Learning is Fragile

Some small ways that learning can go wrong

Here's a really interesting meta-analysis (shared by Zach Groshell) on worked examples. The authors reviewed 43 articles studying worked examples in math. They found (probably not surprising) that worked examples had a positive effect on learning. They also found (probably more surprising) that worked examples worked better when they only included correct examples, and also that including self-explanation prompts had a negative effect on learning.

The self-explanation piece was the most interesting to me. I love self-explanation, but I've seen something similar in my classroom. Ten years ago when the Common Core was new I heard a lot of math teachers say, "if you truly understand something you can explain it." That's not true though! Explanation is hard. I especially see students struggle with explanations early on in the learning progression. I still believe in self-explanation, and there's a lot of evidence that it's valuable for learning, but it makes senes to delay that self-explanation until after students have had time to consolidate and practice with their knowledge.

The other thing this study got me thinking about is that the moment of learning is fragile. By fragile, I mean that very small changes in structure can make a big difference in whether students learn. In that moment when I'm trying to help students understand something new, there are lots and lots and lots of ways it can go wrong, and a pretty narrow path for it to go right. I think this is true across different pedagogical choices, whether I'm trying to facilitate an exploration and discussion, or study a worked example, or deliver an explanation, or anything else. Here are a couple different ways I find learning to be fragile:

Prior knowledge. Learning builds on what students already know. If that prior knowledge isn't there, students are much less likely to learn. Small changes like a quick reteach of a foundational skill can make a huge difference in a lesson.

Distractions. When I want students to learn something new I need them to hold a bunch of different ideas in their working memory at once. That's hard! And it's easy for me to forget how hard that is. A distraction at a key moment can derail the explanation. And I don't just mean distractions from students or announcements on the intercom — this also includes a tangent I go off on or a disjointed discussion summarizing a key idea.

Confidence. Students are more engaged when they feel successful in math class. Making students feel dumb as they're learning something new is a great way to make sure they won't learn it. This is where the result from the study on self-explanations is relevant. Explanations are hard, and asking someone to explain the idea they learned often leads them to think "wow I don't get this at all" even if they do. In the same way, the first few things I ask students to do with any new piece of knowledge should be simple to build confidence. I'm not saying math class should be easy all the time, only that the challenges and self-explanations should come a bit later.

Confusion. This is one I'm guilt of all the time. I'm introducing supplementary angles and I misspeak, saying they add up to 90. That's confusing! Or we're discussing a percent problem, and a student says that we multiply by 100 to convert a percent to a decimal. Those moments when students are learning something new, it's easy for the wrong thing to stick, and precision with language is important. Incorrect examples and common mistakes can be valuable learning experiences, but they’re tricky to get right and might be worth delaying until later.