What's the best way to teach math: explicit instruction or less guided learning? This might seem like an opinion-based question where the proper answer is "to each their own." ... except, it's not. One of these options is supported by decades of research into the science of learning. The other is not. Assuming that we define "best" as "produces measurably superior learning outcomes," then the best way to teach math, hands-down, is explicit instruction. Nobody who knows the science of learning is actually debating this (though unfortunately much of the education world does not know the science of learning, hence the debate). A quote from Clark, Kirschner, & Sweller (2012, "Decades of research clearly demonstrate that for novices (comprising virtually all students), direct, explicit instruction is more effective and more efficient than partial guidance. So, when teaching new content and skills to novices, teachers are more effective when they provide explicit guidance accompanied by practice and feedback, not when they require students to discover many aspects of what they must learn. ... We also have a good deal more experimental evidence [since the 1960s] as to what constitutes effective instruction: controlled experiments almost uniformly indicate that when dealing with novel information, learners should be explicitly shown all relevant information, including what to do and how to do it. We wonder why many teacher educators who are committed to scholarship and research ignore the evidence and continue to encourage minimal guidance when they train new teachers. After a half century of advocacy associated with instruction using minimal guidance, it appears that there is no body of sound research that supports using the technique with anyone other than the most expert students. Evidence from controlled, experimental (a.k.a. "gold standard") studies almost uniformly supports full and explicit instructional guidance rather than partial or minimal guidance for novice to intermediate learners. These findings and their associated theories suggest teachers should provide their students with clear, explicit instruction rather than merely assisting students in attempting to discover knowledge themselves."

Our claims of “learning is not memorization” may sound poetic in the abstract. But when asked to break them down concretely, we find our explanations start to sound more and more like “Tower-building is not nailing wood, not welding steel, not balancing load, it is none of those mechanical things, it is the emergent grace of brushing the heavens.” Which is completely devoid of instructional value, and entirely unhelpful to any aspiring tower-builder, aside from possibly inspiring a fleeting spark of motivation that spends itself searching for a place to spend itself.