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I think for Physics major, and especially Biology major it might be better to live the current classes.

Assuming you mean "take the current classes", what majors should rethink mathematics then?  You say "chemestry" but let me point out a result of that: all the courses need to be very specifically ordered to provide the mathemational foundation for the next classes.
You can't swap courses around that share mathmatical foundations, or else you will miss the foundation.  Like, if one class needs integrals for radioactive deccay, and another class needs integrals for energy released over time in a chemical reaction, the two courses become reliant on each other, even though they don't cover any of the same chemistry materials, and could be taken in any order otherwise.
Your post is much too nebulous.
I can't tell what you're arguing to change or how you want to change it.
Give us some concrete statements that can be discussed. >

You are right. I edit it now >


I think to make better advances it is much better to combine the ideas at both levels and make it a one subject that combines the two, while also keeping a lot more students interested and curious to learn more.

Well, unless those two courses only contain half a courses worth of content you will either have to cut out certain parts of the curriculum or you keep it all, at which point you will need to stretch the course over two semesters. Ideally if you do that you should separate them in a sensible manner, like for example into one theoretical and one applied course. >

I actually study chemistry in Germany, so maybe my input is relevant here.
Maths, as interesting as it can be and as useful as it is, is essentially an annoying side subject for most of us. We take it because we need the basic methods to apply them in theoretical chemistry.
What fascinates me is chemistry, and I have a real passion for it and learning all about it. Maths can be interesting, but I am really not interested in the deeper, logical background.
Many other studies require some basic chemistry, but as exciting as it is for me, I recognize that they don't necessarily need it. >

I agree with you! And it usually is that biology students even like mathematics less than chemistry students, they can actually abhor it.
I actually found chemistry super interesting, and also thought that the mathematics of differential equations are really revealing in physical chemistry. I know also in organic chemistry there are some more quantitative sides - if you want to actually analyse the chance of a reaction to happen. >

I think, for the majority, you’re just going to annoy them because either they ONLY want to do applied or they ONLY want to do theoretical. 
You’re pleasing a minority by mixing the two and I don’t really see why bother. >

I am one person that really appreciate the theoretical side, but really appreciates the idea that I can make something in the world. >

I hate learning the theoretical side. I’ve never been good at it in my life. But it always made sense to be when applied to economics or programming, which I’m very proficient in. 
I just can’t wrap my head around the math and doubt myself when there’s no sniff test for whether the answer is reasonable or not, at least not one that can be applied without math.
Like, I know if I’m doing triangular arbitrage with gold trade, I definitely did something wrong if I made a 2000% profit. >

Wow you seem to know it around Economics! I have no idea what triangular arbitrage is. what kind of mathematical tools are used for this? >

here’s a paper on arbitrage.
Another example of where I use math but for some reason it makes more sense to me is when I’m running loops in R (forgive my sins, I know it’s bad) and due to memory writing to RAM, in large sets for loops I need to create batches of data sets to store. To preallocate the data set so I can use “updates” rather than “adding rows” I need to use math.
This all makes sense to me, but present it as a math problem and I won’t know lol.
Even as a kid, my folks taught me multiplication using coins in pockets. >

I read like 8 pages of the paper, never knew it! the quality of the writing is awful, but the subject interesting. 
I never knew that someone can do such hacking as triangular arbitrage! every time I go abroad and switch currencies I just lose several percentages of money for every transaction. >

You can’t really do it risk free these days because the process is automated by banks so they’re just constantly doing arbitrage which evens out the prices. 
Fun fact, this is actually how bankman-fried actually made his starter money. He was doing arbitrage with crypto between the US and China. >

You'll have to tell me more about how is that possible, it sounds crazy. Or maybe I'll read more sometime!
where are you from? >

I don’t know all the details but I do know he use his effective altruist network to help shuffle the crypto around.
I am from the U.S. >

Was very interesting read about the difference between the observer effect and uncertainty principle! That is actually why I think for example that engineers that deal with quantum system definitely should work harder to wrestle with the mathematics of quantum mechanics! >

Can you give us an example of how a middle-of-the-road maths curriculum might go? I have a physics background, so I understand where your coming from at least in terms of how math is often presented in a theoretical sense or an applied sense.
It seems to me that pure math is done in the theoretical, and applied maths is just pure math in a setting with real life outcomes. You can flip flop back and forth between them in the same class, but I don’t see how you actually directly combine them. >

I edited my post :-) >

The problem here is that most university math classes are obligated to teach fundamental concepts like Calculus I-III, linear algebra, continuous probability etc. under difficult constraints that make it impossible to teach them in an intuitively “beautiful” way. However, most programs in “applied math” fields like physics, economics, electrical engineering etc. have dedicated classes that teach math concepts to students of those disciplines in a contextually relevant way. For example, my school combines linear algebra and fundamental computer engineering into one class - www.eecs16a.org and www.eecs16b.org.
I’m sure you’re familiar with 3Blue1Brown - I think he teaches math in a rather intuitive and applicable way from the get-go. However, I don’t think that it’s possible to take that approach for every topic while also building a rigorous foundation for calculation and proof-writing. It’s already pretty hard to fit something like Calc II into a 14-week course. 
But I’m not convinced that I understand exactly what you’re proposing - could you give an example of how just a single class (maybe something like linear algebra) would be taught under a curriculum like yours? >

