# Math Inspirations Discovery Method™ vs. Traditional Math*

*Traditional math programs included in this comparison: Right Start, Saxon, IXL, Life Of Fred, Khan Academy, Math U See, Teaching Textbooks, Glencoe, Houghton Mifflin Harcourt, etc.

**THE PROCESS**

**Math Inspirations Discovery Method™:**

**1.**Student creates definitions on their own and proves them to a mentor.

**2.** Student solves story and direct operation problems for a target concept/procedure (such as “How to multiply two 2-digit numbers”) on their own with no examples or instruction and proves their answers and reasoning to a mentor.

**3.** Student observes and records patterns and shortcuts from the story and direct operation problems.

**4.** Student teaches a mentor how to solve problems with various constraints (not saying numbers, not using hands, etc.).

**5.** Student creates and writes a hypothesis for the target concept/procedure in their own words from their own observations.

**6.** Student tests and adjusts their hypothesis and proves it true (a theorem) to a mentor.

**7.** Student applies their new theorem in engaging activities and real-world tasks.

**Traditional Method:**

**1.**Teacher/book/video introduces and explains new vocabulary and definitions.

**2.** Teach/book/video models the concept for the student with examples.

**3.** Teacher/book/video models the procedure (i.e steps to solve) with examples.

**4.** Student memorizes the concept model and the procedure which the teacher/book/video introduced.

**5.** Student repeats memorized procedure in practice problems (usually direct operation first and then story problems at the end).

**6.** Student is tested on whether or not they have memorized the previous facts and procedures.

**DEFINITIONS**

**Math Inspirations Discovery Method™:**

Students create their own definitions using a comparison chart and discussion question page. *In this example, students create definitions for positive number, negative number, and integer and use the discussion questions to develop a deeper conceptual understanding.*

**Traditional Method:**

Students are given definitions which they memorize. Definitions are written by the book’s author(s). *In these examples, the books give the student the definitions for positive number, negative number and integer. Note the difference in definitions between authors.*

**DEFINITIONS**

**Math Inspirations Discovery Method™:**

Students create their own definitions using a comparison chart and discussion question page. *In this example, students create definitions for positive number, negative number, and integer and use the discussion questions to develop a deeper conceptual understanding.*

**Traditional Method:**

Students are given definitions which they memorize. Definitions are written by the book’s author(s). *In these examples, the books give the student the definitions for positive number, negative number and integer. Note the difference in definitions between authors.*

**UNDERSTANDING THE CONCEPT**

**Math Inspirations Discovery Method™:**

No examples are given. Instead, students work through word problems using only what they already know and their problem solving skills to create their own models from the contexts of the problems. The goal of this process is for students to identify a model which makes sense to them and build their thinking skills and confidence. *In this example, students work through word problems involving the models of temperature, debt, depth, golf and football.*

**Traditional Method:**

Students are given examples showing a single way (rarely more than one) to model the concept. The goal of this process is for students to memorize an author-chosen model(s). *In these examples, students are given a model (zero pairs, number line, etc.) and shown how to use the model to explain the concept and solve problems.*

**UNDERSTANDING THE CONCEPT**

**Math Inspirations Discovery Method™:**

No examples are given. Instead, students work through word problems using only what they already know and their problem solving skills to create their own models from the contexts of the problems. The goal of this process is for students to identify a model which makes sense to them and build their thinking skills and confidence. *In this example, students work through word problems involving the models of temperature, debt, depth, golf and football.*

**Traditional Method:**

Students are given examples showing a single way (rarely more than one) to model the concept. The goal of this process is for students to memorize an author-chosen model(s). *In these examples, students are given a model (zero pairs, number line, etc.) and shown how to use the model to explain the concept and solve problems.*

**UNDERSTANDING THE PROCEDURE**

**Math Inspirations Discovery Method™:**

Students create their own procedure (called a theorem) in their own words by working through word problems using what they already know and their problem solving skills, collecting data from those problems, observing patterns, teaching another person, and creating and testing a hypothesis of their procedure. The goal of this process is for students to create their own theorem in their own words and in their own unique way of thinking. *In this example, students work through word problems for how to add with negative numbers, observing data, teaching and creating a hypothesis and testing and proving it true.*

**Traditional Method:**

Students memorize procedures through given examples outside of any real-world context. Although there are many different models and procedures for every math concept, books usually present only a single procedure with examples chosen by the author(s). *In these examples, students are given several examples of how to use a procedure which they memorize and replicate.*

