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Precalculus and calculus against the Polish and German school curricula

Level: reference · for anyone who learned mathematics in a Polish liceum or a German Gymnasium and meets American course names, or the other way round

One line: "Precalculus" and "calculus" are American course names, not subjects; the same topics exist in the Polish podstawa programowa and the German Bildungsstandards, but they are cut into different years, the Polish extended matura stops at the derivative, the German Abitur reaches the integral, and only the American course teaches conics, polar coordinates, complex numbers and series at school.

This page is a map, not a lesson: three tables of contents laid side by side, with a link wherever this library has a page on the topic. Its one program treats the three curricula as sets and compares them with set operations, so every claim of the kind "X teaches this and Y does not" is a line of output and can be re-run when a row is corrected. Two short companion pages say the same in Polish, Po polsku, and in German, Auf Deutsch, for a reader who knows those names and not the American ones.

How sure. The Polish curriculum (the 2018 podstawa programowa for the four-year liceum, levels podstawowy and rozszerzony) and the German one (the KMK Bildungsstandards im Fach Mathematik für die Allgemeine Hochschulreife, 2012, with the sixteen Länder's own plans on top) are described from memory: both sites are blocked from this session's network. The American side follows the precalculus reading guide and the AP Calculus AB and BC course descriptions, also from memory. The year a topic first appears is the least certain part, because German Länder differ and Polish textbooks order the liceum years freely; the table at the end says how sure each kind of claim is.

Contents

The three systems

United States Poland Germany
The last school years grades 11 and 12 (ages 16 to 18) liceum ogólnokształcące, four years, classes 1 to 4 (ages 15 to 19), after eight years of szkoła podstawowa Gymnasiale Oberstufe: Einführungsphase then Qualifikationsphase Q1 and Q2 (ages 15 to 19, class 10 or 11 to 12 or 13, depending on G8 or G9), after Sekundarstufe I
The course names Algebra 1, Geometry, Algebra 2, Precalculus, Calculus (AP Calculus AB or BC): one subject per year matematyka, every year, at one of two levels: zakres podstawowy (compulsory) or zakres rozszerzony (extended) Mathematik, every year; in the Oberstufe at grundlegendes Anforderungsniveau (Grundkurs) or erhöhtes Anforderungsniveau (Leistungskurs)
The leaving exam none national; the AP exam is optional and gives university credit matura: podstawowy compulsory for everyone, rozszerzony optional and needed for most mathematics, physics and engineering degrees Abitur: mathematics is compulsory to the end, written at the level of the course taken
What fixes the content state standards (most follow the Common Core) and the College Board for AP one national document, the podstawa programowa (regulation of 2018) one national frame, the KMK Bildungsstandards (2012), and sixteen Länder curricula that implement it
The organising idea one topic per year: precalculus is "the functions calculus needs" thirteen numbered content areas, from liczby rzeczywiste to optymalizacja i rachunek różniczkowy, each at two levels three Sachgebiete (Analysis, Lineare Algebra / Analytische Geometrie, Stochastik) crossed with five Leitideen (Algorithmus und Zahl, Messen, Raum und Form, Funktionaler Zusammenhang, Daten und Zufall)

The first thing the table says is that precalculus has no counterpart. No Polish or German student takes a course by that name, because neither system has a year in which "the functions calculus needs" are gathered; they are taught as they come, mostly in the years before the final two, and calculus (Polish rachunek różniczkowy, German Analysis) starts in the Oberstufe or in the extended liceum course without a separate preparatory year. The second thing is that calculus is compulsory for a German school leaver and optional for a Polish and an American one.

The map

One row per topic. The American column follows the precalculus reading guide's table; the Polish column names the content area (I to XIII) of the podstawa programowa and the level, P for podstawowy and R for rozszerzony; the German column names the stage, Sek I for classes 5 to 10, GK for Grundkurs and LK for Leistungskurs in the Oberstufe. "Earlier" means the topic is taught before the stage the row is about. The last column links the page here, if any.

