Learning math - interactive learning techniques

Learning techniques and methodical feedbacks

EMTeachline Mathematics Software - interactive math problem solving, test authoring

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 tables 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.

Types of tasks

Structure of methodical tasks

Structure of educational tasks

Groups of learning techniques - methodical schemes

Learning technique "To view"
Learning technique "To compare"
Learning technique "To insert"
Learning technique "To learn"
Learning technique "To view and learn"
Learning technique "To relate categories"
Learning technique "To apply"
Learning technique "To select for transformation"
Learning technique "To transform"
Learning technique "To solve independently"

Types of 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 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

●	Ordinary task-books versus training software Apart from standard schoolbooks that are selected by a teacher and are mandatory for a student, many people have to buy additional task-books and tutorials. Prior to visiting a book store, consider new opportunities opened by the world of training software...
●	Organization of an educational material in training software An educational material, both practical and theoretical, can be arranged in different ways, depending on methodical aims...
●	How many math examples should contain an ideal software? There are mathematics programs with no single example at all and there are programs with huge repositories of math examples. Both program types have their advantages and disadvantages...
●	Preparation of tests, variant tests and other methodical materials To prepare various tests, exams, revision exercises and homeworks, math teachers have to either compose examples themselves - which is a long story - or use the existing task-books - which is slow and costly. The EMTeachline mathematics software provides an ideal cost-effective help in these areas...
●	How does the EMTeachline mathematics software help math teachers? Once upon a time, when I was a math teacher I knew little about the world of training programs and was extremely skeptical about training potentials of mathematics software. Although I still firmly believe that no program can replace a good math teacher, I am positive that training programs can help math teachers a lot in the following areas...

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