Germany

The Mathematics Curriculum in Primary and Lower Secondary Grades

Mathematics education at the primary school level currently is regulated across the 16 German states by 13 curricula, which all are informed by the national educational standards. Although 12 states have passed their own curricula, 4 states (Berlin, Brandenburg, Bremen, and Mecklenburg-Western Pomerania) have collaborated in developing and approving a common core curriculum. Exhibit 1 presents an overview of the mathematics topics taught in primary school in North Rhine-Westphalia.

Exhibit 1: Mathematics Curriculum Guidelines for the Primary Level (up to Grade 4) in North Rhine-Westphalia

Content Domains	Content-Based Student Competenciesd
Numbers and Operations
Understanding Numbers	Illustrate the number range up to 1,000,000 using the decimal system Analyze and describe structural relationships between different number systems based on examples Use structures in number systems to understand numbers in extended number ranges Work in the number range up to 1,000,000 by counting in steps, as well as by arranging and comparing numbers according to various characteristics Discover relationships between individual numbers and in complex number systems, and describe these using mathematical terminology and symbols
Understanding Operations	Match basic situations (which require adding and combining or taking away and separating) to the respective basic mathematical operations such as addition, subtraction, or completion Match basic situations (which require repeated addition of the same numbers or repeated subtraction of the same numbers) to the respective mathematical operations, such as multiplication or division (distribution) Switch between different representations of operations (e.g., material, symbolic, figurative, or language-based representations) Discover and describe characteristics of operations and laws of arithmetic based on examples Use mathematical terminology and symbols correctly
Fast Mental Arithmetic	Have sound knowledge and skills of quick mental arithmetic in the number range up to 1,000,000 Repeat all multiplication tables (up to 10) automatically and be fluent in the inverses
Arithmetic	Solve problems using all four basic operations (orally or in a partly standardized written form) by making use of arithmetical laws and analyze strategies using relationships between numbers and arithmetic laws (e.g., distributive law and commutative law of addition) in all four operations Solve problems using multiplication table relationships Describe and evaluate different arithmetic operations based on aspects of arithmetic and demonstrate clear understanding of these structures in writing
Numerals	Explain in writing operations such as addition (with several addends), subtraction (with one subtrahend), multiplication (with multiple digits), and division (using remainder notations with single-digit and important double-digit divisors), describing the steps of calculation logically using examples Calculate fluently, confidently, and in written form using addition, subtraction, and multiplication
Estimations	State approximate results of problems using numbers up to 1,000,000, and round and estimate to the appropriate accuracy
Flexible Calculating	Calculate using individually preferred methods or standard methods, with and without a calculator
Dimension and Form
Spatial Orientation and Spatial Visualization	Trace lines with a pen (eye-hand coordination), name overlapping figures (figure-ground discrimination), and identify forms (visual consistency) Orientation in two-dimensional space using a map Describe spatial relations on the basis of pictures, arrangements, plans, etc., as well as from imagination Visualize the movement of shapes and objects and describe the results of movement in advance
Shapes	Explore, name, and describe shapes using mathematical terminology (e.g., perpendicular, horizontal, parallel, square) Construct shapes by replacing, overlaying, or spreading elements, filling in spaces, and constructing, deconstructing, or continuing patterns Continue, describe, and construct patterns (e.g., band ornaments, tessellations) Name and compare areas of shapes and their perimeters Construct similar shapes from card paper by enlarging or reducing according to scale
Solid Figures	Identify geometrical objects, sort them according to geometrical characteristics, and describe them using mathematical terminology (e.g., area, edge) Construct wireframe and solid models of objects and build more complex cube constructions Find various nets for cubes Identify two- or three-dimensional views of buildings and construct buildings according to a plan Define and compare volumes of objects with unit cubes
Symmetry	Examine shapes for axial (line) symmetry and use their characteristic length preservation and space preservation to explain the symmetry Construct symmetrical figures and use characteristics of axial (line) symmetry (length preservation and space preservation)
Drawing	Construct line segments, simple figures, patterns, curves, and exact parallel or perpendicular lines using instruments like compasses and set squares, and use grid or point patterns to draw shapes and three-dimensional buildings
Measuring and Quantities
Perception and Handling of Quantities	Measure quantities (length, time, weight, and volume) using suitable drawing instruments Compare and organize quantities Name quantities of familiar objects and use these quantities as a reference for estimations Read time from analog and digital clocks Use monetary units (c, €) and units of length (mm, km), time (seconds, minutes, hours), weight (g, kg, t), and volume (ml, l), and convert between units Convert fractional quantities that occur in daily life into the next smaller unit (e.g., 1/4 l = 250 ml ) Calculate with quantities (also using decimals)
Factual Situations	Formulate arithmetical questions for real or simulated situations (also in project-oriented problem contexts) and for contextual problems, and solve them Use aids like tables, drawings, and diagrams to solve problems Reason that estimated values (estimation, evaluation) are sufficient and explain why an exact result is necessary or unnecessary Construct contextual problems (orally and in writing) for mathematical models (equations, tables, etc.)
Data, Frequency, and Plausibility
Data and Frequency	Collect data from real life situations and present it in diagrams and tables Extract data from calendars, diagrams, and tables to solve problems with arithmetic content
Probability	Describe the probability of simple events (using terms such as: certain, possible, impossible, always, often, rarely, never) Name the number of different possibilities in simple combination tasks

