Ellie Casas, a doctoral candidate in the Atmospheric Science department, is working this summer as one of the instructors of the ENcourage Engineering Math Program, which is intended to help incoming first-year students be calculus-ready. Engineering Source talked with Casas about the experience.

**How did you get involved with this program?**

I was recommended for this position after having been a Graduate Teaching Fellow in Fall 2020 for ENGR-101: Grand Challenges of Engineering. This course is designed to help first-year engineering students understand what the differences are between each engineering major at CSU, as well as learn about the largest societal challenges that need new and innovative engineering. ENGR-101 is a great course for those who want to make more informed decisions about which engineering major is right for them, and I found that ENGR-101 is a course that meets the gold standard of effective teaching due in part to its student-led design project. I had a great time learning how to teach first-year engineering students in the most effective ways for both in-person and remote instruction methods, and it was very rewarding to watch as students gained confidence in their engineering and teamwork skills and discover their vocations.

**How is Engineering math different than other math? What is it the students in this camp need to understand?**

We dedicated a morning of camp to discussing some of these differences! In essence, “regular” math is just one step of the 7-Step Problem Solving Process that engineers use to solve problems. Bert Vermeulen (the professor for ENGR-101 who I worked with), came up with the list and teaches it in his class. The 7-Steps are:

- Define a Problem Statement
- This is like writing a prompt for a math “word problem,” where you write a question that needs math to calculate an answer

- Draw a Diagram
- This is like a physics “free body diagram” where you draw and label the relationships of various forces and other variables
- This step often needs a lot of refining as you go about the rest of the design process!

- Research
- This is when you look for useful background information that is pertinent to your problem statement and overarching problem. You need to understand what your stakeholders need, what solutions currently exist, as well as estimates to parameters that you need for your calculations

- Assumptions
- This is when you compile the estimates of numbers that you found from your research, as well as explicitly list what simplifications are needed to solve your problem

- Calculations
- This is the step where engineers use the math that we think of as “normal” math

- Results and Accuracy
- This is where engineers need to critically think about whether their answer makes sense, and how strict tolerances need to be to keep people safe

- Discussion and Conclusions
- This is where engineers interpret their results and distill it into answers to the questions that their stakeholders have. In contrast to many other literature applications, engineers need to work towards being able to communicate complex ideas as briefly and simply as possible.

Typically, the engineering math in Step 5 looks a lot like a physics problem, and while engineers use all sorts of college-level math, most of the math they use will actually be using are Pre-Calculus tools like Algebraic simplification and basic Trigonometry. These are the things that we covered in ENcourage!

**How do you keep a Zoom “room” of 30-40 young people engaged in math during the summer??! ****😊**

This was a big challenge, especially since camp went from 9 am to 4:30 pm each day. However, both Jeff and I had experience with teaching over Zoom, and we both had developed effective teaching methods over the last year. For me, I found that the most effective way to keep students engaged was to make learning as social as possible. Specifically, I emphasized that they are all part of the engineering community and that we all want to help them succeed, gave them opportunities to get to know each other through ice breaker activities, and gave them frequent chances to ask questions and have discussions with each other in Zoom breakout rooms. Jeff found that the most effective way for him was to turn learning into “play time” by designing interactive math activities on Desmos.com that are great for visualizing key math concepts in creative ways that students may not have seen before. Students really enjoyed working in small groups and being able to “tinker” with math equations. Together, we saw students quickly start to open up and get engaged, even though the days were long and chock full of math challenges! We also set up a Slack channel for them to keep in contact throughout the rest of the summer, and some students have used it to set up study groups before the semester starts!

**What are you studying as a doctoral student in Atmospheric Science? How do you use math in that program? Who’s your advisor?**

I am a PhD Candidate in Michael Bell’s research group, and I research hurricanes (I also helped design our group’s semi-new website: https://tropical.colostate.edu). Specifically, I am most interested in the lowest parts of hurricanes that directly interact with the Earth’s surface, because this is the part that directly affects humans. Also, the interactions that a hurricane has with the ocean’s surface go both ways; changes in the upper ocean alter many aspects about hurricanes, but changes in a hurricane’s intensity or size also affect the ocean as well, and the cycle of interactions continue. It’s a very complex problem that many hurricane researchers are interested in better understanding and modeling!

I have used quite a bit of math in my studies. The atmosphere is a fluid like water (it’s just much less dense), and the basis of understanding fluid dynamics is the Navier-Stokes Equations, a famous series of partial differential equations. And like waves at the beach, the atmosphere (including hurricanes) has many different waves as well. (Aside: You can frequently see clouds arranged in stripes that look like barcodes due to mountain waves in Colorado! These mountain waves are like the motion you see in rivers as they go over and past rocks, and clouds form when the air is going up and disappear as the air goes down.) Describing these waves requires sinusoids, exponentials, and imaginary numbers. Modeling these waves on a computer requires discrete mathematics, where you try to approximate a sine wave on a grid of discrete, discontinuous points using Taylor Series and a lot of other unique math. While people typically envision meteorologists throwing darts at a board to forecast the weather, the reality is that we use a lot of math to predict future fluid dynamics and thermodynamics! Many current atmospheric scientists earn a math minor for all of the math that we use, and many early atmospheric scientists were classically trained mathematicians.

**Do you wish you had had this kind of program as an undergrad?**

Absolutely! My biggest regret from my freshman year is that I was too shy to really embrace my meteorology department’s community. While I have made great strides in finding my atmospheric science community since then, I would have very much benefitted from a summer program where I got to meet my classmates before the semester started.

Also, I made sure to set aside time at the end of camp to host a “College AMA (Ask Me Anything)” with the ENcourage Learning Assistants, where students were able to ask and receive concrete answers about all sorts of tips and tricks for thriving in college. When I was about to start college, I remember being very excited but nervous because I had no idea what to expect, and I would have loved to have a chance to ask questions about all of the things that I was worried about.

Finally, I’m also a firm believer that there’s no such thing as “too much math,” and my students who participated in ENcourage are going to have all sorts of math fundamentals fresh in their mind as they transition to college math, and they will be so much more prepared as they embark in Calculus than if they had taken the summer off! I really have to commend all of my students for having the courage to sign up and participate in this camp. It wasn’t easy, but I think ENcourage was a great success for preparing them for college and engineering.