\vspace*{-1.7 true cm}

\parbox{12.2 true cm}{\caption{\label{fig:1-19-ModelingTwoRockParameterEstimationDataPlot} Plot of the data from Table~\ref{table:AlbionData} in which  \% of respondents who said the rock of given mass is more than 100 g is plotted against the mass of that rock.}}
\end{figure}

\begin{figure}[!h]
\centering
\includegraphics[width=12.6cm]{1-19-ModelingTwoRockParameterEstimationOptDataPlot.eps}

%\vspace*{-.3 true cm}

\parbox{12.2 true cm}{\caption{\label{fig:1-19-ModelingTwoRockParameterEstimationOptDataPlot} Plot of the data from Table~\ref{table:AlbionData} in which  \% of respondents who said the rock of given mass is more than 100 g is plotted against the mass of that rock and then compared to a good model on the same axes.}}

\end{figure}

\begin{itemize}
\item [5)] Determine the parameters $r$ and $M_0$ which give rise to the best model fit  for the data in Table~\ref{table:AlbionData} using the logistic equation (\ref{eq:RevisedRockLogisticSolved}). When compared to the actual data  your model should look something like the comparison  offered in Figure~\ref{fig:1-19-ModelingTwoRockParameterEstimationOptDataPlot}.
\item [6)] For the data offered in Table~\ref{table:AlbionData} modify it (we do not recommend doing this with real data - it could get you in trouble!)  (i) so that we have a very poor group of respondents  - worse than our given group, and then (ii) so that we have a very good group of respondents - better than our given group.
\item [7)] Now think hard. What makes them better? Give at least three criteria  you might use to measure good sensitivity from a group in this experiment and apply one to show why your offerings are in line with your criteria.
\item [8)] Go ahead and set up this experiment (we give some practical considerations below) and ask your fellow students who pass by their major and obtain their full responses as to whether or not each rock in your set up has more mass than the 100 g brass mass.   Then determine a model for each set of majors and determine which major is best. Remember you design the criterion. Be sure to compare your best major and see if they are really best in all your criterion for best.
\end{itemize}
 

\subsection*{Practical considerations in setting up a rocks experiment}

\begin{enumerate}
  \item Use  rocks from the same type of rock, if possible. The density of the rocks should all be the same so that appearance is not a part of the decision, e.g., big low density rock offers different sensory possibilities than a large high density rock.
  \item Use smooth rocks, for jagged edges ``dig into'' the skin and alter perception, as the sensation of  the jagged edge on nerves in the hand  triggers a response saying, ``This rock is really heavy.''
  \item  If you are going to set up the collection box for several days take it in at the close of business, for janitorial services might remove it. 
  \item Empty the collection box several times each day.
 \end{enumerate} 


\teacher{
\section*{NOTES FOR TEACHERS}

The  paper \cite{Winkel1998} is devoted to a description and full details of this experiment while the paper \cite{Winkel2012} gives  other sources of data collection as well as  collected data which can be modeled by the logistic equation, including this study on perception of mass of rocks. These include spread of information, technology, and disease.

\subsection*{Starting out}

 Students will always, always plot data first, which is a sign of a good modeling skill that we work hard to instill in them. In our first class period devoted to this
project, we have the students conjecture what the data response might be. Based on their understanding of the possible variables (mass of rock, percentage who say mass of rock is more than 100 g standard mass) and their response to Figure~\ref{fig:1-19-ModelingTwoRockParameterEstimationOptDataPlot} the students are convinced that the data should be S-shaped. Moreover, they quickly convince themselves that the more vertical the central section about the 100 g standard mass is, the better the ability of the respondents to distinguish masses.

Thus the hunt is on for S-shaped curves. This can happen if the teacher lets it. Quite often I let it happen. Notice we do not discuss or pose this  option in our Student Version. Rather we want to move directly and with rationale, into the differential equation model (\ref{eq:RevisedRockLogistic}). However, a  first model students often conjecture is some form of the arctan, perhaps, $f(x) = \arctan (a(x - b))$ with constants $a$ and $b$ to be determined by some best fitting criteria. Incidentally, one could have a discussion about such criteria at this point or offer up minimizing the sum of the squares between the observed data and the predicted data from the model for the data points collected, the so-called least squares approach. We find that students, through minimally guided discussion, arrive at the ``natural'' least squares criteria.

