{"id":750,"date":"2016-08-15T20:32:58","date_gmt":"2016-08-15T20:32:58","guid":{"rendered":"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/the-mathematics-curriculum-in-primary-and-lower-secondary-grades\/"},"modified":"2016-10-18T15:33:43","modified_gmt":"2016-10-18T15:33:43","slug":"the-mathematics-curriculum-in-primary-and-lower-secondary-grades","status":"publish","type":"page","link":"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/the-mathematics-curriculum-in-primary-and-lower-secondary-grades\/","title":{"rendered":"The Mathematics Curriculum in Primary and Lower Secondary Grades"},"content":{"rendered":"<div class=\"carouselwrap\"><div id=\"menu-germany\" class=\"carouselmenu\"><a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Introduction<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/the-mathematics-curriculum-in-primary-and-lower-secondary-grades\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>The Mathematics Curriculum in Primary and Lower Secondary Grades<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/the-science-curriculum-in-primary-and-lower-secondary-grades\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>The Science Curriculum in Primary and Lower Secondary Grades<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/teachers-teacher-education-and-professional-development\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Teachers, Teacher Education, and Professional Development<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/instruction-for-mathematics-and-science-in-primary-and-lower-secondary-grades\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Instruction for Mathematics and Science in Primary and Lower Secondary Grades<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/monitoring-student-progress-in-mathematics-and-science\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Monitoring Student Progress in Mathematics and Science<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/special-initiatives-in-mathematics-and-science-education\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Special Initiatives in Mathematics and Science Education<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/use-and-impact-of-timss\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Use and Impact of TIMSS<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/suggested-readings\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>Suggested Readings<\/span><\/a><\/li>\n<a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/references\/\" class=\"menu-link main-menu-link menu-item menu-item-type-custom menu-item-object-custom\"><span>References<\/span><\/a><\/li>\n<\/div><\/div>\t<style>\n\t\t.carouselwrap{\n\t\t\tbackground: #e5eaef;\n\t\t}\t\t\n\t\t.owl-item a{\n\t\t\tbackground: #ffffff;\n\t\t\tborder: 4px #00b2cc solid;\n\t\t\tcolor: #000000;\n\t\t}\t\t\n\t\t.owl-item a:hover,\n\t\t.owl-item a:focus,\n\t\t.owl-item a.current-menu-item{\n\t\t\tbackground: #00667f;\n\t\t\tborder: 4px #00b2cc solid;\n\t\t\t\n\t\t\tcolor: #ffffff;\n\t\t}\n\t<\/style>\n\t\n<h2>The Mathematics Curriculum in Primary and Lower Secondary Grades<\/h2>\n<p>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.<\/p>\n<p><strong>Exhibit 1: Mathematics Curriculum Guidelines for the Primary Level (up to Grade 4) in North Rhine-Westphalia<\/strong><\/p>\n<table class=\"tableEncyc\" border=\"0\" width=\"100%\">\n<tbody>\n<tr class=\"heading\">\n<td>Content Domains<\/td>\n<td>Content-Based Student Competencies<sup style=\"font-weight: normal;\">d<\/sup><\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Numbers and Operations<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Understanding Numbers<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Illustrate the number range up to 1,000,000 using the decimal system<\/li>\n<li>Analyze and describe structural relationships between different number systems based on examples<\/li>\n<li>Use structures in number systems to understand numbers in extended number ranges<\/li>\n<li>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<\/li>\n<li>Discover relationships between individual numbers and in complex number systems, and describe these using<br \/>\nmathematical terminology and symbols<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Understanding Operations<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>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<\/li>\n<li>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)<\/li>\n<li>Switch between different representations of operations (e.g., material, symbolic, figurative, or language-based representations)<\/li>\n<li>Discover and describe characteristics of operations and laws of arithmetic based on examples<\/li>\n<li>Use mathematical terminology and symbols correctly<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Fast Mental Arithmetic<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Have sound knowledge and skills of quick mental arithmetic in the number range up to 1,000,000<\/li>\n<li>Repeat all multiplication tables (up to 10) automatically and be fluent in the inverses<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Arithmetic<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>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<\/li>\n<li>Solve problems using multiplication table relationships<\/li>\n<li>Describe and evaluate different arithmetic operations based on aspects of arithmetic and demonstrate clear understanding of these structures in writing<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Numerals<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>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<\/li>\n<li>Calculate fluently, confidently, and in written form using addition, subtraction, and multiplication<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Estimations<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>State approximate results of problems using numbers up to 1,000,000, and round and estimate to the appropriate accuracy<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Flexible Calculating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Calculate using individually preferred methods or standard methods, with and without a calculator<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Dimension and Form<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Spatial Orientation and Spatial Visualization<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Trace