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My Final Project

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Accessibility may not always be accessible

Especially in the age of online learning, adaptations are constantly being tested and changed in order to fully meet the needs of the students and the classroom setting. We saw with Covid-19 how quickly instructors had to change their lesson plans in order to create a learning environment online. Everything became distanced and we could no longer rely on teachers being able to explain things at the front of the classroom to their students. This is especially true with STEM courses. Many students lost the ability to have hands-on approaches to their education.

Programs like Blender have the ability to make tools for courses that allow for a more hands-on approach, even in peak pandemic times, however in an article by M.A. Hamid titled Performance efficiency of virtual laboratory based on Unity 3D and Blender during the Covid-19 pandemic, it discussed how the programs could have different results on processor and memory consumption, which highlights how even the accessible tools in a pandemic may not be as accessible as we may think. Along with families that may not have enough computers and electronics in order to get their students connected to their lesson, some families may not be able to have computers that can run these programs well enough. As much as these programs may not hurt a more advanced computer system, we must be aware of everyone.

This week I finished off the last part of the courses I have been working on in order to understand Blender. I think overall the lessons that have been most beneficial to me and that I will use the most are the ones about animation, such as animating a walk sequence of a character, the sculpting, and the work with creating different shapes by playing around with vertices and edges. These all will allow me to make different mathematical shapes and models in order to develop my understanding of some mathematical concepts.

Hamid, M. A., et al. "Performance efficiency of virtual laboratory based on Unity 3D and Blender during the Covid-19 pandemic." Journal of Physics: Conference Series. Vol. 2111. No. 1. IOP Publishing, 2021.

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Math Makes Art Easier

Mathematics and 3D models seem to have a symbiotic relationship. As much as we can learn mathematics through looking at models, we can also understand how to build models better through having a more vast knowledge of mathematics. For instance: UV mapping and planar models.

Planar modeling is an idea from graph theory; we say a graph of vertices and edges is planar if we can move the edges around in such a way that no edge crosses another. In a planar graph, every face can be put into a two dimensional plane without overlap. We see planarity failing in cases where we cannot redraw the graphs without overlapping edges.

As I spent time this week working on building a better understanding of UV mapping in Sebastián Vargas Molano’s Blender course titled The 2023 Blender Primer: 3D Modeling, Animation, & Rendering reminded me of some of the mathematics courses I have taken recently. UV mapping allows for the modeler to put a two dimensional image onto a three dimensional object. Often in order to create a reasonable model, the shape itself must be broken down into smaller sections in order to not twist the two dimensional image around the object in unnatural ways. Having a basic understanding of planarity with graphs and what it looks like to try to rearrange the faces, edges, and vertices of a graph allowed me to better see how a shape could be split up into smaller pieces in order to not warp the images being projected onto the shape.

Kennedy, John W., Louis V. Quintas, and Maciej M. Sysło. "The theorem on planar graphs." Historia mathematica 12.4 (1985): 356-368.

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Throwing Mathematical Shade

Blender and other 3D modeling software are able to demonstrate to mathematics students the concepts in their courses, especially in courses reliant on the mathematics behind shapes, such as in a geometry class. Being able to use visuals in order to demonstrate complex concepts about mathematical models in geometry allows students to supplement their own understanding with a more concrete tool.

In a study on virtual geometry, it was explored how augmented reality is able to work as a supplemental tool for geometry students. Augmented reality works to combine two and three dimensional graphics with the already existing three dimensional world in order to respond to user stimuli. This allows for a much more hands on approach to mathematics, however this research highlights an important aspect of machine and technology based learning. While classrooms in general are an amalgamation of different learning styles and techniques, technological approaches to education relies heavily on the teacher’s ability to use the programs and devices. As much as the student’s interest in technological learning is a key aspect of their education, if the teachers cannot properly use the material, no amount of resources will improve the classroom environment.

Blender, a 3D modeling program, is a common tool for creating supplemental tools in three dimensions. This week I spent my time learning more about the more behind-the-scenes aspects of Blender. I found out how the shading aspects of Blender work at a base level, then moving onto lighting, world and background creation, and more. Using proper lighting is important even in the most mathematical models in order to properly show the objects in their true form and keep every shape distinct.

Sebastián Vargas Molano’s course titled The 2023 Blender Primer: 3D Modeling, Animation, & Rendering

Syafril, S., et al. "Designing prototype model of virtual geometry in mathematics learning using augmented reality." Journal of Physics: Conference Series. Vol. 1796. No. 1. IOP Publishing, 2021.

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Math isn't as universal as I thought...