Learning Processes	Process-Oriented Student Competencies
Problem Solving and Creative Thinking
Select	Find relevant information for solving problems and present it in words
Solve	Try progressively more systematic and result-oriented approaches, and use knowledge of operations to solve problems
Reflect and Check	Check results for adequacy, detect and correct mistakes, and compare and evaluate various approaches
Transfer	Transfer approaches to similar situations
Modify and Invent	Invent tasks and questions
Apply	Select suitable arithmetic rules, algorithms, and tools for problem solving and apply them appropriately
Modeling
Detect	Distill information from problem situations and tasks, and distinguish between relevant and nonrelevant information
Solve	Transfer information from problem situations into mathematical models and solve problems using these models
Validate	Relate solutions back to the problem situation and test plausibility of results
Relate	Define suitable problems for given mathematical models and develop questions related to the models
Arguing
Hypothesize	Make hypotheses about mathematical relationships or irregularities
Test	Test hypotheses using examples and question if assumptions, solutions, statements, etc., are correct
Conclude	Prove or disprove hypotheses based on examples, and develop preliminary conclusions related to these hypotheses
Substantiate	Describe relationships and rules using examples and follow the reasoning of others
Illustrating and Communicating
Record	Record results, procedures, and learning experiences
Present and Exchange	Design and develop suitable means of presentation, such as transparencies or posters, to present solutions, ideas, and results comprehensibly
Cooperate and Communicate	Work on complex problems in groups, organize meetings, and combine opinions
Use Expert Terminology	Use suitable mathematical terminology to present mathematical facts, symbols, and conventions
Change Between Illustrations	Transfer illustrations into other forms of illustrative representation

At the secondary school level, eighth grade mathematics education currently is regulated by more than 40 different curricula, which all are informed by the national educational standards. There is no single or common core curriculum across all the states. In fact, the mathematics curricula differ across grades and school tracks in the details of the content covered and the timing of the introduction of topics: generally, the more demanding a secondary school track, the earlier the students are introduced to advanced topics.

