Multiple Ways of Knowing
Multiple “Ways of Knowing” help students visualise, reinforce, and apply ideas quickly and accurately. Six distinct activity environments provide multiple representations of each concept that integrate with the conceptual sequence.
|Manipulatives||Conceptually understand what the concept “looks like”|
|Mainpulatives & 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|
|Number Bars||Number bars are a tool to represent magnitude. Students will progress to connect their understanding of magnitude to Parts-to-Whole, and Addition, and actively construct number sentences.|
|Dot Cards||Number sequence can be represented through models that show groups of discrete objects such as the dots on dot cards. Students will progress from counting dots to ‘subitising’ (or looking at cards and instantly tell ‘how many’ the card represents), and actively construct number sentences. |
|Number Lines||As students begin to become more comfortable with numbers, the Number Line is introduced. As with Number Bars and Dot Cards, Number Lines are used to construct number sentences. The Number Line is a very common model in math.|
|Fraction Bars||Students are introduced to fractions with simple models that can be divided into equal parts and filled. Students will progress to create the whole that will result from the sum of their parts. Later students will use only symbols, and then word problems, to demonstrate their mastery of addition with fractions.|
Conceptual Sequences of the Most Important Mathematical Ideas
Math is learned in a structured approach, that bridges each concept to the previous one.
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|
|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|
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.
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.
As scaffolding does not directly provide correct answers, students develop long-lasting problem solving skills and reduce their dependence on technology for solutions.
The scaffolding for 8 + 1 = ? is shown below.
|Help Button Activation||Help Provided for the Problem 8 + 1 = ?|
|1st||Show a "near neighbour": 7 + 1 = 8|
|2nd||Show a second "near neighbout": 9 + 1 = 10|
|3rd||Show 8 + 1 using number bars|
|4th||Show that the 9 bar is equal in length to the 8 and 1 bar|
In-Depth Problem Solving
Each stage in Symphony Math® features uniquely designed problems that emphasise 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.
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 to the right represent the progress of two 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.