- Monica VanDieren
- University Professor of Mathematics and Director of the University Honors Program
- Collaborative Research: Improving Conceptual Understanding of Multivariable Calculus Through Visualization Using CalcPlot3D
- https://c3d.libretexts.org/CalcPlot3D/index.html
- Robert Morris University

- Leslie Koren
- Assistant Professor of Media Arts
- Collaborative Research: Improving Conceptual Understanding of Multivariable Calculus Through Visualization Using CalcPlot3D
- https://c3d.libretexts.org/CalcPlot3D/index.html
- Robert Morris University

- Deborah Moore-Russo
- First Year Mathematics Director
- Collaborative Research: Improving Conceptual Understanding of Multivariable Calculus Through Visualization Using CalcPlot3D
- https://c3d.libretexts.org/CalcPlot3D/index.html
- University of Oklahoma

- Paul Seeburger
- Professor
- https://c3d.libretexts.org/CalcPlot3D/index.html
- Monroe Community College

Public

Choice

Choice

Public Discussion

Continue the discussion of this presentation on the Multiplex. Go to Multiplex

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Welcome! We hope you enjoy this video titled “Innovation in Multivariable Calculus Learning with CalcPlot3D.” In it we address some of the challenges that students face when learning multivariable calculus, and we demonstrate how the CalcPlot3D applet can be used to improve student understanding.

CalcPlot3D is a free tool that helps users to visualize mathematical objects and relationships in three dimensions. Not only does CalcPlot3D work across different digital devices, it is available in both English and Spanish, and it uses intuitive drop-down menu functionality. Give it spin at https://c3d.libretexts.org/CalcPlot3D/index.html

This project began 22 years ago when Paul Seeburger (Monroe Community College) began to create dynamic visualizations for his calculus classes. With initial funding in NSF Grant number DUE-CCLI 0736968, he produced a Java version of the CalcPlot3D applet in 2008. In the next phase of funding (NSF Grant numbers DUE-IUSE 1524968, 1523786, and 155216), Paul teamed with Deb Moore-Russo (University of Oklahoma) and Monica VanDieren (Robert Morris University) to continue to develop and disseminate the applet and classroom activities and to research student understanding of multivariable calculus.

The impact of the project is measured across multiple dimensions:

We made the decision to enter this video showcase so more people, both instructors and students, could learn about CalcPlot3D. To create the video, we worked with a Robert Morris University media arts professor, Leslie Koren. Her expertise in visual story-telling helped us describe our project and share it with a wider audience. We plan to continue this cross-disciplinary collaboration and make more use of the footage that Leslie collected to create additional pieces.

Our team welcomes your comments and questions! We also invite you to consider or respond to some of the following questions:

Stephen Alkins Ph.D.

Kristin Flaming

## Paul Seeburger

Co-PresenterProfessor

I want to add another link here to a more general project website that includes links to the CalcPlot3D applet in Spanish as well as the version in English shown in the link above, to a Help Manual and to other visualization tools developed in this project.

Exploring Multivariable Calculus (CalcPlot3D) Project website

## Karl Kosko

This would have been such a wonderful tool when I was taking some of my undergraduate mathematics! I have a very explicit memory of trying to form a mental image of a three dimensional curve and its accompanying equation. Nice work!

Leslie Koren

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

Karl, Hope all is well with you. I'm sure things are winding down at Kent State. Please pass information about CalcPlot3D onto any people you think might find it useful. You might even find a use for it if you are working on either dot or cross products with future high school math teachrs.

## Leslie Koren

Co-PresenterAssistant Professor of Media Arts

Thanks for watching, Karl!

## Cheryl Calhoun

Thank you for sharing your work with CalcPlot3D. I look forward to playing with the tool. I love the ability to rotate the plots in 3D. Do you find students are able to deepen their understanding and grasp the concepts more thoroughly with this interactive 3D modeling approach?

Paul Seeburger

Leslie Koren

## Leslie Koren

Co-PresenterAssistant Professor of Media Arts

Hi Cheryl,

thanks for watching! Monica, Deb and Paul can weigh in, but yes, the impact of the project has yielded positive results. The applet is currently being translated into Spanish as well. Students can see the multivariable calculus concepts come to life and change over time.

