interesting link, Are You Thinking of Majoring in Mathematics?

[wapiti]

I noticed this over at CC this morning and thought it might help someone.  https://www.math.uh.edu/~tomforde/ConsideringMathMajor.html

1. Question #1: Can I get a decent job if I major in mathematics? 
   
   Question #2: What kinds of careers are available to me if I major in mathematics? 
   
   Question #3: Will a mathematics major help me if I want a job in Business or Finance? What if I want to become a Lawyer? 
   
   Question #4: What are the demographics of UH math majors? How many are female? How many are minorities? 
   
   Question #5: Are there any special degree programs or concentrations that the UH Math Department offers? 
   
   Question #6: What is it like being a math major? What is mathematics like beyond calculus? 
   
   Question #7: I'm in Precalculus/Calculus right now, and I'm not the best student in the class. Can I still be a math major? 
   
   Question #8: What about a double major with Mathematics? 
   
   Question #9: Can I talk to someone about being a math major?

    

[Lilaclady]

Interesting, idea this article today about how almost a third of math majors change their major in college

https://www.insidehighered.com/news/2017/12/08/nearly-third-students-change-major-within-three-years-math-majors-most

[Arcadia]

Interesting, idea this article today about how almost a third of math majors change their major in college

https://www.insidehighered.com/news/2017/12/08/nearly-third-students-change-major-within-three-years-math-majors-most

I do think below quoted from the article is possible as a reason given my local public high schools’ generosity with giving good grades for all subjects.

 

“Ed Venit, managing director of the student success collaborative at EAB, which published a study last year showing that students who changed majors graduate at a higher rate than those who don't, said many students who plan to major in rigorous fields like math because they excelled in high school may find themselves "in a little over their head" in the college-level discipline.â€

[luuknam]

Interesting, idea this article today about how almost a third of math majors change their major in college

https://www.insidehighered.com/news/2017/12/08/nearly-third-students-change-major-within-three-years-math-majors-most

 

 

52% of math majors. Almost a third of all majors do. (haven't finished reading the article yet, that just jumped out at me)

[Bootsie]

I don't find this surprising.  Many students who enter college knowing they like math haven't had coursework in some of the areas they may switch to--actuarial science, statistics, economics, or finance.  Once they have a course with a lot of math applications, they may choose to change majors.  

 

Also, large groups of other majors, such as "business" are lumped together.  So, a student who changes from accounting to marketing would not be counted as "changing majors" in this study

[dereksurfs]

This is perhaps my favorite of the Q/A's. It also relates to my prior discussions regarding changing majors and double majoring. It makes a lot of sense appealing to my practical side.

 

Question #8: What about a double major with Mathematics?

 

Answer: Combining a mathematics major with another major can be a great idea. Mathematics can complement the study of many other subjects, and it can make job applications or applications to graduate programs in any subject look much stronger. Employers and graduate school admissions committees know that the study of mathematics develops strong problem solving skills, comprehension of abstract concepts, and creative thinking ability. These are all highly desired qualities in applicants to almost any field or industry. 

 

If you are majoring in science, engineering, finance, economics, political science, or a social science, such as psychology or sociology, then you will find that the coursework in your major relies heavily on math. In order to have the best opportunity to do well in those courses and absorb the material in these subjects, it can be very beneficial to take math courses that have applications to these subjects. In fact, it is often the case that in disciplines such as these the use of mathematics becomes more pronounced as one studies the subject further. Consequently, students in these subjects are often limited by the amount of mathematics they know. The more math you know, the further you can progress in any discipline that uses mathematics. 

 

Besides these majors, it is also common to have double majors who combine their math major with a subject that is very different from math, such as Music, Dance, Art, English, Theater, or Journalism. Mathematics can often serve as a nice counterbalance to majors in the arts or other creative fields. The study of mathematics involves a great deal of creativity, and it is not uncommon for math students to also be interested in other creative endeavors, such as art or music. In addition, since jobs in the arts and many other creative fields are often difficult to get, a double major with math can help diversify your skills and provide greater assurance of getting a job after graduation.

[Janice in NJ]

Regarding the higher change-in-major rate: I believe the part of the reason this happens is imbedded in Question #6. What is it like being a math major? What is mathematics like beyond calculus?

 

Mathematicians seek out patterns and use mathematical structures as models. Mathematicians often want to prove that certain statements about mathematical structures are true, and if the models are good approximations of the real world, mathematical reasoning can provide insight about nature and make predictions about the world around us. Mathematics involves a great deal of logic, abstraction, problem-solving, counting, calculation, measurement, and the systematic study of shapes and motion. Many math majors and students of advanced mathematics tend to use words such as "beautiful", "powerful", and "useful" when describing how they feel about the mathematics they learn.

 

Linear Algebra is probably the first course in which students are universally asked to prove that certain statements about mathematical structures are true. Sure, proofs have been there all the time. Students see them. Teachers explain them. And some courses require students to dig into them more than others. However, before LA I think many successful students get away with tuning out that proof business and just waiting for the algorithm. "Just tell me what to do: What formula do I use? How does it work? Where are the practice problems? (Watch how fast and accurate I am!)" 

 

Sure, the practice problems involve a level of increasing creativity in order to solve them, so that's what a lot of successful students think mentors mean when they make a statement like this (#8): The study of mathematics involves a great deal of creativity, and it is not uncommon for math students to also be interested in other creative endeavors, such as art or music.

 

Although I think in general that last statement can be misunderstood by high schoolers. I think we are failing kids on that one. Few kids come out of high school thinking that musicians make their living playing scales. Musicians aren't technicians. There's more to it. And few artists are painting by number. Kids have a decent idea of what it takes to succeed as a musician or an artist. I don't think they realize what that means when it comes to mathematics. (AND - many music majors peel off during or after college-level music theory because it's their first non-technician musical activity. Now they get it. They discover that music isn't what they thought it was. I'm sure art programs have something similar.)

