Math is learned by doing problems. In the EMTeachline mathematics software, the learning process relies on a set of learning techniques, the socalled methodical schemes. Techniques are a necessary part of any training. Just like in sport one physical exercise trains a particular group of muscles, a separate learning algorithm trains a particular ability or skill in solving math problems and forms a piece of theoretical knowledge related to the chosen topic. All together, learning techniques will equip you with comprehensive theoretical knowledge and durable problemsolving skills. Each learning technique applies to the solution of a math problem and consists of a set of techniquespecific methodical tasks that should be discharged at each step of math problem solution.
Most of methodical tasks are designed to build an understanding of relations between and among math categories underlying the current and the next solution step. The software guides you through problem solving by asking questions and offering multiple choice answers with hints. A pivotal feature of learning algorithms is that they provide an interactive methodical feedback – methodical recommendations for error correction derived from an objective analysis of your work. This is a key component of the learning process.
The list below show what kind of knowledge, abilities and skills you can build using different learning techniques. These tables ground in pedagogics and address mostly math teachers, educators and homeschooling parents.
Sorts of Software tasks
The EMTeachline Mathematics Software offers two sorts of tasks:
 Educational tasks
 Methodical tasks
An educational task is just a math problem to be solved, such as “Solve an equation”, “Evaluate an expression”, “Prove an identity”, etc. Currently, we offer about 40 different types of educational tasks.
A methodical task applies to the selected educational task and represents a set of exercises to be discharged at each solution step. This set makes up a “methodical scheme”. Most of methodical tasks are designed to build an understanding of relations between and among math categories underlying the current and the next solution step. For instance, the educational task “Solve an equation” can be accompanied by the following methodical tasks:
 Review the steps of solution
 Indicate formulas that were used at each step of solution
This algorithm models a situation in classes, when math teacher explains the solution and simultaneously asks theoretical questions.
Structure of methodical tasks
With regard to each step of math transformations, we designate the following math categories:
 The name of the transformation step = a verbal formulation of math transformation
 The objective of the step = the purpose of the carried out transformation
 The formulation = formulation of rules used at the step of transformation
 The definition = definition of math concepts used at the step of transformation
 The formula = mathematical relationship used at the step of transformation
 The proof = derivation of a formula or theorem used at the step of transformation
Most of methodical tasks are designed to build an understanding of relations between and among math categories designated above.
Just an example: the learning technique “To relate the steps of solution to objectives of these steps” implies the following:
 The program provides a step by step solution to a selected math problem
 At each solution step, the program offers you a list with possible objectives
 You should select the appropriate objective
Structure of educational tasks
The database of math problems is structured by:
 topics
 types of educational tasks
 solution methods
 levels of solution complexity
These structural parameters are mirrored in the titles of themes, chapters and sections. All titles are made as short as possible, to avoid inconvenience of working with long names in the lists.
The titles of themes (the first level of the list “Contents”) reflect subjects and general solution methods.
The titles of chapters (the second level of the list) characterize types of educational tasks.
The titles of sections (the third level of the list) correlate with solution methods.
All programs allow sorting math problems by their complexity.
Groups of learning techniques – methodical schemes
Learning techniques – methodical schemes – are designed to help students grasp theoretical material and learn practical problem solving abilities and skills. Each group of learning techniques trains a particular math skill and forms a piece of theoretical knowledge. The table below shows what kind of qualities different methodical schemes exercise. The subsequent tables clear up and detail these general qualities.
Groups of methodical schemes  Methodical aim 
“To view”  shaping math knowledge 
“To learn” “To solve independently”  formation of theoretical knowledge and problemsolving skills 
“To relate” “To insert” “To apply”  formation of abilities to understand problem definition, to reason and apply formulas 
Learning technique “To view” 
Methodical aims  Methodical tasks 
 Building knowledge related to the solution method
 Building knowledge related with the causeandeffect mathematical relationships
 Building knowledge related to theoretical material
 Building knowledge related to application of formulas
  To view the steps of solution
 To view objectives of the steps of solution
 To view rules related to solution
 To view formulas associated with solution

Learning technique “To compare” 
Methodical aims  Methodical tasks 
 Building abilities related to application of rules
 Building abilities related to application of formulas
 Building knowledge related to the problem definition
 Building abilities related to the problem definition
 Building abilities related to the solution method
 Building abilities to select math transformation
  To relate the steps of solution to their objectives
 To relate objectives to the steps of solution
 To relate the steps of solution to rules substantiating the steps
 To relate rules to the steps of solution
 To relate the steps of solution to formulas underlying the steps
 To relate formulas to the steps of solution

Learning technique “To insert” 
Methodical aims  Methodical tasks 
 Building abilities related to application of rules
 Building abilities related to application of formulas
 Building abilities related to the problem definition
 Building abilities related to the solution method
 Building skills related to the problem definition
 Building abilities to select math transformation
 Building abilities to apply formulations and formulas
  To relate the steps of solution to their objectives
 To relate objectives to the steps of solution
 To relate the steps of solution to rules substantiating the steps
 To relate rules to the steps of solution
 To relate the steps of solution to formulas underlying the steps
 To relate formulas to the steps of solution

