The Should I Become a Mathematician? Thread

mathwonk

I liked the beginning ode book by amrtin braun for my class, exactly because it featured applications, and hence entertained and motivated the class. It discussed using ode's to date of paintings and detect forgeries, predict populations of pairs of interacting "predator prey" species like sharks and food fish or hares and wildcats, troop deployment in battles with illustrations from WWII (Iwo Jima), and lots more such as "galloping gertie" the famous tacoma narrows bridge that blew down years ago. It was very well written.

Boyce and DiPrima is a time tested, often used, and well liked standard book at my university too, indeed THE standard book on ode, but I was looking for a good alternative that cost a lot less. Sadly, as soon as a book becomes a standard, the price now shoots above $130. I got my copy of Braun used for $2. Braun is also more entertaining for me, but I think you cannot go wrong with BdP.

I would suggest studying ode sooner than some of the other topics on your list, like reals and complex analysis. Also as wisely mentioned above, it seems prudent to go with the flow, and not be too rigid in your planning at this early stage.

And the calculus book by Sternberg, if you mean Advanced Calculus by Loomis and Sternberg, it is very abstract and advanced, treating calculus essentially as functional analysis. Of course once you have finished Apostol, it will probably be fine, but I suspect the view Loomis gives in the first half of calculus is not essential for an applied mathnematician. I like it though (I took the course from Loomis in the 1960's from which this is the resulting book. The only thing I learned was that the derivative of f at p is a linear map differing from f(x)-f(p) by a "little oh" function, which is of course the main idea.)

There is another newer book by Sternberg and Bamberg, math for students of physics that sounds intriguing, but I have not seen a copy. In the 1960's Bamberg was the absolute most popular and entertaining physics section man in a department which was otherwise bleak and forbidding for its physics instruction. (I still remember his list of useful constants: Planck's constant, Avagadro's number, Bamberg's [phone] number...)

mathwonk

Becoming a mathematician, part 4) College training.

I suspect it does not matter greatly which college you go to, as they all have their strengths and weaknesses. Places like Harvard or Stanford or Berkeley offer famous lecturers on a high level, incredibly advanced courses, and brilliant highly competitive students. For many of us, this can be more intimidating than inspiring. And often the famous professors are simply unavailable for conversation outside of class. In the early 60’s at Harvard, I found the lectures were wonderful, if I got the best professors, and then they walked out and I never saw them again until next time. Office hours were minimal and if I tried to see some of them, they were frequently busy or uninterested. Even intelligent questions in class seemed as likely to be met with sarcasm as a helpful answer. I suspect things have changed now with people like Joe Harris and Curt McMullen there, who are great teachers as well as researchers, and who enjoy students. Of course there were outstanding teachers like Tate and Bott there in the old days too, but not everyone was like them. As a result, I had to go away and get back my enthusiasm for math at a more supportive place.

It is helpful to go somewhere where you will enjoy your time, enjoy the courses and the other students, and get help from professors who think students matter. Today this is more common everywhere, even at famous universities, than it was long ago, but ask around among the student body. And be prepared to work very hard. Some if not most of my own undergraduate frustrations could have been lessened, possibly solved, by better study habits.

As to what courses to take, this is tricky and complicated by the almost worthless AP preparation most kids get today in high school. In general an AP class is a class taught by someone with nowhere near the training or understanding of a college professor, although they may be a fine teacher. But to expect a calculus course taught by an average high schol math teacher to substitute for a honors introduction to calculus taught by Curt McMullen or Wilfried Schmid or Paul Sally, is ridiculous. Nonetheless, so many students have bought this ridiculous idea that Harvard and Stanford do not even offer an honors introduction to calculus anymore for future math majors. There simply are none out there who have not had AP calculus in high school. Thus the student entering from high school is faced with beginning in one of many choices of several variable calculus courses. The most advanced one, the one taught a la Loomis and Sternberg, realistically requires preparation in a very strong one variable course a la Apostol, but which Harvard does not itself offer. So the only students prepared to take it are those elite ones coming in from Andover or Exeter or the Bronx high school of science, but not the rest of us coming in with our inadequate AP courses from normal high schools.

