# 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.

The EMTeachline Mathematics Software offers two sorts of 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.

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

The database of math problems is structured by:

• topics
• 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 methodBuilding knowledge related with the cause-and-effect mathematical relationshipsBuilding knowledge related to theoretical materialBuilding knowledge related to application of formulas To view the steps of solutionTo view objectives of the steps of solutionTo view rules related to solutionTo view formulas associated with solution
 Learning technique “To compare” Methodical aims Methodical tasks Building abilities related to application of rulesBuilding abilities related to application of formulasBuilding knowledge related to the problem definitionBuilding abilities related to the problem definitionBuilding abilities related to the solution methodBuilding abilities to select math transformation To relate the steps of solution to their objectivesTo relate objectives to the steps of solutionTo relate the steps of solution to rules substantiating the stepsTo relate rules to the steps of solutionTo relate the steps of solution to formulas underlying the stepsTo relate formulas to the steps of solution
 Learning technique “To insert” Methodical aims Methodical tasks Building abilities related to application of rulesBuilding abilities related to application of formulasBuilding abilities related to the problem definitionBuilding abilities related to the solution methodBuilding skills related to the problem definitionBuilding abilities to select math transformationBuilding abilities to apply formulations and formulas To relate the steps of solution to their objectivesTo relate objectives to the steps of solutionTo relate the steps of solution to rules substantiating the stepsTo relate rules to the steps of solutionTo relate the steps of solution to formulas underlying the stepsTo relate formulas to the steps of solution
 Learning technique “To learn” Methodical aims Methodical tasks Building knowledge related to the solution methodBuilding knowledge related with the cause-and-effect mathematical relationshipsBuilding knowledge related to theoretical materialBuilding knowledge related to application of formulasBuilding knowledge related to application of formulationsBuilding knowledge related with the cause-and-effect relationships within theoretical materialBuilding skills related to theoretical materialBuilding knowledge related to the problem definitionBuilding skills related to application of formulasBuilding skills related to application of rulesBuilding skills to apply formulations and formulasBuilding skills to substantiate transformations and ground on formulas To learn objectives of the steps of solutionTo learn rules related to solutionTo learn formulas related to solution
 Learning technique “To view and learn” Methodical aims Methodical tasks Building knowledge related to the solution methodBuilding knowledge related with the cause-and-effect mathematical relationshipsBuilding knowledge related to theoretical materialBuilding knowledge related to application of formulasBuilding knowledge related to application of formulationsBuilding knowledge related with the cause-and-effect relationships within theoretical materialBuilding skills related to theoretical materialBuilding knowledge related to the problem definitionBuilding skills related to application of formulasBuilding skills related to application of rulesBuilding skills related to the problem definition To view the steps of solutionTo view objectives of the steps of solutionTo view rules related to solutionTo view formulas associated with solutionTo learn objectives of the steps of solutionTo learn rules related to solutionTo learn formulas related to solution
 Learning technique “To relate categories” Methodical aims Methodical tasks Building abilities related to application of rulesBuilding abilities related to application of formulasBuilding abilities related to the problem definitionBuilding abilities related to the solution methodBuilding abilities to select math transformationBuilding abilities to apply formulations and formulasBuilding abilities to substantiate solutions To relate the steps of solution to their objectivesTo relate objectives to the steps of solutionTo relate the steps of solution to rules substantiating the stepsTo relate rules to the steps of solutionTo relate the steps of solution to formulas underlying the stepsTo relate formulas to the steps of solution
 Learning technique “To apply” Methodical aims Methodical tasks Building abilities related to application of rulesBuilding abilities related to application of formulasBuilding abilities related to the problem definitionBuilding abilities related to the solution methodBuilding abilities to select math transformationBuilding abilities to apply formulations and formulasBuilding abilities to substantiate solutionsBuilding abilities to carry out transformations To select an objective of the current solution stepTo apply a rule substantiating the carried out transformationTo apply a formula underlying the current solution stepTo 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 rulesBuilding abilities related to application of formulasBuilding abilities related to the problem definitionBuilding abilities related to the solution methodBuilding abilities to select math transformationBuilding abilities to apply formulations and formulasBuilding skills to apply formulations and formulasBuilding skills to make transformationsBuilding skills to substantiate transformations and ground on formulasBuilding abilities to substantiate solutionsBuilding abilities to carry out transformations To select a verbal formulation for the current solution stepTo select a math transformation for the current solution stepTo select a math transformation for the next solution stepTo select a rule substantiating the current solution stepTo select a formula underlying the current solution stepTo select an objective for the next step of transformationTo select a rule for the next step of transformationTo select a formula for the next step of transformationTo select a math transformation
 Learning technique “To transform” Methodical aims Methodical tasks Building skills related to application of rulesBuilding skills related to the problem definitionBuilding abilities to select math transformationBuilding abilities to apply formulations and formulasBuilding skills to apply formulations and formulasBuilding skills to make transformationsBuilding skills to substantiate transformations and ground on formulasBuilding abilities to substantiate solutionsBuilding abilities to carry out transformationsBuilding abilities to solve independently To select an objective for the next step of transformationTo select a rule for the next step of transformationTo select a formula for the next step of transformationTo select a math transformationTo select a math transformation together with an objective of this transformationTo select a math transformation together with a rule substantiating this transformationTo 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 formulationsBuilding up skills related to the problem definitionBuilding up abilities related to the choice of transformationBuilding up abilities related to application of formulations and formulasBuilding up skills related to application of formulations and formulasBuilding up skills required for making transformationsBuilding up skills necessary for substantiation of formulas and transformationsBuilding up abilities necessary to carry out substantiationsBuilding up abilities to carry out transformationsBuilding up abilities required for independent solving To select an objective of the next step of solutionTo select a rule justifying the next solution stepTo select a formula that allows making the next stepTo select a transformationTo select a transformation and its objectiveTo select a transformation and rule justifying this transformationTo select a transformation and formula justifying this transformation