This presentation describes a design of a study with prospective mathematics teachers who will be engaged in computational modeling through a series of projects.

The purpose of the study is multifold:

• to develop computational modeling projects for prospective teachers.
• to implement the projects within a course or summer immersive workshop.
• to determine feasible pedagogical approaches (e.g. scaffolding) and potential challenges in the implementation of the projects.
• to explore prospective mathematics teachers’ perceptions of computational modeling, and its role and potential integration in mathematical classrooms.

• Our technological society has become depended on complex systems (Bar-Yam, 2004; Jacobson & Wilensky, 2006; Lesh, 2006).
• Educators have emphasized the importance of students developing new competencies in understanding and dealing with complex systems such as modeling, interpreting, predicting (Gainsburg, 2006)
• Engagement with multi-component modeling projects; and rapidly evolving new conceptual tools such as computational modeling are critical for acquiring such competences ( Lesh & Doerr, 2003; Lesh & Zawojewski, 2007)
• There is growing evidence for doing classroom mathematics inquiry with focus on real world data and computational modeling ( Farris et al., 2019; Fuhrmann et al., 2018).
• However, implementation of computational modeling for solving real-world problems and building mathematical knowledge remains an open field of research (diSessa, 2018).

The study will try to answer the following research questions:

1. What are feasible pedagogical approaches to implementing computational modeling with prospective mathematics teachers?

2. What are potential challenges in the implementation of such projects?

3. What are prospective mathematics teachers’ perceptions of computational modeling, and its role and potential integration in mathematical classrooms?

To gain insight into research questions 1) and 2), we will focus on four qualitative data sources: video recording on the project sessions; researchers’ observational notes, projects’ artifacts, and participants’ written reflections.

Data for research question 3) will be collected through focus group interviews. These interviews will be conducted and transcribed at the conclusion of the project sessions, and the participants will be asked to discuss their views on:

• computational modeling,
• influence of the projects on their views of computational modeling,
• mathematics-related understandings, skills, and practices computational modeling could help learners develop;
• benefits and challenges of computational modeling integration in the mathematics classroom.

Examples of computational modeling projects to be used include the following:

• estimating the volume of an oil spill from flow rate data.
• estimating the area of a lung nodule from its X-ray image.
• fitting mathematical models to astronomical data to rediscover Kepler’s laws of planetary motion.

All projects require the use of real-world data, image processing and scientific computing with R. We plan to translate these projects into Python as well.

Next, we give some more details about the projects.

This picture shows the flow of oil leaking from the Deepwater Horizon platform in the Gulf of Mexico in 2010.

This project uses real data and computational modeling to answer the following question:

• How much oil in total was leaked during the 87 days of the BP oil spill?

Experts estimated the flow rate of the oil spill from pictures like this one.

The idea is to use these flow rate data points and create a continuous mathematical function of the flow rate \(\text{flow}(t)\) as a function of time \(t\), which interpolates the flow rate data linearly.

Then, to find the total volume of the oil leaked during this 87-day period, we just need to numerically integrate this continuous function over the full time period.

\[\text{Volume} = \int_1^{87} \text{flow}(t) dt\] On the next slide, we show the continuous, piece-wise linear model of the flow rate, as a function of time in days.

This is the first major milestone the students should achieve in this project.

To integrate the flow rate, we have to compute numerically the anti-derivative function \(\text{volume}(t)\) of the numerically constructed function \(\text{flow}(t)\).

Mathematically, we have:

\[\text{volume}(t) = \int \text{flow}(t) dt\]

Thus, we can obtain the total volume of the oil spill:

\[\text{Volume} = \text{volume}(87) - \text{volume}(1) = 4.9188616\times 10^{6}\] This is the second and final major milestone the students should achieve.

• The total volume of the oil spill that we obtained is 4,918,862 barrels of oil.

• The report from the US Department of the Interior, cited above, gives an estimate of 4.9 million barrels of oil, with estimated uncertainty of \(\pm 10\%\).

Mathematically, this is a project at the calculus level.

Computationally, we do not implement everything from scratch but we use the mosaic and mosaicCalc R packages, which have many useful functions that implement linear interpolation, numerical anti-derivative function, as well as various plotting functions.

This way we can adjust the computational complexity as needed, and focus on the big picture of computational modeling.

A lung nodule is a round area that is more dense than normal lung tissue. It shows up as a white spot on an X-ray or a CT scan.

Lung nodules are usually caused by scar tissue from a healed infection, and they usually represent a benign tumor, but sometimes a nodule can be an early lung cancer.

The goal of this project is to estimate the area of the tumor from its X-ray image.

We have an X-ray image of the lung nodule as a png image file, and we can use the png R library to read the png image and plot it on a coordinate system.

However, before we can do that we need to convert the image dimensions from pixels into centimeters [cm]. Usually, on a regular X-ray image, one pixel corresponds to 175 \(\mu m\) (microns) and since 1 \(\mu m = 10^{-4} cm\), we can approximate the image dimensions in cm using the scaling:

\[\text{#pixels} \times 175 \times 10^{-4}[cm]\]

The first step for the students is to find the dimensions of the X-ray image, first in pixels, using the png R package, and then in cm, by using the conversion formula above, which they will have to discover on their own. We can then plot the X-ray image on a coordinate system, where the dimensions are in cm.

The next step for the students is to interactively sample some points from the tumor boundary, using RStudio. The manual sampling can be implemented using the locator() R function, which returns a list with the \(x\) and \(y\) coordinates of the sampled points.

Once we sample the points from the boundary of the tumor, we can plot them on the top of the X-ray image and compute the area of the polygon they form.

• Implementing the shoelace formula: We can compute exactly the area of the polygon formed by the vertices of the points sampled from the tumor boundary as the minimum area to be removed. The area of the polygon in complex form is given by the shoelace formula: \(A=\frac{1}{2} \text{Im}\left (\sum_{k=0}^{n-1} z_k \bar{z}_{k+1} \right )\)

• A circle fitting: implementing nonlinear least squares by using a general purpose optimizer to minimize the sum of squared errors between the circle model and the sampled points. We use the surveyor’s method to find good starting guesses for the model parameters, being the radius and the coordinates of the circle center, given by \(\bar{R}\), \(\bar{x}\) and \(\bar{y}\).

• The surveyor’s method for estimating area: The surveyor estimates first the location of the center of the area by computing its center of mass \(C=(\bar{x},\bar{y})\). From this point we measure the distances to each polygon vertex. Then the distances are averaged to obtain \(\bar{R}\).

This project is mathematically and computationally more challenging than the oil spill project, and we need to scaffold it well and split it into many smaller pieces to be manageable for the students, and to better test the development of their computational problem solving skills.