Exhibit 2 presents an overview of mathematics topics covered in the eighth grade curriculum for the Realschulbildungsgang (extensive general education) in North Rhine-Westphalia, and is fairly representative of the 40 eighth grade curricula in place across the German states.¹⁷

Exhibit 2: Mathematics Curriculum Guidelines (Realschulbildungsgang) in North Rhine-Westphalia, Grade 8

Content Domains	Content-Based Student Competencies
Arithmetic and Algebra—Dealing with numerals and symbols
Ordering	Order and compare rational numbers
Operating	Execute basic arithmetic operations for rational numbers (mental arithmetic and written arithmetic techniques) Aggregate terms, multiply them, and factor them using simple factors Solve linear equations by trial and error, as well as algebraically, and check calculations by applying the solution
Applying	Apply knowledge of rational numbers and simple linear equations to solving mathematical and extra-mathematical problems
Systematizing	Give non-mathematical reasons and examples for the extension of the set of natural numbers to the set of rational numbers
Functions—Describing and investigating relationships and changes
Illustrating	Express relationships in words, in tables of values, in graphs, and in mathematical symbols, and shift among different forms of representation
Interpreting	Interpret linear functions in terms of equations and graphs
Applying	Identify proportional, nonproportional, and linear relations in charts, mathematical symbols, and real world situations Apply the characteristics of proportional, nonproportional, and linear relations, as well as simple procedures of the Rule of Three for the solution of mathematical and nonmathematical problems Compute percentages and base values in real world situations
Arithmetic and Algebra—Dealing with numerals and symbols
Conceptualizing	Name and characterize triangles (right, isosceles, and equilateral), parallelograms, rhombuses, trapezoids, and simple prisms, and identify them in real-world situations
Constructing	Draw triangles from given measures of angles and sides Sketch angular illustrations, create nets of cubes and cuboids, and construct geometrical objects
Measuring	Estimate and define the perimeter and surface area of triangles, parallelograms, and figures constituted by these shapes Specify surface areas and volumes of cubes, cuboids, and simple prisms
Applying	Discern and justify attributes of figures by means of symmetry, theorems of angles, or congruence
Stochastic Processes—Working with data and chance
Collecting Data	Plan the collection of data, conduct surveys, and use spreadsheets for data organization
Illustrating	Use median, range, and quartiles for the description of frequency distributions
Analyzing	Use simple experiments of chance to describe stochastic events in everyday situations Use relative frequencies from repeated experiments to estimate probability Use the Rule of Laplace to ascertain probabilities in simple experiments of chance
Evaluating	Use probability to evaluate chance and risk and to estimate frequency Interpret ranges and quartiles in statistical illustrations or descriptions

Learning Processes	Content-Based Student Competencies
Reasoning and Communicating—Communicating, presenting, and reasoning
Reading	Gather, restructure, and evaluate information from simple mathematical figures
Verbalizing	Demonstrate individual problem solving steps using simple mathematical operations, in their own words and using appropriate technical terms
Communicating	Compare and evaluate approaches, solutions, arguments, and illustrations Present solutions in short, prepared statements
Presenting	Find generalizations and specific instances of mathematical facts, and give examples and counterexamples to support findings
Associating	Find generalizations and specific instances of mathematical facts, and give examples and counterexamples to support findings
Reasoning	Apply mathematical knowledge to reasoning in multistep arguments
Problem Solving—Understanding, Investigating, and Solving Problems
Investigating	Analyze patterns and relationships in numbers and figures, and use them to make hypotheses
Solving	Plan and describe approaches to problem solving Use algorithms to solve standard mathematical problems and evaluate answers in terms of practicality Examine multiple ways to solve a problem Use problem solving strategies Apply various forms of representation (e.g., charts, sketches, and equations) when solving problems
Reflecting	Examine and evaluate results by means of plausibility, rough calculations, or sketches to verify solution procedures
Mathematical Modeling—Creating and Applying Models
Modeling	Translate simple real world situations into mathematical models
Validating	Verify results generated by mathematical models by relating them back to the real world situation, and, where necessary, modify the model
Implementing	Match mathematical models (e.g., charts, graphs, and equations) to suitable real world situations
Instruments—Using Technologies and Instruments
Investigating	Use spreadsheets and geometry software to investigate mathematical and nonmathematical relationships
Calculating	Use a calculator
Illustrating	Compile data electronically and present it in spreadsheets
Researching	Use dictionaries, textbooks, and the Internet to acquire information