How My 4-Year-Old Just Broke Down Combining Triangles to Form Polygons

In his four years of existence, my son Thaddeus has developed a knack for building and connecting things.  If  he gets a bag of jumbo Legos, he will create an entire jungle filled with imaginary animals!  If my wife and I give him a new 100-piece puzzle, he usually ends up putting it together within 24 hours. 

If he sees a bunch of his books scattered all over the place, he will either line the books up or stack them neatly in a tall pile. With everything that he creates, he is adamant about making sure it is done with precision.

All that said, it came as no surprise to me when Thaddeus rushed into my bedroom one rainy morning, excited and eager to show me another one of his new creations!  

Using a colorful assortment of triangle magnets, he diligently connected them to form a big multi-colored hexagon. After forming the hexagon, he then disconnected it to show me that the hexagon was formed by six smaller color-coded triangles.

As his father, I was so proud of his discovery that the math teacher in me had to unpack this experience further. Many people may watch our video and sum it up just as a cute little boy playing with magnets.  What may get lost in the viewing experience are three key lessons that Thaddeus unknowingly teaches all of us teachers about how we need to shift agency back to our learners in the math classroom. 

Here are a few of my thoughts:

• Manipulatives must live in the math classroom at all grade levels!
  So often, when we think about the use of manipulatives in the math classroom, we automatically think of them as tools that only apply to our early childhood and elementary math learners.  As our math learners progress through their K-12 schooling, the usage of manipulatives significantly decreases and ultimately gets replaced with the rote memorization of algorithms and formulas.  I don’t know why or how this trend started but we must tell –and show–  our high school and middle school math learners that they aren’t too cool to use manipulatives, whether we’re talking about calculators, fraction tiles, or base ten blocks.     
• Conceptual understanding translates to stronger retention of mathematical skills
  If you noticed, I didn’t share with Thaddeus how the measure of degrees in a triangle is the foundation for the (n - 2)*180 rule for finding the sum of interior angles in any polygon.  Instead, Thaddeus figured out how combining smaller triangles can form bigger polygons.  Long before he made that connection, Thaddeus learned how to identify different geometric shapes visually by associating them with the number of sides that each of them have.  Without that prior knowledge, he probably would not have been able to discover the relationship between triangles and hexagons. Although algorithms and formulas are important for learners to know, I contend that it’s even more important for learners to know how these algorithms and formulas were derived from specific math concepts.
• Exploration and discovery are essential components for mathematical learning
  Thaddeus didn’t receive a pre-lesson on polygon relationships. He just grabbed his triangle magnets and went on his own learning journey. It was through this exploration that he uncovered the polygon relationships with the triangle magnets. Unfortunately, too many of us math teachers, with the best of intentions, adopt a paternalistic approach to mathematical instruction.  We want our learners to succeed so badly that we sometimes hand them the knowledge without giving the time and space to pursue it on their own. Similar to how the great scorers in the NBA keep on shooting until the ball goes through the hoop, the great mathematicians keep on mathing until they get the math right. So what does that exactly mean? That means applying algorithms, exploring different number relationships, testing out theorems, proving conjectures, and making numerous errors before getting to the solution. As math teachers, we must not only model this practice of exploration and discovery but also embody it in the presence of our math learners. 
   

It’s one thing for us to say we want our math learners to take full responsibility for their own learning, but it’s another to ensure they have ample opportunities to take charge of their learning.

If we can instill that confidence in children as young as Thaddeus, imagine how many more learners will enter school with both  a strong interest and  a firm foundation in math.  At any grade level, these kinds of  lessons should serve as the blueprint for meaningful learning in the math classroom.