Elementary school
Foreword
General section
Field of study
Mathematics
This chapter explains the educational value and main purpose of mathematics. It takes into account the six basic elements discussed in Chapter 2, the focus areas in learning discussed in Chapter 6, and the key competences defined in Chapters 9.4 and 18. Competence criteria are presented at the end of grades 4, 7, and 10. On the basis of the competence criteria, teaching methods, learning materials, and assessment methods are selected and shall be explained in the school curriculum of the school in question. At the end of the chapter, assessment criteria are presented at the end of grades 4, 7, and 10. The assessment criteria at the end of grades 4 and 7 are intended to be a guide for schools and are only presented to support assessment at the end of the youngest and middle levels. Schools are required to use the assessment criteria at the end of grade 10 when graduating students from grade 10.
Educational value and main purpose of mathematics
Mathematics has been an important part of culture since the beginning of time, as its task is to find, create, express and explain all kinds of regularities, laws, systems and patterns. It is an important way to reflect, create meaning and understand nature, society, economy, science, technology and digital development.
With knowledge of mathematics, a student can gain better control over situations, increase their ability to take action, communicate, plan, draw conclusions, make more rational and critical decisions, and take responsibility in everyday life, including for their own lifestyle, finances, and consumption.
The main purpose of learning and teaching mathematics is for students to acquire comprehensive skills and understanding to use mathematics as a tool for diverse and varied purposes and in different situations in everyday life, play and work. Students need to be given the opportunity to develop the ability to present solutions and solve problems, to reflect on different methods and models that are useful in solving problems and to evaluate results.
When planning the study and teaching of mathematics, all competence standards for students’ schooling should be taken into account. Students must be encouraged to acquire the expected competence, which includes an overview and the ability to apply the knowledge and skills that have been acquired.
The competency standards in mathematics, as well as in other subjects, are divided into several categories. The categories are not isolated within a subject or field of study, but rather parts of an integrated whole where each category can benefit from the others, within and between subjects and fields of study, which provides opportunities for comprehensive project work and integration.
Mathematics education needs to take place throughout schooling and aims to ensure that students develop and maintain a positive attitude towards mathematical subjects, understand its usefulness and gain confidence in their own abilities. It is important that students are aware that mistakes create opportunities to learn from them, they need to identify what went wrong and thus strengthen their understanding of mathematics.
Students need to be given opportunities to seek solutions, investigate, present issues in a variety of ways, use creative thinking, reflection and reasoning, and express themselves about their issues using mathematical vocabulary and fundamental concepts. The tasks need to be diverse, worthwhile and interesting for students. The role of the teacher is to help students acquire mathematical competence and to create conditions for meaningful mathematical learning where students are active participants in a variety of work, discussions and debates about mathematics.
Students need to be taught and trained in the use of tools, aids and digital devices, software and programming to be able to research, perform calculations, present and interpret data.
Teaching needs to be based on the cultural diversity and history of mathematics so that students understand its value and impact on culture and societies.
Mathematics proficiency standards
The competence standards in mathematics are presented in five categories, namely the workings of mathematics, which should be pervasive in all learning and teaching in mathematics, numbers and arithmetic, algebra, statistics and probability, and geometry and measurement. Although the competence standards are grouped in this way, it is important to view them as a continuous whole. When focusing specifically on one competence standard, it is important to keep in mind that learning is a continuous process and creative activity rather than the accumulation of limited knowledge and training in a specific skill.
