Haven't been up to one my passionate rants in a long time so buckle up...

Technology has enabled educators to reimagine the traditional classroom, how students learn and how we facilitate this process, aka teach. Incredible new opportunities to empower students to take more control over their own learning in a "space-time continuum" sort of way. Not to mention providing powerful tools to analyze data to individualize and maximize learning. Are there any buzzwords I omitted!

BUT...

We have strayed from NCTM'S central message from over 25 years ago:

***BALANCING*** PROCEDURAL LEARNING and CONCEPTUAL UNDERSTANDING.

Technology changes the landscape in a fundamental way but the best source code cannot quite replace the critical dialog and face-to-face interaction that is needed to accomplish the above goal. The spontaneous give-and-take of questions and ideas. Interaction vs interactive...

Oh, Dave, you're so 90s, 80s,70s,... You just don't get it, Dave....

The hexagon/triangle problem in the diagram above can be approached using dynamic software like Geogebra & Desmos. You could develop an extraordinary exploration with carefully crafted questions enabling the student to discover relationships in the figures. I love doing that. I used to do this in the classroom long before it was fashionable. Anyone who follows me knows I'm a techie geek at heart.

BUT I came to realize that there was something missing. If you believe I'm not knowledgeable enough of how these new tools can accomplish the BALANCING I speak of, then challenge my premise! Let the New Math Wars begin...

And I haven't even addressed the myriad of approaches to "solving" this multifaceted geometry problem. Most students/groups will find their own solution paths but it is human nature to CHOOSE THE METHOD THAT FITS YOUR OWN WAY OF THINKING.

To develop the deeper ideas of geometry - symmetry, transformations, dissecting, combining and rearranging pieces of a puzzle, students need to be TAKEN OUT OF THEIR COMFORT ZONE and experience others' ideas and we need to fill in the gaps. That is part of teaching, yes?

And, oh yes, there certainly are algebraic/geometric approaches here with lots of nice formulas like (x²√3)/4...