 

Back to math undergrads: Because of sequencing, it's important to get to Linear Algebra - usually in year two - in order to have time to get to the good stuff in yrs 3 & 4. (See Note below regarding LA.) So most kids take LA prior to things like Probability and Statistics. Those are tucked around the edges. So if LA was unnerving, many students can feel re-grounded when they land back in those plug-n-chug courses like Probability and Statistics; although by then, most are starting to realize that something is starting to shift. Now when they see those proofs, they know that understanding them makes the difference. You can't tune them out and wait for the algorithm. You're going to get lost. Depending on the school and the course, often an algorithm-only student can pass. But they can't land an A. And they know that something is amiss.

 

Then they hit Abstract Algebra and Real Analysis. That's where the road divides, and students are forced to peel off. Mathematics isn't really so much about being a human computer: find a formula and then work through 15 problems using that formula. We have software for that. The subject is really about proving something is true and then using that certainty to prove that something else must be true - and that can be mind-blowingly beautiful. Stunning. AND once you know it to be true, the irony is that you don't need to work an example. It's a moot point. It can't NOT be true. You don't even want to work an example. Why reduce perfection to a simpler form? You just sit there and wallow in the immensity of the always puddle. Feeling the warm wonderfulness of certainty in an uncertain world. 

 

Getting to that moment takes so much time and effort. "Reading" mathematics is a slow and deliberate process. The more advanced the text, the more the reader must bring to the table. No one expects Shakespeare to start with a discussion of vowel phonics and syllabication. The bard assumes you bring a wealth of understanding to his first utterance. Same with mathematics. Each volume is shelved somewhere in the hierarchy, this branching tree that is popping out and expanding at a dizzying pace. People often liken it to music, but the biggest similarity I have found is the way it works in my head. If I'm practicing my scales and my left hand instinctively feels comfortable flying up the keys in G major, then sharing Mozart's ideas certainly becomes easier. It's just quicker. I can do that with little reflection. I understand immediately what he means when he smashes that blur of ink onto the bass clef. Clear intent across time and space. (And if I know what I'm doing, Mozart's blur means something completely different from Chopin's blur. Different intent. Reflect. Process. Play.) In any case, I am free to focus on the bigger idea that Mr. M. is sharing. You can feel your brain shift gears as you move between the mechanics and the art form. That happens when reading mathematics as you reflect on everything that is stored in the various filing cabinets in your brain as you are "reading." Connections. I catch myself handling the ideas that I have come to own. First the recognition. Five symbols on the page, an idea in a language. They mean something. The symbols are a reminder of something I know to be true. Finding that something in my own catalog is the next step. 

 

Sometimes it's just an inventory process. Yes. That. Before. After. Where does it land? Why? Got it. Next idea... (Next building block in the proof.) Sometimes the process grinds to a crawl because I can't quite remember the details of why that must be, so I challenge myself. Can I sketch it out and try to pull that mental file to the top of the stack? After all, I shouldn't really be using this block unless I have confidence in its substance. (Sketching it out yourself is the best way to solidify its certainty within the structure of THIS idea. Convince yourself.) If not? Dang it! Best to go look it up. Find it. Read it. Got it. Close the book. Continue with the intent of doubling back the next day to sketch it out again to solidify it. Back to reading. Where was I? Oh yes. This link in the chain that produces an understanding of THIS proof. Again: Previous link, this link. Got it. Next line of the proof...

 

Sometimes I have to stop reading. I'm lost. Time to back up and fill in the gaps. I slide the book back onto the shelf with the hope that I will make it back before this carcass gives up. It's like meeting someone you are eager to talk to, but then you realize that you aren't ready to listen to them. You can hear the words, but you can't find meaning. There's no resonance. It's all just sounds. You need something else first. So rather than fight the process, you shelve the person and their ideas. Time to grow into your own boots. And then you cast about for the steps between what you know and what you just bumped into. Process. Can takes weeks or years. Not for the faint of heart. Can't rush it. Can't move forward till you are ready. Oh well. Time to enjoy the journey cause the destination just faded into the distance. Time for baby steps. Humbling but beautiful. Facing ones own mortality as you realize that you are definitely going to run out of time before you have a chance to enjoy all the flavors available.  The difference between big ideas and small ones. And ones that are bigger than big. 

 

Sometimes my "reading" slows to a crawl because I make a connection I hadn't made before. You know it when it happens. Now there are two or more ideas in your head that were never part of the same thing. But of course they were all along. You just didn't know it. But now they so obviously fit together. Everything has changed and it's almost as if they have always been together. Except you know they weren't just a moment ago. But now they are. Everything else seems to have sharpened into focus just a tad more. It always was; you just didn't know it. But there it is - as clear as anything you've ever known. When that happens, I usually try to pause to savor the event. It's special. And very satisfying. (I'll admit it - sometimes I've cried. What a rush!) I can feel the spaces inside my own head. It's as if the ideas stored there have a location. A structure. They are connected. And now reconnected in a new way. Such blissful coalescence.  Because so much of my experience with that muscle is spent banging around in there trying to remember if we need toilet paper while standing in the supermarket, it is incredibly satisfying to be reminded that this organ is capable of SO much more. Even at my age. In fact, it's only because of age that there are so many fantastic treasures tucked away in this attic. Such bliss. Such peace. And it's all mine! I'm the only one who will ever be in here. Probably the only thing I can ever truly call my own. Such privacy. 