lines with a pen (eye-hand coordination), name overlapping figures (figure-ground discrimination), and identify forms (visual consistency)<\/li>\n<li>Orientation in two-dimensional space using a map<\/li>\n<li>Describe spatial relations on the basis of pictures, arrangements, plans, etc., as well as from imagination<\/li>\n<li>Visualize the movement of shapes and objects and describe the results of movement in advance<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Shapes<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Explore, name, and describe shapes using mathematical terminology (e.g., perpendicular, horizontal, parallel, square)<\/li>\n<li>Construct shapes by replacing, overlaying, or spreading elements, filling in spaces, and constructing, deconstructing, or continuing patterns<\/li>\n<li>Continue, describe, and construct patterns (e.g., band ornaments, tessellations)<\/li>\n<li>Name and compare areas of shapes and their perimeters<\/li>\n<li>Construct similar shapes from card paper by enlarging or reducing according to scale<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Solid Figures<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Identify geometrical objects, sort them according to geometrical characteristics, and describe them using mathematical terminology (e.g., area, edge)<\/li>\n<li>Construct wireframe and solid models of objects and build more complex cube constructions<\/li>\n<li>Find various nets for cubes<\/li>\n<li>Identify two- or three-dimensional views of buildings and construct buildings according to a plan<\/li>\n<li>Define and compare volumes of objects with unit cubes<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Symmetry<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Examine shapes for axial (line) symmetry and use their characteristic length preservation and space preservation to explain the symmetry<\/li>\n<li>Construct symmetrical figures and use characteristics of axial (line) symmetry (length preservation and space preservation)<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Drawing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>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<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Measuring and Quantities<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Perception and Handling of Quantities<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Measure quantities (length, time, weight, and volume) using suitable drawing instruments<\/li>\n<li>Compare and organize quantities<\/li>\n<li>Name quantities of familiar objects and use these quantities as a reference for estimations<\/li>\n<li>Read time from analog and digital clocks<\/li>\n<li>Use monetary units (c, \u20ac) and units of length (mm, km), time (seconds, minutes, hours), weight (g, kg, t), and volume (ml, l), and convert between units<\/li>\n<li>Convert fractional quantities that occur in daily life into the next smaller unit (e.g., <span class=\"fraction\"><span class=\"fup\">1<\/span><span class=\"bar\">\/<\/span><span class=\"fdn\">4<\/span><\/span> l = 250 ml )<\/li>\n<li>Calculate with quantities (also using decimals)<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Factual Situations<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Formulate arithmetical questions for real or simulated situations (also in project-oriented problem contexts) and for contextual problems, and solve them<\/li>\n<li>Use aids like tables, drawings, and diagrams to solve problems<\/li>\n<li>Reason that estimated values (estimation, evaluation) are sufficient and explain why an exact result is necessary or unnecessary<\/li>\n<li>Construct contextual problems (orally and in writing) for mathematical models (equations, tables, etc.)<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Data, Frequency, and Plausibility<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Data and Frequency<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Collect data from real life situations and present it in diagrams and tables<\/li>\n<li>Extract data from calendars, diagrams, and tables to solve problems with arithmetic content<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Probability<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Describe the probability of simple events (using terms such as: certain, possible, impossible, always, often, rarely, never)<\/li>\n<li>Name the number of different possibilities in simple combination tasks<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"tableEncyc\" border=\"0\" width=\"100%\">\n<tbody>\n<tr class=\"heading\">\n<td>Learning Processes<\/td>\n<td>Process-Oriented Student Competencies<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Problem Solving and Creative Thinking<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Select<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>\u00a0Find relevant information for solving problems and present it in words<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Solve<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Try progressively more systematic and result-oriented approaches, and use knowledge of operations to solve problems<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Reflect and Check<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Check results for adequacy, detect and correct mistakes, and compare and evaluate various approaches<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Transfer<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Transfer approaches to similar situations<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Modify and Invent<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Invent tasks and questions<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Apply<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Select suitable arithmetic rules, algorithms, and tools for problem solving and apply them appropriately<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Modeling<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Detect<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Distill information from problem situations and tasks, and distinguish between relevant and nonrelevant information<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Solve<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Transfer information from problem situations into mathematical models and solve problems using these