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Yes, we see students responses to open-ended and multiple choice questions improve drastically from a pre-test to a post-test after having completed one of the discovery-based activities.

Students also report in semester evaluations that CalcPlot3D is their favorite part of multivariable calculus and that CalcPlot3D was "very important" in contributing to their understanding of the material.

I find that students may struggle at first with thinking about three-dimensional surfaces. But when I bring the 3D printed models into the classroom that they can touch and rotate, the students are able to imagine the 3D image on the screen a little better.

As a teacher I like that I can focus on the graphical interpretations and not emphasize the computations. This seems (although we haven't formally studied it) to level the playing field between those who have strong algebraic and calculus skills with those who may not. It has definitely helped me as a teacher this semester as we moved online because of Covid-19. I was able to continue to use CalcPlot3D in my online presentations and I was able to create online assessments that were difficult to look up on Chegg, wolframalpha, or google.

Stephen Alkins Ph.D.

Paul Seeburger

Leslie Koren

## Paul Seeburger

Co-PresenterProfessor

I can add that my multivariable calculus students have also found CalcPlot3D very helpful in my classes. This has been especially true in my online multivariable calculus course, where many have given feedback that they could not imagine taking the course without CalcPlot3D to help them visualize the concepts.

I find that students can successfully process and respond to more challenging conceptual questions when they are using CalcPlot3D to explore the concepts. It has allowed me to raise my level of expectations for their conceptual mastery.

Leslie Koren

## Bonnie Hall

My department teaches calculus-based physics, and the transition between different formats for the same information is definitely a challenge. I am going to share the info about this tool with our faculty who teach physics--I agree that it could help level the playing field between those who can easily visualize the math concepts and those who need to work with it for it to become part of their mental visualization skills. Thank you for sharing!

Leslie Koren

## Paul Seeburger

Co-PresenterProfessor

Thanks for your posts, Bonnie!

We do have a number of faculty using CalcPlot3D in physics and engineering courses. Interestingly this has been especially true in Mexico where many of the professors that teach multivariable calculus there also teach engineering and physics courses or various kinds.

Note that there are also ways to use CalcPlot3D in other math courses including differential equations (visualizing phase portraits of systems of differential equations and exact equations), linear algebra (visualizing transformations in 3D), single variable calculus (visualizing volumes of revolution and parametric curves) and even introductory algebra courses (e.g. visualizing the intersection of three planes).

We'd love to hear back from you or your colleagues if you find ways to use CalcPlot3D in your courses! Also let us know if you have suggestions or questions on what's possible.

Leslie Koren

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

CalcPlot3D excels at helping students visualize in three dimensions. Besides faculty in mathematics, we've had others in physics and in engineering make use of this tool.

Leslie Koren

## Susan Bateman

I was fortunate enough to work at Monroe Community College briefly and see Paul Seeburger demonstrate CalcPlot3D to a group of professors. I found it to be user friendly and currently use it in teaching multivariable calculus and differential equations at the Rochester Institute of Technology.

Leslie Koren

## Leslie Koren

Co-PresenterAssistant Professor of Media Arts

Wonderful! Thank you for stopping by.

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

Thanks, Dr. Bateman, for stopping by. Please be sure to spread the word about CalcPlot3D.

## Paul Seeburger

Co-PresenterProfessor

Thank you, Susan!

That's wonderful to hear!

I'd love to hear more about how you use CalcPlot3D in your classes at RIT!

## Lisa Dierker

Passing a math course with the goal of leaving it in your rear view mirror is so common. I love

yourfocus on thinking flexibility and developing real fluency. Drawing connections and being able to making decisions is what is truly empowering. Congratulations this great work!Leslie Koren

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

What a great comment! CalcPlot3D does allow for the deep, flexible understanding of calculus (and other mathematical ideas) in three dimensions primarily due to its promotion of visualization. Thanks for your input.

## Vimolan Mudaly

The dynamic possibilities for visualization makes the tool ideal for teaching abstract mathematics. I will certainly share this will colleagues and students.

