Reading "How to Prove It" right before taking Calculus I -- Question

[JoeAllen]

If I were to read up to and including Chapter 3, would I be prepared enough to read Spivak's Calculus or at least some single variable calculus text based on proofs?

I'm asking because I plan on taking Real Analysis later and I'd like to gain a better understanding of Calculus.

I have read "Trig on Tears" and skimmed through a couple Algebra and Trigonometry textbooks, I know the basic Logic, and I've followed through perfectly on Prof. Leonard's Calculus I playlist. Additionally, I've done several practice sets involving related rates, extremely basic differential equations, calculus with inverse trig functions, integration by parts, some introduction to the delta-epsilon definition of limits (and proving limits from this), linear approximations, optimizations, and some additional techniques (like sign analysis, concave up/down, increasing/decreasing, curve sketching).

If I continue to read this text, the next chapters involve Relations, Functions, Proof by Mathematical Induction, and something about series/sequences, in that order. Just thought I'd include what I'd be missing if I stop after this chapter. I don't have enough time to continue once this next term starts, so I may have to hold off until summer for the rest.

I'm not sure if proof by induction will be necessary for Spivak's Calculus, but I will be later taking Discrete Mathematics, anyway.

Or should I trust that my professor will cover enough material and I can wait a couple years to get into actual proof writing?

 

[jedishrfu]

Usually proof writing comes with a college level math set theory course followed by a course on Abstract Algebra with group theory.

Having said that reading a proof book wouldn’t and could give you a deeper understanding of whatever math course you take.

 

[JoeAllen]

Usually proof writing comes with a college level math set theory course followed by a course on Abstract Algebra with group theory.

Having said that reading a proof book wouldn’t and could give you a deeper understanding of whatever math course you take.

Alright. I'll just finish up this chapter for my own enjoyment and continue on with my calculus courses. I'm not sure about my future math courses and their rigor, but I do have multivariable calculus, elementary linear algebra, and an introductory course on differential equations. Though, I do need to check the requirements for the next courses to get an idea of what an entire math degree entails (I do expect abstract algebra, as you mentioned).

I simply feel as if without any proofs, I'm skeptical of new concepts. Sure, I can memorize well and find quick tricks to take a load off the exams, but that's not at all what I expect to rely on. Even if the proofs aren't necessary, I like to read them and most of the time I can follow along. It tends to "click" a whole lot more; the alternative is to do a bunch of challenging problems without any proofs. In other words, I have no qualms leaving them out. This is going to be my second or third time introduced to these concepts, yet will be my first time in a proper course for this stuff. I figured I'd like to try a textbook with some oompf to it. I hope that follows.

 

[jedishrfu]

I was once a physics major who believed my highschool geometry course prepared me for the rigors of serious math courses. In the 1960s geometry was almost totally proof based with some application problems thrown in.

As a college junior, I jumped into taking Abstract Topology and was overwhelmed by the language and the proofs. My prof was very kind realizing my total lack of proof experience, lack of definitions, lack of prereqs and let me slip through with a C after a lot of hard work. In hind sight, i should have audited the course but that wasnt something i did.

Bottomline, its best to know your limits before you get in over your head.

One last thought, have you checked out the 3blue1brown youtube channel. He has sequence on the Essence of Calculus and Linear Algebra which can deepen your knowledge too.

https://www.3blue1brown.com/lessons/essence-of-calculus

 

[JoeAllen]

I was once a physics major who believed my highschool geometry course prepared me for the rigors of serious math courses. In the 1960s geometry was almost totally proof based with some application problems thrown in.

As a college junior, I jumped into taking Abstract Topology and was overwhelmed by the language and the proofs. My prof was very kind realizing my total lack of proof experience, lack of definitions, lack of prereqs and let me slip through with a C after a lot of hard work. In hind sight, i should have audited the course but that wasnt something i did.

Bottomline, its best to know your limits before you get in over your head.

One last thought, have you checked out the 3blue1brown youtube channel. He has sequence on the Essence of Calculus and Linear Algebra which can deepen your knowledge too.

https://www.3blue1brown.com/lessons/essence-of-calculus

3b1b has excellent videos, I'll say that. Hell, I might even slow myself down and quickly ace Precalculus before taking Calculus (I have fears that my pre-algebra and pre-calculus aren't "good enough" for some reason). You seem to have some good advice, and, besides, I'd rather wait to write proofs so I can have someone experienced enough to catch odd mistakes.

