Mathematical education by country

Question

Depending on the university, there are always slight differences in the syllabus and the structure of the standard material undergraduate students learn.

But I also noticed that undergraduate mathematics differs more greatly if one compares for example Germany (my home country) with the U.S. – I do not know this for sure, but I’ve heard rumours that in the U.S., it is common to start with some non-rigorous calculus (not introducing $ε$-$δ$-definitions for continuity, no formal definition of limits etc.) whereas here in Germany is seems to be common to start with more rigorous courses on linear algebra and analysis.

This just now got me interested in how the standard syllabus differs by country.

How does the mathematical education at university look like in your country? What is the standard syllabus/what are the standard courses offered in the first or first two years and in which order do (undergraduate) students undergo them/learn which material?

A quick google got me this discussion which covers only few countries, but has the sort of answer I’m interested in. Maybe we can get something similar going? Each answer a country?

(The question got migrated from MO.)

Answer 1

If this is of any interest here is the case of Romania -Bourbaki's heaven (or hell):

• A year-long course of Analysis: starting with the Peano axioms, construction of Integer, Rational and Real Numbers (as the completion of $\mathbb{Q}$) sequences and series, general topology, differentiation and integration (Riemann) of function of one variable
• A year-long course of Algebra: basic set theory, general group theory and concrete examples (groups of permutations and groups of matrices), rings, vector spaces and linear algebra up to the Jordan normal form (only over $\mathbb{C}$)
• A year-long course in Geometry: Linear Algebra (again) and euclidean spaces , Affine and Euclidean (Analytical and not Axiomatic) Geometry and a little of Projective Geometry (ideally start with the axioms and construct the field of coordinates)
• A semester-long course on Logic and Set Theory -some ZF axiomatics, calculus with predicates, deductive theories (whatever this means)
  • Three semester-long courses in computer programming and Graphs

Students are supposed to have a strong background in Analysis from high-school ($\epsilon-\delta$ definition of continuity, the definition of the derivative as a limit, the definition of the integral as the limit of Riemann sums).

Clarification: This is for the Pure Mathematics. For other majors requiring Mathematics (Physics, Engineering, etc.) the programme is a condensed version of the above plus some ODEs, function of a complex argument and Numerics.

Answer 2

I studied and teach in Finland, but got some exposure to the US system as a graduate student.

Our system is similar to what the OP described. The first year analysis uses $\varepsilon-\delta$. Much to the students dismay I may add.

My educated guesses as to the reasons for differences in the US/Finnish practices. I am prepared to be wrong about them, as I may hold mistaken beliefs about the US system.

• In Finland the kids have their 19th or 20th birthday the year they start college (depending on whether you do conscript duty fresh out of high school or not). In the US it's more like 17-18, so they are younger meaning that you can usually expect a bit less (whether that holds in practice is open to discussion). May be the same is true in Germany?
• In these parts the freshmen apply and are accepted to a math or physics or law or medicine programs right away, in the US you have to wait a year or so to pick a major (law and medical schools are typically for graduate level only), and the freshman year curriculum is comprehensive as opposed to specialized (IIRC freshmen need to take one course in math, one in English, one in social sciences, all depending on the University - at Notre Dame one selective course in theology was compulsory to ALL the undergraduates). Therefore the set of students taking freshman calculus in the US is more diverse. Here we only have math, physics and engineering students sitting in that class. And it is easier to justify that those students actually should learn their $\varepsilon$s. In the US you may have kids in pre-med and such, who will probably use some math tools later on, but don't need to necessarily understand the limitations of those tools.
• And also here, in spite of the above points, we still get some pressure from other departments to offer "tools only" analysis (i.e. calculus) courses, to chemistry, computer science and engineering majors in particular. The physics department is not happy with the speed of progress in math department courses either. Ideally they would like the freshmen to learn multivariable calculus, differential equations and Laplace/Fourier transforms before Xmas. Also the local sorry excuse of high school math no longer prepares students to the world of definitions and proofs, so we may need to convert to US style calculus soon as well.

