### eMath

**eMath** is an advanced environment that enables teaching mathematics in a digital environment. The environment is based on **structured derivations**, a general format for presenting mathematical arguments. The format shows explicitly the logical structure of mathematical argument. It requires that each derivation step is explicitly justified. Structured derivations have been tried extensively in mathematics education in Finland, and also in Estonia and Sweden. It improves students' understanding of mathematical arguments and helps them in finding errors in their own reasoning. It also makes it easier for teachers to explain why and how the mathematics works in an argument.

Structured derivations can describe different kinds of mathematical arguments, like calculations, equation solving, proofs, geometric constructions, and so on. It can be used at any level of mathematics education, and for any topic in mathematics.

### eMath Environment

The eMath environment offers

- **tutorials** on structured derivations

- a complete **textbook series** for high school mathematics based on structured derivations

- an **editor** for writing and editing structured derivations

- a **checker** for checking the correctness of a structured derivation

The eMath environment extends the 4f Studio platform with special tools and content for mathematics education.

eMath tutorials, textbooks and editors can also be used independently of the 4f Studio platform.

### Structured Derivations

Structured derivations make the structure of mathematical reasoning clearly visible. The verbal justifications required for each derivation step helps the reader to understand why each step is correct. Structured derivations make mathematical reasoning much easier to understand, compared to the standard way of presenting mathematical arguments.

### eMath Editor

The **eMath editor** is used to write structured derivations. It will take care of the formatting so that you can concentrate on how to organize the mathematical reasoning.

The editor shows you the editing operations that are allowed at each place in the derivation. The editor uses standard mathematical formula notation in the derivation. Mathematical symbols can be written using LaTeX shortcuts, or by picking the special symbols from a palette.

When making a new calculation step, the editor first creates a copy of the previous term. This makes it easy to make small changes in large formulas, and reduces typing errors in the derivation. **Read more ...**

### Automatic Checking

The eMath checker reads a structured derivation step by step, and checks that each step is mathematically correct. It does this by expressing the correctness of the step as a mathematical theorem, and then using automatic theorem provers to try to prove the theorem. The checker warns the students for each step that has not been proved correct. This helps students find errors in their own derivations, and tells them when they have a correct solution. **Read more ...**

### eMath Textbooks

The **eMath textbooks**, *eMath 1 - eMath 10*, cover all topics in high school mathematics (grades 10 - 12). The books are structured according to the Finnish national curriculum for high school math, but can be used also in other national curriculum, as the content of high school math is more or less the same all over the world. The books are available in three languages, Finnish, Swedish, and English. The books all use structured derivations for presenting different kinds of mathematical arguments, like proofs, calculations, geometric constructions, and general problem solving. The books contain a lot of assignments, which the students can solve directly in the book using the eMath editors available in 4f Studio. **Read more... **

### eMath Studio

Using eMath content and tools in 4f Studio gives a very powerful environment for mathematics education, **eMath Studio**. eMath textbooks can be downloaded as 4f textbooks and used as course books in 4f courses. The students can use 4f notebooks for creating mathematical content and use the notebook tools when solving assignments directly in the course book. Students can check the correctness of their solutions using the eMath Checker. The strong support for mathematical content elements makes 4f Studio particularly well suited for teaching mathematics using structured derivations. **Read more ...**