I might be getting ahead of myself. I'm waiting to take a very first course in calculus while preparing myself with content geared towards people math majors entering their third year. I think finishing the Essence of Calculus playlist would be just fine. I'm watching videos on linear algebra proofs, real analysis, double integrals, divergence/convergence tests, integration in spherical coordinates, et centera. At this point I'm just going to give my mind a break before the term starts. Honestly, I seriously doubt I'd struggle in calculus 1 too much if I already know about "delta-epsilon"; I'm pretty sure some people in calculus 1 are still struggling in certain algebra concepts while I'm worried about upper division courses that my peers likely don't even know about nor care about.

This is really just unnecessary, isn't it? I'm going to chill out for a few weeks ....

 

[MidgetDwarf]

This is what separates math majors (pure) from others. The first time a student is forced to write proofs for a course they are clueless. Takes hours to do a simple problem. People get used to it and enjoy it , or they hit brick wall. Patience is probably the most essential skill.

The most important thing you can do now, is to read your assigned textbooks in detail. Are you doing this?

Play around with Hammock: Book Of Proof. I find it more readable and better formatted than the How To Prove it Book...

 

[JoeAllen]

This is what separates math majors (pure) from others. The first time a student is forced to write proofs for a course they are clueless. Takes hours to do a simple problem. People get used to it and enjoy it , or they hit brick wall. Patience is probably the most essential skill.

The most important thing you can do now, is to read your assigned textbooks in detail. Are you doing this?

Play around with Hammock: Book Of Proof. I find it more readable and better formatted than the How To Prove it Book...

I have to be honest, this is going to be my first time taking a math course since high school, and it's been awhile. Everything I'm doing here is mostly self-taught. I'm not even sure what textbook will be given to me for my calculus 1 course. As a matter of fact, I might not even have the choice to take calculus yet and may be forced to take the precalculus anyway. I'll certainly look over Hammock's book, though.

For this current text, I'll get through 1-3 problems and the answer just seems so obvious that I have something to work with in seconds. Then, the next few questions won't make any sense at all. I was stuck for a few days on a single problem and skipped ahead to the others. Later, I went back to that problem and it all made perfect sense. I read the theorem out loud, and it's like the proof was right there.

I also did very poorly in high school and had no interest in like 90% of it. With the self-taught education after high school, I feel like I'm lost without any peers to compare. Am I supposed to be in college algebra? or trigonometry? or precalculus? or calculus 1? or calculus 2?

I have no point of reference to know what position I'm supposed to be in with math. It feels like my knowledge is so limited, as if my math skills aren't yet good enough. As if I need to learn far more than I already do. To be honest, if I read and understood the entirety of Baby Rudin, I'd still be highly skeptical of my own math skills and try to force myself to start with college algebra.

I'm sorry about this whole mess of a post, but I guess this is what happens when you start reading textbooks, watching lectures, taking notes, and doing dozens of practice problems without any professor or peers. Every time I'd solved a problem incorrectly, I'd just rewrite it and have myself do it correctly without help. If a problem was hard, I'd just assume it had to do with my own lack of math knowledge. If a problem was too easy, I'd assume it was even easier than I thought and look for harder practice.

Let me save all the proof-writing mumbo jumbo for the proof-based courses, and just focus on the matters at hand.

So, I guess I'll just change my question here:

Based on what I mentioned so far, do you think I'm ready for Calculus 1, whether Spivak or Stewart or whoever else they use for a textbook?

 

[PeroK]

I have no point of reference to know what position I'm supposed to be in with math.
I guess this is what happens when you start reading textbooks, watching lectures, taking notes, and doing dozens of practice problems without any professor or peers.

All you have to do is post a few example homework problems in the forums here and ask for feedback. It won't take long for us to figure out roughly what stage you've reached and what you have really understood.