Depending on where the class is in terms of topics, this project might be either an example of how the shape of data might dictate the nature of the model, logistic in this case, or an introduction to a differential equation in response to the shape of the data, namely, if the data  grows slowly at the start and ends while rising rapidly in the middle. One may also present a  summary project to a unit on the minimization of a function of several variables, e.g., the parameters $a$ and $b$ in the model function offered above. The function here is the sum of the squares between the observed data and the predicted data from the model for the data point. Or, if you trust your software sufficiently, it might be an introduction to fitting functions to data with little understanding of the underlying minimization algorithm.
Some of the S-shaped curves include those outlined in Table~\ref{table:candidates}.

\vspace*{-.4 true cm}
\begin{table}[h]
\begin{center}
\begin{eqnarray*}
f_1(s) &=&	\frac{100s}{(1 +a  e^{(b s)})}\\
f_2(s)&=&	\frac{100}{(1 + a s^{(b s)})}\\
f_3(s) &=&a e^{(b s)}\\
f_4(s) &=&a \arctan (b(s-c)) + d)
\end{eqnarray*}

\vspace*{-.4 true cm}
\parbox{8 true cm}{\caption{\label{table:candidates} Types of functions students  use in  search for  elusive S-shape curve to match the data in Figure~\ref{fig:1-19-ModelingTwoRockParameterEstimationOptDataPlot}.}}
\end{center}
\end{table}

\vspace*{-.4 true cm}
It should be noted that students generally go after functions which are asymptotic to 1 as $s$ increases and asymptotic to 0 as $s$ decreases. Some even insist that the function go through the point $(0, 0)$ arguing the fact that it makes no sense to have values of $s < 0$ where $s$ represents mass of sample rock. This insistence is usually dropped in the face of attempting to get a function to do too many things.

If one of these models is selected then a sum of  square errors between the predicting model and the data becomes a function of the two parameters $a$ and $b$ and this can be easily minimized using Mathematica's   {\tt FindMinimum} command ord the Solver command in EXCEL. Then the resulting model after applying the minimizing parameters $a$ and $b$ can be compared to the data. In determining the least sum of square errors for those models in  Table~\ref{table:candidates} along with the logistic differential equation model it turns out that the $\arctan$ function fits the data best, at least in this case with this data, the information is displayed in Table~\ref{table:Compare}.  However, the logistic differential equation function and the first sigmoidal function are in the ball park.  


\begin{table}
\begin{center}
\begin{tabular}{|c|l|} \hline
SSE &\qquad \qquad  Type of model \\ \hline
73.4704 & logistic differential equation function\\ \hline
 60.1385 & first sigmoidal function\\ \hline
 98.4577 & second sigmoidal function\\ \hline
963.021 & exponential function\\ \hline
 53.3795& arctan function\\ \hline
\end{tabular}

\parbox{12 true cm}{\caption{\label{table:Compare} A comparison by  sum of square errors (SSE) for each modeling approach to the data shows that the logistic differential equation function is comparable to the three other reasonable models, exponential growth NOT being reasonable. however, the $\arctan$ function does the best job here.}}
\end{center}

\vspace*{-.8 true cm}
\end{table}

Do the results from Table~\ref{table:Compare}  negate the energy one would put into modeling with the logistic differential equation model?  Not really, for it is the exercise of modeling, parameter estimating, and comparisons that is most important. Moreover, since we are really not sure exactly what the model might be from any first principles, say in perception, then we have three reasonable models with SSE at 73.4704 or under. Further study, further data gatherings, or deeper understanding of perception modeling would be necessary to distinguish which of these, or another model altogether, would actually  work best. 

