Just Keep Thinking, Just Keep Thinking!

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This week in my "Teaching in Mathematics" class we focused on a student's ability and process in problem solving. Problem solving 

questions are most often the hardest questions for student to complete and/or understand. The most common phrases from students during problem solving are "I don't get it!", "What are we doing!", "What's the answer!" and the worst one "I give up!". My teacher presented us with "Skyscraper" math problems and let me tell you, the atmosphere in our classroom full of adult Teacher Candidates was not much different from a class full of students. To give a brief summary of the Skyscraper problem, the students are given a handout with a square grid (can be 2x2 and up) and on the sides of the grid each square is labeled with a number as pictured here in this 4x4 example:

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We were given linking cubes by our teacher and were instructed to find the number of cubes needed on each square in order to represent the numbers on the side of the square. Huh? Exactly my thoughts! Well our only clue was that this problem is similar to Sudoku puzzles where a single number can only be represented once in each row and column. Our group started to build our towers for each square and right away I noticed our different approaches to solving this problem. I had taken the blocks and built the required number of buildings to fill the grid. Based on the Sudoku clue I knew that for a 4x4 grid we needed 4 buildings of each height (4 cubes tall, 3 cubes, 2 cubes, 1 cube). I needed to see what we were working with before I made decisions about where to place these buildings where as my group members were placing cubes right on the grid linking new heights and taking cubes away as they filled the grid. Once our group had reached the point of frustration, we asked our teacher for help and she explained that we should be looking to count the number of "pop out" cubes from the top. We attempted from this perspective for a while with no progress. In the end we did solve the problem by taking on a different view and counting the visible faces of the cubes. Once we knew the key to this puzzle we jumped from the 3x3 grid straight to the 5x5 and by applying our accrued knowledge completed it with ease. Now that we fully understood the problem we also realized that we could add one block to each building and have the same correct answer.

The same process and results can be seen in any problem solving situation, the goal for us teachers to is change the negative phrases I mentioned earlier to more positive and hopeful questions such as "What can I do next?", "What would happen if..?". The issue we should be focusing on is not that the student is struggling; it is how they react and overcome the struggle. My university math professor would say to us "struggle is good" and we would all laugh and possibly cry at the thought of working through some of those calculus problems but in the end I know that he was right. Whenever I had a question I found impossible, it challenged me to think creatively, try new things and not give up. Also for the people reading this saying  "I hate math I would have given up eventually" rest assured I did not solve all of those impossible questions and I did get the answer from my friends or the professor but it was because of my initial struggle and failed attempts that I understood the problem and the solution. Know that this can be applied to any subject or any problem in general. Before students give up or are told the answer they continue to learn as long as they make an attempt, success or fail they are gaining understanding through experience. This message I am sending is deeply related to the Constructivist Learning Theory in which "the core idea is that learners actively construct their own understandings, rather than absorb what they are told or copy what someone shows them" (The Problem-Solving Cycle: Professional Development for Middle School Mathematics Teachers The Facilitator’s Guide, 2003).

This week I leave you with this video that explains the answers are never easy, we have to grow in our experiences and whether it is in something you hate or something you love it is possible for everyone learn. 

References: 

The Problem-Solving Cycle: Professional Development for Middle School Mathematics Teachers The Facilitator’s Guide. (2003) (1st ed.). Retrieved from http://www.colorado.edu/education/staar/documents/FacilitatorsGuide.pdf

Khan Academy,. (2014). You Can Learn Anything. Retrieved from https://www.youtube.com/watch?v=JC82Il2cjqA