25

I think that the proof based on the fundamental group of the punctured plane can be re-wrigged, just by omitting references to homotopies and the fundamental group, to be convincing to students who don't even necessarily know calculus, as long as they have sufficient comfort with the complex plane to be able to think about how a polynomial maps loops about ...


15

You can find a very elementary and direct proof here. Personally, while I think that any of the more sophisticated proofs is very cool the direct proof is valuable since it really shows the fundamental theorem of algebra is not a theorem of algebra but rather of the complex numbers, and that it really is fundamental in the sense that it is very elementary (...


14

I don't think there is any way to get students caught up without expending some effort. If you are willing to expend the effort, you can create a simple "you should remember" pre-class activity for days that require it, and preserve your class time for new concepts. Steps: Look through your upcoming lectures (even just a few in advance), and determine ...


13

Your students might find it useful to see this "visual approach" to proving the FTA: Velleman, D. J. (2007). The Fundamental Theorem of Algebra: A Visual Approach. Link. For a more rigorous approach by the same author, see: Velleman, D. J. (1997). Another proof of the fundamental theorem of algebra. Mathematics Magazine, 216-217. Link. (The latter ...


13

First it's a good sign that you're on this website and keen to make a difference. A passionate tutor is halfway there in my opinion. I've thought about what would have helped me as an undergraduate but everyone is different. 1) Break the theorem-proof-theorem-proof-theorem-proof-lemma-theorem cycle. I'm sure that many will argue that this is the bedrock ...


11

By multivariate analysis, I'm assuming that you're talking about a course that covers the following topics in a rigorous way: Differentiation of functions between vector spaces The inverse and implicit function theorems, and related topics (e.g. rank of the derivative, immersions and submersions). Critical points and regular points of multivariable ...


9

Motivated by Jim Belk's answer, let me mention that a rigorous advanced PDE course builds heavily on the material you mention. In particular, generalizations of integration by parts (Gauss/Green/Stokes..) are used for the weak formulation of PDEs. Technical theorems about Sobolev spaces and traces use elementary facts of basic differential geometry (...


8

You can short-cut the Liouville proof as follows. Assuming that $f$ is a monic polynomial of degree $d>0$ with no roots, define $$ h(r) = \int_{\theta=0}^{2\pi} f(r\,e^{i\theta})^{-1}\,d\theta. $$ If you are willing to differentiate under the integral sign, then it is not hard to show that $h'(r)=0$, so $h$ is constant. When $r$ is large enough the $z^...


8

You can find a proof that goes back to Gauss, which is based only on multivariable calculus (double integrals and partial derivatives) at http://www.math.uconn.edu/~kconrad/blurbs/fundthmalg/fundthmalgcalculus.pdf. I think the proof will come across as rather mysterious even if it can be followed line by line. The last paragraph of the file gives an ...


7

This California college has a no-frills $612$-page textbook available for free PDF download: College of the Redwoods Mathematics Prealgebra Textbook. 2012-13. (weblink). A solutions manual is available at the same link. Because you can download it easily, you can browse through it to decide if it meets your needs before committing to your (admirable) ...


7

I really think this depends on what you intend to cover and what you want students to learn from it. Do you hope for your students to work through the major parts of the books and develop a true appreciation for it? Good luck; you'll need to presume they already have a level of mathematical maturity and enthusiasm that is difficult to quantify and pin to a ...


6

In this video, David Eisenbud (a math prof at Berkeley) explains a proof of the fundamental theorem of algebra that is very enlightening / beautiful, and almost makes the result seem obvious.


6

Social promotion is an interesting topic, and I know my personal views are unpopular in my school. So they probably will be here too, but I'll go ahead. Due to developmental issues with children I think that social promotion is acceptable before high school. It should be coupled with remediation, but I think there are too many issues that can arise from ...


6

There is a point at which material is so far beyond students' skills that they learn less. Imagine a freshman calculus student plopped into a graduate topology seminar: they will learn almost nothing. Consider Csikszentmihalyi's flow model: We'd like students to be in that upper right corner, where they are highly skilled (relative to the task) and highly ...


