Assignment (due Tuesday, December 17). Imagine that you are invited to speak on the first day of MAT 2070, to give advice to entering students.  Write at least three sentences responding to at least one of the following, describing what you would tell them.
- What do you wish that you had been told at the start of this class, to help you succeed?
- Choose one topic in the course that is especially challenging. Identify it, and give advice to students trying to master that topic.
- What is the most important prior knowledge (not taught in the class) that you need in order to succeed?  Why is it important?
Extra Credit. Respond to someone else’s comment.  Do you agree? disagree? Have anything to add?
2. The topic that I thought was the most challenging was the proofs by induction. It was difficult for me to figure out how to make P(k) => P(k+1) for different situations. I know this may sound a bit cliche but an advice I would give to a student is that you should try to think outside the box. For example 3 can be expressed in different ways such as 2+1 or 1+1+1. These elementary ideas look silly and trivial but they are necessary and help you come up with proofs. I would also advise them to be proficient in their algebra such as FACTORING.
Get tons of practice. I can not stress that enough. In learning Proofs and Logic, one is learning a new way of thinking and interpreting mathematical structures and statements in addition to a completely new language. When one does a proof, one practices their ability to reason and their mathematical writing abilities. Both of these abilities need to be strong to write a good proof.
Do not be afraid to make mistakes. Understanding one’s mistakes and misconceptions can lead to a deeper and truer understanding of the material. It can also help one teach others the material, should others have similar misconceptions. Investigating misconceptions can lead to new avenues of knowledge as well.
In terms of prior knowledge, one should know one’s algebra and know it well. Making an algebraic mistake can lead one to think one has proven a proposition falsely or cause one to give up on a proposition that one otherwise could have proven. Some concepts can be tough to wrap one’s head around. Having difficulties with algebra can make the learning process arduous, since one would be trying to overcome both one’s difficulty in understanding the concept and one’s difficulty in understanding the algebra.
Don’t treat this class like your other math classes. There is very little arithmetic to be done in this class. It is unlikely that one will be able to sit down with their homework and just power through it in an hour or two; attempting to do isn’t always very enjoyable. Rather than plan to do the homework in just one sitting, take a look at the questions as early as possible and try to solve them over time, little by little. Play with different approaches and ideas in small bursts, such as while one is one the train or while one tries to fall asleep.
Be responsible about looking at the answers. If one looks at the answers too much, one will rob oneself of the ability to think of original ideas, which are extremely valuable. One may also rob themselves of having a true understanding of a concept. Proving propositions all by oneself without having hints can be a very rewarding experience. However, if one does not look at the answer when one needs to, then one may end up discouraging oneself; in addition, one cannot learn anything when one cannot understand.
I almost forgot.
The only two things you should feel the need to memorize are unproven theorems and notation. I mean only. Everything else inside of this class and in math classes in general should be retained through understanding. There are often many small but important details contained within a proof which make them difficult to memorize. There is almost no reward in memorizing them and eventually, there will simply be far too many statements and proofs to memorize.
I agree with Adam, Algebra Algebra, Algebra! If you have not mastered algebra, you will be in a world of confusion. It is never too late to rectify small but common mistakes and learn not to do them ever again. EVER. I also agree with Adam about going through the proofs in small bursts rather than one sitting. One thing that helped me was to think about what was taught in class after class has ended. I would literally just think about what I learned and it has helped me a lot.
I think that the title of the class gives some ideas of what is it going to be like. There is not a lot of math, even if there is – it is simple. The class is mostly on logic that you should be working with, and the best way to master it is doing the homework, and reading the chapters. Might sound boring for someone, but there is no other way. It is not an easy class, even though it might look like one.
From the begging you should memorize all the connotations, so later on, when you read the problem, you would be able to actually understand the question. So the beginning of the semester is very important.
Do not read the material, which you will not use. Extra information can confuse you, and mix with the information needed; so you will be in trouble.
The topic that was the hardest for me to understand was the Section 10.3 “Fibonacci Numbers”. Someone might think: “If I am here taking this class, I know what is the Fibonacci sequence…” Yes, you are right, but the problem is that you have to prove it, by choosing the right method. No one asks you to give the definition.
For me, it is very helpful, and of course, after listening to the lecture, and reading the chapter, discuss it with the friend. We all speak English, but use different way of understanding. So sharing the ideas will promote your thoughts to combine in one whole picture, named proof.
I totally agree with what Adam said before. And, especially, about not looking at the answers. Of course, it is easy, copy and your homework is done. But, after several home works – comes the exam, where someone would definitely get stock.
Also, all the information in this class is connected to each other. Each topic is like a ring, which at the end can give you a nice necklace, in a meaning of a nice grade. Missing the class means losing the ring, which will not lead to the wanted result.
2. Greetings fellow students. The one topic of this course that truly made my mind boggle was proofs by induction. This one topic is comprised of many proofs, but it was the toughest due to everything it incorporates. When doing proofs, it is as if you are learning to do something new. The interesting thing is that it incorporates concepts from topics learned in high school and middle school and truly challenges how knowledgeable you are in each of them. The most important thing that helped me master this and be more knowledgeable on it would be repetition. It took a good amount of practice, studying and connecting ideas together. Thankfully I am now at a point where I can feel confident with it, and where I am mastering proofs in general.