Whenever we talk about STEM, or science, technology, engineering, and mathematics, we often say that everything boils down to just the “M”- math. It is the foundation of everything else, which is why we see so many STEM majors in college that have to take mathematics courses all the way through the Calculus track, usually adding in courses like Differential Equations, Linear Algebra, or Discrete Mathematics as well. As much as mathematics ties into every other STEM major, is it a universal language? Is it something that can be shared across many cultures, despite language barriers?

Whiteford writes on how English Language Learners may have been brought up in different cultures that have different significant numbers, use different measuring systems, and have different methods for even our most basic functions like adding, subtracting, multiplying, and dividing. As much as our calculators may spit out the same values no matter where we are, how we rationalize why that value is true can be so different depending on where we learned math.

This week I spent time working on understanding the physics tools in Blender. The lessons started off by explaining how mathematics ties into physics and is the foundation of STEM. I learned how to animate a bowling ball rolling towards 10 pins and knocking them over through the use of the physics mechanics in Blender. I also learned how to model cloth falling onto a ball, a wrecking ball knocking into a pile of cubes, and more. Having access to these mechanics allowed for more realistic looking animations that also allowed me to see where mathematics can truly be applied to modeling, rather than applying modeling to the understanding of mathematics.

Whiteford, Tim. “Is Mathematics a Universal Language?” Teaching Children Mathematics, vol. 16, no. 5, 2009, pp. 276–83. JSTOR, http://www.jstor.org/stable/41199460. Accessed 30 Oct. 2023.

Blender lessons from Sebastián Vargas Molano’s course titled The 2023 Blender Primer: 3D Modeling, Animation, & Rendering

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Animation!

Animation is the process of graphics moving and shifting in order to show change over time. It is often used in the context of cartoons, video games, and art work. In its simplest form, we could think of a flip book, where each page represents a “frame” in the animation. This was originally how many comics and movies were made. Each frame was carefully drawn in order to show only a small amount of movement at a time. We still see work like this in claymation, where a model is moved in almost unnoticeable increments in order to create entire movies.

In mathematics, we often discuss how things change over time. One of the first concepts learned when discussing graphs is the slope, something which often in real-world applications relates to a change in our dependent variable based on time, a commonly used independent variable. As the concepts become more complex, it becomes more and more imperative that students are able to understand the graphs and what they represent. Trigonometric functions that describe angles and movement around the outside of a circle are more digestible when shown as the action of tracing your finger along the circle and watching the change in y and x values.

Just in the same way that we see students improve based on a combination of visual, tactile, and auditory stimulus while learning, we also can see improvement in understanding by using a combination of tools, including animation. Animation allows for a stronger representation of that visual material while often bringing in some of the elements of tactile learning by letting students watch the change occur on their screen. In an experiment done to see if animated graphics would increase scores over non animated graphics or text alone, it was shown that the animation group scored higher; animation gives more opportunities to fully understand the scope of a problem. This is something that I will be able to apply to my understanding of Blender. I have started to work on Sebastián Vargas Molano’s course titled The 2023 Blender Primer: 3D Modeling, Animation, & Rendering. It has gone into the details of what creates a good animation, and I worked this week on understanding how a ball would stretch and squish as it bounces through the air. I learned about how keyframes are the individual frames that mark the important poses and positions of items in the animation, and how the in-betweens show the progression from one keyframe to the next. I also worked on creating a walk cycle and learned how to take a small walk cycle and let it repeat for forever (or 500 frames, whichever comes first).

Szabo, Michael, and Brent Poohkay. "An experimental study of animation, mathematics achievement, and attitude toward computer-assisted instruction." Journal of Research on computing in Education 28.3 (1996): 390-402.APA

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Shapes and Colors

In mathematics we often work with shapes, either looking at different functions and analyzing their graphs or working with shapes to define their angles and side lengths. A drawing on a whiteboard of either of these may assist in understanding the concepts, however in many cases, they are not digestible enough. For some, they may not be interesting enough.

Especially in this time after lockdown from the pandemic, students more than ever are invested in technology and their screens. Their attention span is down and their understanding of concepts is even lower. I personally have felt that I have seen students struggle with basic mathematical concepts more this year than in any year prior. 

An issue in any class is making sure that students are engaged; bright colors and flashing lights on a screen are one way to do so. Using technology to demonstrate the concepts taught in class are a way to keep students engaged with the material while using helpful models. As Weng writes in an article titled 3D mathematics-Seeing is believing, “Instruction through a computer is livelier than traditional instruction, and can better elicit student interest,” (Weng 53).

There are many mathematical tools that can be accessed through computers. We use tools like MATLAB, a coding language, or Desmos, an online graphing calculator, to play around with math.