models<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Validate<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Relate solutions back to the problem situation and test plausibility of results<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Relate<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Define suitable problems for given mathematical models and develop questions related to the models<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Arguing<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Hypothesize<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Make hypotheses about mathematical relationships or irregularities<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Test<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Test hypotheses using examples and question if assumptions, solutions, statements, etc., are correct<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Conclude<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Prove or disprove hypotheses based on examples, and develop preliminary conclusions related to these hypotheses<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Substantiate<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Describe relationships and rules using examples and follow the reasoning of others<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Illustrating and Communicating<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Record<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Record results, procedures, and learning experiences<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Present and Exchange<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Design and develop suitable means of presentation, such as transparencies or posters, to present solutions, ideas, and results comprehensibly<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Cooperate and Communicate<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Work on complex problems in groups, organize meetings, and combine opinions<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Use Expert Terminology<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use suitable mathematical terminology to present mathematical facts, symbols, and conventions<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">\u00a0Change Between Illustrations<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>\u00a0Transfer illustrations into other forms of illustrative representation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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.<\/p>\n<p>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.<sup>17<\/sup><\/p>\n<p><strong>Exhibit 2: Mathematics Curriculum Guidelines (Realschulbildungsgang) in North Rhine-Westphalia, Grade 8<\/strong><\/p>\n<table class=\"tableEncyc\" border=\"0\" width=\"100%\">\n<tbody>\n<tr class=\"heading\">\n<td>Content Domains<\/td>\n<td>Content-Based Student Competencies<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Arithmetic and Algebra\u2014Dealing with numerals and symbols<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Ordering<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Order and compare rational numbers<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Operating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Execute basic arithmetic operations for rational numbers (mental arithmetic and written arithmetic techniques)<\/li>\n<li>Aggregate terms, multiply them, and factor them using simple factors<\/li>\n<li>Solve linear equations by trial and error, as well as algebraically, and check calculations by applying the solution<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Applying<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Apply knowledge of rational numbers and simple linear\u00a0<span style=\"font-family: inherit; font-size: inherit; line-height: 1.5;\">equations to solving mathematical\u00a0and extra-mathematical problems<\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Systematizing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Give non-mathematical reasons and examples for the extension of the set of natural numbers to the set of rational numbers<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Functions\u2014Describing and investigating relationships and changes<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Illustrating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Express relationships in words, in tables of values, in graphs, and in mathematical symbols, and shift among different forms of\u00a0representation<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Interpreting<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Interpret linear functions in terms of equations and graphs<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Applying<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Identify proportional, nonproportional, and linear relations in charts, mathematical symbols, and real world situations<\/li>\n<li>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<\/li>\n<li>Compute percentages and base values in real world situations<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Arithmetic and Algebra\u2014Dealing with numerals and symbols<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Conceptualizing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Name and characterize triangles (right, isosceles, and equilateral), parallelograms, rhombuses, trapezoids, and simple prisms, and identify them in real-world situations<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Constructing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Draw triangles from given measures of angles and sides<\/li>\n<li>Sketch angular illustrations, create nets of cubes and cuboids, and construct geometrical objects<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Measuring<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Estimate and define the perimeter and surface area of triangles, parallelograms, and figures constituted by these shapes<\/li>\n<li>Specify surface areas and volumes of cubes, cuboids, and simple prisms<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Applying<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Discern and justify attributes of figures by means of symmetry, theorems of angles, or congruence<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Stochastic Processes\u2014Working with data and chance<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Collecting