Learning Techniques and Methodical Feedbacks

Math is learned by doing problems. In the EMTeachline mathematics software, the learning process relies on a set of learning techniques, the so-called 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 problem-solving skills. Each learning technique applies to the solution of a math problem and consists of a set of technique-specific 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:

  1. The name of the transformation step = a verbal formulation of math transformation
  2. The objective of the step = the purpose of the carried out transformation
  3. The formulation = formulation of rules used at the step of transformation
  4. The definition = definition of math concepts used at the step of transformation
  5. The formula = mathematical relationship used at the step of transformation
  6. 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 problem-solving 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 cause-and-effect 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 cause-and-effect 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 cause-and-effect 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 cause-and-effect 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 cause-and-effect 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