Tools and object-oriented data
By the end of 4th grade, a student can:
utilize tools and object-based data needed to find solutions to mathematical problems,
By the end of 7th grade, a student can:
use tools and object-based data to investigate and find solutions to mathematical problems,
By the end of the 10th grade, a student can:
select and use different aids and tools for mathematical practice, including computer technology and artificial intelligence,
Conversation and expression
By the end of 4th grade, a student can:
participate in discussions about mathematics, its concepts and solutions,
By the end of 7th grade, a student can:
express themselves about mathematics, explain their thinking to others and ask questions to find solutions,
By the end of the 10th grade, a student can:
express themselves and ask questions about mathematical topics, explain and justify their solutions to others,
Symbols and concepts
By the end of 4th grade, a student can:
use the correct names for mathematical symbols and use them correctly in calculations,
By the end of 7th grade, a student can:
use the correct names for mathematical symbols, use them accurately in calculations, and transfer between sign language and everyday language,
By the end of the 10th grade, a student can:
use mathematical concepts and symbols to present, represent and solve everyday and academic problems,
Method development
By the end of 4th grade, a student can:
develop suitable methods based on one's own understanding of arithmetic with natural numbers,
By the end of 7th grade, a student can:
develop suitable methods based on one's own understanding of arithmetic with spoken numbers,
By the end of the 10th grade, a student can:
develop, explain and justify suitable methods based on one's own understanding of arithmetic with spoken numbers,
Proofs
By the end of 4th grade, a student can:
By the end of 7th grade, a student can:
evaluate whether a proof or simple argument based on mathematical properties is valid,
By the end of the 10th grade, a student can:
present, interpret and evaluate simple mathematical arguments and proofs,
Puzzle solutions
By the end of 4th grade, a student can:
use methods and approaches to solve problems and justify their answers,
By the end of 7th grade, a student can:
solve puzzles and justify answers,
By the end of the 10th grade, a student can:
use a variety of calculation operations and mathematical contexts to solve problems and justify their answers,
Research work
By the end of 4th grade, a student can:
work, alone and in collaboration, on solutions to mathematical problems by exploring, investigating and formulating hypotheses,
By the end of 7th grade, a student can:
work systematically, alone and in collaboration, to find solutions to mathematical problems by investigating, researching, analyzing and evaluating,
By the end of the 10th grade, a student can:
work systematically, alone and in collaboration, to research, present, analyze, interpret and evaluate mathematical subjects and models,
Presentations
By the end of 4th grade, a student can:
prepare and give short presentations on their own work with mathematics,
By the end of 7th grade, a student can:
prepare and give presentations on their own work with mathematics,
By the end of the 10th grade, a student can:
prepare and deliver presentations and write texts about their own work with mathematics,
Computational thinking and programming
By the end of 4th grade, a student can:
create simple algorithms and express them using variables, conditions, and loops.
By the end of 7th grade, a student can:
use programming to examine data in tables.
By the end of the 10th grade, a student can:
investigate mathematical properties and relationships using programming.
Ordering numbers and sets of numbers
By the end of 4th grade, a student can:
arrange natural numbers and simple fractions by size,
By the end of 7th grade, a student can:
arrange ordinal numbers by size and explain their relationship to integers,
By the end of the 10th grade, a student can:
explain properties of the sets of natural numbers, integers, rational numbers, and real numbers,
Rounding
By the end of 4th grade, a student can:
use rounding with integers,
By the end of 7th grade, a student can:
use rounding when calculating with whole numbers and decimals,
By the end of the 10th grade, a student can:
use rounding with appropriate precision in calculations,
Seating system
By the end of 4th grade, a student can:
use decimal notation and demonstrate understanding of the place value system when writing natural numbers,
By the end of 7th grade, a student can:
use decimal notation and demonstrate understanding of the place system when writing decimal fractions,
By the end of the 10th grade, a student can:
use decimal notation and demonstrate understanding of the place value system when writing standard form numbers,
Basic arithmetic operations
By the end of 4th grade, a student can:
use the four basic arithmetic operations and calculate with natural numbers,
By the end of 7th grade, a student can:
use the four basic arithmetic operations and calculate with real numbers,
By the end of the 10th grade, a student can:
calculate with real numbers and use the relationships and context of arithmetic operations in calculations,
Ratios
By the end of 4th grade, a student can:
work with simple fractions and ratios using an area model, on a number line and on a fraction bar,
By the end of 7th grade, a student can:
express percentages as common fractions, decimals and percentages and apply them to real-world problems,
By the end of the 10th grade, a student can:
explain the relationship between common fractions, decimals and percentages and apply them to real-world topics,
Arithmetic rules
By the end of 7th grade, a student can:
use arithmetic rules, the relationship between arithmetic operations, and the order of precedence of operations in calculations,
By the end of the 10th grade, a student can:
use the context and relationships of arithmetic operations and their knowledge of number theory in calculations,
Roots and squares
By the end of the 10th grade, a student can:
calculate with squares and square roots in simple cases,
Finance
By the end of 4th grade, a student can:
use basic arithmetic operations to find solutions to and deal with everyday life and financial problems and understand the value of money.
By the end of 7th grade, a student can:
use mathematics to tackle tasks related to the main concepts of personal finance and manage your own finances.
By the end of the 10th grade, a student can:
use the main concepts of finance, in relation to personal finances, payroll, taxes and investments, and use them in calculations and solving everyday tasks.