 

That happens very infrequently. Most of the time reading math is just work. I do it cause I'm waiting for the next "high." (It's THAT good!) However, to build all that? Takes discipline. Diligence. Hard work - you can feel the stress it puts on the muscle. Glorious! But not the same thrill you get from doing something repetitive like cranking out a page of examples. The hand just moves - with only casual mental effort. And it feels so productive - especially when it's easy for you and other people think it's hard. VERY rewarding. What an ego pump! But that's not REALLY what mathematics is about. Mathematicians are tasked with creating something new. Pure mathematicians prove new things. Applied mathematicians find new ways to use those new certainties. Neither spends her time cranking out examples and then checking to see if she got the right answer. There are NO answers in the back of the book. That thrill of being "right" that kids who are good at math in high school feel - the little endorphin rush that the smart kids get 20+ times every night doing their homework? That's over. A lot of math majors discover that they love arithmetic, not mathematics. They peel off. 

 

An example:

 

1 = 1

1+3 = 4

1+3+5 = 9

1+3 +5+7 = 16

1+3+5+7+9 = 25

 

Did you notice that 1, 4, 9, 16, and 25 are all perfect squares?

 

Arithmetic asks, "What about the next one? Is that a perfect square too?" Viola! Yes, 36. But maybe the next one will fail... nope still a square. Then +13, +15, +17... Endorphins. Speed. Rush. Faster. Dang, I'm good at calculating things in my head. I feel so smart right now. Next one! The irony is that no matter HOW fast you go, you're never closer to being done. And you are no closer to answering the question.

 

Yup. That.

 

Mathematics asks, "Is this a coincidence? Will this always work? Will the next sum always be a perfect square? How can I be certain? (And I'm not talking about a pile of examples where I give up and decide that it probably always works? I'm talking about how can I be sure!)"

 

Cranking out the next example is easy. Certainty is harder; but it's worth finding. And the journey is glorious!

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

 

Regarding Undergrad Linear Algebra: Keep in mind that a good undergrad LA course makes it clear to students that they are only seeing the tip of the iceberg. These waters are deep. And the more you know, the better. Ask Sheldon Axler. He thinks we are doing students a great disservice by letting them even HAVE the determinant as undergrads. 

 

 

Edited December 9, 2017 by Janice in NJ

[clementine]

A bit off topic, but two of dd's college friends are majoring in math to become actuaries. They are pretty much guaranteed jobs after graduation & pay is good. It's a difficult program, but they both love it.

[Janice in NJ]

A bit off topic, but two of dd's college friends are majoring in math to become actuaries. They are pretty much guaranteed jobs after graduation & pay is good. It's a difficult program, but they both love it.

 

And yes, actuarial science is one of the peel off locations for kids who discover that they don't really like mathematics even though they are majoring in it. According to industry-standard info: To prepare for an actuarial career, you should take three semesters of calculus, two semesters of probability and statistics, two semesters of economics, one or two semesters of corporate finance, business communications and a well-rounded group of liberal arts courses.

 

The actuarial exams are know for being speed-rounds. It's a well known fact that you have to be fast in order to be successful. And I believe that most people who end up there understand the difference between what they are doing and what a mathematician does. They have taken Real Analysis; they know about Baby Rudin - they have taken a math class that wasn't about numbers. They can tell you that they aren't the same thing. 

 

I am suggesting that we could do a better job of helping kids understand the differences BEFORE they start down this road. In my experience, it's not uncommon for kids to land in actuarial science because they either want to make the money or because they found out they were being forced off the road. The former is fine. So is the latter - but the latter can make kids doubt their aptitude and their bent. It chose them instead of vice versa. And that can be tough when you're 21. I don't want to start a war, but it's a thing in undergrad math departments that the kids get divided up and sectioned in their 3rd and 4th year. Some of them are encouraged to become mathematicians. And even among that group, there is a pecking order that happens in grad school. Sometimes for kids who have found their identity in their aptitude, it can be a tough lesson. No, you can't always do everything you try to do. Sometimes you need to do the things you like to do. And the things you are naturally good at. 

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

Edited December 9, 2017 by Janice in NJ

[elegantlion]

Thank you for the link, I'll share that with ds, the math major. 

 

And thank you, Janice. This is beautiful, this is what I think my son sees in math. He's been looking for patterns most of his life. He's taking Linear Algebra next semester. :)

 

 

 

Regarding the higher change-in-major rate: I believe the part of the reason this happens is imbedded in Question #6. What is it like being a math major? What is mathematics like beyond calculus?

 

Mathematicians seek out patterns and use mathematical structures as models. 

 

 

 

Mathematics asks, "Is this a coincidence? Will this always work? Will the next sum always be a perfect square? How can I be certain? (And I'm not talking about a pile of examples where I give up and decide that it probably always works? I'm talking about how can I be sure!)"

 

Cranking out the next example is easy. Certainty is harder; but it's worth finding. And the journey is glorious!

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

 

 

 

[OnMyOwn]

 

I noticed this over at CC this morning and thought it might help someone. https://www.math.uh.edu/~tomforde/ConsideringMathMajor.html

Thanks for the links. This is wonderful information.

[Arcadia]

My oldest is doing LA now so I am reading this long article with vested interest. The article is more detailed than what I quoted. LA and MVC is offered in some high schools here after AP calculus BC and dual enrollment options for public school students has those courses as well.

 

“More Linear Algebra, Please

Posted on September 19, 2016 by Ben Braun

By Drew Armstrong, Associate Professor of Mathematics, University of Miami

 

Three Fundamental Problems

Linear algebra is being undersold. Linear algebra is the common denominator of mathematics. From the most pure to the most applied, if you use mathematics then you will use linear algebra. This is also a fairly recent phenomenon, historically speaking. In the 19th century, linear algebra was at the cutting edge of mathematical research. Today it is a universal tool that every user of mathematics needs to know. This becomes more true every year as algorithms and data play a bigger role in our lives.

... I believe that a two-semester linear algebra sequence in the first year will be a more honest representation of how mathematics is used today.