Sorry, that last bullet turned into a rant - I need to blow off some esteem every now and then. Reminiscing the good ole days when middle school here had proof based Euclidean geometry as a compulsory topic.

Answering the question. Locally the freshmen take

• Analysis I - II: elementary functions, limits (1st sequences, then functions, then continuity, then derivatives) in the Fall, and integration and series in the Spring. Presumable a lot of this is review of high school (chain rule, improper integrals and power series are new), but now we bring in the epsilons.
• Linear algebra: ${\bf R}^n$, linear systems of equations, linear independence, determinants and matrices in the Fall, and
• First course in abstract algebra in the Spring: congruences, intro to elementary number theory, a very quick intro to groups, rings, fields, polynomials and abstract vector spaces. This is in two parts and many postpone the latter part to their second year.
• Ordinary differential equations (the basic stuff).

In the second year it becomes more diverse, except that multivariable calculus in some form is a nearly universal choice. Number theory and combinatorics are popular choices as are first courses in metric spaces and complex analysis.

Answer 3

Jyrki Lahtonen is correct that in the US students do not specialize into a major subject until (usually) the end of their second year at University. For that reason, in each of the first two years at University students typically take only two math courses -- one in each semester -- so by the time they begin their third year, they have only had four math classes. The default sequence of these classes is, at most institutions:

• 1st semester, year 1: Calculus I (differentiation, some integration; see below)
• 2nd semester, year 1: Calculus II (integration, sequences & series)
• 1st semester, year 2: Calculus III (multivariable calculus)
• 2nd semester, year 2: Differential equations

I think it is correct to say that most US institutions treat the notions of limit somewhat informally in this two-year sequence; the $\epsilon-\delta$ definition may be briefly presented but it is not emphasized, and overall the course is not proof-focused. After a student chooses a mathematics major, he or she will take an "Analysis" course that covers the content of Calc I-II again but with a focus on rigorous definitions and proof.

Although I referred to this above as the default sequence, there are some variations in this scheme that are so common as to be almost standardized:

• Many high school students -- and certainly the ones who are most likely to become Mathematics majors (but remember, that decision is not made until the end of the second year) -- will take a Calculus course in high school, and a large proportion of those students will take one of the two (optional) Advanced Placement exams in Calculus. Depending on the outcome of that exam, the University may grant them college credit for one or both of the first two semester courses. In that case, students may begin right away with the higher-level courses; on the other hand many students choose to forgo their AP credit and instead take Calc I again, reasoning that it is likely to be an "easy A".
• Some Universities offer in the first two years an alternative "Honors Calculus" track that gives more attention to rigorous definitions and proof. (My alma mater the University of Michigan has four different Honors Calculus sequences with varying degrees of rigor.) Depending on University policy and student enthusiasm, a student with AP credits for the first year might decide to forgo those credits and take an Honors Calculus sequence, not because it is going to be an "easy A" but because they want an intellectual challenge and to lay the foundation for more advanced coursework.
• The fourth semester Diff Eq course may be offered in several different versions, depending on whether the student is heading toward a major in pure mathematics or theoretical physics (in which case the course will be more theoretical and, typically, will include some introductory linear algebra material) or something in the applied sciences or engineering (in which numerical methods might get more emphasis).

After the student completes this two-year sequence of four prerequisite courses -- typically at the beginning of the third year, to coincide with declaring a major, but possibly sooner if the student uses AP credit to skip one or more courses -- then a student will begin with the "core courses" of the math major: Linear Algebra, Abstract Algebra, an advanced Geometry or Topology course, and so on.

Answer 4

The first two years in the mathematics program in my university in Brazil goes as follows:

• First semester: Calculus I, comprised of a review of real functions and inequalities (triangle inequality and $|x| - |y| \leq |x-y|$, limits of one variable functions, continuity (all with $\varepsilon-\delta$ arguments), definition of derivatives, computations with derivatives, important theorems (Weierstrass's theorem, mean value theorem, intermediate value theorem, rolle's theorem), chain rule, drawing a sketch of a function's graph using calculus, applications of derivatives, antiderivatives, definite integrals, all integration techniques (substitutions, partial fractions, integration by parts, trigonometric substitutions), improper integrals, applications of integration; Analytic geometry comprised of vectors in two and three dimensions, basis, lines in two and there dimensions, parametric representations of figures, conic sections, matrix operations, Gaussian elimination (I don't remember much of this one by head).