Thus the jump from high school to college has been made harder by the existence of AP courses. So in my opinion, even with AP calculus preparation, it is often helpful for a prospective mathematician to try to begin college in an introductory, but very challenging, one variable calculus course, modeled on the books of Spivak or Apostol, if you can find them. These do exist a few places, such as University of Georgia, and University of Chicago, which still offer beginning Spivak style calculus honors courses. To quote the placement notice from Chicago: “The strong recommendation from the department is that students who have AP credit for one or two quarters of calculus enroll in honors calculus (math 16100) when they enter as first year students. This builds on the strong computational background provided in AP courses and best prepares entering students for further study in mathematics.”

(I am not positive, but I assume that 16100 is the spivak course. But do your own homeworkl to be sure.)

The point is that AP preparation provides no theoretical understanding, so plunging students into advanced and theoretical calculus courses of several variables, as they do at Harvard and Stanford, by beginning in Apostol vol 2, or Loomis and Sternberg, without background from Apostol volume 1 or Spivak, is academic suicide even for most very bright and motivated students.

If you go to a school where there is no Spivak or Apostol vol. 1 type course, where the calculus preparation is from Stewart, or some such book, you are perhaps getting another AP course, only in college. Then you have to choose more carefully. Many such college courses will indeed be no more challenging than a high school AP course, and should not be repeated. Just ask the professor. They know the difference, and will help you choose the right level course. Either get in an honors section, or an advanced course suitable to your background. And join the math club. Try to find out who the best professors are, and do not be scared off if weak students say a certain professor is tough. You may not think so if you are a strong student. Once you get there, try to sit in on courses before taking them, to see which professors suit you. Student evaluations are notoriously hard to interpret correctly. The professor with the worst reputation among students, Maurice Auslander, was in my grad school days at Brandeis my absolute favorite professor. He cared the most, offered the most, and taught us the most. He also worked us the hardest.

Once you get a semester or two under your belt, it will get easier to find the right class, as hopefully the colleges own courses prepare you for their continuations, although this is not guaranteed! There is no way to force one professor to included everything the enxt one expects, nor to exclude material he/she loves that is outside the curriculum. Do your own investigating. Ask the professor what is needed for his/her course and try to get it on your own if necessary. After leaving the honors program temporarily as an undergraduate, I got back in by studying on my own over the summer from an advanced calculus book (David Widder), to make up my theoretical deficiencies and survive the next course.

Everyone should study calculus, linear algebra, abstract algebra, ode, and some basic topology. If you have no background in proofs from high school, you will need to remedy that as soon as possible. It is best to do this before entering, even if they offer a “proofs and logic” course. Such courses are often offered to junior math majors, whereas they are needed to understand even beginning courses well. For this reason it is extremely helpful to read good math books on your own that contain proofs. Today especially it is important to know some physics even if if you only plan to do math. Much of the inutition and application of math comes from physics. Even if you only want to do number theory, sometimes viewed as the purest and most esoteric branch of math, many of the deepest ideas in number theory come from geometry and analysis and even statistics, so nothing should be skipped. Work hard, read good books, seek good teachers, and try to have fun. College is potentially the most exciting and fun time of your life, and the one where, believe it or not, you have the most freedom and free time.

Hurkyl

When I thought I was going to look for a programming job, my plan was to go back to school and learn more math.

But since I'm actually employed as a mathematician (and have become fairly good at self-study), I don't feel as much need. OTOH, my employer will pay for some full-time schooling (both the classes, and giving me my full pay!), so I really ought to take advantage of it. My buddies keep trying to tell me to go and get a masters in logic.

mathwonk

Some recommended undergraduate books for future mathematicians.