 

Complex Numbers. Complex numbers are currently an orphan in the undergraduate curriculum. According to the Common Core Standards, students are supposed to learn about complex numbers in high school; however, from my experience with US college students I know that they are not learning this material. At the University of Miami (where I teach), the basic ideas of complex numbers including de Moivre’s Theorem appear in our pre-pre-calculus remedial course, so it is not reasonable that a student who takes this class will have time to complete the math major in four years.

...

Linear Algebra remediation eats up other courses. At the University of Miami we require all math majors to take MTH 210 (Introduction to Linear Algebra). As is typical in many departments, this is a one-semester course that is usually taken in the second year, as it has Calculus II as a pre-requisite. Most of the students have never seen vectors or matrices before, so our goal is to get them from the basic ideas to the useful applications in one semester. This course is then required as a pre-requisite for many upper-division courses. However, most instructors find that the students’ subsequent linear algebra background is very shaky because one semester is not enough to absorb all of the material.

...

In summary, by the time I had completed a Bachelor of Science degree in mathematics with a physics minor, I had been required to take four semesters of general-purpose linear algebra. And you want to know something funny? As I proceeded to graduate school in pure mathematics at Cornell, I shortly came to feel that the most serious deficiency of my undergraduate education was that I had not seen enough linear algebra! Not only did I find my multilinear algebra background weak when I learned representation theory, I was also shocked when I learned about the Perron-Frobenius theorem and its amazing applications (e.g., to ranking webpages): Why had no one told me about this theorem?

...

Undergraduate programs in the US do not introduce linear algebra early enough and they do not place it beside calculus at the center of the curriculum. Other countries (such as Canada) have done a better job with this, and I hope that we in the US can learn from their example.â€

https://blogs.ams.org/matheducation/2016/09/19/more-linear-algebra-please/

[OnMyOwn]

Regarding the higher change-in-major rate: I believe the part of the reason this happens is imbedded in Question #6. What is it like being a math major? What is mathematics like beyond calculus?

 

Mathematicians seek out patterns and use mathematical structures as models. Mathematicians often want to prove that certain statements about mathematical structures are true, and if the models are good approximations of the real world, mathematical reasoning can provide insight about nature and make predictions about the world around us. Mathematics involves a great deal of logic, abstraction, problem-solving, counting, calculation, measurement, and the systematic study of shapes and motion. Many math majors and students of advanced mathematics tend to use words such as "beautiful", "powerful", and "useful" when describing how they feel about the mathematics they learn.

 

Linear Algebra is probably the first course in which students are universally asked to prove that certain statements about mathematical structures are true. Sure, proofs have been there all the time. Students see them. Teachers explain them. And some courses require students to dig into them more than others. However, before LA I think many successful students get away with tuning out that proof business and just waiting for the algorithm. "Just tell me what to do: What formula do I use? How does it work? Where are the practice problems? (Watch how fast and accurate I am!)"

 

Sure, the practice problems involve a level of increasing creativity in order to solve them, so that's what a lot of successful students think mentors mean when they make a statement like this (#8): The study of mathematics involves a great deal of creativity, and it is not uncommon for math students to also be interested in other creative endeavors, such as art or music.

 

Although I think in general that last statement can be misunderstood by high schoolers. I think we are failing kids on that one. Few kids come out of high school thinking that musicians make their living playing scales. Musicians aren't technicians. There's more to it. And few artists are painting by number. Kids have a decent idea of what it takes to succeed as a musician or an artist. I don't think they realize what that means when it comes to mathematics. (AND - many music majors peel off during or after college-level music theory because it's their first non-technician musical activity. Now they get it. They discover that music isn't what they thought it was. I'm sure art programs have something similar.)

 

Back to math undergrads: Because of sequencing, it's important to get to Linear Algebra - usually in year two - in order to have time to get to the good stuff in yrs 3 & 4. (See Note below regarding LA.) So most kids take LA prior to things like Probability and Statistics. Those are tucked around the edges. So if LA was unnerving, many students can feel re-grounded when they land back in those plug-n-chug courses like Probability and Statistics; although by then, most are starting to realize that something is starting to shift. Now when they see those proofs, they know that understanding them makes the difference. You can't tune them out and wait for the algorithm. You're going to get lost. Depending on the school and the course, often an algorithm-only student can pass. But they can't land an A. And they know that something is amiss.

 

Then they hit Abstract Algebra and Real Analysis. That's where the road divides, and students are forced to peel off. Mathematics isn't really so much about being a human computer: find a formula and then work through 15 problems using that formula. We have software for that. The subject is really about proving something is true and then using that certainty to prove that something else must be true - and that can be mind-blowingly beautiful. Stunning. AND once you know it to be true, the irony is that you don't need to work an example. It's a moot point. It can't NOT be true. You don't even want to work an example. Why reduce perfection to a simpler form? You just sit there and wallow in the immensity of the always puddle. Feeling the warm wonderfulness of certainty in an uncertain world.

 

Getting to that moment takes so much time and effort. "Reading" mathematics is a slow and deliberate process. The more advanced the text, the more the reader must bring to the table. No one expects Shakespeare to start with a discussion of vowel phonics and syllabication. The bard assumes you bring a wealth of understanding to his first utterance. Same with mathematics. Each volume is shelved somewhere in the hierarchy, this branching tree that is popping out and expanding at a dizzying pace. People often liken it to music, but the biggest similarity I have found is the way it works in my head. If I'm practicing my scales and my left hand instinctively feels comfortable flying up the keys in G major, then sharing Mozart's ideas certainly becomes easier. It's just quicker. I can do that with little reflection. I understand immediately what he means when he smashes that blur of ink onto the bass clef. Clear intent across time and space. (And if I know what I'm doing, Mozart's blur means something completely different from Chopin's blur. Different intent. Reflect. Process. Play.) In any case, I am free to focus on the bigger idea that Mr. M. is sharing. You can feel your brain shift gears as you move between the mechanics and the art form. That happens when reading mathematics as you reflect on everything that is stored in the various filing cabinets in your brain as you are "reading." Connections. I catch myself handling the ideas that I have come to own. First the recognition. Five symbols on the page, an idea in a language. They mean something. The symbols are a reminder of something I know to be true. Finding that something in my own catalog is the next step.