Pattern
By the end of 4th grade, a student can:
explore, create and comment on rules in numbers and graphic patterns and predict the continuation of patterns,
By the end of 7th grade, a student can:
explore and create geometric patterns and use algebraic symbols to work with them,
By the end of the 10th grade, a student can:
work with number sequences and patterns in a systematic way and describe them using algebraic symbols,
Hnitakerfi
By the end of 4th grade, a student can:
use a grid with cardinal directions in a real-world context, draw and position points in a coordinate system,
By the end of 7th grade, a student can:
draw and position points in a coordinate system and use coordinates to draw planar figures,
By the end of the 10th grade, a student can:
interpret the graphs of straight lines in a coordinate system in a real-world context, draw them on paper and digitally, and use the drawings to solve equations,
Fell
By the end of 4th grade, a student can:
use mathematical symbols to evaluate truth values and describe relationships between quantities,
By the end of 7th grade, a student can:
research and present number sequences and describe them with pictures, words and function quantities,
By the end of the 10th grade, a student can:
work with the concept of function in different forms and under different circumstances, both on paper and with digital tools,
Unknown sizes
By the end of 4th grade, a student can:
calculated with unknown quantities in simple cases,
By the end of 7th grade, a student can:
used letters to represent unknown quantities and calculated with them,
By the end of the 10th grade, a student can:
simplify algebraic problems and solve equations with one unknown quantity,
Equations and inequalities
By the end of 4th grade, a student can:
find one unknown quantity in an equation with integers and justify their solutions.
By the end of 7th grade, a student can:
find solutions to equations using common arithmetic rules.
By the end of the 10th grade, a student can:
express equations and inequalities based on given assumptions and interpret the meaning of their solution in a real-world context,
Equanimity
By the end of the 10th grade, a student can:
solve equations with more than one unknown quantity,
Polynomial
By the end of the 10th grade, a student can:
find points of intersection with axes and explain the relationship to factorization and standard form for first and second degree polynomials.
Data processing
By the end of 4th grade, a student can:
collected data in their environment and about their own field of interest and interpreted it in a simple way,
By the end of 7th grade, a student can:
collected and processed data, communicated information about it in a variety of ways,
By the end of the 10th grade, a student can:
use statistical concepts to present, describe, explain and interpret data,
Statistical surveys
By the end of 4th grade, a student can:
counted, classified and recorded, read and interpreted results and presented in simple graphs, with and without digital aids,
By the end of 7th grade, a student can:
conducted simple statistical surveys and performed simple statistical calculations,
By the end of the 10th grade, a student can:
planned and conducted statistical surveys and drew conclusions from them,
Diagram
By the end of 4th grade, a student can:
read from simple pictures,
By the end of 7th grade, a student can:
read, explain and interpret data and information given in tables and graphs,
By the end of the 10th grade, a student can:
read, understand and evaluate statistical information that is presented publicly,
Probability experiments
By the end of 4th grade, a student can:
conducted simple experiments with probability and put them in the context of cards.
By the end of 7th grade, a student can:
draw conclusions about probability based on their own experiments and compare them with theoretical probabilities,
By the end of the 10th grade, a student can:
conducted experiments involving probability and chance and interpreted their results,
Calculus of Probabilities
By the end of 7th grade, a student can:
Calculate probabilities in simple cases.
By the end of the 10th grade, a student can:
use concepts in probability theory, such as conditional probability, independent events, and use simple calculations to calculate and interpret the probabilities of events.
Language
By the end of 4th grade, a student can:
use concepts from geometry to describe objects in their environment,
By the end of 7th grade, a student can:
use geometric concepts to describe the properties of everyday and theoretical phenomena,
By the end of the 10th grade, a student can:
use geometric concepts and methods to describe the properties of theoretical phenomena,
instrument
By the end of 4th grade, a student can:
research, conduct experiments and draw in a simple way, e.g. by using geometry programs and object-based data,
By the end of 7th grade, a student can:
research, conduct experiments and draw drawings using geometry programs and object-based data,
By the end of the 10th grade, a student can:
a new geometry program for making drawings, investigating geometric properties, and presenting arguments,
Pattern
By the end of 4th grade, a student can:
mirrored and flipped flat images when researching patterns,
By the end of 7th grade, a student can:
flip, mirror or rotate flat images, for example when studying patterns,
Diagrams
By the end of 4th grade, a student can:
conducted informal research on two- and three-dimensional shapes, drawing diagrams of them and objects in their environment,
By the end of 7th grade, a student can:
investigate and analyze two-dimensional and three-dimensional shapes, draw simple two- and three-dimensional figures and use scales and uniform shapes,
By the end of the 10th grade, a student can:
draw diagrams and work with other people's drawings based on given assumptions, e.g. about scale, uniformity and proportions, investigate and describe the relationship between a drawing and an object,
Time and clock
By the end of 4th grade, a student can:
read and use different representations of time, use a dial and digital clock and read time tables,
By the end of 7th grade, a student can:
apply equivalence between different time units to accurately determine elapsed time,
By the end of the 10th grade, a student can:
applied fractions of time units and calculated between time zones,
Units of measurement
By the end of 4th grade, a student can:
