The Students Will Cry Out For Algorithms
And I'll look out and whisper, "No".
I fully deserve this welcomed curse. For almost a decade now I have been teaching Computer Science. In the first week of classes I introduce the core concept of the subject: algorithms. I soften my voice and lean in as if I am telling my students a vast and insidious secret. They already know what algorithms are, you see, because they have been lied to for their entire academic career! That class they had been calling "Math" for years? No. That has really been algorithm class all along!
And I stand by that! Finding slopes, completing the square, applying the quadratic formula to find roots - all of these are processes that have strict steps and expected outcomes. Identifying triangles, hypothesis testing, power/product/quotient rule, the list of algorithms students practice could fill up years of math, which is exactly what it does. These are essential skills for critical thinking, they can't just be skipped because they are algorithmic. They are the conditioning the mind needs so that it can approach the real problems that need to be addressed. An artist can't rely on their creativity alone, they need the technical skills to execute on their imagination. That is exactly why our students need algorithm class, even if we call it math.
Practicing an algorithm is satisfying. There is, by definition, an end result and you can even check your answer! Which is what makes the transition into real mathematics frustrating to so many students. Students move on from practicing known strategies to deciding which ones to use, or even designing their own processes to approach new problems. With that comes a new skillset to develop and a plethora of new uncertainties. Which is where my welcomed curse comes back into play: aside from Computer Science I am fortunate enough to typically teach the courses where students have to dive deep into this struggle like Physics or Calculus. (The AP Computer Science Principles content doesn't dive deep enough into code for this to be apparent, which is a whole different issue.)
I am not a master of developing inquiry but I have found some fairly effective strategies. The most effective of which has been a seemingly innocuous warmup at the start of class. Bite sized prompts like Hewitt's [Next Time Questions](tab: https://www.arborsci.com/pages/next-time-questions) and assorted math puzzles I have accumulated over the years through different sites like Tanya Khovanova's Math Blog and digests like the Carnival of Mathematics. Even though my students quickly pick up on the "tricks", (by the end of the year the class has near 100% accuracy on the NTQs because they know to go against their intuition...) these questions are intriguing enough and rich with relevant content where they go along with my antics in a productive way.
But sometimes a warmup needs to be just a warmup, especially when students' minds start to focus on grades and upcoming assessments. Dylan Kane has some great structures for how to use warmup routines to balance short and long term learning that can bring much needed confidence to students, especially compared to the riddles I default to.
Which brings me to my call to action! Non-algorithmic problem solving needs to be worked into curriculum through a student's STEM studies! I concede this seems like a given, but with the immense volume of algorithmic work that needs to get covered in early secondary it is a steep ask. But the resources are out there! I have always been in awe of Dan Meyer's work, especially his [Three Act]{tab:https://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/) math lessons and organizations like the Mathematics Vision Project provide entire curricula to work from.
Beyond all of this, though, will be the structure of a class. Specifically in the way we frame "solutions" and the problem solving process. A great example of this is in how we frame the states of matter in a Science progression. Young learners are given a straightforward set of three categories: solid, liquid, and gas (and a pregnant pause for a student to chime in about plasma). But as students develop their argumentative writing skills they can start to wrestle with things like non-Newtonian fluids and Jell-O and how there are twenty-two types of ice. In Chemistry, students can fully dive into their understanding of intermolecular forces where the original three categories no longer matter and have instead served their purpose as an essential foundation for learning.
This ambiguity is everywhere but it is rare for an academic setting to acknowledge it even though it can lead to rich discussions. When I taught 9th grade Math the curriculum would often force students into binaries - functions would be increasing or decreasing, or linear or not. These make nice flashcards and straightforward answer keys, but betray the beauty of math. For example, go into Desmos and plot y=x|sin(x)|. Would you consider it increasing? What if you zoom in from just [0, pi]? And what if you keep zooming out towards infinity? Increasing and decreasing can turn into limits and high degrees of abstraction, just like the states of matter to intermolecular forces, but if students rely only on algorithms, that thread might be lost.