Even more challenging is applying those procedures to solving problems. Imagine a student who is still shaky on the skills for multiplying and dividing fractions having to solve a problem that involves choosing which of the two operations is appropriate. The cognitive load there is overwhelming compared to the student who has automatized (chunked!) those skills, and can focus solely on the method selection embedded in the problem. (Method selection is the ability to decide what arithmetic is going to apply to the problem at hand. I first heard the term in this podcast with Mark McCourt. Problem sets with problems from previous lessons mixed in give students a chance to practice method selection.)
This doesn't really change the student landscape of my classroom, but somehow this totally transforms the way I look at them. I have always been uncomfortable with providing exemplars (worked examples for reference) during tests to students who were struggling with procedures, because then what am I testing? It should have been obvious before, but it suddenly is now that I am testing two things: procedural fluency and the bigger concepts that we applying the fluency to. So I need to separate those and assess and provide feedback on both. Perhaps a small quiz on one day, offering students the choice to solve procedural problems outright (fraction division, solve an equation), or with steps missing, or in an example - problem pair. On another day, with exemplars available, assess method selection and conceptual knowledge. Also, in my room I plan to post exemplars for students to use on the days when we are doing more conceptual work and those skills are required, and just be more thoughtful overall about when and how I can most effectively provide those supports.
Also how I teach those skills. I see three strands now when I used to see one. Take fraction multiplication.
First there is understanding the operation itself... what does 1/2 x 3/5 mean? Using manipulatives and models, making sure that students build an understanding of what multiplying by a fraction means concretely. That may be a task without end, but I can establish some particular goals.
Second, there are the arithmetical mechanics of how you get the answer efficiently. It has been a long evolving idea of mine that these first two strands appear closely related, but are perhaps only distantly related, and I still am unsure exactly how. The goal of this second strand clearly is chunking. But what leads to lasting chunking? Providing meaning is one way: if the last four digits of someone's phone number are 1812, I can just remember the War of 1812. But it doesn't seem that the conceptual understanding of how dividing fractions works provides meaning for remembering the procedure. You might think that it would, but maybe the cognitive load of re-imagining the procedure and re-establishing the link between the procedure and the symbol manipulation does not provide a stable hook for memorization. We can prove the Pythagorean Theorem, but retracing the path from geometry to proof to formula is not how experts remember and apply it. Math gurus/curriculum often say that when students learn conceptually, it helps them to remember the procedures longer. I am just not convinced. It seems that whenever the conceptual understanding leading to a procedure is a chain of reasoning two or more steps long, a significant number of students will have forgotten the procedure within days, if not hours. I think part of it is their limited understanding or experience with what it means to justify something by reasoning. Proof is a concept we are still establishing.
So how to best support this second strand? Of course, an obvious answer is rehearsal. But seeing clearly that this problem of chunking the mechanics is its own issue, I can ask also if there are other ways to chunk? I have been exposed to other ideas that I have discarded and mostly forgotten, ie mind maps, that I might go back and take another look at. Example-problem pairs are something that Craig Barton frequently brings up with his guests, and I plan to take a deeper look into those. With just the knowledge I have so far, however, they make a lot of sense, and seem to combine some of the features we have seen so far that seem effective. Students see a problem either presented or demonstrated, and then do a similar problem alongside. There are lots of scaffolding variations: a complete example, an example with blanks present, examples with successively fewer steps included, etc. I can see how you can continually adjust the complexity of the examples and ebb and flow the scaffolding from full to minimal, full to minimal each time a level of complexity is added. The procedures for things like solving equations are not entirely rote, there are still decisions that can be made, so the gradual addition of complexity might be beneficial. With example-problem pairs, we combine sense-making and pattern-seeking with rehearsal, and the research apparently shows this is effective.
The third strand is method selection, being engaged in problem-solving where the particular skill is needed but the student has to recognize that without clues or prompting from the teacher or the materials. I have not thought as much about this, but I recognize that providing these kinds of problems for students are an important part of being able to effectively use mathematics outside the context of a structured lesson.
Cognitive load theory provides an actionable explanation for some of the things I have observed and struggled with in math class. It has given me a number of ideas of new things to experiment with (providing exemplars, structuring practice with example-problem pairs, thinking more about effective ways to differentiate students and instruction or integrate instruction better based on who has chunked the prerequisite skills and who has not). I will also be looking into what kinds of evidence there is for the theory or against. But for now, it has changed the way I am thinking about my classroom significantly.