Saturday, May 30, 2015

Alphabet Blocks


According to Piaget, cognitive growth occurs
through physical maturation coupled
with firsthand experience. 
Blocks further these processes.
Dr. Elizabeth Hirsch
City University of New York

   Adults provide alphabet blocks to children to help them learn the 26 letters of the English alphabet and the ten numerals of the number system. However, if one observes children playing with alphabet blocks, they will sooner than later notice that the blocks are also used to build structures, especially towers.
   Action knowledge is what a learner learns (knows) by doing. Action knowledge is easily observed in preschool and older children by watching them stack blocks or build towers.
   Research indicates that even infants develop an understanding that an unsupported object falls. Toddlers learn how to place an 
object on a support, like a table top, so that it doesn’t fall.
   Every parent knows that this action knowledge takes many spilled cups of juice or milk for the toddler to master.      

    In the diagram, the center of the cube is the center of gravity (and in this case, the center of mass) of the cube. Expert knowledge accepts that, for all practical purposes, all of the mass of the cube can be considered to be concentrated at that point.

   Since we, as adults, have expert knowledge, we can use the center of gravity of the cube-shaped block to think about the stability of a cube.
   In the diagram, a cube (a) is shown in front view. The base of the cube is colored red. The center of gravity (cg) is the small circle and the arrow indicates the direction of the force of gravity (down, towards the center of the earth). 
   If the cube is tilted to the left, but the cg still falls within the red baseline (b), the cube will fall back to its original position (c). In this scenario, the cube is stable. However, if the cube is again tilted to the left but the cg falls outside of the red baseline (d), the cube is unstable and will tumble to the left (e).
   We can use the center of gravity concept to examine what happens if one block is stacked on top of another block. Essentially, there are only two possibilities; (1) the top block is stable and rests on the bottom block or (2) the top block is unstable and tumbles off of the bottom block. Again, examining the position of the cg of the top block determines which of the two events will happen
   If the cg of the top block is over the top face of the bottom block, the top block will be stable (a). If the cg of the top block lies off the face of the bottom bloc (b), the top block will be unstable and will tumble to the left.

   Through extended play, toddlers and preschoolers learn (action knowledge) to stack a tall tower of stable blocks.  The height of the tower can be taken as a measure of the youngster’s block-building skill.
   In the next post I will share a block-building session with Emma, Kate, and Asher, three grand kids visiting from their home in Georgia.

Sunday, May 17, 2015

Another Visit to My Mathematical Zoo

   In my previous post, My Mathematical Zoo,  I said that I would reveal the geometric shape that has a never-changing area but a perimeter that one can make as long as desired. Here is the shape!
   How was this shape born?  It evolved from a square.
   Gather 16 small squares (cut from paper) and form a 4x4 square as seen in this figure (color is not important). Let the side of the square equal one. Therefore, each edge of a small square is one-fourth the length of the side of the square.
   Starting at the top of the square, shift the second square to the right to a position above the first square along the top edge. Continue this process in the clockwise direction.
   Now, let’s compare the area of the original square and its offspring, the funny-looking shape seen above.
   The original square has an area of 16 square units and a perimeter of 4 units. After shifting the squares, the area of the shape is still 16 square-units but the perimeter has increased to 8 units!
   Let’s build a table.
   We could continue on with area tiles but it takes a lot them and it’s easy to make mistakes. So. Let’s focus on the perimeter of one side.  In Figures 1 and 2, I started with a line one unit long and then divided the line segment into fourths.

   Then building on Figure 2, I duplicate the shape of the top edge of the area tiles shape, as shown in Figure 3.
   Obviously, I can do the same to the other three sides to get the following shape as seen in Figure 4.  Note that the red line is identical to the shape we obtained by shifting the area tiles.
   The advantage of the line method is that it is easily extended. Each straight-line segment in Figure 4 can also be divided into fourths and the same figure as in Figure 1, at a reduced scale, can be drawn on each of the eight red line segments that make up the top edge. The result is shown in Figure 5.
   If the above construction is applied to all four sides of Figure 4, then the following shape, Figure 6, emerges. For comparison, I’ve colored the original square yellow.
   Those with an algebraic bent will appreciate the following generalization for the perimeter of this shape.
   Now do you understand why this shape is in my Mathematical Zoo?
   Isn’t geometry a wonderful construct of the human mind and imagination? Watch this short video and see how the curve I call the Squareflake is constructed.
   To view the Scratch program used to make the video click on the following link.