 

Sometimes it's just an inventory process. Yes. That. Before. After. Where does it land? Why? Got it. Next idea... (Next building block in the proof.) Sometimes the process grinds to a crawl because I can't quite remember the details of why that must be, so I challenge myself. Can I sketch it out and try to pull that mental file to the top of the stack? After all, I shouldn't really be using this block unless I have confidence in its substance. (Sketching it out yourself is the best way to solidify its certainty within the structure of THIS idea. Convince yourself.) If not? Dang it! Best to go look it up. Find it. Read it. Got it. Close the book. Continue with the intent of doubling back the next day to sketch it out again to solidify it. Back to reading. Where was I? Oh yes. This link in the chain that produces an understanding of THIS proof. Again: Previous link, this link. Got it. Next line of the proof...

 

Sometimes I have to stop reading. I'm lost. Time to back up and fill in the gaps. I slide the book back onto the shelf with the hope that I will make it back before this carcass gives up. It's like meeting someone you are eager to talk to, but then you realize that you aren't ready to listen to them. You can hear the words, but you can't find meaning. There's no resonance. It's all just sounds. You need something else first. So rather than fight the process, you shelve the person and their ideas. Time to grow into your own boots. And then you cast about for the steps between what you know and what you just bumped into. Process. Can takes weeks or years. Not for the faint of heart. Can't rush it. Can't move forward till you are ready. Oh well. Time to enjoy the journey cause the destination just faded into the distance. Time for baby steps. Humbling but beautiful. Facing ones own mortality as you realize that you are definitely going to run out of time before you have a chance to enjoy all the flavors available. The difference between big ideas and small ones. And ones that are bigger than big.

 

Sometimes my "reading" slows to a crawl because I make a connection I hadn't made before. You know it when it happens. Now there are two or more ideas in your head that were never part of the same thing. But of course they were all along. You just didn't know it. But now they so obviously fit together. Everything has changed and it's almost as if they have always been together. Except you know they weren't just a moment ago. But now they are. Everything else seems to have sharpened into focus just a tad more. It always was; you just didn't know it. But there it is - as clear as anything you've ever known. When that happens, I usually try to pause to savor the event. It's special. And very satisfying. (I'll admit it - sometimes I've cried. What a rush!) I can feel the spaces inside my own head. It's as if the ideas stored there have a location. A structure. They are connected. And now reconnected in a new way. Such blissful coalescence. Because so much of my experience with that muscle is spent banging around in there trying to remember if we need toilet paper while standing in the supermarket, it is incredibly satisfying to be reminded that this organ is capable of SO much more. Even at my age. In fact, it's only because of age that there are so many fantastic treasures tucked away in this attic. Such bliss. Such peace. And it's all mine! I'm the only one who will ever be in here. Probably the only thing I can ever truly call my own. Such privacy.

 

That happens very infrequently. Most of the time reading math is just work. I do it cause I'm waiting for the next "high." (It's THAT good!) However, to build all that? Takes discipline. Diligence. Hard work - you can feel the stress it puts on the muscle. Glorious! But not the same thrill you get from doing something repetitive like cranking out a page of examples. The hand just moves - with only casual mental effort. And it feels so productive - especially when it's easy for you and other people think it's hard. VERY rewarding. What an ego pump! But that's not REALLY what mathematics is about. Mathematicians are tasked with creating something new. Pure mathematicians prove new things. Applied mathematicians find new ways to use those new certainties. Neither spends her time cranking out examples and then checking to see if she got the right answer. There are NO answers in the back of the book. That thrill of being "right" that kids who are good at math in high school feel - the little endorphin rush that the smart kids get 20+ times every night doing their homework? That's over. A lot of math majors discover that they love arithmetic, not mathematics. They peel off.

 

An example:

 

1 = 1

1+3 = 4

1+3+5 = 9

1+3 +5+7 = 16

1+3+5+7+9 = 25

 

Did you notice that 1, 4, 9, 16, and 25 are all perfect squares?

 

Arithmetic asks, "What about the next one? Is that a perfect square too?" Viola! Yes, 36. But maybe the next one will fail... nope still a square. Then +13, +15, +17... Endorphins. Speed. Rush. Faster. Dang, I'm good at calculating things in my head. I feel so smart right now. Next one! The irony is that no matter HOW fast you go, you're never closer to being done. And you are no closer to answering the question.

 

Yup. That.

 

Mathematics asks, "Is this a coincidence? Will this always work? Will the next sum always be a perfect square? How can I be certain? (And I'm not talking about a pile of examples where I give up and decide that it probably always works? I'm talking about how can I be sure!)"

 

Cranking out the next example is easy. Certainty is harder; but it's worth finding. And the journey is glorious!

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

 

Regarding Undergrad Linear Algebra: Keep in mind that a good undergrad LA course makes it clear to students that they are only seeing the tip of the iceberg. These waters are deep. And the more you know, the better. Ask Sheldon Axler. He thinks we are doing students a great disservice by letting them even HAVE the determinant as undergrads.

This is great information. My dd is spending a bit of time each day exploring different college majors to get an idea of what each is all about. Recently, she's been talking about double majoring in the classics and math. She's taking calculus in her sophomore year and we've started exploring opportunities for her for next year at the local cc or university. However, she has only been exposed to the plug and chug sort of math and she does enjoy getting the problems correct, lol. Right now, she is using Saxon and one career option she has looked at is being an actuary. Maybe that would be perfect for her, but I've been wondering about how to expose her to more problem solving, The obvious would be to use AOPS to see if she is interested in growing in this area, but would AOPS still serve that purpose if she has already finished calculus? Any other suggestions for exploring the creative side of math at this point?