Another way in which we could further demonstrate math on computers could be through mathematical modeling, such as in Blender. I have been working for the past month on understanding Blender and what I can do with it. I still have many more lessons to learn, however I have now finished the first course I decided to work through. I now know more about animation, shading, lighting, creating shapes, and more. 

This past week, I worked through the final lessons to sculpt a head and paint it. What started as a sphere turned into a face, and when the clock struck midnight on Friday night, I had a villain-like character.

Weng, Ting-Sheng. "3D mathematics-Seeing is believing." International Journal of e-Education, e-Business, e-Management and e-Learning 1.1 (2011): 52.

All of this was thanks to Grant Abbitt’s course on Udemy with GameDev, titled Complete Blender Creator: Learn 3D Modeling for Beginners.

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Simplification!

Sometimes I wonder about the efficacy of mathematical diagrams. I spend my time TAing in precalc looking at the diagrams that act more as a baseline rather than an accurate representation of the work and think about whether it really helps at all if the picture isn’t true to the problem. Any time I sit down with a student, the first thing I have them do in the triangles and trigonometry unit is make them draw a picture. It seems to help, but how much can a diagram help?

Cooper, Sidney, and Alibali write on the pros and cons of diagrams in mathematics. They state that “extraneous features can also be detrimental to learning, particularly when they are alluring or distracting” (Cooper et al. 25). It is understandable that adding too many details that the student can rely on would stop them from fully understanding the concept or if the details stop their focus. Finding that line between providing enough detail for the student to work on without leaving them with too much or too little information is an important step in mathematics. In the article, it further states that more simplified diagrams will work the best.

Something that I learned this week while learning about animating in Blender is the art of simplification. Seeing that one frame will be a flipped of another in order to minimize the work done, looking at the fall and bounce of a ball by tracking one point in a line, and using mirrors all make the work more manageable and understandable.

This week, I animated a ball bouncing using a graph editor to map the movement of the ball falling down and bouncing back up repeatedly, gradually decreasing in height between each bounce. I then moved into creating a TV man and animated a walk pattern, adding in an animated video for the screen.

I think being more proficient in animation would allow me to make mathematical models feel more tactile to a viewer; seeing the model spin around will add to the diagram.

Cooper, Jennifer L., Pooja G. Sidney, and Martha W. Alibali. "Who benefits from diagrams and illustrations in math problems? Ability and attitudes matter." Applied Cognitive Psychology 32.1 (2018): 24-38.

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Dinosaurs, Planes, and Learning Styles

There is a fault in our current learning system, and it is learning styles. Often when we think of learning styles, we go straight to auditory, visual, and kinesthetic. People use this to explain and excuse any academic faults they may have. Maybe they don’t take notes because they are an auditory learner. Maybe a student will blame their failing grade on the lack of visual aspects in the classroom. This is a danger that can arise from building learning on a discrete basis when we lack empirical evidence that any student is only one kind of learner or another. This is something developed in Consequences of Endorsing the Individual Learning Styles Myth: Helpful, Harmful, or Harmless? by Veronica X. Yan and Connie M. Fralick.

When we put students into categories based on how they learn, this pushes them towards a fixed mindset in the classroom. If students believe that they are either a smart person or not, then they will not believe that they can improve. Similarly, if they believe they are only a visual learner, and the course they are currently taking relies more heavily on other styles, then that student may put in less effort. The connection between learning styles and the fixed mindset is a difficult one that we need to pull away from. (Fralick and Yan)

It is a constant conversation in mathematics classrooms that the downfall of many students is a fixed mindset; if a student believes that they are not a “math person,” then they will struggle to push themselves or see any improvement. Many professors of mathematics take time at the beginning of their courses to explain the importance of having a growth mindset. If students believe that anyone can understand math if they put in the effort, and that “smartness” is not a fixed value that varies between each person, then they will often see improvement in their math skills. If they also understand that they can learn math through a variety of methods and styles, then they will only give themselves more opportunities to grow.

We often use visuals in mathematics. One of the first steps to solve many problems is to draw a picture or a diagram. When the concepts become more complex, however, then the harder a diagram will be to draw, and it will be even harder to follow. Regardless of how accurate a picture may be, it also lacks that tactile form that many students use to understand a problem. The usage of 3D modeling programs can make it so that diagrams are easier to follow and they can be manipulated similarly to a 3D figure a student could hold in their hands. This allows mathematics to span across multiple learning styles in order to help students grow in multiple ways.

In my continued efforts to learn Blender, a free 3D modeling program, I spent this past week on two separate lessons. In the first one, I created a dinosaur and a background. I learned more about different textures and patterns, which then was further developed in the UV Mapping section. Further into that section, I also started the process of learning how to animate a plane. Animation will be especially useful in mathematics to show multiple sides, slices, and more of different graphs and shapes.