Learning technique “To learn” 
Methodical aims  Methodical tasks 
 Building knowledge related to the solution method
 Building knowledge related with the causeandeffect mathematical relationships
 Building knowledge related to theoretical material
 Building knowledge related to application of formulas
 Building knowledge related to application of formulations
 Building knowledge related with the causeandeffect relationships within theoretical material
 Building skills related to theoretical material
 Building knowledge related to the problem definition
 Building skills related to application of formulas
 Building skills related to application of rules
 Building skills to apply formulations and formulas
 Building skills to substantiate transformations and ground on formulas
  To learn objectives of the steps of solution
 To learn rules related to solution
 To learn formulas related to solution

Learning technique “To view and learn” 
Methodical aims  Methodical tasks 
 Building knowledge related to the solution method
 Building knowledge related with the causeandeffect mathematical relationships
 Building knowledge related to theoretical material
 Building knowledge related to application of formulas
 Building knowledge related to application of formulations
 Building knowledge related with the causeandeffect relationships within theoretical material
 Building skills related to theoretical material
 Building knowledge related to the problem definition
 Building skills related to application of formulas
 Building skills related to application of rules
 Building skills related to the problem definition
  To view the steps of solution
 To view objectives of the steps of solution
 To view rules related to solution
 To view formulas associated with solution
 To learn objectives of the steps of solution
 To learn rules related to solution
 To learn formulas related to solution

Learning technique “To relate categories” 
Methodical aims  Methodical tasks 
 Building abilities related to application of rules
 Building abilities related to application of formulas
 Building abilities related to the problem definition
 Building abilities related to the solution method
 Building abilities to select math transformation
 Building abilities to apply formulations and formulas
 Building abilities to substantiate solutions
  To relate the steps of solution to their objectives
 To relate objectives to the steps of solution
 To relate the steps of solution to rules substantiating the steps
 To relate rules to the steps of solution
 To relate the steps of solution to formulas underlying the steps
 To relate formulas to the steps of solution

Learning technique “To apply” 
Methodical aims  Methodical tasks 
 Building abilities related to application of rules
 Building abilities related to application of formulas
 Building abilities related to the problem definition
 Building abilities related to the solution method
 Building abilities to select math transformation
 Building abilities to apply formulations and formulas
 Building abilities to substantiate solutions
 Building abilities to carry out transformations
  To select an objective of the current solution step
 To apply a rule substantiating the carried out transformation
 To apply a formula underlying the current solution step
 To apply a rule and formula substantiating the current solution step

Learning technique “To select for transformation” 
Methodical aims  Methodical tasks 
 Building abilities related to application of rules
 Building abilities related to application of formulas
 Building abilities related to the problem definition
 Building abilities related to the solution method
 Building abilities to select math transformation
 Building abilities to apply formulations and formulas
 Building skills to apply formulations and formulas
 Building skills to make transformations
 Building skills to substantiate transformations and ground on formulas
 Building abilities to substantiate solutions
 Building abilities to carry out transformations
  To select a verbal formulation for the current solution step
 To select a math transformation for the current solution step
 To select a math transformation for the next solution step
 To select a rule substantiating the current solution step
 To select a formula underlying the current solution step
 To select an objective for the next step of transformation
 To select a rule for the next step of transformation
 To select a formula for the next step of transformation
 To select a math transformation

Learning technique “To transform” 
Methodical aims  Methodical tasks 
 Building skills related to application of rules
 Building skills related to the problem definition
 Building abilities to select math transformation
 Building abilities to apply formulations and formulas
 Building skills to apply formulations and formulas
 Building skills to make transformations
 Building skills to substantiate transformations and ground on formulas
 Building abilities to substantiate solutions
 Building abilities to carry out transformations
 Building abilities to solve independently
  To select an objective for the next step of transformation
 To select a rule for the next step of transformation
 To select a formula for the next step of transformation
 To select a math transformation
 To select a math transformation together with an objective of this transformation
 To select a math transformation together with a rule substantiating this transformation
 To select a math transformation together with a formula underlying this transformation

Learning technique “To solve independently” 
Methodical aims  Methodical tasks 
 Building up skills related to application of formulations
 Building up skills related to the problem definition
 Building up abilities related to the choice of transformation
 Building up abilities related to application of formulations and formulas
 Building up skills related to application of formulations and formulas
 Building up skills required for making transformations
 Building up skills necessary for substantiation of formulas and transformations
 Building up abilities necessary to carry out substantiations
 Building up abilities to carry out transformations
 Building up abilities required for independent solving
  To select an objective of the next step of solution
 To select a rule justifying the next solution step
 To select a formula that allows making the next step
 To select a transformation
 To select a transformation and its objective
 To select a transformation and rule justifying this transformation
 To select a transformation and formula justifying this transformation