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

Thanks, Vimolan, I hope that those in the South African mathematics community will find this helpful.

Leslie Koren

## Vimolan Mudaly

The dynamic possibilities for visualization makes the tool ideal for teaching abstract mathematics. I will certainly share this will colleagues and students.

## Carolina

Great work, loved learning about your project and Leslie did a fantastic job with the video!! Congrats!

Leslie Koren

## Leslie Koren

Co-PresenterAssistant Professor of Media Arts

Thank you Carolina for watching.

## Kristin Bass

Wow. This is a very powerful, impressive, and accessible visualization tool. How does it support students who may be underrepresented in mathematics and engineering, or have trepidations about calculus? In addition to Paul's wonderful feedback in this discussion thread, do you have any stories you can share? I'm especially wondering if you've ever assessed confidence, which can be a predictor of knowledge and skills.

Stephen Alkins Ph.D.

Paul Seeburger

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Hi Kristin,

Thanks for your questions. We do have a Spanish version of the applet available here: https://c3d.libretexts.org/CalcPlot3D/index-es.html

It would definitely be interesting to examine if CalcPlot3D changes student's attitudes towards mathematics.

## Paul Seeburger

Co-PresenterProfessor

Hi, Kristin,

I would add that CalcPlot3D provides valuable support for students who have difficulty visualizing mathematical objects both in 2D and especially in 3D, while also increasing the confidence of stronger students who may have been able to visualize these things better in their minds.

In the context of a series of pre-test/exploration/post-test activities (exploring the dot product, the cross product, velocity and acceleration, etc.), stronger students who were able to answer the pre-test questions correctly almost always indicated (in a qualitative essay answer at the end of the post-test) that they were now more confident and clear about why these were the correct answers after completing the visual exploration of the concepts in CalcPlot3D.

Another aspect of using this (or any other) visualization tool in the context of a lesson is that it provides a way to make the mathematical concepts more intuitive and accessible to students, helping them to see not only the right steps to solve a problem, but why the solution makes sense visually in a graphical context. And hopefully to be able to see why an incorrect solution does not make sense visually in this context. For example, we can visually check that the equations of a line of intersection of two planes is really contained in this intersection in 3D. As another example from differential equations, we can visually verify that an initial value problem solution curve fits correctly through the phase portrait corresponding to the given system of differential equations. This would also correspond to showing that a flowline fits correctly in its corresponding vector field.

And, in addition to verifying solutions visually, we can also examine a series of visual examples to help students discover relationships and restrictions. For example, seeing that motion along a curve will have a constant speed when the acceleration vector of the motion is always orthogonal to the velocity vector of the motion (and the speed will not be constant when the acceleration vector is not always orthogonal to the velocity vector).

Stephen Alkins Ph.D.

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

I have used this with math majors who were future and practicing teachers. They often comment on the ability to visualize (and really understand) what tasks were asking for. The most common remarks are that they had used memorized formulas to make it through multivariable calculus successfully.

We have not assessed confidence, but that is a fantastic idea! Do you have any suggestions for instruments, especially that target postsecondary students, that we might use to study this?

Leslie Koren

## Stephen Alkins

FacilitatorDiversity, Equity, and Inclusion Officer

Great job with developing this tool!

This would have definitely helped explain the derivation of many formulae:

This is certainly a great supplement to learning. Thank you for sharing the video!

Paul Seeburger

Leslie Koren

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Hi Stephen,

Thank you for watching our video and for your thoughtful questions! I'm sure that my colleagues will have more to add, but here are some initial responses to your questions.

thank you!

Monica

Stephen Alkins Ph.D.

Paul Seeburger

Leslie Koren

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

I agree with everything that Monica said. In relation to her first and fourth bullet points, I believe that having common options pre-populated in drop down menus (for example, equations that yield interesting curves) is one of the key aspects that make both instructors and students find CalcPlot3D so easy to use.

Stephen Alkins Ph.D.

