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. 

Reflection 3- Personal Experience with Calculus

I have a twin brother who had a very similar start to math as I did; we both absolutely sucked at it at first (referring to math more than just talking about what a fraction is or decimals, so essentially an entry into Algebra). Since then, we have diverged on two different roads that proves to be a perfect example of where attitude and math can lead.

Middle school was about where my school unit began to show different level math classes; there was a lower level math class, a middle level, and an upper level. This is where effort first altered fate, where in our exam that would determine where we land, I tried a little harder than my brother, but we both tested into the same level of math, except he decided he should be in a lower level.

I never thought I was good at math and still don't think I'm great at it, but effort truly made a difference from this point on; even though I was in a higher level math class and of what was probably very literally the same intelligence level as my brother, I excelled in my class, and my brother struggled with the lower class.

This trend continued through high school, where we started off with a clean slate and were placed into the same level of math (Algebra 1). He failed the class by half a point and I passed with a B.

Failing by half a point, I think, is what truly lit a fire in my brother to begin putting a bit more effort into math. Having to retake a class in high school while his twin brother was moving on probably wasn't the greatest feeling, and so he quickly caught himself back up to me through geometry and algebra 2, where he finished just points behind me and receiving an A.

Sadly though, his experience with math for the majority of his years as a student plagued him into calling it quits there; he was no longer required to take math, and so he took a class that simply reviewed the lessons learned from previous years, while I took AP Statistics. Come college, he'd attempt to take statistics and to my surprise, fail, and I'd find myself taking the most math intensive major available at my school.

We're two different people with different interests, but I think that if he had a better experience with math earlier on, where he got into a higher level math class or felt as though he was capable of understanding it from an earlier time, perhaps he'd have a much better time wrestling with math. By all means, he's very capable, and an understanding of upper level math would probably be very helpful to him in his chosen fields of law enforcement and forensic psychology, but the effort doesn't seem to be worth it to him.

Since he's no longer required to take math, this is probably his end point with it, whereas my point is apparently a limit and therefore I seemingly can't surpass it, though I can get pretty close to never encountering math again.

Exploration 3 Continuity

(Answer to question 5)

A point that lies between a series of negative output regions and positive output regions(example, where A<0, and B>0) will point to a limit, or an area of discontinuity if it exists. The negative output regions could be any number approaching the limit/discontinuity from the left, and the positive output regions could be any number approaching the limit/discontinuity from the right.

SHP and Groupwork

The Sustainable Harvest Project went very well; although we waited until the last possible day to do the project (mostly a result of a lack of time and difficulty in finding time where we were all available) we feel that it was completed successfully and well understood. The first few questions were fairly easy and didn't take long to solve, but the last question proved to be a bit more difficult for us, but nonetheless was completed without much trouble (creating the spreadsheets to find answers was honestly the most time consuming part).

Our success may have been due to our ability to work well together; communication was strong and despite being in a pinch, we got everything done within a few hours (including an hour gym break!). In the future, I hope that we work together again, or at least with similar hard working people. We corrected each other where it was needed, worked hard, and divided work equally. Neither of us had a particular "role" either; with each question, we were typically on the same direction of thought and if there was any issue with the result of a question, we'd all work toward the answer individually to insure its correctness or check each other's work. The write-up was about the only area where we had assigned roles; we each took a section and answered it.

At this point, I think I'm confident enough with discrete time modeling that if given a problem similar to this project, I could work it out. It now comes relatively easy to me, though I would still need to reference my notes when it comes to putting equations into general solution form when dealing with linear or difference equations, though linear difference equations I can do without much difficulty since I've been working with them more recently.

Reflection 2: Clear and Fuzzy

Thus far, I think I've done fairly well with learning the information in the course. However, my mind does a brilliant job of pushing one thing out while absorbing another, particularly with concepts in classes such as this and Chemistry; as soon as a new concept is learned, it's almost like I have to re-learn the last unless it is incorporated into whatever the current lesson is. My highest point of confidence with each lesson is anytime between when the homework is finished and the next lesson starts; after that, I really start to question whether or not I truly remember how to solve those problems. Thus far, every quiz has been open notes and so remembering hasn't been such a problem, so I wonder how well I'd perform without my notes.