Introductory calculus.
1. Calculus (ISBN: 0521867444)
Spivak, Michael Bookseller: Blackwell Online
(Oxford, OX, United Kingdom) Price: US$ 53.66
Shipping within United Kingdom:FREE

Book Description: Cambridge University Press, 2006. Hardback. Book Condition: Brand New. 3Rev ed. *** CONDITION NEW COPY *** TITLE SHIPPED FROM UK *** Pages: 672, Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis. Preface; Part I. Prologue: 1. Basic properties of mumbers; 2. Numbers of various sorts; Part II. Foundations: 3. Functions; 4. Graphs; 5. Limits; 6. Continuous functions; 7. Three hard theorems; 8. Least upper bounds; Part III. Derivatives and Integrals: 9. Derivatives; 10. Differentiation; 11. Significance of the derivative; 12. Inverse functions; 13. Integrals; 14. The fundamental theorem of calculus; 15. The trigonometric functions; 16. Pi is irrational; 17. Planetary motion; 18. The logarithm and exponential functions; 19. Integration in elementary terms; Part IV. Infinite Sequences and Infinite Series: 20. Approximation by polynomial functions; 21. e is transcendental; 22. Infinite sequences; 23. Infinite series; 24. Uniform convergence and power series; 25. Complex numbers; 26. Complex functions; 27. Complex power series; Part V. Epilogue: 28. Fields; 29. Construction of the real numbers; 30. Uniqueness of the real numbers; Suggested reading; Answers (to selected problems); Glossary of symbols; Index. Bookseller Inventory # 0521867444

2a. Calculus. Volume I. One-Variable Calculus, with an Introduction to Linear Algebra. Second Edition
Apostol, Tom M Bookseller: Paper Moon Books
(Portland, OR, U.S.A.) Price: US$ 20.00
Shipping within U.S.A.:US$ 4.50
Book Description: New York John Wiley & Sons, Inc. 1967., 1967. Fine. 666pp. Clean and bright book. No previous owner's markings. 2nd.Edition. Binding is Hardback. Bookseller Inventory # 068435

2b. Calculus. Volume II. Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probabil
Apostol, Tom M Bookseller: Paper Moon Books
(Portland, OR, U.S.A.) Price: US$ 20.00
Shipping within U.S.A.:US$ 4.50
Book Description: New York John Wiley & Sons, Inc. 1969., 1969. Fine. 673pp. Clean and bright book. No previous owner's markings. 2nd.Edition. Binding is Hardback. Bookseller Inventory # 068436

3a. Introduction to Calculus and Analysis (Volume I)
Courant, Richard; Fritz John
Bookseller: Harvest Book Company
(Fort Washington, PA, U.S.A.) Price: US$ 9.95
Shipping within U.S.A.:US$ 3.95
Book Description: Interscience Publishers/ New York 1965, 1965. First American Edition, 1st Printing Hardback in Decorated Boards. 661p. Very good condition. Very good dust jacket with one small closed tear and sunned jacket spine. Satisfaction Guaranteed. Bookseller Inventory # 515288

3a, alt. Introduction to Calculus and Analysis Volume 1 (ISBN: 0470178604)
Richard Courant
Bookseller: Frugal Media Corporation
(Austin, TX, U.S.A.) Price: US$ 10.00
Shipping within U.S.A.:US$ 3.70
Book Description: Wiley, John Sons. Hardcover. Book Condition: VERY GOOD. USED Ships within 12 hours. Bookseller Inventory # 873302

3b. Differential and Integral Calculus Volume 2
R. Courant
Bookseller: Pioneer Book
(Provo, UT, U.S.A.) Price: US$ 13.50
Shipping within U.S.A.:US$ 3.50
Book Description: Interscience Publishers, 1947. rebound Hard Cover Good. Bookseller Inventory # 481571

4. ANALYSIS 1
Lang, Serge
Bookseller: The Book Cellar, LLC
(Nashua, NH, U.S.A.) Price: US$ 39.99
Shipping within U.S.A.:US$ 4.00
Book Description: Addison-Wesley 1968., 1968. Fine in Good dust jacket; Light shelf wear to book. Heavy wear to DJ. 460 pages. Binding is Hardcover. Bookseller Inventory # 374309

5. Calculus of One Variable
Joseph W. Kitchen, Jr. Bookseller: Antiquarian Books of Boston
(Winthrop, MA, U.S.A.) Price: US$ 150.00 [sorry]
Shipping within U.S.A.:US$ 3.50
Book Description: Addison-Wesley Publishing, Reading, Mass., 1968. Hard Cover. Book Condition: Very Good. No Jacket. 8vo. xiii, 785 pages. Tightly bound and clean. No writing in book. The book also deals with plane analytic geometry and infinite series. Bookseller Inventory # 7620

also Honours Calculus*(ISBN: 0965521117) $24. from the author.
Helson, Henry
http://members.aol.com/hhelson/