I completely agree with Ricky on this one proof by induction was also hard and still a bit hard for me to master. The more I practice this topic though the better I got at it.
I disagree I found that induction was the easier type of the proofs. I do agree repetition helps with practicing proofs. I would like to take like another class of proofs just so I can get more practice I like proof and logic.
2.Choose one topic in the course that is especially challenging. Identify it, and give advice to students trying to master that topic.
Generally speaking, proofs are not so easy. Although it’s the main topic of the class, they get some getting used to. I would say overall, proofs were the more difficult part of the class, perhaps with proofs by induction more specifically being the most difficult. What has helped me the most in these proofs has been a) practice and b) reason / examples. The homework was helpful in forcing us to practice proofs. I think if I hadn’t done the homework I would have felt behind or confused, especially as we moved onto the next topic. And I think the homework assigned was just enough to get that practice – not too much, not too little. Also, I found that before starting any proof, I try to think it out and scribble a few examples. I try a few numbers/figures to see if the proof is indeed true. And if it does seem to be true by the few quick examples I run on a side sheet of paper or my head, and I see if I notice a pattern of some sort before diving into the proof. I feel pretty good about proofs now! Of course, Prof Reitz really helps!
1.What do you wish that you had been told at the start of this class, to help you succeed?
I wish that I have been told that I will be dealing with the majority of concepts that I already knew. I find myself at the end of semester’s never picking up my books until the start of the new semester. Math will be mastered with constant practice and for the future students coming into proofs and logic, the course is a little challenging at first but manageable after you have some experiences with the proof. In a way, you get to see why some stuff works instead of being spoon feed of why it works. The course was a good experience for me and my fellow peers, practice, practice, and when you’re done practicing practice some more. That is how you will succeed and do well in the course.
Good advice Jean, the constant practice works for every class not only Math. when you practice constantly, you don’t need to put a lot of effort in studying for the finals because you find yourself remembering every thing from the very beginning of the class.
What I wish was told to me at the beginning of the class to help me succeed.
I think prof Reitz does a good job at given you enough time and resources in order to succeed in class. I would say first and foremost listen to him and you will do great and second of all I would say make sure to take advantage of the group work that’s done in class . There were a lot of time where I did not fully understand an idea or concept and by working with one of my classmates in class I understood the material a bit more. Sometimes it’s easier to learn from a classmate , they may phrase things in a way that’s easier to understand allowing you to grasp a concept better. The second advice I would give to future students of this course is to practice practice. Someone once said to me math is not a spectator sports . If you do the work and put in the effort you will succeed.
I could not have said it better myself. Firstly I agree on Professor Reitz giving out enough time and resources for the class. Secondly, the importance behind group-work activities could not be emphasized enough. Like you, there were many times that working with a group helped me understand things a lot better. Your classmates are such a useful resource, and each student needs to take advantage of this.
I want to echo this entire post, since I agree with it all. It’s not enough to simply understand a concept when you hear it. You have to be able to apply it as well. Practice (which is easily done by just doing the homework) is a great way to help yourself understand anything in this class. And group work is beneficial for helping someone understand material that they just didn’t get.
Hi everyone, first of all, I would like to tell you something that I wish I had been told at my first day in this class, which is: after taking each course, do some homework _other than the assigned ones by the professor_ to make sure that you understand the topic even though it seems easy. so if you find one of the homework is difficult, don’t let it go and ask questions until you get it. I’m telling you this, because if I did that, I would have understood the challenging topic for me which is the counting. While professor Ritz was explaining that topic, it seemed super easy, but when I couldn’t do the homework and I didn’t know which formula to use in which problem, I let it go, so I couldn’t do the counting problems in the exam either even though I knew the formulas. So to grasp the course don’t hesitate to do the homework and ask questions if you don’t get it. About the prior knowledge that are needed for this class is the basic algebra which I think, since you are all in this class, every body has it since calculus II is prequisite or corequisite of this class. And good luck to every body.
Hi, one thing that will kick you in the *** are the proofs. You have to really practice the proofs. They take a lot of patience, but the more practice you get, the better you’ll do on the exams. Also make sure you do the homeworks, which really helps to practice all the materials. I wish I had done more homework, but yeah do all your hw. Also come to class because a day you missed, is an important topic you missed. Plus, Prof. Reitz is amazing so its not hard to understand anything , even if math isn’t your thing. All in all, this class is great and if you put in effort, trust me, you wont regret it… not even a bit.
Proofs are difficult, I will agree with everyone else ( I am pretty good at them though… Thank GOD!!) but what really got me frustrated was counting. For some strange reason, I could not solve any of the problems when it came to how many elements in a subset can be chosen from a set with n amount of elements. I honestly don’t know why I cant get it but all I can say is I can keep trying.
I will leave advice:
STUDY, DO HW and TIME MANAGEMENT!!!