The algebra under everything

Topic United States Poland Germany Here
Exponent laws, roots, rational exponents Algebra 2, reviewed in Precalculus I Liczby rzeczywiste, P Sek I (Klasse 9–10, Potenzen und Wurzeln) a definition is a test proves the laws
Quadratic equations, the quadratic formula Algebra 1 and 2 III Równania i nierówności, P Sek I (Klasse 9, quadratische Gleichungen) catastrophic cancellation, circles for completing the square
Polynomials: zeros, factoring, division Precalculus II–III, R (wielomiany, twierdzenie Bézouta) GK, inside Analysis (ganzrationale Funktionen) not here
Rational functions, asymptotes Precalculus V Funkcje, R (funkcja wymierna, mostly homographic) LK (gebrochenrationale Funktionen, not in every Land) not here
Absolute value, intervals, inequalities Algebra 2 I and III, P Sek I rectangular coordinates on intervals; the glossary entry on absolute value
Systems of linear equations Algebra 1, matrices in Precalculus IV Układy równań, P (two unknowns) Sek I; Gauß elimination in the Oberstufe (lineare Gleichungssysteme) linear equations and their solutions
Matrices and determinants Precalculus not in the podstawa LK in some Länder (Matrizen, Übergangsmatrizen) the linear algebra reading guide

Functions

Topic United States Poland Germany Here
Function: domain, range, graph, composition Algebra 2, Precalculus V Funkcje, P (composition only in R) Sek I (Funktionsbegriff), composition in the Oberstufe relations and functions, function katas
Shifting and stretching a graph Precalculus V, P (przesunięcie wykresu), reflections in R Sek I (Klasse 9–10, Parameter) not here
Inverse functions Precalculus V, R (funkcja odwrotna appears with logarithms) GK (Umkehrfunktion, with ln and eˣ) multiplication can be undone is the idea for one operation
Exponential functions Algebra 2, Precalculus V, P Sek I (Klasse 10, exponentielles Wachstum) velocity equals position, the doubling-time example
Logarithms, logarithmic functions Algebra 2, Precalculus I, P (definition and laws); V, R (the function) Sek I introduces log for solving equations; GK (ln, Logarithmusfunktion) maps that keep the laws
Sequences and series, Σ notation Precalculus VI Ciągi, P (arithmetic and geometric) Sek I (Folgen, lightly; many Länder skip Σ) power series for what series become
Limit of a sequence, sum of a geometric series Precalculus (informally) VI, R (granica ciągu, szereg geometryczny) GK (Grenzwert, as the entry to the derivative) convergence, with ε and N
Mathematical induction Precalculus (some books) II, R (indukcja matematyczna) not in the Bildungsstandards; LK in a few Länder induction
Binomial theorem Precalculus II, R (dwumian Newtona) GK, inside Stochastik (Binomialkoeffizient) not here

Trigonometry

Topic United States Poland Germany Here
Right-triangle trigonometry Geometry VII Trygonometria, P Sek I (Klasse 9–10) congruent and similar triangles, the ratios' reason
Unit circle, radians, trigonometric functions of any angle Precalculus VII, R (miara łukowa, wzory redukcyjne) GK (Sinusfunktion, Bogenmaß) radians
Identities and trigonometric equations Precalculus VII, R LK, lightly not here
Law of sines, law of cosines Precalculus VIII Planimetria, P (R in older editions) Sek I (Klasse 10, Sinus- und Kosinussatz) the glossary entry on the law of cosines

Geometry

Topic United States Poland Germany Here
Pythagoras, similar triangles, area and volume Geometry, reviewed in Precalculus szkoła podstawowa, classes 7–8 Sek I (Klasse 8–9) all of 10_Geometry
Plane geometry with proof: triangles, circles, similarity Geometry VIII Planimetria, P and R (twierdzenie o kątach w okręgu, czworokąty wpisane) Sek I the Pythagorean theorem and its converse, what a proof is
Solid geometry: prisms, pyramids, spheres, volumes Geometry X Stereometria, P (angles between lines and planes in R) Sek I area and volume formulas
Lines, slope, distance, midpoint in coordinates Algebra 1, Precalculus IX Geometria analityczna, P Sek I (lineare Funktionen) 08_Analytic_Geometry, lessons 1 to 5
The circle in coordinates Precalculus IX, P (equation of a circle), R (circle and line) GK, as a vector equation in some Länder circles
Vectors in the plane Precalculus IX, P (coordinates, sum, scalar multiple); R (dot product in some editions) GK, as the start of Analytische Geometrie distance in n dimensions, the length of a vector
Vectors in space, lines and planes, dot product not at school (multivariable calculus) not in the podstawa GK and LK, the whole second Sachgebiet (Geraden und Ebenen im Raum, Skalarprodukt, Abstände) distance in n dimensions is the one piece
Conic sections Precalculus not in the podstawa (parabola only as a graph) not in the Bildungsstandards not here
Parametric equations, polar coordinates Precalculus not in the podstawa LK, parametric curves in a few Länder multiplication rotates is polar form for one kind of number
Complex numbers, polar form, de Moivre Precalculus not in the podstawa (removed from the rozszerzony in the 2008 reform) not in the Bildungsstandards; an elective in some Länder all of 03_Complex_Numbers