Calculus of several variables.
6. CALCULUS ON MANIFOLDS: A MODERN APPROACH TO CLASSICAL THEOREMS OF ADVANCED CALCULUS
Spivak, Michael Bookseller: BRIDGEWAY ACADEMIC BOOKSTORE, ABA
(TAOS, NM, U.S.A.) Price: US$ 25.00
Shipping within U.S.A.:US$ 6.50
Book Description: W. A. Benjamin, NY, 1965. PAPERBACK COPY. Book Condition: Very Good. VERY GOOD CONDITION, PAPERBACK, 146pp. Bookseller Inventory # 001874

7. Mathematical Analysis: A Modern Approach to Advanced Calculus
Apostol, T. M. Bookseller: Textsellers.com
(Hampton, NH, U.S.A.) Price: US$ 12.50
Shipping within U.S.A.:US$ 3.50
Book Description: Addison Wesley, 1957. Book Condition: Good. Dust Jacket Condition: Fair. 8vo - over 7¾" - 9¾" tall. Hardcover, 559 pp. Notes, jacket has edge chips. Bookseller Inventory # 011916

8. Functions of Several Variables.
Fleming, Wendell H. Bookseller: Significant Books
(Cincinnati, OH, U.S.A.) Price: US$ 12.00
Shipping within U.S.A.:US$ 3.50
Book Description: 337 pp. Addison Wesley (1965) (Hardback) Good condition, ExLib. Glue Spot on cover. Bookseller Inventory # MATH10273

9. Advanced Calculus
Loomis and Sternberg
free download from Sternberg’s website.

Linear Algebra:
10. Linear Algebra : A Geometric Approach (ISBN: 071674337X)
Malcolm Adams, Ted Shifrin Bookseller: www.EMbookstore.com
(Flushing, NY, U.S.A.) Price: US$ 67.98
Shipping within U.S.A.:US$ 3.25
Book Description: W. H. Freeman; (August 24, 2001), 2001. Book Condition: New. Free Delivery Confirmation!! Brand New Hardcover, US Edition, Quality Paper Printed in USA. Bookseller Inventory # 071674337X-2

11. Linear Algebra.
Hoffman, Kenneth, & Ray Kunze Bookseller: Zubal Books
(Cleveland, OH, U.S.A.) Price: US$ 11.46
Shipping within U.S.A.:US$ 6.50
Book Description: Englewood Cliffs: Prentice-Hall 1965, 1965. 1st edition, fourth printing (1965) 332 pp., hardback, wear to spine & covers, previous owner's name to front free endpaper else textually clean & tight. Bookseller Inventory # ZB471098

Ordinary Differential Equations
12. Ordinary Differential Equations (ISBN: 0262510189)
V. I. Arnold Bookseller: A1Books
(Netcong, NJ, U.S.A.) Price: US$ 28.77
Shipping within U.S.A.:US$ 4.95
Book Description: Brand new item. Over 3.5 million customers served. Order now. Selling online since 1995. Few left in stock - order soon. Code: M20060602184422T0262510189. SKU: 0262510189-11-MIT. Bookseller Inventory # 0262510189-11-MIT

13. Lectures on Ordinary Differential Equations.
Hurewicz, Witold. Bookseller: Significant Books
(Cincinnati, OH, U.S.A.) Price: US$ 7.00
Shipping within U.S.A.:US$ 3.50
Book Description: Book Condition: Good condition, no dj. 122 pp. Wiley (1958 ) Hardback. Bookseller Inventory # MATH12978


Topology
14. First Concepts of Topology
Chinn, W. G. & Steenrod, N.e. Bookseller: aridium internet bookstore
(Cranbrook, BC, Canada) Price: US$ 8.32
Shipping within Canada:US$ 8.95
Book Description: SInger, 1966. Trade Paperback. Book Condition: Very Good. First Printing. Usual library markings in and out. non-circulating. very light use, clean crisp pages. edge rub/wear. A solid copy. Ex-Library. Bookseller Inventory # 010917