choose the appropriate unit of measurement for length, mass and volume and know the relationships between common units of measurement in the metric system,
By the end of 7th grade, a student can:
apply the relationships between the metric system's units of length, area, volume, time and mass,
By the end of the 10th grade, a student can:
explain the relationship between units of measurement in the metric system and interpret them in relation to other systems,
Measurements
By the end of 4th grade, a student can:
estimated and measured mass, length, volume, time and temperature using standard and non-standard measuring instruments,
By the end of 7th grade, a student can:
estimated and measured mass, length, volume, speed, time and temperature using appropriate scales and drew conclusions from the measurements,
Two-dimensional form
By the end of 4th grade, a student can:
estimated and measured perimeter and area using standard and non-standard measuring instruments,
By the end of 7th grade, a student can:
estimated, measured and calculated angular sizes, perimeter and area of polygons and degrees in a sector of a circle,
By the end of the 10th grade, a student can:
apply known properties of lines, triangles and other polygons to calculate side lengths and angles, calculate and explain relationships between basic properties of a circle, explain and apply the Pythagorean theorem,
Three dimensional form
By the end of 4th grade, a student can:
identify a cube, sphere, cone, cylinder, pyramid and triangle.
By the end of 7th grade, a student can:
estimated and calculated volume and surface area of triangles, rectangles, and pyramids.
By the end of the 10th grade, a student can:
measured dimensions of three-dimensional shapes, estimated and calculated volume and surface area, and understood relationships to two-dimensional shapes, symmetry, and centers of rotation.
Assessment criteria for mathematics
Assessment criteria are set for all subjects and areas of study at the end of grades 4, 7 and 10 and are a description of how well a student has mastered a defined competency. Schools are required to set assessment criteria for subjects and areas of study for other grades and explain them in the school curriculum.
Assessment criteria are presented on an A–D assessment scale. A describes excellent competence, B describes good competence, C describes fair competence, and D describes competence that does not meet the criteria described in C. It can be assumed that the majority of students will achieve the competence specified in B, competence achieved, as they are based on the competence criteria for the year.
A is awarded to those who demonstrate outstanding competence and C to those who do not fully meet the requirements set out in B criteria. There are no criteria set for D, but this evidence is used when a student does not meet the requirements set out in C criteria. The school then makes a separate report on the competence of the student in question.
Schools are required to use the A–D grading scale and assessment criteria when graduating students from grade 10.
The A–D assessment scale and assessment criteria at the end of grades 4 and 7 are intended to be a guide for schools and are only presented to support assessment at the end of the junior and middle levels.
The assessment criteria for each study period should be based on the competency criteria and the learning objectives that have been worked on. The most important thing is to assess at the end of the study period what was expected according to the assessment criteria.
Assessment results need to be based on reliable assessment data and diverse assessment methods so that students, parents, and schools receive the most accurate information about their status.
The student can express themselves about mathematical topics and present them in a clear and varied manner . Use mathematical concepts and symbols confidently to present, represent and solve more complex problems. Select and use different aids for mathematical work, analyze, interpret and evaluate mathematical models and present their results in a clear and concise manner. Use real numbers and calculate confidently with spoken numbers, confidently analyze the relationship between numbers in different sets of numbers. Be a leader in developing solutions, utilize the context and relationships of mathematical operations and use this knowledge flexibly in calculations and evaluations. Investigate organized patterns and generalize about them, solve equations confidently , use variables flexibly , and describe their relationships with positions and functions. Use basic geometric concepts confidently and in a variety of ways. Use unit forms, angle rules, and coordinate systems to draw, analyze, and create geometric objects. Present precise geometric arguments, confidently measure and calculate length, area, and space, and use computers for these objects. Use statistical concepts confidently to plan, conduct, and interpret statistical investigations. Perform and draw conclusions independently and critically from experiments involving probability and chance. Use probability concepts and counting confidently to calculate and interpret probabilities of events.
The student can express themselves about mathematical topics and present them clearly. Use mathematical concepts and symbols to present, represent and solve problems. Use different aids for mathematical work, analyze and interpret mathematical models and present their results clearly. Use real numbers and calculate with real numbers, analyze the relationship between numbers in different number systems. Participate in developing solutions, utilize the context and relationships of arithmetic operations and use that knowledge in calculations and evaluations. Investigate patterns and generalize about them, solve equations, use variables and describe their relationship with positions and functions. Use basic geometric concepts in a variety of ways. Use uniform shapes, angle rules and coordinate systems to draw, analyze and create geometric objects. Present geometric arguments, measure and calculate length, area and space and use computers for these objects. Use statistical concepts to plan, perform and interpret statistical research. Conduct and draw conclusions from experiments involving probability and chance. Use probability concepts and counting to calculate and interpret probabilities of events.
