I am thinking hard about this, and took some time to write down my current thoughts, some of them arriving just now.
Many years ago
Students can’t remember (or use flexibly) math procedures because they don’t understand them. If they explore and learn the concepts, they will remember the procedures that come from understanding.
Problem: they still forget the procedures
A couple of years ago
Memory and recall of facts and procedures happens in a different part of the brain than theory and conceptual thinking. We should be talking to the two parts in different ways. Once we arrive at the procedure through conceptual work, discussion and synthesis, then we follow that up with practice to engrain the procedure. The cognitive knowledge has been established and should remain, because it “makes sense.” Maintaining the fact/procedure part just requires repetition. Revisiting the theory (through reminders and mini-lecture, not re-engagement in thoughtful questions and problems) during practice uses up time and cognitive load, making practice harder.
Problem: That doesn’t seem to move knowledge forward overall. Eventually the procedural knowledge becomes disjoint from the conceptual knowledge. Also, I suspect that the way I revisited the theory, in the form of reminders from my knowledge, also tended to nurture “facts” (chunks of conceptual knowledge) that were disjoint from real understanding.
More recently
Maybe both procedural* and conceptual knowledge need active maintenance. And they should be integrated. My scales practice certainly benefits from thinking about the theory behind the scales – not at the same time, but consecutively… what is the major Ionian scale? If I don’t know that, that becomes my lesson for today... then I run a few scales based on that. A few minutes thinking, a few minutes practicing. If I do remember the scale, then I can go a little further… practice in C, now figure out how to practice in D… then practice in D.
*[I think we need to distinguish procedures (how to multiply fractions or get the area of a triangle) from facts. And maybe distinguish particular types of facts: multiplication facts, labels for mathematical objects (angles, denominators), labels for mathematical concepts or more abstract objects (integers, linear equations, commutative property)… worth thinking more about.]
So what are best practices for maintaining procedural knowledge? This goes back to the fact that symbols and quantity are processed in different parts of the brain. We want the symbols to engage thinking, not shut if off. How to do we practice procedures with the concepts still engaged, feeding the retention and integration of both?
For conceptual knowledge? Do we practice conceptual knowledge? Or are we really practicing a third type or a hybrid sort of knowledge, factual knowledge about concepts/relationships (multiplying by numbers between 1 and 0 have a common effect – making the other factor smaller)? Should practice of procedures be included when we practice concepts? Or how about the same type of short burst – re-engage the concept in a thoughtful way, then have students summarize in a way that is meaningful to them. Now what we are demonstrating and practicing is generalization, the existence of conceptual knowledge itself, which the students probably understand less than I think, and the variety of ways to do that: with language, with pictures and diagrams, with algebraic symbols, with examples, etc.
Do slogans and posters have a role in helping students to chunk and integrate new conceptual knowledge and help, or do they tend to divorce the fact from the understanding? Answer: they won’t be divorced from the facts if the students make them up. Give the students a chance to make their own procedure cards, posters, slogans, reminders, etc. (Embarrassed to be just now seeing the actual value in that, but there you go. I guess I had to figure that out for myself - meta; and isn't it cool that "meta" is a cultural concept now?).
Appendix A - Ideas for maintaining procedural knowledge-
I like the idea of little chunks… pick an idea behind a procedure, revisit that in a thought-engaging way, with a clothesline or what do you notice or open middle/open beginning (create the problem)… then quick chunk of repetitive practice of that idea… if possible with reference to the idea… maybe the procedure is written on the board next to the work that we just did.
Represent a concept visually and have students write the symbols for the problem and the answer.
Practically, what happens when some students already have learned the procedure and understand the knowledge? Perhaps just tweaking my attitude a bit: rather than asking them to explain it or go further with a little bit of chip on my shoulder, saying out loud “You can always go deeper” while really thinking “Why can’t they all just be in the same place for once?!” or wanting to prove that the student doesn’t really know it all – I can get really excited about what they know, and maybe figure out a better type of extension than representing it a different way, or explaining their thinking…
What is a multiplication problem you can’t do? What is a multiplication problem you think I can’t do? What do you notice, what do you wonder? … Is there some category of activity (3 or 4) that they might look forward to doing after a quick session of practice that would extend the concept while I help other students who need conceptual engagement earlier in the process? (I have decided this year to try to avoid the term "struggling").
BECAUSE… procedural practice should be concept driven… students should only be practicing the procedures that come out of the concepts they understand.