Combinatorics, probability, statistics

Topic United States Poland Germany Here
Permutations and combinations Precalculus (one chapter) XI Kombinatoryka, P (reguła mnożenia), R (wariacje, kombinacje) GK, inside Stochastik counting partitions is the one counting page
Probability: events, conditional probability Precalculus, or a statistics course XII, P (classical probability), R (prawdopodobieństwo warunkowe, twierdzenie o prawdopodobieństwie całkowitym) Sek I, then GK (bedingte Wahrscheinlichkeit, Baumdiagramm) probability zero for what the measure is
Binomial distribution, expected value AP Statistics, not calculus XII, R (schemat Bernoulliego) GK (Binomialverteilung, Erwartungswert) not here
Hypothesis tests, normal distribution AP Statistics not in the podstawa GK (Hypothesentest), LK (Normalverteilung) not here
Descriptive statistics: mean, median, deviation middle school and Algebra 1 szkoła podstawowa, then XII, P (odchylenie standardowe) Sek I mean, average, arithmetic mean, a share above a cutoff

Calculus

Topic United States Poland Germany Here
Limit of a function, continuity Calculus AB XIII Optymalizacja i rachunek różniczkowy, R GK (Grenzwert, mostly intuitive) continuity by ε and δ
The derivative and its rules Calculus AB XIII, R (polynomials and rational functions only) GK (Ableitung, Ableitungsregeln: Potenz-, Summen-, Faktor-, Produkt-, Kettenregel; Quotientenregel in LK) the derivative is a velocity
Chain rule Calculus AB XIII, R GK related rates uses it
Curve sketching, extrema, optimisation Calculus AB XIII, R (optymalizacja: the heart of the Polish section) GK (Kurvendiskussion, Extremwertprobleme) not here
Related rates Calculus AB not in the podstawa not in the Bildungsstandards related rates
Derivatives of eˣ, ln x, sin x, cos x Calculus AB not in the podstawa GK (e-Funktion), LK (trigonometrische Funktionen) velocity equals position, radians
Antiderivative, definite integral, fundamental theorem Calculus AB not in the podstawa GK (Stammfunktion, Integral, Hauptsatz) not here
Area between curves, volumes of revolution Calculus AB not in the podstawa GK (Flächen), LK (Rotationskörper) not here
Integration by substitution, by parts, partial fractions AB (substitution), BC (the rest) not in the podstawa LK (Substitution, partielle Integration) not here
Differential equations: separable, growth and decay Calculus AB not in the podstawa LK (Wachstumsmodelle, Differentialgleichungen in some Länder) velocity equals position is y′ = y
Taylor and power series Calculus BC not in the podstawa not in the Bildungsstandards power series
Parametric, polar and vector-valued functions Calculus BC not in the podstawa not in the Bildungsstandards not here

What the program shows

The program holds the map above as a dictionary, topic to the courses that first teach it, closes each system upward (what the podstawowy teaches, the rozszerzony also knows), and then asks set questions.

Verified output of curriculum_sets.py — regenerated by tools/run_examples.py, never hand-typed.

1. THE COURSES AS SETS
   49 topics in the table. How many each school-leaving course has met:
      US-PRE   30   US precalculus
      US-AB    40   US precalculus + AP Calculus AB
      US-BC    43   US precalculus + AP Calculus BC
      PL-P     20   Polish matura, zakres podstawowy
      PL-R     33   Polish matura, zakres rozszerzony
      DE-G     35   German Abitur, Grundkurs
      DE-E     42   German Abitur, Leistungskurs