15. Differential Topology: First Steps
Wallace, Andrew Bookseller: Books on the Web
(Winnipeg, MB, Canada) Price: US$ 30.25
Shipping within Canada:US$ 5.50
Book Description: NY: W.A. Benjamin, 1968, 1968. paper bound, 1st edition, illustrated in colour, 130pp including bibliography and index. As new. Bookseller Inventory # 16779

16. An Introduction to Algebraic Topology
Wallace, Andrew H. Bookseller: BOOKS - D & B Russell
(Shreveport, LA, U.S.A.) Price: US$ 12.00
Shipping within U.S.A.:US$ 4.00
Book Description: Pergamon Press, New York, 1963. Book Condition: Very Good hard cover/ no dust. Octavo, 198 pp., Last name of prior owner inside front cover. One of a series of the International Series of Momographs in Pure and Applied Mathematics. Bookseller Inventory # 013208


Abstract Algebra.

17. Algebra (ISBN: 0130047635)
Artin, Michael Bookseller: DotCom Liquidators / DC 1
(Fort Worth, TX, U.S.A.) Price: US$ 44.50
Shipping within U.S.A.: US$ 3.50
Book Description: Bookseller Inventory # NA/DC8/T999/*114552

Abstract Analysis

18. Foundations of Modern Analysis. Pure and Applied Math., Vol. 10
Dieudonne, J. Bookseller: Zubal Books
(Cleveland, OH, U.S.A.) Price: US$ 9.49
Shipping within U.S.A.:US$ 6.50
Book Description: Academic 1960, 1960. 361 pp., hardback, ex library, else text and binding clean, tight and bright. Bookseller Inventory # ZB472982

mathwonk

notice there is a dearth of books listed for elementary diff eq since few of them inspire much admiration among people. on the other hand i have found some amazing bargains for you, including courant, apostol, hurewicz, hoffman/kunze, and dieudonne, at prices about 1/5 to 1/10 those often seen. sorry about kitchen. its a nice book but at that price it is absurd to buy it, given that copies of fleming, dieudonne, courant, etc... exist for so much less. almost any one of these books will give you an enormous amount of education. i have also shortchanged complex analysis, but you will find another example on henry helson's website. he is a student of loomis i believe, and former berkeley professor who writes excellent books and publishes and sells them himself at reasonable rates, with some written by others. he has a linear algebra book too but i have not seen it.

mathwonk

here are two more really good, really cheap books:

Elementary Theory of Analytic Functions of One or Several Complex Variables
Henri Cartan
Format: Paperback
Pub. Date: July*1995
B&N Price: $13.95
Member Price: $12.55
Usually ships within 2-3 days

also: Differential Forms
Henri Cartan
Format: Paperback
Pub. Date: July*2006
NEW FROM B&N
List Price: $12.95
B&N Price: $11.65*(Save*10%)
Member Price: $10.48

courtrigrad

For any mathematicians (pure or applied) did you guys intern anywhere during your summers? I am trying to find places where an applied mathematics major could go and intern during the summer (freshman). Maybe I could go abroad? Typically, does a math major do research over the summer or intern if he opting for a pHd? Does it have to be necessarily math related? Also, for an applied mathematician, what would you say is the most important area to know? Would it be ODE's / PDE's. I might be interested in going into quantitative finance, or something like biological math. This summer, I want to try to focus on learning a range of math rather than a depth of math (i.e. only studying Apostol, but not studying other areas of math like probability theory). Sure, I may not be a scholar in the end in any of the particular fields, but I can always go ahead and brush up later when the time calls for it (i.e. if I do a pHd). I find that the internet offers me the most versatility in learning different fields of math.

mathwonk

I did not intern myself. Today there are several programs for math types in summer funded by VIGRE grants from NSF. Some schools also offer sumemr research opportunities but these are often voluntary activites by faculty, hence may fall short of volunteers. I.e. we are asked to do it for free, and that is something hard to sustain for long.

mathwonk

hurkyl, i do not see how you can resist getting paid plus free tuition to study something interesting. how can you lose? it also adds to your resume for pay increases, new job opportunities, etc. i say grab it. you will do it easily. you are really strong mathematically. I am sure of this.

mathwonk

Here are some foundational graduate books for future professionals. * means an especially high level recommended book.

