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, 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.

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.

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.

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.

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.

Wonderful! Thank you for stopping by. 

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

Thank you, Susan!

That's wonderful to hear!

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

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. 

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

Thank you Carolina for watching. 

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.  

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).

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.

• Concerning reticence, I don't believe we have. In fact during many of the professional development sessions that we have run at conferences, we have seen enthusiasm for using the software.  There is great flexibility in how instructors can incorporate CalcPlot3D into their classes so I think instructors choose what works for them.  Some faculty use it solely for in class demonstrations, while others set aside time in class for students complete lab assignments.  Some faculty will use the software to create 3D printed models to bring to class and others will simply screen shot the graph to add to a paper worksheet.  
  
  
• Concerning the static textbooks, Yes!  In fact Paul Seeburger has worked to get CalcPlot3D integrated in Libretext so that the dynamic images can be added to Open Educational Resources. I've also added dynamic figures generated into CalcPlot3D into WeBWorK mathematics homework problems.  
  
  
• Our work on measuring student success is ongoing.  Early on, Paul created four discovery-based learning modules using CalcPlot3D on the dot product, cross product, velocity and acceleration of parametric position functions, and lagrange multipliers.  These activities were given to students after they completed a pre-test and then they were given a post-test.  Deb and I took this data from over 300 students from several two and four year colleges and analyzed changes in student answers between the pre test and post test.  For instance, with the cross product activity, students were asked about the geometric relationship between the cross product and the two vectors that formed it.  In the pre-test students usually focused on just on aspect of the cross product (that it was perpendicular to the two vectors that form it), but by the post test students were adding to their response features like the length of the cross product in relation to the length of the two vectors that form it, and also the right hand orientation.  But, no, we have not done a comparison between students who used the tool versus those that didn't. 
  
  
• Students are very quick to learn the applet.  In fact, many times students will discover on their own features of the program that I had not yet introduced to class.  I think this is because of the intuitive menu-driven navigation.  Also, the built in examples are easy to click on and explore.

thank you!

Monica

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.

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. 

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 Online book.  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.

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.

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.

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

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

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.

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

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.

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:) 

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.

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!

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

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. 

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.