[Janice in NJ]

Arcadia,

 

Yes, complex numbers are another fascinating puddle to play in. I'll never forget the first time I was introduced to the Riemann Sphere. How intensely beautiful!

 

I have much to learn within those waters and am very motivated to play there. 

 

Peace,

Janice

[Janice in NJ]

This is great information. My dd is spending a bit of time each day exploring different college majors to get an idea of what each is all about. Recently, she's been talking about double majoring in the classics and math. She's taking calculus in her sophomore year and we've started exploring opportunities for her for next year at the local cc or university. However, she has only been exposed to the plug and chug sort of math and she does enjoy getting the problems correct, lol. Right now, she is using Saxon and one career option she has looked at is being an actuary. Maybe that would be perfect for her, but I've been wondering about how to expose her to more problem solving, The obvious would be to use AOPS to see if she is interested in growing in this area, but would AOPS still serve that purpose if she has already finished calculus? Any other suggestions for exploring the creative side of math at this point?

 

Hi! I always recommend this one as a starting point. Not because it's easy but because it isn't. 

 

Courant's What is Mathematics?

https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

 

It's the kind of book that I will never give up. If forced to flee to a desert island with only a small pile, this one would make the cut for me. Not just because of what it says but how it says it.

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

[Arcadia]

Maybe that would be perfect for her, but I've been wondering about how to expose her to more problem solving, The obvious would be to use AOPS to see if she is interested in growing in this area, but would AOPS still serve that purpose if she has already finished calculus?

My oldest is enjoying the AoPS WOOT class. Your daughter can try the AMC10 in February 2018 since she is in 10th grade and the AMC12 until February 2020. The past year questions and solutions are on AoPS website so she can try those without taking the exams in February if she doesn’t want to.

 

Hi! I always recommend this one as a starting point. Not because it's easy but because it isn't. 

 

Courant's What is Mathematics?

https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

I read that at Barnes & Noble as my library didn’t have a copy.

[Janice in NJ]

I read that at Barnes & Noble as my library didn’t have a cop

 

Hi Arcadia,

 

I would love to hear what you thought about the book.

 

Thanks,

Janice

 

Edited December 9, 2017 by Janice in NJ

[Kassia]

Hi! I always recommend this one as a starting point. Not because it's easy but because it isn't. 

 

Courant's What is Mathematics?

https://www.amazon.com/Mathematics-Elementary-Approach-Ideas-Methods/dp/0195105192

 

It's the kind of book that I will never give up. If forced to flee to a desert island with only a small pile, this one would make the cut for me. Not just because of what it says but how it says it.

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

 

 

Thank you!  I ordered this for DH for Christmas.  He's an engineer who loves math and I'm sure he will be sharing it with one of our sons (also an engineer who loves math even more).  My dd is considering majoring in math and I hope she will look at it, too.

[Arcadia]

Hi Arcadia,

 

I would love to hear what you thought about the book.

 

Thanks,

Janice

  

I am rereading the book now from internet archive :)

 

This is perhaps my favorite of the Q/A's. It also relates to my prior discussions regarding changing majors and double majoring. It makes a lot of sense appealing to my practical side.

Derek.

 

This link would probably interest you on the practical side. Many double and triple majors with math and where they are working now.

 

“What Can One Do With a Major in Mathematics?

 

Recent Cornell graduates who majored in mathematics have embraced a variety of careers, professions, and possibilities for post-graduate study. Listed below are some of the postgraduate activities of Cornell mathematics majors during the last few years†https://www.math.cornell.edu/m/Undergraduate/Major/whatcando

[dereksurfs]

  

I am rereading the book now from internet archive :)

 

Derek.

 

This link would probably interest you on the practical side. Many double and triple majors with math and where they are working now.

 

“What Can One Do With a Major in Mathematics?

 

Recent Cornell graduates who majored in mathematics have embraced a variety of careers, professions, and possibilities for post-graduate study. Listed below are some of the postgraduate activities of Cornell mathematics majors during the last few years†https://www.math.cornell.edu/m/Undergraduate/Major/whatcando

 

The short answer is: a lot!  :D

[Arcadia]

The short answer is: a lot! :D

I was looking at what are the possible triple majors in that link

 

Other interesting news articles

“At the end of year academic year, Dean of Students Jerry Price, Ph.D., and his office celebrate remarkable achievements among the Chapman University undergraduate student body at the Campus Leadership Awards. This year senior Taylor Lee Patti, triple physics, math and Spanish major with a chemistry minor, won the prestigious Cecil F. Cheverton Award, the highest student honor at Chapman University.†https://blogs.chapman.edu/scst/2017/05/08/taylor-lee-patti-triple-physics-math-and-spanish-major-wins-prestigious-cheverton-award/

 

“Curious about his next step after graduating, Hopkins spent last summer interning at Google to see what an industry job would be like. He worked on a programming language design team alongside PhDs, which was a perfect fit. “Sam immediately grasped very subtle concepts,†recalls Vijay Menon, a Google staff software engineer who supervised Hopkins. “Since he left, the team has…[been] building upon his initial design and implementation.â€

 

Menon was equally impressed with Hopkins’ communication skills. “Sam could articulate his work and the overall effort better than most full-time Google engineers,†comments Menon. “We were confident enough in Sam to ask him to present intermediate work internally, to one of our vice presidents, and externally, to the broader engineering community in a webcast. His end-of-project write-up was also exceptional. He evaluated different options, made recommendations, and found hard data (even when we did not ask for it) to back up his conclusions. We have gone back to his write-up several times since he left to guide our further decisions.â€

...