Yan, Veronica X., and Connie M. Fralick. "Consequences of Endorsing the Individual Learning Styles Myth: Helpful, Harmful, or Harmless?." Learning Styles, Classroom Instruction, and Student Achievement. Cham: Springer International Publishing, 2022. 59-74.

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30 minutes x 4 mistakes= ???

A.K.A, week 2 of learning Blender so that I can understand math better.

Blender and many other 3D modeling tools are art. Often, art is used to share a message or an idea; while these same ideas can often be expressed through writing, art often spreads messages faster. Being able to look at something and understand it is powerful. This can be applied to mathematics and the importance of visual tools. Especially in calculus, many of the concepts are about shapes, their areas, and their volume. By the time students are in their third calculus class, they are asked to understand complex 3D shapes and equations, which are difficult to draw on paper and even more difficult to try to picture in their heads.

This is why creating physical or virtual representations of mathematical structures is vital. Knill, Oliver, and Slavkovsky wrote about mathematics and 3D printing; we see with this how visual and tactile learning comes together. Especially with mathematics, giving students multiple pathways towards understanding a concept is key for allowing the most students to have the most success. Being able to move and manipulate a model in your own hands allows you to understand the fine details and the big picture.

I am looking forward to learning more of the ins and outs of blender in order to be able to create physical representations of my favorite math problems. This week, I worked through a tutorial on how to build a dungeon in Blender. I was able to test my memory on the previous shortcuts and tools in the program, and there was an introduction to the “knife” feature, beveling, and mirroring. Features like this allow the modeler to create more faces in the shapes, which is something that could be used to analyze graph theory in a more hands-on approach.

These are images of the work that I did over this week. I have learned that even with thirty minutes of tutorials left, I might have four times that amount of work to do before I can get it to work properly. This can be frustrating, but it makes it feel more rewarding when I finally get it right.

All of this was thanks to Grant Abbitt’s course on Udemy with GameDev, titled Complete Blender Creator: Learn 3D Modeling for Beginners.

Knill, Oliver, and Elizabeth Slavkovsky. "Illustrating mathematics using 3D printers." arXiv preprint arXiv:1306.5599 (2013).

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Why I'm learning Blender

In mathematics, we often work in detail with shapes and lines. We analyze slopes, intercepts, and confined areas in order to create a mathematical picture. Because of this, from the second we start to teach mathematics, we use images and graphs. To show that the sum of two and three is five, we might draw a representation of the numbers as cookies or apples. To show that the slope of a parabola at its vertex is zero, we might plot some points and connect the dots.

As a senior in college studying mathematics, I have found myself drawing countless graphs and diagrams of the shapes and formulas I have been working on. Even when dealing with abstract concepts, I use a combination of colors and points to explain what is happening. When tutoring precalculus, my first method of explaining a concept is always a drawing.

Bettina Rosken and Katrin Rolka did research through the University of Duisburg-Essen, Germany, that shows the importance of visualization in mathematics regarding calculus. They state that “visualization allows for reducing complexity when dealing with a multitude of information,” (Bettina and Rolka 458). Being able to see a graph and label every important point allows students to keep their work clear and concise. The authors, however, make a point that a drawing or representation of the mathematical concept only does so much; a mathematical model cannot stand by itself without an understanding of what it is and its rules.

While math is best learned in a classroom, Blender, a 3D modeling program, can be one of those tools that can help assist in understanding concepts outside of class time. As much time as I can spend on drawing a graph of a 3D shape like a donut, more commonly known in the math world as a torus, if I can instead learn a program that allows me to hold two keys down and click through some interface to place a torus into a world, then I can push my understanding of mathematics even further in less time.

And that is what I have done. So far, I have spent time learning how to place shapes, edit the world around them, and their materials. While I will not need to necessarily be able to create a sphere that is reflective in order to understand a math problem, the more I understand the entire interface and abilities of Blender, the better I will be able to use it. Understanding both what to use and what not to will allow me in the future to focus further on the mathematical side of this program and how I could have used it in the courses I have taken. This week, the final project I made was a lighthouse on a rock with some small houses around it. I learned how to sculpt different shapes, create different light sources, and assign different colors to different faces. All of this was thanks to Grant Abbitt’s course on Udemy with GameDev, titled Complete Blender Creator: Learn 3D Modeling for Beginners.

Rösken, Bettina, and Katrin Rolka. "A picture is worth a 1000 words–the role of visualization in mathematics learning." Proceedings 30th conference of the International Group for the Psychology of mathematics education. Vol. 4. Prague: Charles University, 2006.