Post # 32 Grasping at Possibilities & Validating Curiosity

I was observing a teacher working with arrays in multiplication and noted that the student quite spontaneously began creating perfect squares with the manipulatives.  To the teacher's credit, she mentioned to the student how wonderful it was that he was exploring a topic he would study later.  She described the square he had created and that both factors were the same.  She created one more and compared the similar attributes for him, continually telling him that he would study these much later on.

The lesson plan has boundaries.  It does stipulate that a teacher cover specific material and try to accomplish the day's goals.  That said, we should always be open to curiosity and possibility.  If the student is intrigued by some aspect of math and ready to investigate, we can instill in the student a sense of adventure by noting when he or she touches on something which will be important for later study.  I would suggest that if the student appears curious, we occasionally stop planned work and allow students to investigate patterns and possibilities which will inform later work.  Allowing a student to occasionally follow curiosity validates their interest, intellect and observations.  I would have been quite happy if the instructor, who was not formally being observed, had stopped the lesson and allowed the student to create several perfect square arrays.  She did use the term "perfect square" and told him that it would be important later on.  For this I commend her.  I would have liked it if she had explored the pattern and even said the word "square root." Five minutes of following the student's interest may well be rewarded with increased attention and interest.

I have demonstrated this same lesson with slightly older students.  They construct the square numbers with unit cubes on a multiplication chart or foam squares on an array.  We transition to the representational after constructing squares to about sixteen. By this I mean drawing the external boundaries of the squares on the times table chart.  After the students have drawn all the square numbers to one hundred twenty one or one hundred forty four and identified the square roots, we are ready to learn the coding of the squares and roots at the abstract level.  The link to multiplication provides the bridge and the transition is exceedingly easy. If you consider another link, you might consider the square numbers to be similar to the “doubles facts” in addition/subtraction activities.  When both addends or both factors are the same, the memory load is reduced for association and students learn valuable patterns which will serve them at higher levels of math.