“I do math for aesthetic reasons, and I want to share that with people,†says Hopkins. “A lot of K-12 math makes it such a grinding bore. I want to open people’s minds to see that math is beautiful—complex and interesting and challenging. I’d love to create an environment for elementary school students to see the same wonderment and discovery that I feel when doing a research project.â€

 

Hopkins also served as a teaching assistant for a UW computer science course and worked as a tutor in the Philosophy Writing Center for two years. “The Socratic approach is a big part of philosophy,†says Hopkins. “That means that, as a tutor, you have to be on your toes. You have to be as engaged with the material as the student is. When you can be, it’s really fun.†This fall, Hopkins heads for Cornell University as a graduate student in computer science. Asked whether he will choose a career in academia or industry, he just smiles. “Two years ago, if you asked me to predict what I’d be doing now, I’d give all the wrong answers,†he says. “It’s the same thing now. If I were to predict where I’ll be in five years, I’d probably be wrong again.†“ https://artsci.washington.edu/news/2013-07/triple-threat-math-philosophy-and-computing

Sam Hopkins webpage https://www.cs.cornell.edu/~samhop/

 

“On Saturday, May 11, Walter will graduate with a bachelor of science degree in mathematics, physics and economics, a triple major in the J. William Fulbright College of Arts and Sciences. He is accomplishing the feat at age 18.

 

What makes his accomplishment even more impressive is that he has a severe form of muscular dystrophy that forces him to use a motorized wheelchair.

 

“I really do just love learning,†said Walter, who was just 14 when he graduated from high school. “I like to learn as much as I can. I am willing to work and I want to work and learn. There’s an element of ability, for sure, but it wouldn’t mean much at all if I didn’t work as much as I do.â€

...

Raymond will continue his studies at the U of A in the Graduate School as a Distinguished Doctoral Fellow, which provides a minimum of $30,000 annually for up to four years. He also won a highly competitive $30,000 National Science Foundation Graduate Research Fellowship for the forthcoming academic year.

 

“Distinguished Doctoral Fellowships allow us to keep great minds in Arkansas, and Raymond is a perfect example,†said Todd Shields, dean of the Graduate School and International Education. “We are thrilled that he chose to continue his studies as a graduate student at the University of Arkansas.â€

 

Raymond Walter will pursue separate doctoral degrees in math and physics, with an expected graduation date in 2017.“ https://news.uark.edu/articles/21204/eighteen-year-old-finishes-triple-major-will-pursue-doctoral-degrees-in-math-and-physics

[catz]

Great thread - from a BS math/BS comp sci major.  :lol: Higher level math is much more dynamic and creative than most people percieve it.  I LOVE problem solving.  And the double major got me toe dipping into some grad level classes. 

[quark]

Then they hit Abstract Algebra and Real Analysis. That's where the road divides, and students are forced to peel off. Mathematics isn't really so much about being a human computer: find a formula and then work through 15 problems using that formula. We have software for that. The subject is really about proving something is true and then using that certainty to prove that something else must be true - and that can be mind-blowingly beautiful. Stunning. AND once you know it to be true, the irony is that you don't need to work an example. It's a moot point. It can't NOT be true. You don't even want to work an example. Why reduce perfection to a simpler form? You just sit there and wallow in the immensity of the always puddle. Feeling the warm wonderfulness of certainty in an uncertain world. 

 

[...]

 

Sometimes I have to stop reading. I'm lost. Time to back up and fill in the gaps. I slide the book back onto the shelf with the hope that I will make it back before this carcass gives up. It's like meeting someone you are eager to talk to, but then you realize that you aren't ready to listen to them. You can hear the words, but you can't find meaning. There's no resonance. It's all just sounds. You need something else first. So rather than fight the process, you shelve the person and their ideas. Time to grow into your own boots. And then you cast about for the steps between what you know and what you just bumped into. Process. Can takes weeks or years. Not for the faint of heart. Can't rush it. Can't move forward till you are ready. Oh well. Time to enjoy the journey cause the destination just faded into the distance. Time for baby steps. Humbling but beautiful. Facing ones own mortality as you realize that you are definitely going to run out of time before you have a chance to enjoy all the flavors available.  The difference between big ideas and small ones. And ones that are bigger than big. 

 

Sometimes my "reading" slows to a crawl because I make a connection I hadn't made before. You know it when it happens. Now there are two or more ideas in your head that were never part of the same thing. But of course they were all along. You just didn't know it. But now they so obviously fit together. Everything has changed and it's almost as if they have always been together. Except you know they weren't just a moment ago. But now they are. Everything else seems to have sharpened into focus just a tad more. It always was; you just didn't know it. But there it is - as clear as anything you've ever known. When that happens, I usually try to pause to savor the event. It's special. And very satisfying. (I'll admit it - sometimes I've cried. What a rush!) I can feel the spaces inside my own head. It's as if the ideas stored there have a location. A structure. They are connected. And now reconnected in a new way. Such blissful coalescence.  Because so much of my experience with that muscle is spent banging around in there trying to remember if we need toilet paper while standing in the supermarket, it is incredibly satisfying to be reminded that this organ is capable of SO much more. Even at my age. In fact, it's only because of age that there are so many fantastic treasures tucked away in this attic. Such bliss. Such peace. And it's all mine! I'm the only one who will ever be in here. Probably the only thing I can ever truly call my own. Such privacy. 

 

That happens very infrequently. Most of the time reading math is just work. I do it cause I'm waiting for the next "high." (It's THAT good!) However, to build all that? Takes discipline. Diligence. Hard work - you can feel the stress it puts on the muscle. Glorious! But not the same thrill you get from doing something repetitive like cranking out a page of examples. The hand just moves - with only casual mental effort. And it feels so productive - especially when it's easy for you and other people think it's hard. VERY rewarding. What an ego pump! But that's not REALLY what mathematics is about. Mathematicians are tasked with creating something new. Pure mathematicians prove new things. Applied mathematicians find new ways to use those new certainties. Neither spends her time cranking out examples and then checking to see if she got the right answer. There are NO answers in the back of the book. That thrill of being "right" that kids who are good at math in high school feel - the little endorphin rush that the smart kids get 20+ times every night doing their homework? That's over. A lot of math majors discover that they love arithmetic, not mathematics. They peel off. 

