Reflection 12: Final Goals

Honestly, I can't believe I'm almost done Calculus. It was a lot of work, but I still can't believe it.

Over the past semester, I have definitely acquired more knowledge of mathematics than I had before; I've learned about and gained a pretty decent understanding of concepts like limits and derivatives, as well as an idea as how to model certain things in the natural world. I certainly feel more proficient in some subjects than others, particularly in growth models as compared to things like continuity and derivatives (though I still understand them, I'm just not quite so good at solving their problems). I held an A in the class most of the semester, which I'm very proud of and somewhat surprised at, because as I mentioned in one of my blog posts before, I am not the best mathematician and certainly my middle school self would be astonished. Things do change (ha-ha calculus ha-ha mathematics of change).

Achieving these goals were probably most helped by in class examples followed by homework problems that were achievable/able to be checked. I often found that homework problems that couldn't be checked/compared to something often were fairly useless in my learning because I had nothing to compare them to, and so it was pretty easy for me to accidentally go down a wrong path. Additionally, guidance in class was also very useful, and projects often helped apply the knowledge to what would likely be real world scenarios.

I would definitely advise incoming students that if they want to perform well on exams and quizzes, homework is probably the best way to study, but not to stress on problems that are profoundly difficult- sometimes it just takes good judgement as to what constitutes good studying and what is no longer productive.

I would probably give myself an overall grade of around a high B to a low A. I've put in quite a lot of time to this course and have definitely learned a lot, but obviously there have been some concepts that I've needed working on, and there were times when I just couldn't do homework to gain a better understanding of the concepts. Overall, I did understand most of the material though (at least so I think), so I feel as though I'm pretty "alright" with calculus concepts, hence the B to a low A reasoning.

Reflection 11: Walk or Run in the Rain

In this article, the modelling of a common problem- whether or not we should walk or run in the rain- was created to get an idea of what factors constitute which form of movement. This was designed by integrating parameters into the original equation to dictate what may effect the ultimate outcome; things like surface area of your body, the density and angle of the rain, the distance to your front door, etc. The derivative of this equation was used to find what may be the optimum running velocity (an optimization of speed in this scenario, as related to the rain coming down). All of these concepts have been ideas covered in class, and can be applied to other scenarios of interest.

For example, I used to be an avid airsofter (essentially paintball for definition purposes), and a lot of situations can be modeled/optimized in the sport. Optimizing time spent on objectives, modelling the accuracy/shot placement of the BBs, and modelling the velocity/FPS speed of a BB leaving the barrel based on things like length, power of the spring in the gun, compression of the air, etc. are all possibilities. For this assignment, however, I will be modelling the optimal team size for succeeding at an objective (let's say capturing a base or rescuing a "hostage").

There are quite a few factors to take into account for. First off, our output from the equation should be a probability of success of sorts (above or below a certain value will indicate the probability of success or failure), which will be referred to as a Success/Fail Ratio (S/F). As for parameters, there should be three main portions to the equation; parameters relating to your team, parameters relating to the opposing team, Parameters relating to your team and the opposing team could be as follows; team size (x=number of individuals on team, or may be thought of as "number of weapons" since each player is only considered if they have a weapon), number of medics on team that can heal players back into game (typically game rules allow for one per ten players, so x/10), a constant assigned based on team skill (perhaps something on a scale of 1-10, ten being most skilled, 1 being least skilled, represented as a_skill), and a coefficient which would represent the gear being used by the players, which may include the probability of weapon/gear malfunction, which can be assigned a constant (a_malfunction, which is about 1 in every 15 rifles, so 0.067) and the availability of spare gear (how many spare batteries, gas, weapons, ammo, etc. is available), which can be represented by Gp- a coefficient of the supplies needed for the situation and the supplies currently held (S_h/S_n).

a_team skill(0.067(x_t+x_t/10)+Gp)-a_opponent skill(0.067(x_o+x_t/10)+Gp)

