I love stories. They capture my attention in pretty much any situation in which I find myself, and I don't think I'm the only one. Story telling has always been an easy way for me to capture the attention of students in my classroom. Each story I tell usually shares a snapshot of my life outside of the classroom. In calculus, I tell the story connecting integral calculus to the way I pester my mom by drinking glasses of chocolate milk with increasingly smaller spoons. I tell graduating seniors my story of never feeling like high school was actually over until I experienced eye-lid twitches immediately after my flight took off at the end of the summer en-route for my new college home.
I use stories as pedagogy too; there is something satisfying about talking about number sets as a series of successive inventions introduced to address the mathematical needs of humanity. Counting sheep, signed integers for money, measurement and immeasurable numbers. My intention in doing so has been just, or at least it has felt just up through fifteen years of teaching. The idea is that we start with the most basic elements of mathematics by counting, and then just add complexity as we attempt to account for the rest of what we see in the universe. If we start with the basics, and if we build up our understanding continuously from the basics all the way to the highest levels of mathematics, we are doing right by our students. As our age or number of years in school increases, the complexity increases alongside to match. This, after all, is often what traditional mathematics course sequences have always done.
The concept I confront fairly frequently is how much I disagree with the inverse of this progression. That if we do not start with the basics, then we will not get students to understand increasingly more complex material. This is most often the conversation around students that have gaps in their understanding. We can lament those gaps as teachers, and though my colleagues have always sought ways to help students across them, they can lead to conversations that make me uneasy. "Student A is not ready for X. He can't factor a quadratic."
The main reason I object to this argument is that it leads to a number of issues around course offerings and their structure. I believe curriculum most frequently becomes bloated because of the demands of courses that come after them. Colleges demand X from high schools, so high schools add to their course offerings to match those expectations. This means that high schools demand Y from middle schools, middle schools expect Z from elementary, and so on. I've argued about this mismatch of expectations about the basics across the levels in a previous post.
I am not just concerned about this in the context of mathematics.
My high school US history class twenty years ago made it to the civil rights movement in the 1960s. One could argue that understanding the 1960's requires that we understand the entire story of what happened before it. Maybe we could have moved faster over the entire year so that we could have made it closer to present day, but I don't think my teenage brain would have been able to handle a higher speed.
Must a physics class always start with one dimensional kinematics? Must we do projectile motion algebraically to really understand it?
Stories usually have a beginning, a middle, and an end. If the story of school mathematics always starts with algebra, has geometry and more algebra in the middle, and the end is calculus, students will always be waiting for us to push them forward through the curriculum because as teachers we aknow the story. In traditional timeline based history, we know what happens next because it's the next day or month in time, or the next page in the book. In chemistry, everything is made of atoms, so we have to first build atoms from subatomic particles, then combine elements into compounds, then combine compounds into reactions, and so on.
The other thing that really good stories do, however, is start exactly when and where they need to start. This is not always at the beginning of the action, when things are simple and easy. Good stories expect the audience to trust the medium to provide necessary details along the way. There is backstory, there is foreshadowing and detail and confusion - all deliberately baked in to capture our attention. This is not to say that our job is about entertaining our students. I believe that making sense of what students see in front of them is more important than adhering to a traditional notion of what is basic.
We don't have to construct a car from bolts and sheet metal in order to learn to drive it.
We don't have to understand that water is a polar molecule to understand that it freezes into ice. At some point though, understanding this might help us understand why it ice floats. It's our job to make that knowledge necessary.
The pathways we craft for our students do not have to start at the very beginning of all knowledge or content. They can start with an interesting starting point that leads to questions. They will be confused at first to figure out where they are, what is going on. This is an opportunity to teach knowledge to help students work their way out of this confusion. We can start at the big picture level and dig deeper as increasing complexity demands it. I think teachers broadly understand this on the individual class level, but that this often gets lost in conversations of curriculum or course sequence. We need to be doing more to build our courses to have more experiences like this.
I really think those of us in content based subjects need to talk to our colleagues that teach art. Their courses are often equally dense and skill based, but they take an approach to learning and analyzing that is much more along the lines of being plopped down in an alien environment, shown something novel and unique, and being expected to ask questions. What do you see? How does this make you feel? What questions do you have about what you see? Why did the artist make this choice? How did the artist achieve this result?
There are basics to be taught, but they rarely need to be the starting point.
This is where the biggest shifts in my understanding of this job have occurred over the past fifteen years. Stay tuned.