 

[MidgetDwarf]

I have to be honest, this is going to be my first time taking a math course since high school, and it's been awhile. Everything I'm doing here is mostly self-taught. I'm not even sure what textbook will be given to me for my calculus 1 course. As a matter of fact, I might not even have the choice to take calculus yet and may be forced to take the precalculus anyway. I'll certainly look over Hammock's book, though.

For this current text, I'll get through 1-3 problems and the answer just seems so obvious that I have something to work with in seconds. Then, the next few questions won't make any sense at all. I was stuck for a few days on a single problem and skipped ahead to the others. Later, I went back to that problem and it all made perfect sense. I read the theorem out loud, and it's like the proof was right there.

I also did very poorly in high school and had no interest in like 90% of it. With the self-taught education after high school, I feel like I'm lost without any peers to compare. Am I supposed to be in college algebra? or trigonometry? or precalculus? or calculus 1? or calculus 2?

I have no point of reference to know what position I'm supposed to be in with math. It feels like my knowledge is so limited, as if my math skills aren't yet good enough. As if I need to learn far more than I already do. To be honest, if I read and understood the entirety of Baby Rudin, I'd still be highly skeptical of my own math skills and try to force myself to start with college algebra.

I'm sorry about this whole mess of a post, but I guess this is what happens when you start reading textbooks, watching lectures, taking notes, and doing dozens of practice problems without any professor or peers. Every time I'd solved a problem incorrectly, I'd just rewrite it and have myself do it correctly without help. If a problem was hard, I'd just assume it had to do with my own lack of math knowledge. If a problem was too easy, I'd assume it was even easier than I thought and look for harder practice.

Let me save all the proof-writing mumbo jumbo for the proof-based courses, and just focus on the matters at hand.

So, I guess I'll just change my question here:

Based on what I mentioned so far, do you think I'm ready for Calculus 1, whether Spivak or Stewart or whoever else they use for a textbook?

If you are wondering whether you need to be in college algebra, trig, pre calculus etc. Then you are not ready for basic calculus imo. There are many levels to understanding. Let's take Calculus for example. There is the low level as seen in something like Stewart's book. The middle level, say something like Spivak/Courant/Apostol, or a very high level (Loomis).

So in mathematics, we start somewhere, generalize it a bit more, rinse and repeat. The importance is to have a strong foundation. You say you did not do math since high school. Maybe purchase Cohen: Pre-Calculus and Lang: Basic Mathematics. Work through these two books in earnest. If repeated trouble occurs, you may need to go down to algebra 1, and proceed upwards. Everyone reviews material at one point or another. Focus on this, and ignore all the calculus stuff right now. It is not a race.

You're post are a bit hard to read in certain places. What I gathered is that you are enrolled in college? If so, you should have taken a placement exam. This is a good indicator of where you should be. A bit of caution, since California has pushed to remove remedial math/English courses from community colleges, and universities are following suite. Not sure if other states are doing this, but something to look into. I have seen students thrown into a pre-calculus course at a university I am attending for my graduate degree. Let's just say, the results are disastrous for the majority of students.The important thing is that you start doing. Videos are a good supplement, but not a replacement for an actual textbook.

I also have an unconventional school background. I dropped out of 9th grade, received a GED at 21, then started CC. It is possible, but one must be honest with themselves, work hard, and not take short cuts.

 

[jtbell]

Are you planning to major in math itself, or in a math-using field like physics or engineering?

Whatever college/university you attend should have a pathway for prospective math majors to get "up to steam" with proof-writing and the mathematicians' way of thinking about math, even if you're lacking in those areas when you enter. Except maybe for really elite schools which probably wouldn't admit you as a math major anyway.

At the small non-elite college where I taught for many years, math majors and non-math (e.g. physics) majors took the same freshman calculus sequence. It couldn't be too "mathy" and rigorous because of the non-math majors who were the majority of those taking those courses.

In order to get the math majors ready for the more rigorous courses, the math department taught a course called "Transition to Advanced Mathematics" which was pre-requisite for those courses.

Larger schools probably have separate calculus sequences for prospective math majors and non-math majors, with different levels of rigor and proof-writing. But even the courses for prospective math majors shouldn't assume a lot of previous experience with proof-writing, except maybe at the the elite schools.

When you're applying to colleges, look at their math curricula and see which approach they take.