Grad math books:

Algebra:
1. *Lang, Algebra,

2. Jacobson, Basic algebra 1,2.

3. Dummit - Foote, Algebra

4. Hungerford, Algebra

Reals

5. Measure and Integral: An Introduction to Real Analysis
Richard L. Wheeden, Antoni Zygmund

6. Royden, Real Analysis

7. Rudin, Real and complex analysis

8. * Functional Analysis, Riesz - Nagy

Complex

9. Ahlfors, Complex analyhsis

10. Conway, complex analysis

11. *Hille, Complex Analysis

12. Complex Analysis in One Variable, R. Narasimhan,

Topology
13. Fulton, Algebraic topology

14. *Spanier, algebraic topology

15. Hatcher, algebraic topology.

16. Vick, Homology theory.

mathwonk

A remark for graduate students that they do not always seem to understand: Your instructor in a basic graduate course is often an expert in the field, at least on a level with many authors, although perhaps not all, of basic books. Hence it is not to be expected that the instructor will slavishly plow through a standard book on the topic, but may well merely present the material as best suits him or her. Do not be automatically disappointed if your instructor lectures from his/her own notes as they are often actually superior to what is found in many books. At the elast the lecturer will probably select from the best presentations available for each topic.

This is a plus for the student. I am having difficulty citing here standard books for each subject, since at this level the presentation given in class is normally better than that found in any one book, for one thing as it is more up to date, being given by a practicing professional. I.e. at this level the best instruction is often obtained in person rather than from books.

mathwonk

Another remark: You will notice that all the books I have cited are theoretical ones, on specific bodies of theory, rather than being say problem books. This is the way I was taught, proving theorems. We were expected to find and work problems on our own.

But in Russia e.g., there is a wonderful tradition of problem solving and problem teaching. This type of activity was what brought me to math in high school but was slighted in my college instruction. Nonetheless it is gresat fun, and leads well toward the experience of doing a PhD and solving open problems.

Thus it would be good to list some books of problems, but I will have to do some research to find them.

Sisyphus

Hey mathwonk, would you mind doing a little comparison between pure math and applied math? As in the types of classes you'll take in each major, their differences, what you can do with each degree, etc

I am starting my undergraduate studies in September. While I don't have to decide on my major until I am done by first year, I'm still kind of curious as to how it all works.

Awesome thread, by the way. I've been reading it since you've started it.

mathwonk

Well I really don't know squat about applied math, but i gather you should go heavy on the ode, partial de, and numerical analysis courses.

Thanks for the feedback.

I have been dominating the discussion but I want to explicitly solicit reports from other math people on their experiences in school, getting ready, what helped, what was a problem, what led to productive results at work, etc,...

Perhaps Matt could shed some light on his journey to a math PhD, and Hurkyl on his path to gainful employment, and J77 on his life as an applied math guy. Also physics guys like Zapper and others could help us with input on what math you really need if you might want to get into physics, or mathematical physics.

My friends in physics have emphasized group representations, but that was a long time ago. More recently it has been Riemann surfaces and algebraic geometry.

mathwonk

My older son was a math major with numerical emphasis at Stanford, and now does web based internet stuff for Arriba. He likes it. He also needs some business skill, as in a company you have to manage people who work for you, motivate, sell, service, hire and fire, and educate customers and clients.

mathwonk

My wife was also a math major and is now a pediatrician. Math is not her main resource but all dosages require mathematics to scale them to suit each child by weight. I may be trivializing her math usage, but math majors can do a lot of things because they can reason and calculate well. She also needs to manage people and service customers.

Besides her ability to deal with all people she meets, her main skill that impresses me is her terrific diagnostic ability. She actually saves lives when she detects a serious infection by its outward signs. This is deductive ability appield to real life emergencies.

mathwonk

One thing i can guarantee, everyone needs to take linear algebra, pure applied, whatever. The thing that is so frustrating about the AP courses in high school is their focus on calculus instead of linear algebra. I.e. linear algebra is easier than calculus, more important for more people than calculus, and even a prererquisite for understanding calculus.

So it sems odd to make calculus the focus of high school AP courses instead of linear algebra. Unfortunately no one listens to math professors when planning math education curricula.