3-C-2: Logic & Reasoning

The most valuable information in this module is Bloom’s Taxonomy.  As an educator, understanding the taxonomy is very  important because in addition to educating our students, we must know how to inspire them to think critically, make conjectures, and demonstrate their understanding.  More importantly, we have to know how to assess our student’s understanding, or lack thereof.  If our students aren’t able to analyze and synthesize the mathematics that’s taught within the classroom, then the education is useless because they’re unable to apply it to the real-world experiences they will encounter outside of the classroom.  Therefore, I believe the biggest value of Bloom’s Taxonomy is that it assists educators with enhancing our student’s intellectual growth.






Mathematics education today focuses more on standardized testing than logical and lateral thinking skills.  Therefore, unfortunately, our instruction is dictated by the previous year’s test results, not on whether or not students know how to engage in complex reasoning.  However, in an attempt to incorporate more logical/lateral thinking activities within my classroom, I plan to post a Problem of the Week on my classroom bulletin board.  Those students who answer the problems correctly will receive extra credit.  I also plan to infuse logical/lateral thinking problems into student’s journal writing.

3-B-2: Solving Lateral Thinking Problems

Eggs in a Basket Response

I think there is one egg left in the basket because one of the eggs was returned to the basket.   For example, six people took an egg out of the basket, one at a time, and they kept their egg.  However, one of the six people removed the egg from the basket, then returned it to the basket; thus, one egg was left in the basket.






Manhole Covers Response

Manhole covers are round because the manholes are round, and round covers are easier to remove and replace.  For example, round covers don’t have angles, so they can be replaced easier than a square cover with four angles.   Similarly, since circular covers don’t have angles, they can be rotated or reflected for removal purposes.  Squares, on the other hand, could only be reflected open.





As I think about the manhole covers question, I think it’s a Level 6 evaluation question on Bloom’s Taxonomy.   Level 6 comes to mind because the question requires one to make judgements and defend a position.   Nonetheless, the question can also fall under Levels 4 (Analysis)  and 5 (Synthesis).  For example, at Level 5, one is required to predict and imagine.  Therefore, in attempting to answer the question, one must predict why the covers are round, as well as imagine what would happen if the covers were square.  Similarly, at Level 4, one  would have to compare and contrast the properties of circles and squares.  Clearly, the properties associated with these figures have something to do with the fact that manhole covers are circular and not square.

If I were to present this question to my students, I would first make sure my students know what manhole covers are.  Therefore, I would print-out a few images from the internet, then proceed to ask them what they notice about the shape of all the various manholes.  Subsequently, I would have them discuss the question within their cooperative learning groups.  I would encourage them to create a manhole by cutting a circle out of a piece a paper, then I would provide them with square and circular manipulatives to use as their manhole covers.  The higher-order thinking questions I’d pose include the following:

Create a manhole by cutting a circle out of a piece of paper.  Use your diagram and the manipulatives  to explain what would happen if the manhole cover was a square. (Synthesis)

Analyze your “manhole” and both “covers”.  Which cover would be easiest to remove & replace?  Justify your answer using your prior knowledge of square and circles. (Evaluation)