As for the environment, things such as area of cover as a proportion to area where combat is likely (Cover/Combat Area= Co/Ca) (integrated later) and desired amount of time to complete vs amount of time available in minutes (Tc/Ti).

a_team skill(0.067(x_t+x_t/10)+Gp)-a_opponent skill(0.067(x_o+x_t/10)+Gp)*(Tc/Ti)

Finally, none of this matters if your accuracy is poor, and you can't perform better than your accuracy except by chance, so it will act as the "carrying capacity" for the equation. The likelihood of hitting a target vs the likelihood of getting hit could be modeled as average accuracy at any distance in the combat area per 100 shots, represented as a percentage or a ratio related to the number of your team members versus the number of your opponents team members. Accuracy may be thrown off by cover, so cover available to your team vs cover available to the enemy team is now in consideration.

So, finally:
S/F=

((a_team skill(0.067(x_t+x_t/10)+Gp))/(Cot/Ca*Accuracy(x_o)*100))-((a_opponent skill(0.067(x_o+x_t/10)+Gp))/(Coo/Ca*Accuracy(x_t)*100))*(Tc/Ti)

Given my experience with modelling anything like this, this equation is probably not set up quite right and likely needs things to be shuffled around. However, it definitely delves into the ideas of what may control the outcome of certain things, which would lead to another portion of my model; what numbers constitute success, and which ones failures.

From this point one may optimize the equation to decide what the optimum time would be to complete an objective, what the optimum team size would be, the optimum amount of cover, the optimum amount of gear, the optimum area of combat, etc., depending on which variables are known and which ones are desired for manipulation.

Reflection 10

Calculus makes its way into the news in subtle but noticeable ways; in the article, an example is provided explaining how the ozone may be healing itself because the amount of ozone damaging chemicals in the air is lowering. The article, however, appeared to confuse the idea of "amount of damaging pollutants" with "the rate that damaging pollutants are being placed into the air". The amount of pollutants are not depleting, however, the rate of increase is decreasing and leveling off, which is regardless good news, though it'd be better to hear that the ozone was actually healing itself and not taking lighter punches.

This idea of rate relates to our derivatives. The derivative, if graphed, was likely increasing rather quickly, and as it hit 1989 it began leveling out toward a slope of 1. If things keep going well this way, the graph will hopefully become concave down as the rate of pollutants produced becomes negative (i.e. pollutants being taken out of the air) and the slope takes on an increasingly negative rate until the ozone is once again completely intact.

http://www.msnbc.com/rachel-maddow/watch/flint-home-posts-highest-lead-reading-ever-672191555650

This article and many others like it detail the environmental catastrophe of Flint, Michigan, where water quality is being destroyed. In this article, lead was found to be twice the previous record found in tap water. Though there isn't a whole lot of "calculus" to this statement, there is calculus to the story; the amount of lead in water should be below a certain point (below a "carrying capacity"), the rate of change of water pollutants indicates the pollution levels of the new water source (due to the sudden increase in pollutants coinciding with fracking), and the rate of change needs to be negative in order for the water to be cleaner.

Reflection 9

Learning is a life process that can occur through failure and success, which was described in The Practicing Mind. Failure has the ability to teach a lesson, perhaps better, than learning through success, as offered through this article. It's certainly relatable for some of our greatest life lessons; it taught us how to balance on a bike and what you shouldn't say in a relationship, and it can be applied to academic settings.

Calculus can integrate itself into this rule; through struggle we can learn in a way that'd get these lessons to stick. But, as the article mentions, the real goal is to propose a challenge that is reasonably achievable through the student's ability. The article used a bit of "clickbait" in proposing an idea that seems outlandish and leads the reader to believe that letting students learn on their own in difficulty leads to a more efficient lesson. Obviously, it ends in saying that with the ability to fail followed by success afterward, learning is perhaps more efficient, which is much more plausible.

