The South African Way
Millroy (1992) studied a group of carpenters in Cape Town, South Africa to learn how they utilized mathematics in their everyday job duties. These carpenters stopped all formal mathematics training from the ages of 12 to 16 years old and did not consider their
work to have much to do with “real” mathematics. One day the researcher observed one of the carpenters finding the center of a box lid, so that a star design could be placed at the center of a piece of furniture. The scenario of the carpenter finding the center of the box lid is as follows:The carpenter took a box lid and drew diagonals (See figure 1). He measured diagonally across the frame from corner to corner and recorded the length. He then measured the length diagonally between the other two corners. According to the carpenter, if the lengths are slightly different, then the cupboard or table frame is not a square. (Millroy, 1992).
The carpenters referred to all rectangular shapes as “squares.” This is an incorrect
statement mathematically and should be pointed out to the students. All squares are
rectangles, but not all rectangles are squares. The carpenters measure both parts of each
diagonal. If all four parts are the same length, then the table frame is a rectangle. The
intersection of the two diagonals is the “center” of the box. The center of a rectangle is defined as a point inside the rectangle equidistant from the four corners (vertices). The researcher asked the carpenter why this method of finding the center worked, and the carpenter replied, “Because it is the center” (Millroy ,1992). Many times the carpenters
gave explanations similar to this when asked why things were done a certain way.This group of carpenters has learned to find the center of a box lid through everyday job experiences and not from geometry principles learned in school. What this means is that these carpenters were taught certain processes to follow to perform job tasks, but they can not tell you the specific geometric principles that enable them to be successful with their job tasks. There are three geometry principles that the carpenters understand that enable them to correctly find the “center” of a box (Millroy, 1992). First, the opposite sides of a rectangle are equivalent in length. Secondly, the lengths of the diagonals of a rectangle are equal, which is not the case for other parallelograms.Thirdly, the diagonals of a rectangle bisect each other, which allows the carpenters to find the “center” from the point of intersection of the diagonals. From the point of intersection of the diagonals, it is the same distance to each one of the corners or vertices,which is why we can call that point the “center.” With other parallelograms that are not rectangles, the intersection of the diagonals does not produce a “center.” Since the non rectangular parallelograms have diagonals that are not congruent, then the distance from the point of intersection to each of the 4 corners or vertices would not be equal; therefore, that point would not be a true “center” by definition. This activity illustrates how South African carpenters utilize mathematics in their everyday lives to find the center of a rectangular side of a piece of furniture.
Now you will try to recreate how the South African carpenters found the center of the rectangle. First move the vertices of the square to A = (1,6) B = (5,6) C = (5,0) and D = (1,0). This is your table top. Take point G and move it to point A and point H and move it to point C. This creates one of the diagonal lines. Repeat the process with point E to point B and point F to point D. This creates the other diagonal line Where do these two diagonal lines meet? What is the name of that point? Remember it for later. Now read on...
work to have much to do with “real” mathematics. One day the researcher observed one of the carpenters finding the center of a box lid, so that a star design could be placed at the center of a piece of furniture. The scenario of the carpenter finding the center of the box lid is as follows:The carpenter took a box lid and drew diagonals (See figure 1). He measured diagonally across the frame from corner to corner and recorded the length. He then measured the length diagonally between the other two corners. According to the carpenter, if the lengths are slightly different, then the cupboard or table frame is not a square. (Millroy, 1992).
The carpenters referred to all rectangular shapes as “squares.” This is an incorrect
statement mathematically and should be pointed out to the students. All squares are
rectangles, but not all rectangles are squares. The carpenters measure both parts of each
diagonal. If all four parts are the same length, then the table frame is a rectangle. The
intersection of the two diagonals is the “center” of the box. The center of a rectangle is defined as a point inside the rectangle equidistant from the four corners (vertices). The researcher asked the carpenter why this method of finding the center worked, and the carpenter replied, “Because it is the center” (Millroy ,1992). Many times the carpenters
gave explanations similar to this when asked why things were done a certain way.This group of carpenters has learned to find the center of a box lid through everyday job experiences and not from geometry principles learned in school. What this means is that these carpenters were taught certain processes to follow to perform job tasks, but they can not tell you the specific geometric principles that enable them to be successful with their job tasks. There are three geometry principles that the carpenters understand that enable them to correctly find the “center” of a box (Millroy, 1992). First, the opposite sides of a rectangle are equivalent in length. Secondly, the lengths of the diagonals of a rectangle are equal, which is not the case for other parallelograms.Thirdly, the diagonals of a rectangle bisect each other, which allows the carpenters to find the “center” from the point of intersection of the diagonals. From the point of intersection of the diagonals, it is the same distance to each one of the corners or vertices,which is why we can call that point the “center.” With other parallelograms that are not rectangles, the intersection of the diagonals does not produce a “center.” Since the non rectangular parallelograms have diagonals that are not congruent, then the distance from the point of intersection to each of the 4 corners or vertices would not be equal; therefore, that point would not be a true “center” by definition. This activity illustrates how South African carpenters utilize mathematics in their everyday lives to find the center of a rectangular side of a piece of furniture.
