Lev Landau

Russian Theoretician and Teacher.
I believe that the teaching of mathematics is ripe for a most serious reform. Those who undertake this most important and difficult task will deserve the sincere gratitude of the already trained physicists, but in particular of the numerous members of future generations. - Lev Landau

My son was practicing the Moonlight Sonata the other day.That was a pleasant change of pace from the video games. However, like any good parent, I needed to check that actual homework wasn't being sacrificed ... So, how was the homework going?
It turns out that for "science" class, the students had to build, and launch, a water rocket. Standard fare for a school that is in the Kennedy Space Center penumbra. Among other requirements, the students had to measure the height the rocket reached.
The problem with these types of activities is that at this level, it can be quite challenging to teach serious science. Why? The formulation of physical law of necessity must ultimately be done mathematically. But the fifth graders are only studying fractions and decimals in their math class. It would require the rocket project instructor to devise a special approach tailored to that math level. Such an approach would be time consuming both for the kids and the instructor.
At a minimum, I personally would not attempt to teach a rocket problem without at least a mathematical formulation of Newton's laws in one dimension. But it could be done. The great Richard Feynman was able to teach his theory of Quantum Electrodynamics, a subject normally reserved for graduate students in theoretical physics, with almost no math, by the device of just drawing arrows on a piece of paper. He could do that because he invented the theory, and because he was, well, Feynman.
So Feynman could do it, but for the rest of us, pedagogically, it is a difficult thing to teach science with limited math. The language of science is indeed mathematics. How then does one teach science when the student struggles to comprehend the language it is written in?
The great Russian theoretical physicist Lev Landau had an essential insight on the matter. His students were tested on all branches of theoretical physics in his school, but the very first test, which Landau always administered himself, was on mathematics. Landau insisted his students learn the necessary mathematical techniques before embarking on his rigorous course in theoretical physics. Experience had taught Landau that once a student became engrossed in a research problem, they often lacked the inclination to study any new mathematics.
Anna Livanova, in "Landau: A Great Physicist and Teacher", gives examples of Landau's admonition to study math first:
First was always the test in mathematics. Landau constantly emphasized and expounded the necessity of this sequence in taking the theoretical minimum. In almost every one of the reply letters already quoted, we find the same words and the same arguments. "Bear in mind that a mastery of mathematics is especially important." "You must first master properly the techniques of theoretical physics. . . . Mathematical techniques are the foundations of our science." "You must start with mathematics which, you know, is the foundation of our science. The scope is indicated in the programme. Bear in mind that by 'knowledge of mathematics' we mean not just all kinds of theorems, but a practical ability to integrate and to solve in quadratures ordinary differential equations, etc." "As you have yourself understood, a theoretician must above all know his mathematics. What is needed is not all kinds of existence theorems, on which mathematicians lavish so much praise [Landau called this 'mathematical lyrics' — A . L . ] , but mathematical techniques, that is, the ability to solve concrete mathematical problems."
I hope it is clear. If you wish to understand nature, or if you wish to enable your students to understand nature, you have no choice but to learn mathematics. The solution is always more math. Never less. Nature is a harsh mistress. She insists you speak her language before she shares her secrets.
A Method of Math Instruction for Young Students
Taking our cue from Landau and our observation that Nature is undecipherable without math, we can assert some principles of math instruction:

  • Math First : As much as is possible, the math education must come first. More is better.
  • Diversions into physics (or other) topics to illustrate the power and relevance of the mathematical technique is allowed but must be brief lest the student's math progress comes to a halt.
  • There is a Pareto like effect in play. Roughly 80% of the benefit comes from studying math. In other words, a new math technique may be applied to many different science subjects, but a new science principle will often not be applicable outside its field of origin.
  • Emphasis is on mathematical technique, not on mathematical nuance and generality. There may come a time for Bourbaki, but this is not it.
  • Deliberate practice must be used. (cf. Peak: Secrets from the New Science of Expertise)
  • Because math skill is proportional to the amount of time invested in studying math using deliberate practice, the parent/teacher must find ways to keep the student motivated.
Public vs Homeschool
In the United States, homeschool is often the better choice for mathematics. A motivated student can make faster progress at home. A public school is unlikely to understand the "math first" principle. Further, the public school curriculum will have been created by bureaucrats, not mathematicians, whose motivations can be different and at odds with those of the parents.
A small caveat: If the student's parents are not university educated, it is likely tutors will eventually be needed. This is not to say that the time-honored method of going to the library, checking out a math book, and burning the midnight oil will not work. It is just that learning can be more structured and directed if an expert is teaching.
Which Math Should Be Studied
If you wish to study pure math and have no interest in applications, then this question is one of personal mathematical taste. If you find the Separable Galois Theory of Commutative Rings interesting, for example, then have at it. But you will make no progress in understanding science and nature. However, if you wish to understand nature, then the choice of topics is not arbitrary, and has been laid out by the Landau School.
From the Landau Institute for Theoretical Physics website, we can ascertain the minimum mathematics required:
Math 1: Freely take indefinite integrals and solve ordinary differential equations. Be proficient in vector algebra and tensor analysis.
Math 2: Theory of functions of a complex variable, calculation of integrals by residues, solution of equations using contour integrals (Laplace's method), calculation of asymptotics of integrals, special functions (Legendre, Bessel, elliptic, hypergeometric, gamma functions).
If those topics are unfamiliar, they would be the math one might study after finishing calculus, which implies mastery of calculus is a must.
In addition to those topics, you must study Linear Algebra. This is essential, for example, in the Dirac formulation of Quantum Mechanics, and also for eigenvalue problems. You must also study group theory and the representations of groups, which will be used in Quantum Mechanics. It is a mistake to wait until you are enrolled in Quantum Mechanics to learn the math. It can and should be learned earlier. It will make the Quantum Mechanics much more palatable. You must learn probability and statistics, which will be valuable in Thermal Physics and Statistical Mechanics.
Someone once said "clarity is power". So let me be clear on what a young student should be aiming at: Mastery of calculus. Mastery of calculus is the foundational mathematical skill.
Incidentally, you may wonder how my fifth grader handled the rocket height question. After all, he was subjected to the same stultifying ninety minutes of fractions and decimals, 5 days a week as the other fifth graders. Why, he wrote down a differential equation and solved it, of course.