• Second semester: Multivariable calculus (Calculus II), comprised of functions of two variables, limits of functions of two variables, partial derivatives, differentiability, directional derivatives, gradient of a function, directional derivatives as dot product of the gradient of a function with a direction, applications of partial derivatives, optimization using the Hessian criterion and Lagrange multipliers, double and triple integrals in cartesian, cylindrical, spherical and general coordinates using the Jacobian, line integrals, conservative vector fields, fundamental theorem of line integrals, Green's, Gauss's and Stokes's theorem in $\mathbb{R}^2$ and $\mathbb{R}^3$; Linear algebra, comprised of vector spaces over $\mathbb{R}$ or $\mathbb{C}$, subspaces, bases and dimension, coordinates, computations, linear transformations, isomorphisms, transformations as matrices, linear functionals (sometimes), eigenvalues and eigenvectors, inner products, unitary and normal operators.

• Third semester: Calculus III, comprised of ordinary differential equations using integration factors, constant coefficients, Cauchy-Euler's equation, Riccati's equation, Bernoulli's equation, Euler's method, method of undetermined coefficients, variation of parameters, Wronskians, Laplace's transform definition, application to solving ODEs, Dirac's impulse function, unit step function, convolutions, sequences, numerical series, convergence of numerical series, Taylor series, solving ODEs using Taylor series, Frobenius's series method of solving ODEs, function series, Fourier series, solving the heat and wave equation using Fourier series.

• Fourth semester: Complex calculus, comprised of complex functions of one complex variable, definition of limit and continuity using $\varepsilon-\delta$, differentiability of complex functions, Cauchy-Riemann equations in cartesian and in polar equations, definition of harmonic functions, analyticity, the various Cauchy theorems of complex analysis, Jordan curves, rectifiable curves, complex sequences, complex series, Laurent series, computation of said series, complex integrals, computation of real integrals using residues, conformal mappings; Analysis I, comprised of the usual analysis topics: basic set theory, natural, integer, rational and real numbers, sequences and series, real line topology, limits of functions, continuity of functions, differentiability and important theorems cited above; and Advanced Linear Algebra (graduate course), comprised of a review most of the previous linear algebra course now in general fields, a more thorough investigation of eigenvalues and eigenvectors, dealing with invariant subspaces, primary decomposition theorem, cyclic decomposition theorem and other forms, special emphasis on Jordan's form, a reinvestigation of inner products, dealing with all types of operators, sometimes the min-max theorem, and tensor algebra, the universal diagram, symmetric tensors, exterior product, and the rest of tensor constructions.

  I apologize for the space, I tried to be thorough. There are other simultaneous courses non-math related I left out, like basic physics, experimental physics and the like.

Answer 5

Italy (the situation is not completely standard and may vary a little from one university to another):

Analisi I (calculus) Real functions of one variable + real line topology. Usually split in two: 1st semester $\varepsilon-\delta$ limits, derivatives, 2nd semester integration+ series.

Geometry I 1st semester: Linear algebra (vector spaces, matrices, linear systems, eigenvectors) Applications to affine geometry (time permitting classifcation of conics and quadrics). 2nd semester: Euclidean spaces (i.e. scalar products), orthonormal basis, dual space, norms, applications to Euclidean geometry (problems concerning metric issues between lines adn planes in 3d space)

Algebra Basics of abstract algebra: Sets, functions, polynomials, groups (+actions, symmetric group, finite cyclic groups), rings, UFD. May include elements of Logic.

Physics I Elementary mechanics, some thermodynamics

Probability Theory (1 semester - not everywhere) Basic probability theory

This is for math majors, of course.

