Two sides here:
1. Discovery learning is bad. Students’ working memory is focused on figuring out the task and not becoming fluent with important skills. Some students never figure out efficient methods of problem solving. Plenty of evidence for this, for instance here
2. Direct instruction is bad. Students aren’t learning to think, they’re learning to do what the teacher tells them, and they can’t transfer those skills to a novel context. I’ve seen less formal evidence for this, but my visceral reaction to this video is plenty for me.
1. Discovery learning taught poorly is definitely a bad thing. Discovery learning pushed onto teachers from the administration is also a bad thing.
2. Direct instruction taught poorly is definitely a bad thing. Direct instruction pushed onto teachers from the administration is also a bad thing.
3. Students need practice. This is, I think, the most common pitfall of discovery learning–we hear students figure out the answer and say yes! they’ve got it! but they need practice — practice in context, mixed practice, spaced repetition — in order to retain it.
4. Students need to believe in their efficacy as problem solvers. This is, I think the pitfall of direct instruction — students believe that knowledge comes from the teacher, and if they haven’t been taught it, they can’t do it. That will hamstring them for their entire lives.
5. There’s a false dichotomy here, on the spectrum of guided vs less guided instruction. This ranges from the video above (nearly 100% guided), to largely guided instruction (I/We/You or similar), to partially guided instruction (inquiry with significant scaffolding and summarizing by the teacher) to minimally guided instruction where students are discovering entirely on their own. For my purposes I’m using direct instruction as shorthand for teaching methods on the more-guided end of the spectrum, not for 100% guided instruction.
There are two types of standards in mathematics instruction. I teach 8th grade, and I wouldn’t send an 8th grader to high school without an ability to manipulate expressions, solve linear equations with whole numbers, graph functions, apply the Pythagorean Theorem in context, know how to read a scatterplot, and be comfortable working with exponents and square roots. There are plenty more, but those are (I think) the heart of the 8th grade curriculum, at least in terms of what I want my students to remember a few years from now.
Type two are standards like the Triangle Inequality Theorem, solving systems of linear inequalities, point-slope form of a function, completing the square, formulas for the volume of cones and spheres, and the difference between rational and irrational numbers. Not that these are unimportant, but they won’t be spiraled often in the next few years of their mathematics education and don’t fit as well into the rest of the content they are learning.
Type one, students have to learn. I would advocate for a mix of inquiry, problem-based learning, and direct instruction — giving students an opportunity to figure as much out for themselves as they can, but also using direct instruction to make sure all students — especially those with lower skills — can solve these questions efficiently and apply the concepts appropriately. But if direct instruction shows stronger results for students across the learning spectrum, I’m willing to support it here.
Type two, I’m not too worried about. I’m happy to teach them, but if my students forgot how to complete the square I won’t lose sleep over it. These are, however, incredible opportunities for more inquiry-based teaching. Students will retain what they figure out for themselves longer — which is the only way they will retain completing the square — and because these standards are lower stakes, they can become playgrounds for the Standards for Mathematical Practice.
I don’t mean to say that these standards are unimportant — they are important for exactly this reason. We don’t teach math so that students can pass a test, or so that they can have high-paying jobs, or so they can get into college. We teach math to exercise their brains, their ability to solve puzzles, to think logically, to apply rules and analyze patterns. These standards provide exactly those opportunities to learn, and those skills are what they should be retaining in 2 or 3 or 5 years — not how to turn a polynomial into vertex form. So let’s teach those skills!