Data<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Plan the collection of data, conduct surveys, and use spreadsheets for data organization<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Illustrating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use median, range, and quartiles for the description of frequency distributions<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Analyzing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use simple experiments of chance to describe stochastic events in everyday situations<\/li>\n<li>Use relative frequencies from repeated experiments to estimate probability<\/li>\n<li>Use the Rule of Laplace to ascertain probabilities in simple experiments of chance<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Evaluating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use probability to evaluate chance and risk and to estimate<\/li>\n<li>frequency<\/li>\n<li>Interpret ranges and quartiles in statistical illustrations or descriptions<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table class=\"tableEncyc\" border=\"0\" width=\"100%\">\n<tbody>\n<tr class=\"heading\">\n<td>Learning Processes<\/td>\n<td>Content-Based Student Competencies<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Reasoning and Communicating\u2014Communicating, presenting, and reasoning<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Reading<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Gather, restructure, and evaluate information from simple\u00a0mathematical figures<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Verbalizing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Demonstrate individual problem solving steps using simple mathematical operations, in their own words and using appropriate technical terms<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Communicating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Compare and evaluate approaches, solutions, arguments, and illustrations<\/li>\n<li>Present solutions in short, prepared statements<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Presenting<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Find generalizations and specific instances of mathematical facts, and give examples and counterexamples to support findings<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Associating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Find generalizations and specific instances of mathematical facts, and give examples and counterexamples to support findings<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Reasoning<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Apply mathematical knowledge to reasoning in multistep arguments<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Problem Solving\u2014Understanding, Investigating, and Solving Problems<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Investigating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Analyze patterns and relationships in numbers and figures, and use them to make hypotheses<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Solving<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Plan and describe approaches to problem solving<\/li>\n<li>Use algorithms to solve standard mathematical problems and evaluate answers in terms of practicality<\/li>\n<li>Examine multiple ways to solve a problem<\/li>\n<li>Use problem solving strategies<\/li>\n<li>Apply various forms of representation (e.g., charts, sketches, and equations) when solving problems<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Reflecting<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Examine and evaluate results by means of plausibility, rough calculations, or sketches to verify solution procedures<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Mathematical Modeling\u2014Creating and Applying Models<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Modeling<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Translate simple real world situations into mathematical models<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Validating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Verify results generated by mathematical models by relating them back to the real world situation, and, where necessary, modify the model<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Implementing<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Match mathematical models (e.g., charts, graphs, and equations) to suitable real world situations<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom headingInside\" colspan=\"2\">Instruments\u2014Using Technologies and Instruments<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Investigating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use spreadsheets and geometry software to investigate mathematical and nonmathematical relationships<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Calculating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use a calculator<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Illustrating<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Compile data electronically and present it in spreadsheets<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<tr>\n<td class=\"borderBottom borderRight headingInside\" valign=\"top\">Researching<\/td>\n<td class=\"borderBottom\">\n<ul>\n<li>Use dictionaries, textbooks, and the Internet to acquire\u00a0information<\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>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 &hellip; <a href=\"http:\/\/timss2015.org\/encyclopedia\/countries\/germany\/the-mathematics-curriculum-in-primary-and-lower-secondary-grades\/\">Continued<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":137,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":"1"},"class_list":["post-750","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/pages\/750","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/comments?post=750"}],"version-history":[{"count":37,"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/pages\/750\/revisions"}],"predecessor-version":[{"id":6910,"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/pages\/750\/revisions\/6910"}],"up":[{"embeddable":true,"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/pages\/137"}],"wp:attachment":[{"href":"http:\/\/timss2015.org\/encyclopedia\/wp-json\/wp\/v2\/media?parent=750"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}