6

The more the students can reinvent theorems themselves, the better. This is the philosophy of math circles, and it's also how R.L. Moore ran his classes (http://legacyrlmoore.org/reference/mahavier1.html). He had a record number of PhD's come out of his classes I believe. (Though his legacy is problematic due to his racism.) You can work somewhere in ...


5

I appreciate the question and the problems it is trying to solve. I have the feeling that it is trying to "attack the symptoms rather than the cause", and that what is really needed is a more foundational approach. However, my perspective is firmly grounded in my experience as a student in America; the problems being mentioned may actually be different in ...


4

The US has different courses at different schools. Sometimes with same name but differences in content or prereqs. It would be more meaningful to sketch this tree for a given school. Or do a few schools. That should give you some feel for the general lay of the land. And I suggest to sketch it yourself, using a course catalog. You'll learn more doing a ...


4

Indeed it is futile to expect students to successfully catch up with the prerequisite mathematical understanding for a course if they are too far behind. This is because many students think that they would rather focus on getting their grades for the current course despite not having any proper foundation in prerequisite courses. (This is real and happens ...


4

Following this answer, students can experiment with Velleman, D. J. (2007). The Fundamental Theorem of Algebra: A Visual Approach using interactive versions of the paper's plots. (Disclosure: I am a dev for the site.)


4

Here is an elementaryproof which considers $ p(z) $ as a function of $ (x,y) $ where $ z=x+iy $. The only assumptions are that $z^{n}=r $ has a solution for all integer $n$ and real $r$ and that intermediate value theorem holds for continuous real valued functions along any continuous curve on the complex plane. The second assumption can be proved by ...


3

I admit I'm a bit late to the party, but for completeness let me add a link to an interactive version of one of the visual proofs: The fundamental theorem of algebra - a visual proof (Disclaimer: I'm the author of that page.)


3

The easiest method would be to issue an addendum to the syllabus that highlighted specific chapters of the Calculus I material that would be helpful as prerequisites to the new material. Since you indicated in the comments that you are using a separate textbook, this doesn't seem as feasible, since it's tough to ask students to shell out another $150 on a ...


3

$\def\RR{\mathbb{R}}\def\CC{\mathbb{C}}$ Here is a proof via ODE. I imagine it's been found before, but I hadn't seen it before. The essential idea is that, if $z: \RR \to \CC$ is an inverse function to $f$ then we should have $z'(t) = 1/f'(z(t))$ (formula for the derivative of the inverse function, in an exotic setting). Therefore, we can solve this ODE and ...


3

If you don't mind something a bit old-fashioned, you can use C. V. Durell's arithmetic and algebra textbooks. These were standard in good schools in England at one time, and continued to be widely used into the fifties and sixties. They're concise and have good problems, with answers. The explanations on fractions are a bit shorter than one would like (I ...


3

Art of Problem Solving has a pre-algebra text that gets glowing reviews. I would expect it to be complete and rigorous.


3

There are no Amazon reviews. Doesn't seem like a popular title. (Just an indicator, not a Euclidean proof...but a negative note.) The preface says that it is approaching teaching all of graph theory via this one graph as motivation. Seems rather non-standard. Again, not Euclidean proof, but a negative indicator. If you are self studying AND a weak ...


2

The UK A-levels are widely studied outside the UK. The new syllabus is designed to have all the exams at the end, rather than the modular system that was in place before. So while there might be more than one exam (I don't remember the details), they could be taken in one block.


2

Here is a thought that might help in your development. Write up a sample lecture/class period to describe the material and how it goes. Get two others to help you run through a trial and videotape it. (Or should I say "Cell-phone it"?) Publish on the net the written material and representative samples from the video of how the class might go. Include ...


1

I have two suggestions... 1) for the students that are having the most difficult time, you might ask them to get a copy of "Hurricane Calculus". This book is immensely helpful in simplifying calculus concepts quickly and succinctly. It is also less intimidating to students because it reads less like a text book and more like a how-to guide. You can get ...


Only top voted, non community-wiki answers of a minimum length are eligible