This follows some basic ideas of learning and psychology; aversive results (failures and struggling) are avoided, and pleasurable results are seeked out (success, rewards). However, in the spirit of using science to discuss learning, it is determined that using rewards is twice as effective as resulting in a learned behavior than punishment; this is because one simply learns what to avoid and not what to seek when facing punishment. Some people fall into a cyclic-punishment where they simply are faced with so much challenge that they become frustrated, and oftentimes don't find solutions (or at least the right ones). Eventually, they just associate the whole practice of calculus with punishment-like results, and the present-day hate for math is born.

The ability to fail correctly comes with a lot of weight; it needs to be done in such a Goldilocks fashion. The challenge needs to be reasonably within the student's grasp (if they can't reach it, then it just becomes failure punishment), guidance needs to be available, and the failure shouldn't be something too devastating (because why would you try it again if there's a huge risk?). The sensitive nature of it is just why some people head down a path of hating certain subjects when at one point they were on the verge of unlocking its secrets, but due to the fault of a crappy failure setup, it went awry.

This is something I have seen in this class (as I've mentioned in other blogs) in homework problems that have kept me thinking for hours to no success, or in classes like Chemistry where equilibrium problems are now my least favorite when I used to be extremely efficient with the concept in high school.

Sometimes, I feel like this concept of letting students fail can be used as an excuse for a highly inefficient learning system at the hands of the wrong teacher where students just perish, and other times it can be orchestrated perfectly in unexpected scenarios.

Sometimes, it takes just getting back on the bike, because we know that in the end, it will be worth it. Salvaging that intrinsic worth can be what pulls us back into action when we feel that all we are capable of doing is falling off the bike.

Reflection 8: Demography

The Powerpoint shown was almost identical to lectures I had in Population and Community Ecology, where we discussed demographics of a population and how to calculate them, This also included discussions of growth/survivorship types and parameters that may impact a population, such as the carrying capacity or probability of surviving interactions with other individuals (assuming a sort of aggressive or predatory interaction). Some of these concepts have also been touched in Conservation Biology, Herpetology, North American Wildlife, and Intro to Wildlife Conservation.

In Calculus, we have discussed modelling functions and what goes into them, namely with the concept of population dynamics in mind. Demographics serve as model parameters that are implemented into the function, which can include things like birth rate, death rate, immigration and emigration, etc.

The idea of limits are implemented in that the population's limit is the carrying capacity, which is somewhat related to density dependence, where the density of a population can change the rates of birth/death rates, so as the number of individuals in the population approaches the limit, rates change. Logistic growth, a form of growth discussed in population ecology, was also discussed in class, which is essentially what's described above; the population approaches an upper limit, but does not surpass it. Exponential growth, another form of growth discussed in population ecology, is much simpler, in that a population grows without parameters working on it (and as such it is normally a less accurate model, since there is always a limit to growth).

Reflection 7 What is Calculus

Calculus was "invented" through the 17th century primarily by Isaac Newton and Gottfried Lebniz, though throughout time it has seen many attempts of understanding certain mathematical concepts. As early as 1820 BC in Egypt saw attempts to understand areas and volumes of objects, and by the mid 1600s, Isaac Newton and Gottfried Lebniz began to take the reigns of calculus and develop theories and ideas that we still reference today.

Calculus has historically been referred to as "Calculus of infinitesimals", also coined by Gottfried Lebniz, which refers to the calculations of things so infinitely small that they might as well be zero (though they are not zero), which could reference things such as limits, continuity, or derivatives. The concept of infinitesimals are used to manipulate or understand the behavior of certain lines, such as finding their slopes or their limits.

The concept of infinitesimals connects differential calculus and limits in that both deal with concepts that require an infinitely small definition of some feature of an equation. Limits are an infinitely small way of showing that an equation will approach a certain number but will never achieve it, though the difference between the equation and that number will grow infinitely smaller as the equation approaches that number. In differential calculus, derivatives can be taken at any point of a continuous line to find the slope between any two points, which can have an infinitely small difference between them, indicating another limit of sorts, just in this case the limit is the slope between two points, infinitely close to each other.