Many years ago
Students can’t remember (or use flexibly) math procedures because they don’t understand them. If they explore and learn the concepts, they will remember the procedures that come from understanding.
Problem: they still forget the procedures
A couple of years ago
Memory and recall of facts and procedures happens in a different part of the brain than theory and conceptual thinking. We should be talking to the two parts in different ways. Once we arrive at the procedure through conceptual work, discussion and synthesis, then we follow that up with practice to engrain the procedure. The cognitive knowledge has been established and should remain, because it “makes sense.” Maintaining the fact/procedure part just requires repetition. Revisiting the theory (through reminders and mini-lecture, not re-engagement in thoughtful questions and problems) during practice uses up time and cognitive load, making practice harder.
Problem: That doesn’t seem to move knowledge forward overall. Eventually the procedural knowledge becomes disjoint from the conceptual knowledge. Also, I suspect that the way I revisited the theory, in the form of reminders from my knowledge, also tended to nurture “facts” (chunks of conceptual knowledge) that were disjoint from real understanding.
More recently
Maybe both procedural* and conceptual knowledge need active maintenance. And they should be integrated. My scales practice certainly benefits from thinking about the theory behind the scales – not at the same time, but consecutively… what is the major Ionian scale? If I don’t know that, that becomes my lesson for today... then I run a few scales based on that. A few minutes thinking, a few minutes practicing. If I do remember the scale, then I can go a little further… practice in C, now figure out how to practice in D… then practice in D.
*[I think we need to distinguish procedures (how to multiply fractions or get the area of a triangle) from facts. And maybe distinguish particular types of facts: multiplication facts, labels for mathematical objects (angles, denominators), labels for mathematical concepts or more abstract objects (integers, linear equations, commutative property)… worth thinking more about.]
So what are best practices for maintaining procedural knowledge? This goes back to the fact that symbols and quantity are processed in different parts of the brain. We want the symbols to engage thinking, not shut if off. How to do we practice procedures with the concepts still engaged, feeding the retention and integration of both?
- Short bursts – engage a concept, follow up with short practice session
- Students should only be practicing the procedures that come out of the concepts they currently understand
For conceptual knowledge? Do we practice conceptual knowledge? Or are we really practicing a third type or a hybrid sort of knowledge, factual knowledge about concepts/relationships (multiplying by numbers between 1 and 0 have a common effect – making the other factor smaller)? Should practice of procedures be included when we practice concepts? Or how about the same type of short burst – re-engage the concept in a thoughtful way, then have students summarize in a way that is meaningful to them. Now what we are demonstrating and practicing is generalization, the existence of conceptual knowledge itself, which the students probably understand less than I think, and the variety of ways to do that: with language, with pictures and diagrams, with algebraic symbols, with examples, etc.
Do slogans and posters have a role in helping students to chunk and integrate new conceptual knowledge and help, or do they tend to divorce the fact from the understanding? Answer: they won’t be divorced from the facts if the students make them up. Give the students a chance to make their own procedure cards, posters, slogans, reminders, etc. (Embarrassed to be just now seeing the actual value in that, but there you go. I guess I had to figure that out for myself - meta; and isn't it cool that "meta" is a cultural concept now?).
Appendix A - Ideas for maintaining procedural knowledge-
I like the idea of little chunks… pick an idea behind a procedure, revisit that in a thought-engaging way, with a clothesline or what do you notice or open middle/open beginning (create the problem)… then quick chunk of repetitive practice of that idea… if possible with reference to the idea… maybe the procedure is written on the board next to the work that we just did.
Represent a concept visually and have students write the symbols for the problem and the answer.
Practically, what happens when some students already have learned the procedure and understand the knowledge? Perhaps just tweaking my attitude a bit: rather than asking them to explain it or go further with a little bit of chip on my shoulder, saying out loud “You can always go deeper” while really thinking “Why can’t they all just be in the same place for once?!” or wanting to prove that the student doesn’t really know it all – I can get really excited about what they know, and maybe figure out a better type of extension than representing it a different way, or explaining their thinking…
What is a multiplication problem you can’t do? What is a multiplication problem you think I can’t do? What do you notice, what do you wonder? … Is there some category of activity (3 or 4) that they might look forward to doing after a quick session of practice that would extend the concept while I help other students who need conceptual engagement earlier in the process? (I have decided this year to try to avoid the term "struggling").
BECAUSE… procedural practice should be concept driven… students should only be practicing the procedures that come out of the concepts they understand.
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