# 3 : InstructionSupport students who need help in math

The best practices in developmental psychology and cognitive science are the cornerstones of Symphony Math®. Our uniquely designed delivery methods ensure that students – regardless of learning styles or knowledge levels – fully grasp fundamental mathematical ideas, even for difficult-to-explain and abstract concepts. The result is a solid foundation for acquiring higher math skills, as well as a positive learning experience.The primary elements in the Symphony Math® approach, which cover Common Core Math standards in grades K-5, include:

- Conceptual Sequences
- Multiple Ways of Knowing
- Instructive Feedback
- Active Scaffolding
- In-Depth Problem Solving
- Dynamic Branching
- Guided Practice Offline Worksheets
- Teacher Overviews

### Conceptual Sequences of the Most Important Mathematical Ideas

A tightly connected progression forms the conceptual sequence of Symphony Math®. These underlying “big ideas” provide the foundation for mathematical learning. As students master each big idea before moving on to the next, they learn to succeed with more complicated math later on.

Mathematical Topic | Underlying Big Idea |

Sequencing, Number | Quantity |

Addition and Subtraction | Parts-to-whole |

Place Value | Hierarchical grouping |

Multiplication and Division | Repeated equal grouping |

Multi-digit Addition and Subtraction | Hierarchical grouping coordinated with parts-to-whole |

Fractions | Repeated equal grouping coordinated with parts-to-whole |

### Multiple Ways of Knowing

Six distinct activity environments provide multiple representations of each concept and integrate with the conceptual sequence.

Activity | Purpose |

Manipulatives | Conceptually understand what the concept “looks like” |

Manipulatives & Symbols | Explicitly connect symbols to visual representations |

Symbols | Understand concepts at abstract levels |

Auditory Sentences | Learn the formal language of math |

Story Problems | Apply learning to real life problem solving |

Mastery Round | Develop immediate recall of number relationships |

#### Manipulatives

By “seeing” mathematical concepts, students develop mental models for meaning. In the screen below, a student must place a card with 2 dots to show that 1 + 1 = 2 (the part-to-whole concept).

#### Manipulatives and Symbols

Students learn the meaning of the symbols by explicitly connecting them to visual representations. In the screen below, a student must use symbols to construct the number sentence, “4 = 1 + 3.”

#### Symbols

Manipulatives automatically appear to help students working at this abstract level.

#### Story Problems

Story problems deliver real-life applications of the concepts and help students who learn better through narratives and examples.

#### Mastery Round

This learning environment fosters fluency once a student has demonstrated understanding of a concept using manipulatives, symbols, language, and story problems.

### Instructive Feedback

Instructive feedback encourages independent thinking by revealing the nature of each incorrect response. For example, if a student answers 3 + 2 = ? with a 6, the program immediately shows that a 2 bar combined with a 3 bar is not the same length as a 6 bar.

This approach helps students deduce for themselves why an answer is incorrect. This also is preferred to saying, “That’s not quite right, try again,” which often leads to guessing and no meaningful explanation of why the response was incorrect.

### Active Scaffolding

The “Help” button provides scaffolding that leads the student closer to the solution, but does not give the answer immediately.

For example, if a student is working on 8 + 1 = ?, she can press the “Help” button to activate scaffolding that will help her connect 8 + 1 with her knowledge of concepts and number relationships. Pressing the “Help” button again provides additional scaffolding. Hints include the addition of visuals onscreen, re-ordering of possible solutions, and inclusion of one of the elements of the correct solution.

As scaffolding does not directly provide correct answers, students develop long-lasting problem solving skills and reduce their dependence on technology for solutions.

### In-Depth Problem Solving

Each stage in Symphony Math® features uniquely designed problems that emphasize comprehension and problem solving.

For example, to master place value concepts, students solve a series of problems to understand the base ten system. Students combine numbers of different place values, such as “30 + 400 + 7 = ?”. They also create number sentences for which the sum is provided but the addends are missing, such as “? + ? + ? = 286″. Each addend must correspond to the ones, tens, and hundreds place value (e.g. 200 + 80 + 6 = 286). At the most difficult level, students provide three different solutions to this type of problem.

Students also learn to problem solve by connecting current problems to similar easier ones they answered previously.

### Dynamic Branching

The dynamic branching of Symphony Math® allows students to learn at their own levels. As the program illuminates an area of need, progress slows until the student achieves the necessary understanding.

For example, the graphs below represent the progress of two third grade students who used Symphony Math® for the same amount of time. The student on the left required relatively little practice to demonstrate mastery, whereas the student on the right needed more practice to fully grasp the concepts.

### Guided Practice

#### Offline Worksheets to Frame Conversations

Each concept in Symphony Math has accompanying worksheets that allow students to practice and master the most important ideas in mathematics. Each set of worksheets follows the same approach:

- Manipulatives
- Bridge (to Symbols)
- Symbols
- Apply (Story Problems with models)

Help indicators in Symphony reports suggest Guided Practice by comparing student performance to a nationally representative sample. For every suggestion, a simple click allows easy access to Guided Practice worksheets. Links to Latest Award certificates have also been added so that you can celebrate student achievement in the Symphony Math program.

#### Teacher Overviews

Every stage of Symphony Math now has a Teacher Overview available. The Teacher Overview provides in-depth information for teachers as they support their students, including:

- A full description of the “Big Idea” for the Stage
- Worksheet instructions
- Answer Keys for student worksheets
- Common Core State Standards reference