Though it's often difficult and the most frustrating form of learning, I learn the most from homework. I often have to look through the chapter to figure out how to solve whatever is being asked if I can't figure it out, and thankfully some answers are in the back of the book, so I can assure that I am doing things right. Sometimes, the ability to see the answer allows me to work backwards to understand the process of going through the problem (this reminds me of how in Animal Training we discussed that learning how to perform a sequence of things is best done backwards). It definitely takes a huge amount of my time to run through the homework, but thus far it has been very valuable and has asserted why I feel as though I'm doing well. Sometimes, my notes can be supplemental to solving the homework problems, but surprisingly I find that it often isn't as useful as the book; I've yet to have an "a-ha" moment led by my in-class notes when faced with a difficult question, but sometimes it can help me reach that point.

Perhaps the best way I learn is through practice, but when applying new concepts to problems, having a step-by-step instruction/flow chart has always been something I've always found very useful when trying to figure out something new. However, since Calculus isn't always necessarily a "step-by-step" process, it's somewhat difficult to design a technique for learning the concepts in any other way than through practice. Perhaps more guided practice during class would be useful, particularly since there are some problems encountered in the homework that don't get covered in class; some of them have different rules or techniques for solutions that I never realized existed until spending some time trying to find the solution in the book. It'd certainly speed up my process, but it may also be possible that since I'm not learning it on my own it may not stick as well and I may find myself somewhat lost when I have to do things on my own (though I suppose that's what homework would cover).

Discrete Time Modeling

Discrete time modeling is a way of determining population sizes at certain time intervals. The equations involve growth rates, a pre-determined population size (such as X_0), and the input of a time-step, which results in the output of the population size at that time interval. Growth rates greater than 1 signify growth, rates less Essentially, your x is a time-step, and your y is the population at that given time.

Perhaps one of the most interesting examples, or rather ideas, from discrete time modeling is the ability to predict population growth or decline through time. The ability to predict how much time is needed/left before populations hit a certain point is incredibly valuable to my future profession, in that it can help develop management plans for over-abundant species, or how to go about implementing conservation measures for species in decline.

I am, however, slightly confused by two topics under this umbrella of discrete time modeling. Primarily, the implementation of multiple different formulas that describe the same thing -yet are used in different contexts- are confusing to me. Remembering and knowing when to use them will be a challenge. Additionally, measuring multiple variables at once went over my head a bit, but I'm sure with some practice I'll actually understand it.

Reflection 1- Predicting Limits

In looking at the difference equation (x_n+1=(lambda)xn, x_0), I've noticed a correlation with how the sequences behave long term and the value of lambda. When lambda equals 1, the long term outcome is no growth, yet no decline. If the value falls below 1 in the form of a decimal/fraction of one, the outcome is a decrease, with lower values indicating faster decreases. Anything above 1 indicates growth, with higher values indicating more extreme cases.

As an example, if lambda was a 0.5, a population of 100 deer would decrease to 50 by x_1. By x_2, the population would only be 25, assuming the rate stayed the same.

Alternatively, with the same example in mind but with a different lambda, such as 1.5, the population would increase by one and a half times per generation. Therefore, by x_1 (with x_0 population still being 100), the population will be 150.

Additionally, the limits of these equations can depend a lot on what x_0 is compared to what the lambda value is. For example, the higher the original population is, the lower lambda would have to be to decrease the population to its limit of zero. However, this equation doesn't have a limit for increase; therefore, the original population could be literally any number, as this equation assumes infinite growth is possible (whereas in the real world, limits such as carrying capacities exist, and therefore the carrying capacity would have to be implemented into the equation to add a limit of increase).

EDIT: After actually doing the activity this time, a few more rules were noticed. This includes the fact that a lambda less than 1 when dealing with a negative X_0 still results in a limit of zero, so technically the population is increasing in this instance, though it will never go beyond zero. However, a lambda greater than 1 with a negative X_0 will result in the decrease of the population infinitely. A negative lambda greater than -1 (-1<x<0) with a positive X_0 results in the graph "pinching" inward to the limit of zero from both a negative and positive side. A negative lambda less than -1 results in the population decreasing indefinitely. These rules were added to the list below.

The ultimate list of  "rules" that I mentioned are as follows:
1. When lambda = 1, there is no growth or decrease.
2. When lambda > 1, the population is growing.
3. When lambda < 1, the population is decreasing.
4. The limit of a function with a lambda < 1 is 0.
5. The limit of a function with a lambda > 1 is infinite (DNE).
6. The size of the population at x_0 can determine how fast a population grows/decreases in proportion to lambda.
7. A lambda less than 1 with a negative x_0 results in a limit of zero.
8. A lambda greater than 1 with a negative x_0 does not have a limit as it decreases.
9. A lambda greater than -1 with a positive x_0 causes the graph to "pinch" inward to the limit of zero from a positive and negative end.
10. A negative lambda less than -1 causes a population to decrease infinitely.