Now you will try to recreate how the South African carpenters found the center of the rectangle. First move the vertices of the square to A = (1,6) B = (5,6) C = (5,0) and D = (1,0). This is your table top. Take point G and move it to point A and point H and move it to point C. This creates one of the diagonal lines. Repeat the process with point E to point B and point F to point D. This creates the other diagonal line Where do these two diagonal lines meet? What is the name of that point? Remember it for later. Now read on...
The American Way
Two American carpenters were asked individually how they would proceed to place a design at the center of a rectangular side of a piece of furniture.The American carpenters informed us that cabinet doors and other rectangular pieces of wood are usually ordered from catalogs and come with exact measured
dimensions. Thus, when the carpenters receive the wooden boards for making furniture, they do not have to generally verify whether or not the boards are rectangles. First, they measure the length of a pair of the parallel sides (AB &CD) and mark the center of each side, points Y and Z. Secondly, they connect the center points of each parallel side with a line segment YZ. The American carpenters point out that segment YA is a perpendicular bisector for segments AB and CD and verify this by measuring angle BYZ and angle YZD and show me that both angles measure 90 degrees.Thirdly, the carpenters repeat the same
process with the other parallel sides, segment BD and segment AC. Next, the center points of each parallel side (X) and (W) are connected with line segment WX, which is the perpendicular bisector of AC and BD (figure 5). Finally, through years of experience, the carpenters realize that point O, the intersection of the perpendicular bisectors, is the center point of the rectangle.
From the method followed by the American carpenters, it is easy to verify that point O is the center of the rectangle. Again, let me emphasize that the center of a rectangle is defined as an interior point equidistant from the four vertices. Since point O lies on the perpendicular bisector of segments BD and AC, then point O is equidistant from the endpoints of both segments (Figure 6). Thus OB=OD and OA=OC. Also, point O lies on the perpendicular bisector of segments CD and AB, and point O is equidistant from the endpoints of both these segments (Figure 6). Thus OB=OA and OD=OC. The lengths of the four
segments are equal, which makes point O the center of the rectangle.
Now you will try to recreate how the American carpenters found the center of the rectangle. First move the vertices of the square to
A = (1,6) B = (5,6) C = (5,0) and D = (1,0). This is your table top. Find the midpoint of segment AB, move point G to this point. Now, find the midpoint of segment CD, move point H to this point. This line is one of the perpendicular bisectors. Repeat this process for segments AD and BC. This creates the other perpendicular bisector. Where do these bisectors cross? What is the name of the point? Do you remember where you have seen it before? Yes, it is the same point the African Carpenters found for their center !
dimensions. Thus, when the carpenters receive the wooden boards for making furniture, they do not have to generally verify whether or not the boards are rectangles. First, they measure the length of a pair of the parallel sides (AB &CD) and mark the center of each side, points Y and Z. Secondly, they connect the center points of each parallel side with a line segment YZ. The American carpenters point out that segment YA is a perpendicular bisector for segments AB and CD and verify this by measuring angle BYZ and angle YZD and show me that both angles measure 90 degrees.Thirdly, the carpenters repeat the same
process with the other parallel sides, segment BD and segment AC. Next, the center points of each parallel side (X) and (W) are connected with line segment WX, which is the perpendicular bisector of AC and BD (figure 5). Finally, through years of experience, the carpenters realize that point O, the intersection of the perpendicular bisectors, is the center point of the rectangle.
From the method followed by the American carpenters, it is easy to verify that point O is the center of the rectangle. Again, let me emphasize that the center of a rectangle is defined as an interior point equidistant from the four vertices. Since point O lies on the perpendicular bisector of segments BD and AC, then point O is equidistant from the endpoints of both segments (Figure 6). Thus OB=OD and OA=OC. Also, point O lies on the perpendicular bisector of segments CD and AB, and point O is equidistant from the endpoints of both these segments (Figure 6). Thus OB=OA and OD=OC. The lengths of the four
segments are equal, which makes point O the center of the rectangle.
Now you will try to recreate how the American carpenters found the center of the rectangle. First move the vertices of the square to
A = (1,6) B = (5,6) C = (5,0) and D = (1,0). This is your table top. Find the midpoint of segment AB, move point G to this point. Now, find the midpoint of segment CD, move point H to this point. This line is one of the perpendicular bisectors. Repeat this process for segments AD and BC. This creates the other perpendicular bisector. Where do these bisectors cross? What is the name of the point? Do you remember where you have seen it before? Yes, it is the same point the African Carpenters found for their center !
The South African carpenters and American carpenters both found the center of a rectangle, but they utilized different methods. The South African carpenters centered their approach around their knowledge of diagonals, and the American carpenters approached the task by building upon their knowledge of perpendicular bisectors. This comparison demonstrates that in other cultures individuals sometimes use different problem solving methods. This is an important concept because students must realize that there is usually more than one problem solving strategy to arrive at the correct answer.