2. THE COMMON CORE: taught to every school leaver on the stronger track
   US-BC ∩ PL-R ∩ DE-E (28)
      - Pythagorean theorem, similar triangles, area and volume formulas
      - absolute value and intervals
      - binomial theorem
      - chain rule
      - combinatorics: permutations and combinations
      - curve sketching, extrema, optimisation
      - exponent laws, roots, rational exponents
      - exponential functions
      - functions: domain, range, graph, composition
      - inverse functions
      - law of sines and law of cosines
      - limit of a function, continuity
      - limit of a sequence, sum of a geometric series
      - lines, slope, distance and midpoint in coordinates
      - logarithms and logarithmic functions
      - polynomials: zeros, factoring, division
      - probability: events, conditional probability
      - quadratic equations and the quadratic formula
      - rational functions and asymptotes
      - right-triangle trigonometry
      - sequences and series, Σ notation
      - shifting and stretching a graph
      - systems of linear equations
      - the circle in coordinates
      - the derivative and its rules
      - trigonometric identities and equations
      - unit circle, radians, trigonometric functions of any angle
      - vectors in the plane

3. TAUGHT BY ONE SYSTEM ONLY (strongest track of each)
   US only: US-BC − (PL-R ∪ DE-E) (6)
      - Taylor and power series
      - complex numbers, polar form, de Moivre
      - conic sections: parabola, ellipse, hyperbola
      - parametric, polar and vector-valued functions in calculus
      - polar coordinates
      - related rates
   Poland only: PL-R − (US-BC ∪ DE-E) (0)
      (none)
   Germany only: DE-E − (US-BC ∪ PL-R) (2)
      - hypothesis tests and the normal distribution
      - vectors in space, lines and planes, dot product

4. WHERE THE AMERICAN PRECALCULUS YEAR LANDS
   Of the topics American precalculus teaches, how many the other systems
   teach before the final two years, in them, or not at all:
   Poland: 1 before, 24 during, 5 never, of 30
   Germany: 13 before, 13 during, 4 never, of 30
   Precalculus is not a course anywhere but America: its content is
   spread over the years before the Oberstufe or the liceum, and the
   rest is either in the final years or absent.

5. HOW FAR EACH SYSTEM GETS INTO CALCULUS
      PL-P     0 of 13 AP Calculus topics
      PL-R     4 of 13 AP Calculus topics
      DE-G     7 of 13 AP Calculus topics
      DE-E    10 of 13 AP Calculus topics
      US-AB   10 of 13 AP Calculus topics
      US-BC   13 of 13 AP Calculus topics
   in AP Calculus AB but not in the Polish rozszerzony (11)
      - antiderivative, definite integral, fundamental theorem
      - area between curves, volumes
      - complex numbers, polar form, de Moivre
      - conic sections: parabola, ellipse, hyperbola
      - derivatives of exponential and trigonometric functions
      - differential equations: separable, growth and decay
      - integration by substitution
      - matrices and determinants
      - parametric equations
      - polar coordinates
      - related rates
   in the Polish rozszerzony or German Grundkurs calculus but not in AP AB (0)
      (none)
   Poland rozszerzony has, Germany Grundkurs has not (3)
      - mathematical induction
      - rational functions and asymptotes
      - trigonometric identities and equations
   Germany Grundkurs has, Poland rozszerzony has not (5)
      - antiderivative, definite integral, fundamental theorem
      - area between curves, volumes
      - derivatives of exponential and trigonometric functions
      - hypothesis tests and the normal distribution
      - vectors in space, lines and planes, dot product

6. CHECKS
   every code in the table is a known course: True
   every topic is taught somewhere: True
   each system's tracks nest, lower inside higher: True
   |only one| + |shared by two or more| = |union|: 8 + 41 = 49: True

Three lines are worth reading twice.

  • "Poland only: none." Every topic the Polish extended matura teaches, either the American or the German strong track teaches too. The Polish course is not smaller in kind, only in reach: it stops at the derivative of a rational function and never integrates.
  • "Poland: 1 before, 24 during, 5 never" against "Germany: 13 before, 13 during, 4 never." The German Sekundarstufe I has already done half of American precalculus by Klasse 10, which is why a German Oberstufe can start Analysis at once. The Polish liceum starts at fifteen with much of the same material still ahead, so its first two years are, in American terms, Algebra 2 and precalculus at once, and the extended course gets to the derivative only in the fourth year.
  • "Germany Grundkurs has, Poland rozszerzony has not: 5." The integral, the derivatives of eˣ and the trigonometric functions, three-dimensional vector geometry and hypothesis testing. A German school leaver on the ordinary track has met more calculus than a Polish one on the extended track; the Polish one has met induction, trigonometric identities and rational functions, which the Grundkurs has not.