Proofs are difficult, I will agree with everyone else ( I am pretty good at them though… Thank GOD!!) but what really got me frustrated was counting. For some strange reason, I could not solve any of the problems when it came to how many elements in a subset can be chosen from a set with n amount of elements. I honestly don’t know why I cant get it but all I can say is I can keep trying.
I will leave advice:
STUDY, DO HW and TIME MANAGEMENT!!!
I wish I was told not to take 5 other classes while taking proofs. And to do the homework it will help with understanding the content. ALL the Proofs are challenging the direct proofs, proof by example, and surprisingly proof by induction are the easiest proofs but the other proof you need to practice.
Some prior knowledge you need is n+1 rule and having an open mind that sometimes the simple answers to the next steps.
1. What I wish I was told at the start of the semester? I guess I wish I knew how difficult it could be to tackle certain proofs. It’s easy to prove simple things, but it gets much harder as you go along. I still have some trouble with strong induction and any proof that tends to involve a proof within a proof. It would help to know what exactly you should be thinking about when it comes to solving proofs. How can you turn any single example for proposition into a universal proof? I’d like to know the answer to that question.
2. I’m going to talk about proofs again here. Proofs in general can be hard to solve. Whether it be direct proofs or proof by induction, it’s easy to get stuck on a step and not know where to go. My advice would be to work in groups. Not everybody thinks the same way, so it is always helpful to get insight on a problem from your peers. Where you might be stuck, others may have an idea on how to proceed. While it isn’t a good idea to simply copy off of others, learning how other people solved something you couldn’t is a good way to learn how to solve problems on your own in the future.
3. Most important prior knowledge? Study. Study, work hard, and look at the optional homework problems if you can. If a problem gets you stuck for half an hour, it’s all the more satisfying when you solve it. And the methods used to solve such problems remain in your head as well. Personally, I would say that doing your homework is necessary in order to pass this class (or most math classes). I would have been lost for the entire semester if I never did my homework. And I probably wouldn’t have done the homework if Mr. Reitz didn’t collect it…
I definitely agree with you when you say to work in groups, I think having the point of view of someone else helps a great deal when trying to solve the proofs. This technique worked for me when it came to trying to prove by contradiction.
1. What do you wish that you had been told at the start of this class, to help you succeed?
2. Choose one topic in the course that is especially challenging. Identify it, and give advice to students trying to master that topic.
3. What is the most important prior knowledge (not taught in the class) that you need in order to succeed? Why is it important?
1. I wish that I actually researched what a proof and logic class consisted of before I entered the class. I mean the name pretty much tells you what the class is about. I also wish I paid more attention to my Probability and Statistics because someone of the topics definitely did pop up early to mid semester. I also think it’s beneficial if students know that in order to succeed in this class students must must must do their homework’s. I made the mistake of stop doing homework for my Linear Algebra class and my grade suffered. I told myself, this semester I wouldn’t make the same mistake twice. That is why I made it my business to do every homework assignment because it helped a great deal.
2. The one topic that was challenging was proof by contradiction. OMG I hate this proof. I never know what my c is and then the not c. It was helpful when I worked on the problem with another student from class who was kind enough to allow me to figure it out on my own but helped to guide me through the steps (THANK YOU ADAM, I now can prove that radical 5 is irrational). I think this was very useful; it’s also useful to use board space when trying to come up with your proof, sometimes writing it down on paper won’t cut it.
3. Brush on your algebra. I made the silly mistake and stated on my first exam that 2*1= 1 LOL. Well obviously I probably was rushing but in terms of brushing up on algebra, when we get to function and induction proofs it’s really essential to know how to add and subtract fractions and how to factor out probably. Also if you take Probability and Statistics prior to taking this class definitely pay attention to the part where your professor talks about unions and intersections it will help a great deal.
The only real advice I have is to PRACTICE, PRACTICE, PRACTICE!
Overall, this was a great class and I think you did a great job at teaching the course material. You made it extremely enjoyable and I think that’s what helped me learn the concepts easier. The homework assessments were helpful and didn’t feel as thought they were punishment as they are in other classes. The daily sheets with the topics were definitely helpful and I think they will help in our other classes.
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After I read all the comments roughly, I feel that the former peers had difficulty in proofs and counting. I guess it might also be my weakness. However, I do believe I will be fine if I work hard enough. I like ibrahimadam193’s saying, “Understanding one’s mistakes and misconceptions can lead to a deeper and truer understanding of the material.” The reason I think it is more relevant to me because I did learn from my personal mistakes I made as well as others. Everyone makes mistakes. If a person does not realize he or she made mistakes, then he or she is going to make the same mistakes again and again. In other words, for people who know how to deal with their mistakes, they are going to be the winners. I like to share my work with other peers and discuss with them. I would be happy if they could point out the mistakes that I made. Also, I feel glad to help others check their assignments if I could. In the near future study, I think I have to know how to get along with other classmates well because I can learn more from them. Also, in order to do that, I need to do the homework assignments on time. I cannot study with people if I do not have my work with me. If I have questions, I need to think briefly before discussing with other classmates. Just do not be afraid of asking questions because it can prevent us from making mistakes on the exams. Overall, always try to work hard is the way to do well in this class! Never feel bad of making mistakes because it is going to be the path that everyone is going to step on.