# How Student Learning in DreamBox Transfers Skills Beyond the Computer

The short answer to this question is that we always have DreamBox lessons in which students must answer problems without using the manipulatives and visuals. Our lesson progressions are very carefully crafted to be sure students move from quantitative reasoning with manipulatives to more abstract reasoning without them. To illustrate with one example, I’ll use our DreamBox Open Array virtual manipulative (which is well known for its “googly eyes” that are fun to interact with). This manipulative is used for students to make sense of the distributive property, partial products, and 2- or 3-digit multiplication. Teachers can use this manipulative on an Interactive Whiteboard by clicking here, and anyone can try a lesson as a student here.

In early lessons, students use this DreamBox manipulative to build individual partial product arrays however they wish. They use a “zipper” to compose the entire array from smaller ones. They start by being able to build up to six partial products, but eventually must create no more than four. In early lessons, students aren’t required to solve the partial products because they need to focus on the array model instead of the computation. There are virtually an infinite number of possible ways for students to compose the array, so DreamBox is able to capture each student’s unique strategies.

In later lessons, the “zipper” interaction is replaced with a “button” interaction, where students are given only one point (looking like a coat button) that they drag around inside the array. The button is at the center of two axes, and manipulating the button moves a vertical axis and a horizontal axis that automatically divides the array into 2 or 4 pieces. The partial products are displayed on the left and live update as students move the button around. Students are looking for optimal partial products that are easy to compute mentally. So for a problem such as 47×53, an optimal group of partial products would be 40×50, 40×3, 7×50, and 7×3.

In the last lessons with this DreamBox open array, students don’t interact with the virtual manipulative at all. By this point, students have had a significant amount of time with concrete interactions and now encounter problems requiring more abstract reasoning. Instead of manipulating the array, students are simply shown the array and are asked to enter an equation that will define their first partial product. That first product defines the other partial products similar to how the button interaction defined two or four partial products. So for 47×53, if a student entered 40 × 50 = 2000 as their first equation, then they would have to solve 7×50, 40×3, and 7×3 before calculating the sum of all four partial products. Students see what they built represented on the array, but there isn’t any interaction with the array. And if a student typed 7×3 =21 first, it would be just as valid because it determines the same four partial products but in a different order.

There are a many other subtle adjustments within the learning progression with this open array virtual manipulative, but these are the “big” moves from quantitative to abstract reasoning where students stop interacting directly with the manipulative and are required to work only with the numbers in DreamBox. Students also later work with another array that serves as a bridge to the standard algorithm with estimation. In the algorithm lesson, we’re using the paper-pencil representation, but with the added step of requiring an estimate before computing so that students aren’t simply following steps.

In conclusion, to demonstrate long-term proficiency in all of our Pre-K through grade 6 DreamBox content, students must eventually complete problems and lessons without ever actually seeing the manipulatives or workspaces. We ensure students can perform and transfer in mathematics without the concrete representations because they have had plenty of time to develop strong mental models of those representations. Thanks for taking some time to gain some insights into how we ensure student learning in DreamBox transfers beyond the computers and concrete manipulatives.

### Tim Hudson

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