Leslie Koren

## Paul Seeburger

Co-PresenterProfessor

It appears that Monica and Deborah have addressed most of your questions, Stephen, but I wanted to include a couple links to OER textbooks that incorporate dynamic figures created with CalcPlot3D here. This will show you one way we have been able to do better than "static" textbooks to represent several three dimensional problem types.

See the following page from my OER Calculus III texbook on the LibreText platform for several examples of dynamic figures in a section on lines and planes in space: Section 11.5

See Figures 11.5.1, Figure 11.5.5, and Example 11.5.10 (a rotatable intersection of two planes).

In Section 6.2 of my online OER Calculus II textbook there are four separate examples where CalcPlot3D is used to generate dynamic volumes of revolution. See Figures 6.2.8e, 6.2.12e, 6.2.13c, and 6.2.14c, A representative rectangle can be moved back and forth through the region in the xy-plane that is rotated about an axis. This representative rectangle can be revolved about the axis of revolution. And the region itself can be revolved about the axis of revolution to show how the solid of revolution is formed. Finally, the corresponding shells or washers can be displayed and revolved about the axis of revolution. In the last example, the user can even change the axis of revolution to any horizontal or vertical line to see how this affects the corresponding solid and its washers or shells.

Stephen Alkins Ph.D.

## Michael I. Swart

A great curricular tool for both teachers and students. And what a wonderful partnership to have a developer willing to implement design revisions. As a supplement to instruction, could you expound on the research questions you are addressing. In one response, "discovery-based learning modules" was mentioned. Is this the underlying theoretical model as to how this software enhances instruction? Providing digital manipulatives? 3D printing of problem spaces? The usage of a dynamic digital system? Variable isolation strategies? Impact of feeback? Scaffolds? Individual vs. Collaborative learning?

Seems also that you could produce a great corpus of causal data A/B testing different versions of the software with and without various features to isolate contributions of specific factors, as well as UX design comparisons and feature design comparisons. Thanks for sharing.

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Hi Michael,

Thanks for visiting our page!

Yes, Paul has been great at being open to both practical and research-based design changes to CalcPlot3D. He implemented many user-requested features such as polar Riemann sums, options to graph multiple curves or regions of integration simultaneously, having the ability to restrict the domain of the function graphed, making the 3D-printing stl file compatibility work with regions of integration etc.

But Paul has also been receptive to suggestions that resulted from the research we conducted on student responses to exercises. For instance, Deb and I found that several students were giving a common incorrect response in a cross product activity that Paul designed. We realized that this may be due to the fact that in that particular activity, students manipulated vectors in R^3, but two of the vectors were always on the xy-plane. We suggested that Paul program the activity so that students could move the vectors freely in three-dimensional space. We haven't yet done a formal analysis of the results of student responses after this change, but anecdotally students are not reporting the same misconception as before.

We wrote a self study examining some of the UX design changes in a paper that will shortly appear in the

Teaching and Learning Mathematics Onlinebook. The title of our paper is "Technological pedagogical content knowledge for meaningful learning and instrumental orchestrations: A case study of a cross product exploration using CalcPlot3D."Concerning the underlying theoretical model, we are informed by Marton and Booth's Variation Theory among others. This framework fits this setting particularly well since we encourage students to explore mathematical relationships in the software by fixing one feature while varying an other. We have developed a validated model of the critical features of vectors and are using this as the basis for developing an assessment tool to use in future research.

Unfortunately currently it is difficult to create an A/B test for this since there is not an assessment tool for measuring student understanding of multivariable calculus (like for instance the Calculus Concept Inventory for single variable derivatives). Creating such a tool is one of the long term goals of this ongoing research project.

Paul Seeburger

## Paul Seeburger

Co-PresenterProfessor

Thanks for your thoughtful questions and suggestions, Michael!

As the developer, I want to add that in addition to responding to the suggestions I have gotten from other calculus professors and my project collaborators, I am also a math professor myself, teaching multivariable calculus and differential equations, so that I am often motivated to create new visualization features to be able to better teach or represent the concepts I am teaching my own students.

For example, this past fall, this desire caused me to add a feature to visualize the curl of a vector field, something I had hoped to implement for a long time.