Maybe I should come to Berkeley! It looks great, I'll sure check it out. Explanation of Tomography in first Linear algebra exposure, brilliant idea.
Let's take analysis, as I am more familiar with it than linear algebra. Calculus usually gives all the methods, going very fast over all of them. There are ideas, like I heard explanation in class why the jacobian between euclidean and spherical is what it is, but it is by no means rigorous. Than analysis makes idea more precise:

continuous at D if for every point in d for every e>0 exists delta>0 abs(f(x)-f(a)) ball around f(a) close).

etc. So start with this but instead of dwelling on that, show this idea of continuity and move to where exactly a function fails in non-continuous case (maybe not from physics, but in engineering a model of piecewise continuous is the model of some acceleration system and control, where you hit the brake then stop, then hit acceleration - give this is an example of non-continuous system that occurs but is still piecewise. what are the problems in the boundary as far as system analysis? good point in a hybrid class).
It made become somewhat dense, but I think it will make many students appreciate the ideas behind the math. (Minimization and maximum problems in engineering and Lagrangian mechanics are also a great way to present the idea of maxima/minima in a concrete way that you can relate to).
For linear algebra what you showed me looks perfect, and also what I saw in a course I've done in Coursera - Mathematics for machine learning - linear algebra from imperial college London, where the assignments are to flip an image and to try to recreate Google's page rank algorithm are great example. If you intertwine it with some proofs of dimension and determinant properties, you can actually combine the theoretical with the practical. >

One clarifying question: Do you consider mathematics as a tool for solving real life problems or is it a field of study that tries to logically analyze mathematical objects like numbers, functions, vectors and the relationships between them? >

That is my main point - it is both! It is equally both in my opinion, and that's the greatest charm of it. >

High level math is going to be boring and hard, these teachers that teach at a high level are not cheerleaders for you, they are not going to organize a scavenger hunt or send teams out into the world solving equations in an ARG mystery, to keep the fun levels up. >

It can be changed! also the curicculum is designed in a way to push a lot of material, so no time left for this. The change should also be in the adminstrative level. And also trying to assign courses to professors that enjoy teaching, not ones that just want to get along with teachings and invest everything in their research. >

The responsibility is on the student, it is getting you ready for the real world.
Your boss is not going to plan fun activities to keep you happy, he is going to assign you work and go do his work. >

Ah yes, the real world, most frequently invoked by people who live in a bubble.
Nothing completely prepares you for actually entering the professional world. But also in a job you have many liberties that you do not as a student. Also life doesn't begin when you get a job, having fun learning might actually make people passionate about their work.
I wore cat ears to a meeting in December. People either thought it was strange or they loved it. It was very fun. Work is only a drag when people are in the business of policing each other's joy. >

I agree but putting the responsibility on others to provide that joy and passion is what I have a problem with. >

I would be happy to lead people who asked that of me, who gave me the opportunity to engage them on a deep level.
Someone teaching a classroom or managing a team should be expected to support their juniors. That's an example we set for the people who take our place >

I hope you realize that is not doable in most situation and to expect that will only lead to disappointment. As I understand it some professors are not even there to teach but they have to do it as it is the only way to have proper funding for their research. Some classrooms can be huge. Some teams can be huge. Time is money, deadlines need met. Sorry I did not meet the deadline, or my students are way behind on the curriculum but we are having a lot of fun and sparking all kinds of interest. >

It's an ideal to work toward, not an expectation. As much as you seem to enjoy condescension, having opinions about better ways to do things comes from having been in some horrific and unforgiving workplaces. We're not discussing what is, we're discussing what can be.
If you want to offer an argument as to why this is unattainable that would be fun to engage with but a response with terse tautologies like "time is money" really is quite irritating because it conveys that you can't even be bothered to articulate a point of view, presumably because you consider yours to be self-evident.
Also strawmen, especially the last sentence are really tiring, because you seem to be more interested in telling me what I propose so you can argue against yourself. If you'd like to explain why what I've talked about necessitates the dynamic you bring up, that's an interesting argument. >

Am I reading this correctly: you are arguing about teaching math in context of it's uses, rather than as abstract concepts and then told how to apply it?  Am I understanding you correctly? >

I am arguing to motivate math by its uses, then go over its theoretical proofs and contexts completely intertwined with methods for solving the problem, including numerical concepts and sketches of algorithms. I want a holistic math degree. >

So, right now you can say "this course requires Calc II and linear algebra".  And that can be the case for both "Physics" and "biology".  In your proposed system, how would you handle the fact that the same math is needed in the "physics" and "biology" course?  Will they both cover the same math information, or will the courses be required to be taken in a very specific order? >

I think for Physics major, and especially Biology major it might be better to live the current classes. Though I might think that biology major can gain from more mathematical skills, those that are interested in computational sides usually go for bioinformatics, with computer science or math minor or double major.