 

[...]

 

Arithmetic asks, "What about the next one? Is that a perfect square too?" Viola! Yes, 36. But maybe the next one will fail... nope still a square. Then +13, +15, +17... Endorphins. Speed. Rush. Faster. Dang, I'm good at calculating things in my head. I feel so smart right now. Next one! The irony is that no matter HOW fast you go, you're never closer to being done. And you are no closer to answering the question.

 

Yup. That.

 

Mathematics asks, "Is this a coincidence? Will this always work? Will the next sum always be a perfect square? How can I be certain? (And I'm not talking about a pile of examples where I give up and decide that it probably always works? I'm talking about how can I be sure!)"

 

Cranking out the next example is easy. Certainty is harder; but it's worth finding. And the journey is glorious!

 

Thank you Janice. Beautiful. Moving. Written only as a mathematician/musician who feels the love in their bones could.

 

My 2 cents is that I see many kids love math because they do well in math competitions in middle and high school and while that problem solving practice is fantastic, I want to see more kids do well in problem solving because they love math and for some kids that can only happen when the speed and competition factors are taken away.

 

When we took speed and competition away in our home school, it offered more time for studying and thinking about math, leading to:

• 6 years' practice in proof writing (from high school geometry onwards) prior to starting as an undergrad. Every other math class after geometry was proof-heavy.
• 3 years' worth of calculus.
• 2.5 years' worth of abstract algebra and group theory at high school to undergrad upper div level - quite an effective indicator to whether or not this kid has the stamina for higher math.
• A lot of faith that the subjects we left by the wayside to accommodate math would eventually be learned somewhere, somehow, enough to get kid a foot into a college for more math. This last factor is a huge luxury isn't it? So many kids just don't have that access or trust or reassurance that they can spend their time on something they love without worrying too much about the other subjects.

That some kids find math to be their calling while in college *after* being exposed to the really hard math? Genuine awe and wonder here and so very happy for them.

 

For my kid, double majoring in something else would take time away from immersing more deeply into math. The practical side of me would be happy if A took on another double major. The idealistic side wants that to be because A truly wants to and not because it's a good plan B.

 

PS: Those bold sentences are often repeated here. :001_wub:

[Janice in NJ]

 

• A lot of faith that the subjects we left by the wayside to accommodate math would eventually be learned somewhere, somehow, enough to get kid a foot into a college for more math. This last factor is a huge luxury isn't it? So many kids just don't have that access or trust or reassurance that they can spend their time on something they love without worrying too much about the other subjects.

 

 

Hi Quark!

 

This really resonated with me. Our youngest had a strong musical bent. While he was in high school, we allowed him to put that big rock - that very time consuming rock - into his jar FIRST before anything else was added. A luxury? Yes. At the time, I guessed that it was the right thing to do for this kid. We managed to fit the other important things in around the edges, but music always came first.

 

He's in his final year of undergrad, and he is doing well. We did the right thing. 

 

I see that your son is at UC Berkeley. He is SO lucky to have you as a parent. I know a number theorist who studied there back in the 80's, and he is a brilliant guy who received a fantastic education. What a world you have helped him find. Nicely done!!

 

Peace,

Janice

 

Enjoy your little people

Enjoy your journey

[quark]

Thank you for your kind words, Janice! You are very right. I call it a luxury but I think in many ways it was a necessity. What a time consuming rock it was!

 

Sometimes, truth and beauty take time. Such excruciatingly long time (for a type A mom who wishes output was less molasses and more warm honey). And that was true in many ways for my child.

 

Berkeley is a good choice (maybe our only choice for the present come to think of it) both logistically and financially as well. Very grateful to be living close by.

 

Our only regret? That kiddo entered Cal after Ken Ribet retired.

 

One more point therefore, to potential math majors: if at all possible, attend public lectures or math festivals organized by local universities. Talk to the professors or speakers there. The people-professor connections make such a huge difference. Kiddo's LA prof is such a character but kiddo so enjoys her as a person and is hoping to ask her for a rec to join an undergrad-grad mentoring program. Kiddo's abstract algebra professor was so spacey but is so respected that just mentioning kiddo took a class with this man to a faculty advisor gave kiddo an advantage in having a math major requirement waived. I understand now what posters used to say about the school-within-a-school culture in larger universities. Make those connections early if you can!

Edited December 10, 2017 by quark

[daijobu]

 

Linear Algebra is probably the first course in which students are universally asked to prove that certain statements about mathematical structures are true. Sure, proofs have been there all the time. Students see them. Teachers explain them. And some courses require students to dig into them more than others. However, before LA I think many successful students get away with tuning out that proof business and just waiting for the algorithm. "Just tell me what to do: What formula do I use? How does it work? Where are the practice problems? (Watch how fast and accurate I am!)" 

 

 

 

 

This reminds me of an AMC 8 class I was teaching a couple of days ago,  We were solving a problem that requires knowing the formula for the arc delimited by an inscribed angle in a circle.  Since many of them hadn't studied geometry yet, I was trying to remember how to derive the formula, but I couldn't remember the proof.  

 

My helpful students had a great suggestion:  "Use a protractor!"  Yes, that would work in a finite number of instances, but how does one prove it generally?  I'm glad to have had that moment with the students, not because it is important to know that the measure of an inscribed angle is half the arc, but because I think it was a start to teaching them that in mathematics, it isn't enough to measure it to be true, but to prove that it is always true.  That was fun.  (Also, my circles aren't very circular so even a protractor wouldn't do much good.) 

 

Now I need to go look up that proof.