Umami is a loan word from Japanese, which means "pleasant, savory taste." Examples of Umami are when you drink a soda on a hot, sweaty afternoon, and you say "Ahhh!" Or perhaps you walking in nature and reach a favorite spot to breath in the fresh air, see the wonderful sights, and take it all in. Ahhhhh!
The notion of umami applies to teaching. Instead of Ahhhh that was savory, it's more like an Aha! "That makes sense!"
"Beautiful idea!"
"Great solution -- I never would have thought of that!"
In math courses where students primarily memorize rote skills and learn procedures developed by someone else, it is hard to provide an opportunity for students to have an Aha moment. The ideas are dull and foreign. The emphasis is on getting to answers quickly, and there's little time for unpacking deep concepts and enjoying the experience of thinking about beautiful ideas.
Thinking about or discovering for oneself beautiful ideas can result in Umami. This is another way to look at authenticity of teaching. By authenticity I mean that students are doing "real mathematics" and are not just memorizing say Gauss-Jordan elimination. When students do authentic mathematics that allows them to grow intellectually, that's a good thing. And if the students buy-in and find the experience of discovery an aesthetically pleasing and fulfilling one, then I believe this is a highpoint in these students' intellectual development. In IBL classes, the opportunity exists to provide regularly authentic math tasks that engage students in figuring out how things work and why things work. Doing math, then becomes an enriching, pleasant, savory experience.
Umami.