Reflection 6- Midterm Reflection

Course Learning Goals;

1. Understand sequences, limits, and continuity algebraically, numerically, visually, and verbally.
2. Understand derivatives algebraically, numerically, visually, and verbally.
3. Be able to model simple scenarios of change through either difference equations or differential equations.
4. Be able to apply principles of derivatives to optimization and relative change.
5. Recognize limits and derivatives in the practical and professional world, particularly in environmental and life science.
6. Be able to use a computer algebra system and spreadsheet system to investigate or evaluate given problems.
7. Work in groups to investigate problems and communicate solutions on an introductory level.
8. Practice time management and discipline in self-paced study.

Thus far, I feel as though I'm doing fairly well with meeting these goals. Goal 1 I feel I understand fairly well; the concepts generally make sense to me and I feel as though if presented with a problem in regards to those topics, I could solve them with little to no review. This is the same for Goal 3 in relation to Goal 1.

Goal 2, however, I'm still trying to grasp, for it isn't coming to me as easy as Goal 1. I feel like I'm learning things, but I'm learning them on a false base, like I'm building strong structures over mud or something like that. I feel as though I understand derivatives, but at the same time I feel as though I really don't; I often ask myself "but what IS a derivative", and though I can describe it as a "rate of change", I'm often questioning as to when and where to take the derivative. Do I simply just find the derivative at a point that I'm interested in? Should I take multiple derivatives to determine the overall average rate of change, or would it be just as okay to take one derivative of the entire equation to get my average rate of change that way? Additionally, the limit definitions of derivatives are very challenging to me; something about square roots and exponents get me, I'm not too sure why.

Goal 4 I don't believe we've delved deep enough into for me really to speak on it, especially since I'm not entirely sure what is implied by "optimization". As for the rest of the goals 5-8, I feel as though I'm succeeding fairly well. I work well with my group on projects, I can use Excel and calculators to help solve problems, can apply this knowledge to the natural world, and I can manage my time fairly efficiently (most of the time). There are occasions where I simply do not have the time to do everything that's required for all of my classes, and so learning where there is room for error has been fairly helpful, particularly in our weekly assigned problems. I used to do every last problem assigned, but as the semester ramped up, I'd find myself up until three or four AM trying to figure out problems that made no sense to me, and because of that situation, grades in my other classes would feel that- the combined stress impacts all of them, and I've found that this semester has just been learning to know when to quit and when to keep going. Thankfully, since homework isn't a graded assignment, I do have the ability to give up on certain conditions (a. other stuff needing to get done, b. it being past 3AM, or c. I am confident that I understand the homework enough to pass up the remaining questions, which requires a quick look through of what's left) so that I don't lose my mind.

This, though, does not mean that skipping homework entirely does me any good. The only time I didn't do homework (the homework that we were quizzed on last week), I did pretty horrifically, and nonetheless on the subject I don't understand. This wasn't necessarily by choice, I should add; the last few weeks have been insanity between homework, exams, setting up my internship, trying to find funding for next semester, and two jobs. I'm hoping that the next few weeks calm down a bit and I can keep myself on track, because skipping assignments is not what I want, but I do need sleep and I do need time for academics.

Even though it doesn't sound like it, I really do think that I'm succeeding at goal 8; if I did literally everything to the fullest extent possible for every class while meeting all sorts of other requirements for work and internships and scholarships, I'd be struggling with everything. I'm learning when to cut corners and when to go through with the hardest effort. I'm learning what's efficient use of time, and what's not. That is what I consider to be successful time management, not the idea of giving everything 200%, because that's simply not feasible.

Besides my recent fumble with the last quiz/homework assignment, I think I'm really doing well with meeting what's going to give me a solid final grade. I participate in class, do my weekly blogs (though I often forget to do them until sunday, like now, though I kinda like ending my weekend with a blog assignment... hm), I contribute to and complete projects, and I study like crazy for exams and do alright on them, so I think I'll end this class on a much stronger foot than what I could have ever imagined myself doing with a college level calculus class.