Skyscraper Activity: Mathematical Processes and Constructing Your Own Learning

This week in class we participated in an activity called Skyscraper. For this activity students are given a grid with numbers along the outside (see example below) and a handful of unit snap cubes.  The purpose of the game is to fill the grid with stacks of cubes, each of which represents a building.  The buildings must be placed in a way that each number on the outside of the grid represents the number of buildings (or stacks of cubes) someone standing there would be able to see.  Similar to the game Sudoku, each column and row must have one each of each building height.
  

Example Grid

Solution to Example Grid

When we first were given the activity, the rules of the game were not explicitly given to us.  We almost immediately began the activity, with our facilitator Amy only quickly giving us a brief explanation.  These limited instructions and examples resulted in almost the entire class misinterpreting the rules of the game.  Instead of thinking that the outside numbers represented the number of buildings (or stacks) we can see, we were under the impression that the numbers represented the number of floors (or blocks) we could see.  This led to confusion and frustration when trying to solve the problem.  Amy walked around the room, giving hints to certain groups (for example, she told us that it is similar to Sudoku). However this initial miscommunication of the rules of the game caused a lot of problems when trying to solve the problem.

Misinterpretation (how many floors you see):
Front = 3; Back = 3

Correct (how many buildings you see):
Front = 1; Back = 2

Misinterpretation (how many floors you see):
Front = 3; Back = 3

Correct (how many buildings you see):
Front = 3; Back = 1

 

Misinterpretation (how many floors you see): 

Front = 3; Back = 3

Correct (how many buildings you see):
Front = 2; Back = 2

Initially for this activity, I was going to discuss the importance of being clear with the instructions you give to your students so they are fully aware of what is expected of them.  However, after reflecting on the activity in its entirety, I believe that the purpose of this activity was to experience the many different mathematical processes, as well as realizing the importance of letting students discover and construct their own learning.  If we were given the full and correct rules of the game, less mathematical processing would have occurred.

 

(image created by Lindsey Crawford)

Reasoning and Proving

Through misinterpreting the rules of the game, we were able to reason with each other and prove why either the rules given to us were incorrect, the grid numbers were impossible, or there was something else we were missing.  For example, I proved that with using the incorrect rules it would be impossible for numbers opposite to each other on the grid to be different.  This is because no matter which side of the buildings you are standing on (front or back) you will still see the same number of floors (or blocks).  This started to get us thinking that maybe we misunderstood the rules.

Communicating

Through discussing with my peers as well as the teacher, we were able to realize that the rules of the game were misinterpreted.  When Amy communicated to my group the hint about Sudoku, we were able to problem solve and realize the correct rules of the game.  Once we determined that the numbers represented the number of buildings we could see as opposed to the number of floors, we solved the problem together, communicating our ideas and reasoning.  When we determined where a stack of blocks would go, we would explain our reasoning to the other members in order to prove that it is correct, as well as contribute to our whole group's understanding.

Connecting

Connecting occurred in this activity through relating the rules of the game to a game that most of us would likely be very familiar with, i.e. Sudoku.  Making this connection assisted in our understanding that each row and column should have one of each sized building.  Connecting mathematics to previous experiences is very efficient for furthering one's understanding of a new problem and helping with one's approach to problem solving.

Reflecting

After completing a puzzle, we would look at the solution making sure it was correct.  We looked at all the numbers to ensure they accurately described how many buildings we could see.  We also made sure that each row and column had one of each building height.  This reflection of our solution would help us find any mistakes we may have made.  Reflecting is an important step in solving math problems.  When you reach an answer to a problem, it is important to reflect on it and make sure your answer makes sense before dubbing it your final answer.

 
 

Other mathematical processes occurred during this activity, but these four were the most prevalent in my opinion.  I realized many different things during the activity about being a math teacher.  One thing I learned is that as a math teacher, it is important to sometimes simply step back from the problem and let students problem solve on their own or with their peers.  Discovering mathematics is the best way to truly understand and appreciate it.  I also learned that the final result of a math problem is not nearly as important as how you got there.  The mathematical processes that occur during the problem solving and the discovering of mathematics are most important.