**UNDERSTANDING THE PROCEDURE**

**Math Inspirations Discovery Method™:**

Students create their own procedure (called a theorem) in their own words by working through word problems using what they already know and their problem solving skills, collecting data from those problems, observing patterns, teaching another person, and creating and testing a hypothesis of their procedure. The goal of this process is for students to create their own theorem in their own words and in their own unique way of thinking. *In this example, students work through word problems for how to add with negative numbers, observing data, teaching and creating a hypothesis and testing and proving it true.*

**Traditional Method:**

Students memorize procedures through given examples outside of any real-world context. Although there are many different models and procedures for every math concept, books usually present only a single procedure with examples chosen by the author(s). *In these examples, students are given several examples of how to use a procedure which they memorize and replicate.*

**PRACTICE**

**Math Inspirations Discovery Method™:**

Students practice using their new theorem in a game or activity. Every topic in each unit includes at least one activity. *In this example, students use their new adding, subtracting, and multiplying with negatives to play the games Integer Football, and 999 to -999 and explore Consecutive Number Problems.*

**Traditional Method:**

Students regurgitate what they memorized from the examples using a worksheet or set of problems. Most problems are repetitive and few word problems are given. *In these examples, students repeat the procedures they memorized in the examples in a series of repetitive problems.*

**PRACTICE**

**Math Inspirations Discovery Method™:**

Students practice using their new theorem in a game or activity. Every topic in each unit includes at least one activity. *In this example, students use their new adding, subtracting, and multiplying with negatives to play the games Integer Football, and 999 to -999 and explore Consecutive Number Problems.*

**Traditional Method:**

Students regurgitate what they memorized from the examples using a worksheet or set of problems. Most problems are repetitive and few word problems are given. *In these examples, students repeat the procedures they memorized in the examples in a series of repetitive problems.*

**ASSESSMENT**

**Math Inspirations Discovery Method™:**

Student assessment is done through the process of proving their theorems. Students prove theorems by solving each test problem in two different ways, one using their hypothesis steps and the other using a model or manipulative to prove each answer true. Students record their final definitions and theorems in their own blank book called Definitions And Theorems. *In this example, students prove their new How To Add With Negative Numbers hypothesis using the given testing problems and records their new theorem in their own math Definitions And Theorems book.*

**Traditional Method:**

Students are assessed using free-response and multiple-choice quizzes and tests. The book provides the test answers and a student is scored on the number of correct answers they provided regardless of whether or not the student understands the concept. *In these examples, students use what they have memorized and practiced in free-response or multiple-choice quizzes and tests..*

**ASSESSMENT**

**Math Inspirations Discovery Method™:**

Student assessment is done through the process of proving their theorems. Students prove theorems by solving each test problem in two different ways, one using their hypothesis steps and the other using a model or manipulative to prove each answer true. Students record their final definitions and theorems in their own blank book called Definitions And Theorems. *In this example, students prove their new How To Add With Negative Numbers hypothesis using the given testing problems and records their new theorem in their own math Definitions And Theorems book.*

**Traditional Method:**

Students are assessed using free-response and multiple-choice quizzes and tests. The book provides the test answers and a student is scored on the number of correct answers they provided regardless of whether or not the student understands the concept. *In these examples, students use what they have memorized and practiced in free-response or multiple-choice quizzes and tests..*

**APPLICATION**

**Math Inspirations Discovery Method™:**

Students work through each of the 6 tasks given at the end of each topic. A task is a culminating problem in which students use their theorems from the topic to explore real applications of the topic as well as fun challenges. *In this example, students use their theorems for adding, subtracting, multiplying and dividing with negative numbers to explore real applications of integers and fun challenges with integers.*

**Traditional Method:**

Students apply the memorized procedures in a few word problems at the end of each problem set. Few traditional math programs offer a single task at the end of chapters. Most real-life applications are simply told to the student with limited exploration. *In these examples, students apply the memorized procedures on a few word problems (and, if provided, a task) after practicing the procedure on the given set of problems.*

**APPLICATION**

**Math Inspirations Discovery Method™:**

Students work through each of the 6 tasks given at the end of each topic. A task is a culminating problem in which students use their theorems from the topic to explore real applications of the topic as well as fun challenges. *In this example, students use their theorems for adding, subtracting, multiplying and dividing with negative numbers to explore real applications of integers and fun challenges with integers.*

**Traditional Method:**

Students apply the memorized procedures in a few word problems at the end of each problem set. Few traditional math programs offer a single task at the end of chapters. Most real-life applications are simply told to the student with limited exploration. *In these examples, students apply the memorized procedures on a few word problems (and, if provided, a task) after practicing the procedure on the given set of problems.*

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