Stephen Alkins Ph.D.

Leslie Koren

## Wendy Smith

FacilitatorResearch Professor

CalcPlot3D looks like a fantastic tool that allows for connections between graphs and algebraic representations. It sounds like you have provided technical professional development to help instructors learn how to use the software. Do you also have professional development that helps instructors design lessons that actively engage students (i.e., students engage in exploring 3D graphs rather than the instructor using the software in a lecture/demonstration)?

Stephen Alkins Ph.D.

Monica VanDieren

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Hi Wendy,

Yes, we held two 3-day CalcPlot3D Workshops (once in 2016 and another 2019) dedicated to professional development to help instructors use the software, but also to develop activities and material that they can bring back to their classrooms and share with others. Additionally all of our professional development involves providing instructors with a library of interactive modules that they can bring directly into their classes or adapt as necessary. These modules are student centered and are designed for the students to be actively engaged with the software, exploring graphs.

Paul can also talk about his work in Mexico with professional development workshops.

Stephen Alkins Ph.D.

## Paul Seeburger

Co-PresenterProfessor

In addition to the two 3-day CalcPlot3D summer workshops Monica mentioned above, we also presented two 4-hour minicourses on CalcPlot3D at the Joint Math Meetings (2018 and 2019) and two 1.5-hour minicourses at ICTCM (2018 and 2019) (the International Conference on Technology in Collegiate Mathematics). At each of these mincourses, we helped faculty learn to use CalcPlot3D to demonstrate concepts visually in both multivariable calculus and differential equations. We also showed participants how to generate scripts in CalcPlot3D that can be loaded back into CalcPlot3D for use as either a dynamic classroom demonstration of pre-created dynamic plots or a guided exploration for students to use (inside or outside class) to gain intuition about various topics in these courses.

As Monica mentioned above, I also had the privilege to present four 5-day workshops in Mexico on CalcPlot3D in the summers of 2011-2014. Those workshops were rich experiences, where professors who teach a wide range of STEM subjects all came together to find ways to use CalcPlot3D to demonstrate topics in their courses. These workshops not only provided these participants with professional development, they provided the project with many new applications and feature requests that helped accelerate developments in directions requested by these faculty.

Stephen Alkins Ph.D.

## Omar Ashour

Great work!

Visualization definitly improves learning abstract math concepts.

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

Thanks, Omar. I definitely agree with you on that.

## Hong Liu

Hi, Paul, thanks for your team to present the nice app for helping our multivariate Calculus students. I am interested in embedding your applet into the web pages of online learning materials. Do you have instructions to share to public?

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Hello Hong Liu,

I'm sure Paul will have more informative instructions, but I believe this can be done with iframes. If your online learning materials allow for that. I've successfully embedded dynamic interactive examples from calcplot3d into WeBWorK problems, some with a full set of controls, others without. And I know Paul has done this for his OER books.

Which online learning materials are you thinking of adding CalcPlot3D to?

Best regards,

Monica

## Paul Seeburger

Co-PresenterProfessor

Hi, Hong!

Here is a link to my online Help Manual where I explain how you can embed CalcPlot3D figures into online pages. I do usually use an iframe, as Monica described.

Creating Dynamic Figures using CalcPlot3D

Note that there are instructions in this manual to generate the URL query string you will need to use along with a slightly different HTML file on our server to display a dynamic figure instead of the whole app with the plot saved in the URL query string. See this link in the manual.

See my post above at 5:10 pm on May 6 for a couple links to see examples of these dynamic figures in my online textbooks for Calculus II and III.

## Luann Richardson

This is not my field but I was amazed by what you are doing--visualizing the concept gave it a whole new meaning. Great work!!

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

Thank you, Luann! I think it's important in any field to provide students with multiple avenues for learning.

## Feng Liu

FacilitatorSenior Researcher

Thanks for sharing this video as well as the great tool to learn multivariate calculus! This reminds me that when my 8-year old daughter was learning multiplication and division she was introduced a modeling approach that is to use some 2D graphs to represent the equations. I think the underlying theory is same as CalcPlot3D, though the target level is different.

