Connecting ideas, knowledge, and experiences from biology, chemistry, and calculus to inspire questions, insights, or answers:
Over the course of the semester, we studied various topics in biology, chemistry, and calculus. The overarching theme that stood out to me the most was the theme of antibiotic resistance. We conducted labs in biology and chemistry and mathematically analyzed the results in calculus, creating graphs to make conclusions about the trends seen in the data we gathered. In order to start out the study of antibiotic resistance, one of our first labs, Lab 2, investigated the effects of temperature on bacterial growth. The goal of this experiment was to determine the optimal growth temperature for Escherichia coli, Agrobacterium tumefaciens, and Vibrio natriegens. In this lab, we grew bacteria in growth media that contained nutritional components integral to microbial growth. We inoculated cultures for each bacteria and then incubated them for a period of time to allow the bacteria to grow. We learned how to do turbidity measurements using the spectrophotometer and acquire optical density (OD) values; the OD of a culture increases as the number of organisms in the culture increases. We then plotted our data on graphs with time on the X-axis and OD on the Y-axis.
In calculus, we were able to determine the generation time and growth rates for our bacteria at each temperature and show their relationship. I believe getting this experience to determine optical growth temperatures and corresponding growth rates can be extremely useful in the future when needing to test the growth rates of various other bacteria. Due to antibiotic resistance, many bacteria are finding ways to thrive despite the constant use of antibiotics. Knowing key facts such as a bacteria’s optimal growth temperature and rate is a major step in learning more about how these dangerous bacteria work and how we can create solutions to combat resistance. For example, exposing a bacteria that cannot survive past 45 degrees to 50 degrees to hopefully kill it before it has a chance to grow resistance. If heat ends up acting as a selection pressure and causing resistance mutations, the growth rate could possibly be used to disrupt the work of enzymes to slow down the rate of growth. These types of hypotheses and more could be made through the strong connection between biology, chemistry, and calculus.
Developing a growth mindset and learning from mistakes:
While I have taken a semester of Calculus before in high school and thus haven’t experienced too many struggles so far, there have definitely been times when I found myself questioning my accuracy and correctness. This class was a prime example that one can infinitely continue improving in math, with harder and more complex problems always presenting a new way of thinking about a topic or an angle/perspective one hasn’t considered before.
One topic, seemingly not too complex or difficult, particularly confused me when I repeatedly got it wrong on the Learning Target quizzes. This was LT14 or L’Hospital’s rule. Originally, I was confident of my answer, so I was genuinely confused as to what was wrong with my work. I made the mistake of not asking Dr. Torres after the first time I didn’t receive a check for LT14. I finally asked her the second time, realizing my mistake and where exactly I was lacking in understanding. Namely, I had not fully grasped the fact that if there were equal degrees on both the numerator and denominator, I only had to simplify the coefficients on the x’s. Another key point I was missing was the writing of “as x approaches infinity, the limit is…” in algebraic form within every step of my work. I didn’t show I was still taking the limit and just kept simplifying. While my answer may have turned out to still be mathematically correct, my work was inherently wrong. The work I did technically should not have resulted in the answer I got, purely because I didn’t show work.
This Learning Target showed me the importance of showing my work in any math I am doing. Especially in college, where mathematics will reach increasingly heightened levels of difficulty, the work between problem and answer is very crucial. It is a good habit to start writing all my work out, no matter how confident I am. This experience also showed me that it isn’t always a sign of weakness or lesser intelligence to ask for help and that there is always room for improvement and growth. This point, now looking back, was shown to us in Journal 1. After watching the video about Michael B. Jordan, I specifically concluded that “Growth is more about how much you learn from repeated failures than it is about how many times you do something correctly.” Learning Target 14 showed that moving forward, I need to work on further shifting my perspective to growing and learning instead of correctness.
Growth in self-efficacy
In calculus this year, the homework assignments have been the most challenging part of the course. Homework was a great way to apply the basic concepts from lectures to increasingly complex problems and see how far I could go without stressing out and losing my grasp on the concept. Throughout this extremely busy semester, there were several times when I had to do my homework during my other extracurricular commitments, whether it be during my work shifts, Bonner shifts, club meetings, or other organization meetings. While this caused issues like turning in homework late, I knew learning the concept was the most important part. Especially because this class was based on mastery, I prioritized learning the material as best as I could.
A prime example of a time when I struggled with a concept was finding the integral of complex expressions, especially those that included trigonometric identities. Homework 29 was particularly tough and something I had to go back and revise. I struggled with trig identities in both pre-calculus and calculus during my time in high school, and I realized this semester that I needed to strengthen my knowledge of trig identities to be successful in future math classes in college. As I stated in Journal 2, I watched YouTube videos and did practice problems, trying my best to improve. I knew I was not actually bad at it; I needed to trust myself and keep practicing until I got it. I ended up understanding, although late, the antiderivatives of all the trig functions. This experience, where I had to do most of the learning and understanding completely on my own under incredibly stressful circumstances, made me confident in my ability to be an efficient learner on my own. College is a time when I am juggling academic, personal, and extracurricular pursuits, and being self-efficient and managing my time properly is pivotal in succeeding. While I still have to work on this next semester, this experience was the beginning of a journey of becoming a better learner under stress.
Contribution to community learning and problem-solving:
While I cannot show exactly the times I contributed to community learning and problem-solving, there were a few moments when my collaborative teaching/learning skills were tested and improved. I was able to help one of my peers, Thomas, how to do various homework problems, especially when it came to using the limit definition of a function. Since I had taken Calculus in the past, I was very used to using shortcut methods like the Power Rule, but helping Thomas with using the limit definition made me realize this was also a weak spot for me. So, while explaining it to him, I was able to reteach it to myself as well and improve my own understanding of the concept. Seeing his perspective on the problem taught me that everyone has a different way of learning math, and no one way is the “right” way. The way I learned the limit definition was not exactly the way he learned it, so I learned to use the right calculus terms to properly explain the concept. I definitely still have to work on the way I explain things so I can help my peers more efficiently.
On the other hand, when we started learning about integrals, it was all new information to me. I had to ask my peers questions often about how and why certain problems were solved. Adley, who sat next to me, was extremely helpful whenever I asked questions and gave me hints to point me in the right direction. For the last topic we learned, I was slightly confused during Team Quiz 18, but comparing my work with Adley’s cleared my confusion. I realized that the coefficient in the original function is converted to its reciprocal in the integral. She also explained to me how to know which expression within the function would be the “u” in U-substitution. Overall, collaborative learning taught me the differences in learning and how to be confident with being a learner and asking questions to my peers.
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