What each system has that the others lack

Only the American course teaches conic sections, polar coordinates, complex numbers in polar form and infinite series at school. All four are old university material that moved down into precalculus and Calculus BC; the other two systems leave them for the first university year. This library has pages on three of the four: 03_Complex_Numbers, power series, and polar form inside multiplication rotates.

Only the German course treats vector geometry in space as a full Sachgebiet, lines and planes in ℝ³ with the dot product, distances and angles, and makes hypothesis testing part of every Abitur. An American student meets the first in multivariable calculus or linear algebra at university, and the second in AP Statistics, which is a different course from calculus; a Polish student meets neither at school.

The Polish course has no topic of its own in the strong-track comparison, but its emphasis is distinct: proof. The podstawa lists dowód as a skill from class 1, the rozszerzony has induction and a geometry section built on proving theorems about circles and quadrilaterals, and the matura rozszerzona asks for proofs every year. The American course states and practises; the German Bildungsstandards name mathematisch argumentieren as a competence but the Abitur asks for argument more than for proof. The nearest pages here are 11_Logic, above all what a proof is and induction.

What all three skip is what this library spends most of its pages on: sets as a subject (the Polish podstawa of 2018 kept only intervals; the German Sek I uses set notation without a topic on it), precision, measure zero, algebraic structures, the axioms of set theory and metric spaces. Those are the first university year in all three countries, which is where the map stops.

How sure

Claim How sure Why
The names of the courses, levels and exams in each country high stable for decades, and the owner can confirm the Polish ones
The thirteen content areas of the Polish podstawa and the P/R split of each row medium to high the 2018 document is recalled by its section titles; the P/R split of a few rows (law of cosines, composition, dot product) has moved between editions and textbooks
The three Sachgebiete and five Leitideen of the KMK standards high the 2012 document's structure
Which German topics are GK and which LK, and in which Klasse Sek I teaches a topic medium to low sixteen Länder, G8 and G9, and the GK/LK split is partly each Land's; "in some Länder" marks the rows known to vary
The American rows high for precalculus and AP Calculus AB/BC, medium for the Algebra 1/2 and Geometry placement the precalculus reading guide and the AP course descriptions; the earlier courses vary by state
Complex numbers removed from the Polish rozszerzony in 2008 medium from memory of the reform; they were in the pre-2008 extended course and are not in the 2018 podstawa
Everything the program prints exact, given the table it is set arithmetic on the rows above; correct a row and re-run

Po polsku, w skrócie

„Precalculus" i „calculus" to nazwy amerykańskich kursów, nie działów matematyki. W polskim liceum nie ma roku, w którym zbiera się „funkcje potrzebne do analizy": wielomiany, funkcja wymierna, logarytm, trygonometria i ciągi są rozłożone na pierwsze trzy klasy, a rachunek różniczkowy (dział XIII podstawy programowej) pojawia się dopiero w zakresie rozszerzonym i kończy się na pochodnej wielomianu i funkcji wymiernej, bez całki. Niemiecka Oberstufe dochodzi do całki i do geometrii wektorowej w przestrzeni nawet na poziomie podstawowym (Grundkurs); amerykański kurs jako jedyny uczy w szkole krzywych stożkowych, współrzędnych biegunowych, liczb zespolonych i szeregów. Polska matura rozszerzona wyróżnia się czym innym: dowodem, indukcją, tożsamościami trygonometrycznymi. Program na tej stronie traktuje trzy programy nauczania jak zbiory i liczy ich część wspólną i różnice, więc każde zdanie „X tego uczy, a Y nie" można sprawdzić i poprawić, zmieniając jeden wiersz tabeli. Krótsza wersja tej strony po polsku: Po polsku.

Auf Deutsch: Stichwörter

Precalculus und Calculus sind amerikanische Kursnamen; dieselben Themen stehen in der polnischen podstawa programowa und den deutschen Bildungsstandards, anders auf die Jahre verteilt; die ganze Seite auf Deutsch ist Auf Deutsch.

Stichwörter: Lehrplan, Bildungsstandards, Sekundarstufe I, Oberstufe, Grundkurs, Leistungskurs, Abitur, matura, Analysis, Analytische Geometrie, Stochastik, Precalculus, AP Calculus.

See also