Several students will suggest a polynomial (a cubic at least), but others will try to argue them out of this in lieu of the flattening which they deem important at either end of the mass of rock scale, a flattening which a polynomial does not possess. Thus polynomials are usually not assumed. One reason for pushing a polynomial is the fact that in software such as Mathematica it is easy to fit a polynomial, say of degree 3, to a data set data. One uses the command {\tt p[x\_] = Fit[ data, \{1, x, x\char'136 2, x\char'136 3\}, x]} to find a third-degree polynomial $p(x)$ which will fit the data.

\section*{Moving to differential equations}

The introduction of the logistic differential equation (\ref{eq:RevisedRockLogistic}) may be a bit backward approach, as it might appear that this model is based solely on   the sigmoidal shape   of its solution. This then causes a discussion about the ``physics'' of the equation. This is where the issue can get sticky, for the students do not see a rationale for the logistic equation. However, if one addresses question (3) about the rationale for (\ref{eq:RevisedRockLogistic}) students can see even in the formative stage of a differential equation, not just in the final S-shaped curve of its solution, that this is a reasonable model for the relationship between the percentage who say a rock of mass $s$ has more mass (or weighs more) than the 100 g brass mass and the actual mass used as the comparison in all cases.

One rationale for the flat regions, on either side of the sharp rise at 100 g, in a perceived model is that for a small change in the mass of the rock the percent of those who guess correctly does not change much. Howewever, around the 100 g mark a slight change in the mass might cause one to change or flip an opinion because it is mostly  real guessing that goes on here.

\subsection*{Timeline and student action issues}

This activity can be done as a long-term project over several weeks, from formulation through collection and on to analysis or as a short term exercise if the collection devices are in place or if you use the data we offer in Table~\ref{table:AlbionData}.  We have used groups all the time and we permit the groups to go their own way and develop their own analysis within the context of class sharing and discussion so that all benefit from each group's wisdom. The class time (perhaps two short 10 minute class conversations over two consecutive days to permit some reflection) devoted to this project is a wonderful time to bring misconceptions and issues out in the open. Indeed, we have found doing such complex problems with students to be a rich time to assess students' capabilities and attitudes.

This is a sure starter, as students form quick convictions as to what they believe will happen and they like the idea of designing a data collection station and collecting data from their classmates. There is the slowdown when they come to the following: ``We have the data. Now what do we do?'' This is where the teacher comes into play (perhaps a half hour discussion after data collection) guiding the discussion to bring out as many ideas as possible from the students. Collectively the students have the necessary ideas to do an excellent job. They will need technical guidance, technology assistance, and continual encouragement to complete an analysis and summarize their conclusions.

\subsection*{Ending up}
A very lively conversation takes place with students when it comes to  developing criteria to determine which of the groups (if they have noted differences in groups, e.g., majors), say geology majors and philosophy majors, are better at this activity.

Here are three criteria that seem to emerge naturally from the student perspective. 

\begin{enumerate}
  \item  Compute the sum of square error between the given population's percentages and the perfect knowledge model and then compare all the populations as to which has the least sum of square error. This always comes up. 
\item Integrate the absolute value of the difference (or difference squared) between the continuous model over the whole interval of masses, and   then compare all the populations as to which has the lowest integral value. This usually comes up. 
\item Compute the slope at mass 100 g for the model for each population and whichever population's model has the highest slope at 100 g in their model becomes the winner. The thought here is that the steeper the curve the more it approximates the ``jump'' in the perfect model and the more discerning the population is where it counts, namely, near the 100 g known mass.
\end{enumerate}

\subsection*{One more resource}

We provide a Mathematica notebook, 1-019-T-Mma-RocksInHead-TeacherVersion.nb and accompanyling pdf versinfor those who cannot read Mathematica files, with much of the analysis in a free flow format to address many of the issues in this scenario. 

\begin{thebibliography}{99}

\bibitem{Winkel1998} Winkel, B. J. 1998. Rocks in Your Head: Data Analysis Project. {\it International Journal of Mathematical
Education in Science and Technology\/}. 29(5): 773-779.

\bibitem{Winkel2012} Winkel, B. J. 2012. Sourcing for Parameter Estimation and Study of Logistic Differential Equation. {\it International Journal of Mathematical Education in Science and Technology\/}. 43(1): 67-83. 

\end{thebibliography}


}



\end{document}
