### So…. what are some of these activities and conversations we can have with young children to thrust them upward on a powerful learning trajectory in mathematics?

Let’s begin by looking into a “tool bag,” containing several types of concrete materials. These materials can provide topics for power-chats between an adult and a young child or group of children. The materials in these bags can be purchased commercially on the Internet, but my Power-Start!Mathematics bags (pictured here) are special because they also include written instructions about some conversations I believe are particularly important for building strong math foundations.

Power-Start! Mathematics Tool Bags

I’m hoping in 2018 to get my message about the importance of early experiences with mathematics, not only through this website but by making presentations. At these presentations, I will have my Power-Start!Mathematics tool bags available for sale. Included in the bags are the following 4 sets of concrete materials, sometimes referred to as “math manipulatives.”

- A set of 20 poker chips of two different colors (10 of each color)
- A set of pattern blocks of 6 different colors and shapes: orange square; green equilateral triangle; red trapezoid; blue rhombus; ivory rhombus (thin); and yellow hexagon.
- A set of 100 stacking cubes, 10 stacks of 10 different colors
- A set of 36 one-inch square tiles of two different colors (18 of one color and 18 of another).

**Poker Chips** A fellow doctoral student and I hit on the idea of using poker chips for counters when we were interviewing preschoolers about counting and cardinality. (Follow the link to another page “The Cardinality Principle.”) We found them to be easy for children to pick up, and the interviewer could slide them under a piece of paper as a hiding place. Concrete materials such as poker chips, counters, or other small objects can be used to model counting, as well as the four operations of addition, subtraction, multiplication, and division.

**Stacking Cubes ** While children are learning to count to 20 and beyond, we can prepare them for discovering the great power of our base-ten number system, by introducing them to the idea of ten as a unit. With stacking cubes, a great visual image for any number up to 100 can be constructed. For example, 74 can be modeled with seven stacks of ten and a stack of four. Stacking cubes can also be used to model operations and even to construct geometric shapes.

**Pattern Blocks** It’s truly amazing all the things that can be done with sets of these six color-coded shapes! Children can make pictures, work puzzles, construct tiling patterns; and make initial explorations into quantifying and comparing areas of shapes. Just google “pattern blocks” on the Internet to begin exploring the possibilities for creativity, fun, and learning about shapes.

**Square Tiles ** Each of these flat tiles measures one inch (1 in.) on a side and has an area of one square inch (1 sq. in.). Preschool experiences filling rectangles with square tiles can lead to early understandings of area measurement.

**The Great Value of Early Experiences with Concrete Materials**

As children move up through elementary school, counters, tiles, and stacking cubes will be replaced by symbols for representing number ideas. Therefore, I can’t emphasize enough the importance of concrete experiences with math ideas in preschool and primary grades. For example, an understanding of the equation “3 + 4 = 7” should be grounded in the visual memory of composing groups of four and three objects and knowing there are seven all together. Furthermore, a key third grade expectation, a stepping stone along the path to world-class math proficiency, is knowing from memory the multiplication tables. However, before third graders commit these facts to memory they should have deep conceptual understandings for the operation of multiplication (when it is used) and its operational properties ( order doesn’t matter and division is the reverse the multiplication process). For example, 3 x 8 = 24 should be associated visually in a child’s mind with eight groups having three things in each group; 3 groups having 8 things in each group; an array of 24 dots with three rows and eight columns; or a rectangle measuring 8 inches by 3 inches filled with 24 square tiles one inch on a side.

Here are links for two activities using poker chips.