Answer 6

Czech Republic

Bachelor Study: 1st year Mathematical analysis 1 (8 hours a week): Logic, Sets, Continuity, differentiation, Taylor series. Algebra 1 (6 hours a week): Groups, Fields, Vector spaces, Bases, Linear mappings, Matrices, equations, determinants. Foundations of programming I(2 hours) Mathematical analysis II.(8 hours a week): Newton-Leibniz integral. Riemann Integral, techniques of integration. Algebra II(6 hours a week): Matrix decompositions, linear forms, bilinear and quadratic forms. Multilinear algebra. Foundations of programming II(2 hours)

2nd year: Mathematical analysis III: Calculus on vector spaces, Gateux derivative, Frechet derivative, derivatives of higher order. Lagrange multipliers. Fourier series and transform. Geometry: Manifolds, basics of topology, connections, curvature, Riemannian manifolds, Lie Derivative Mathematical analysis IV: Measure theory, Lebesgue integral, Fubini theorem, manifolds, differential forms, chains, Stokes theorem. Numerical mathematics: Basic interpolation, aproximations, SVD...

3rd year: Topology: Definition, open and closed sets, factor spaces, Tychonoff theorem, Nets, Filters. Ordinary differential Equations: clasification, methods of solution, stability os solutions, manifold of solutions. Functional analysis 1: Topological vector spaces, Hahn-Banach theorem and other great theorems, functionals, operators. Functional analysis 2: F-spaces, Frechet spaces, convex analysis, Hilbert spaces. Diff. Geometry 1: Manifolds, connections, Covariant derivative, geodesics. Diff. Geometry 2: Lie groups and algebras Partial differential equations 1: Classification of PDE, classical theory.

Graduate: Partial differential equations II: Distributions, LAx - Milgram lemma, Sobolev spaces, FEM. Algebraic topology I: Category theory, fundamental groups, simplices Algebraic topology II: Homology and cohomology

And much more

Answer 7

Here in South Africa, I'm not sure how other universities work, but I'm in the second year of a maths degree at UCT.

It's worth noting that in South Africa we do 12 years of school and students entering university are usually 18. The level of mathematics education in our high schools is pretty shocking, with nothing further than power rule being covered, unless you do AP maths and then you do some integration, although I'm not sure of the details since I did the CIE system. Also, a bachelors degree is 3 years and a fourth "honours" year has to be completed before moving on to your masters.

In the first year of university we take a whole year course, which starts off with a review of high school maths and then moves on to limits, treated informally, and then moving on to calculus. By the end of the first semester, integration by parts has been covered; I'm not sure where this falls in relation to Calc I or II because we don't have such notions. Some methods of proof have also been covered (induction and contradiction). In second semester we do some applications of integration (applications of derivatives are in the first semester) and then move on to complex numbers, basic linear algebra, and basically how functions of multiple variables work in preparation for multivariate calculus. Also solving some basic DEs.

Maths majors are also expected to take a "fundamentals of mathematics" course at some point in their degree. It's a first year course but many, such as myself, choose to continue to take it in second year, or even third. In first semester the course covers set theory and logic, in second semester it covers group theory and basic analysis along with some other miscellaneous topics (complex numbers, combinatorics) in preparation for second year maths.

In second year first semester the course splits up into modules. Each module consists of 30 lectures which span 45 minutes each.

Semester 1:

• Linear Algebra
• Multivariate Calculus (up to divergence theorem)

Semester 2:

• "Introductory Algebra"
• Real Analysis

Students can also take a module on differential equations, but for maths majors that's not recommended as it would mean playing catch-up in third year.

Year 3:

Semester 1: Students pick two of the following modules:

• Metric Spaces
• Logic & Computation
• Algebra

Semester 2: Students pick two of:

• Complex Analysis
• "Topics in Algebra"
• "Topics in Analysis"

There is also an optional research project which can be taken in third year.

That's the pure maths track. We also have applied maths modules which can be taken concurrently and cover nonlinear dynamics, ODEs, numerical analysis, boundary value problems (second year), and general relativity, fluid dynamics, methods of mathematical physics, "advanced numerical methods", and methods of functions of complex variables (third year).

In the honours programme (fourth year) there's a broad range of modules to choose from and a research project must be completed in that area before the end of the year.