The student can express themselves fairly well about mathematical topics. To a certain extent, use mathematical concepts and symbols to present, represent and solve simple problems. Use different aids for mathematical work, present mathematical models and present their results in an appropriate manner. Use real numbers and calculate with spoken numbers, record their answers using decimal notation and participate in developing their own solutions. Use the context and relationships of mathematical operations and use common calculation rules. Investigate patterns, solve simple equations, calculate with variables and use the rules of algebra in calculations. Use basic geometric concepts to a certain extent . Use unit forms, angle rules and coordinate systems to draw, analyse and create geometric objects. Present simple geometric arguments under guidance , measure and calculate length, area and space and use computers for these objects. Use statistical concepts to plan, carry out and interpret statistical research under guidance. Perform and draw simple conclusions from experiments involving probability and chance. Use simple probability concepts and calculations to calculate and interpret probabilities of events.
The student can confidently apply concepts that are characteristic of mathematics in everyday life, discuss their solutions and express themselves about them. Confidently and independently apply objective data to solve problems. Take the lead in developing suitable methods for solving problems. Confidently and independently apply different arithmetic operations and use their context and relationships in calculations and evaluations. Work with, investigate and describe patterns, solve equations and inequalities. Use letters for unknown quantities and confidently use the rules of algebra in calculations. Confidently and in a variety of ways apply basic concepts of geometry. Confidently draw, analyse and create geometric objects. Demonstrate a good grasp of measurements and units of measurement. Collect information and carry out statistical research alone and in collaboration with others, read from it, communicate and draw conclusions from it.
The student can use concepts that are characteristic of mathematics in everyday life, discuss their solutions and express themselves about them. Apply objective data to solve problems. Participate in developing suitable methods for solving problems. Apply different arithmetic operations and use their relationships in calculations and evaluations. Work with patterns and solve equations and inequalities. Use letters for unknown quantities and use the rules of algebra in calculations. Apply basic geometric concepts in a variety of ways. Draw, analyze and create geometric objects. Demonstrate mastery of measurements and units of measurement. Collect information and conduct statistical research alone and in collaboration with others, present them graphically and explain their results.
With support, the student can apply and express himself/herself about concepts that are characteristic of mathematics in everyday life and discuss his/her solutions. Work systematically on mathematical tasks with guidance. Apply simple arithmetic operations and use object-based data to solve tasks. Work with patterns and solve equations with assistance or following instructions . Use letters for unknown quantities and use the rules of algebra in calculations with assistance or following instructions . Apply basic concepts of geometry. With assistance, draw and analyze geometric objects. Demonstrate a fairly good grasp of measurements and units of measurement. With assistance, collect information and conduct statistical research, present them graphically and explain their results.
The student can confidently express themselves about mathematics, use mathematical symbols, object-based data and concepts. Easily calculate and apply natural numbers in an object-based and abstract way. Use the decimal system confidently and present solutions correctly. Easily identify patterns in numbers and figures and express themselves about their rules. Solve simple equations. Work independently with geometric concepts, measure, draw, analyze and create geometric objects. Apply different arithmetic operations and units of measurement to some extent and use their context and relationships in calculations and evaluations. Conduct research from everyday life in collaboration with others and read from them. Collect information, communicate it and draw conclusions. Conduct experiments with probability and express themselves about their results.
With assistance, the student can express themselves about mathematics, solve mathematical problems, use mathematical symbols, object-based data and concepts. Calculate and apply natural numbers in an object-based and abstract way. Use the decimal system and present solutions correctly. Analyze patterns in numbers and figures and express themselves about their rules. Solve simple equations. Work with geometric concepts, measure, draw, analyze and create geometric objects. Apply different arithmetic operations and units of measurement to some extent. Conduct research from everyday life in collaboration with others and read from them, collect information and communicate it. Conduct simple experiments with probability and record their results.
With support, the student can express themselves about mathematics, solve mathematical problems, discuss them and use appropriate diagrams. Calculate and apply natural numbers in a concrete and abstract way. With assistance, recognize the decimal system and present solutions. Analyze simple patterns in numbers, figures and solve simple equations. With assistance , work with geometric concepts, measure, draw, analyze and create geometric objects. Conduct simple research with guidance and/or in collaboration with others, collect information and record it. Participate in experiments with probability.