As the research described, there are currently two versions of this applet: English and Spanish. I wonder whether you are also considering to integrate other languages such as Chinese, French, etc. into system. Regarding the impact evaluation of this applet on student learning, you might want to consider a quasi-experimental design approach (if not a randomized control trial) including a matched comparison group to make it a more rigorous study. Also, I would be also interested to see the impact of this applet on teacher/instructor-level outcomes such as teacher efficacy and/or confidence of teaching multivariate calculus.

## Paul Seeburger

Co-PresenterProfessor

Thanks for your thoughtful post, Feng!

We had targeted Spanish as a second language partly because of the large Spanish speaking population in the U.S. and Puerto Rico, but I'd love to generate versions of CalcPlot3D in other languages too! If you have interest in helping with one, let me know. Now that I have all the text in a key-coded external file, I can fairly easily switch these out for additional language translations of the text phrases and menu items.

We appreciate your suggestions for approaches to our future research! I really like the idea of documenting the impact of CalcPlot3D on teacher efficacy and confidence in teaching multivariable calculus!

I know it has greatly increased my efficacy and confidence, and I have heard many testimonials from other professors that indicate this is true for them as well.

## Feng Liu

FacilitatorSenior Researcher

Thanks for your responses, Paul! Having a Spanish version within the U.S. education system totally makes sense to me. When feasible, I would like to explore the possibility of generating version of the CalcPlot3D in other languages such as in Chinese since I am a Chinese myself:)

## Paul Seeburger

Co-PresenterProfessor

Hello, Feng!

Yes, I'd be glad to work with you on trying to create a translation of CalcPlot3D to Chinese! Just let me know, and I can send you a file of phrases to translate to Chinese for me. =) I also have other Chinese friends that I could ask.

## Mark Bealo

This takes me way back to my days in CALC I, II, and III. It would have been far less of a struggle had these kinds of tools been available back then.

## Paul Seeburger

Co-PresenterProfessor

As a professor trying to teach multivariable calculus, I saw the need to be able to show my students the concepts visually. That's what motivated me to create CalcPlot3D!

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

Thanks for the comment, Mark. CalcPlot3D is especially helpful with the multivariable concepts that typically arise in Calculus 3 classes.

## Diane Savoie

Thank you so much for sharing about Calc PLot 3D. I can definitely see that students will be able to make great connections using this applet. I cannot wait to share it.

## Paul Seeburger

Co-PresenterProfessor

Thanks for your post, Diane! We'd love for you to share our project with anyone who could benefit!

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

Thanks, Diane, we'd love to get the word out about this resource. It could be particular helpful for any math and science instructors who are introducing vectors, vector dot product, and vector cross product.

## Dave Miller

This is SO cool. Thanks for sharing. Takes me back to my high school college algebra and calculus classes and makes me think how valuable this would have been. We had to create the spacial "maps" and representations in our heads - or draw them on paper as best we could. You mention the value of novice --> expert development, and that made me wonder if you might have plans for a connected archive of real-world applications of where the concepts are applied. This could be a value-add for students and teachers. Thanks!

## Paul Seeburger

Co-PresenterProfessor

Thanks for your post, Dave!

This is also an excellent suggestion! Adding real-world applications for specific concepts has been one of our long-term goals. One example where we have done this so far is by adding topographical maps in the context of Lagrange Multiplier optimization.

## Deborah Moore-Russo

Co-PresenterFirst Year Mathematics Director

I remember those days too, Dave. We don't currently have plans for such an archive, but that is a fantastic idea. It would definitely have value to extend the impact of this tool to more of the sciences and engineering. Thanks for sharing!

Leslie Koren

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

We have also added examples of involving the mathematics behind ray-tracing animation and Bezier curves in design. The example cited by the student at the very end of the video involves visualizing Bezier curves and connected the symbolic representation with the geometry.

## Bridgette Scott

Awesome video! Great work!

Leslie Koren

## Monica VanDieren

Lead PresenterUniversity Professor of Mathematics and Director of the University Honors Program

thanks, Bridgette!

To post to this discussion go to