Conceptual and Procedural Knowledge

“Just plugging and chugging numbers into a formula while solving a problem is NOT learning.” This is a common refrain you might hear from a teacher in a class that requires some quantitative work. Closely related to this is the oft-quoted dictum that “knowledge does not equal understanding”. We’ve been hearing this more and more in the age of the internet where knowledge seems “easy” to acquire, while understanding remains elusive. This leads to setting up a distinction between conceptual knowledge and procedural knowledge; the former being the holy grail of learning, and the latter being its mindless robotic plug-and-chug counterpart.

 

As an instructor of chemistry, I don’t think the two can be easily separated, if one desires to learn the material (at least in General Chemistry and Physical Chemistry, the two classes I teach most often). Sometimes my P-Chem students will say “I get the concept, but I get lost in the math.” I beg to differ, especially when it comes to quantum chemistry – if you don’t comprehend the math, you likely don’t have a firm grasp on the concepts. I teach the students conceptual material alongside problem-solving, weaving back-and-forth between two sides of the same coin.

 

Furthermore, a good way to check if you really understand the material is to work on a variety of problems, i.e., to make your knowledge more flexible! Conceptual material is always fuzzy when first encountered, and to sharpen both the heart of the matter and its boundaries, one needs to try and answer questions that probe the understanding in a variety of ways. Sometimes this requires a calculation. Sometimes it requires drawing a structure. Sometimes it requires comparing two calculations or two structures. Sometimes it requires making arguments and constructing explanations. I think conceptual material in chemistry isn’t intuitive, and that’s why it’s a challenging subject to learn. But it can be illuminated by practicing problem-solving.

 

There’s an interesting review article by Rittle-Johnson and colleagues titled “Not a One-Way Street: Bidirectional Relations Between Procedural and Conceptual Knowledge of Mathematics” (Educ. Psychol. Rev. 2016, 27, 587-597). The title pretty much tells you what the story will be. There has been a trend in mathematics education in the U.S. to focus on conceptual material first before getting to the procedural parts. The article investigates whether there is evidence that the conceptual-to-procedural approach leads to superior learning outcomes. Broadly speaking, the answer is no. But that’s because setting up experiments that can compare conceptual-to-procedural versus procedural-to-conceptual are not so easy to set up without other confounding factors intruding on the experimental design.

 

The article also discusses the evidence in favor of the two-way mutual support between conceptual and procedural knowledge acquisition. It points out some problematic prevalent beliefs: (1) that “conceptual knowledge has sometimes been used to refer to knowledge that is richly connected while procedural knowledge [was] sparsely connected”, (2) that “procedural fluency seems to refer only to an end state of well-developed knowledge, while conceptual knowledge can refer to a variable amount of knowledge”, and (3) that culturally in the U.S. “practice is not believed to aid the development of understanding” while in many other countries, “practice is viewed as a route towards understanding”. In Asia for example, mathematics education often involves learning procedural knowledge first (being able to plug-and-chug efficiently) before addressing some of the trickier conceptual bits.

 

In G-Chem, there has been a shift in which topics we cover first, and we can see this by comparing textbooks from the last decade or two with earlier ones. Stoichiometry (lots of plug-and-chug) has been moved to later in the first semester. Gases (often also requiring calculations) are typically encountered at the tail end of G-Chem 1, sometimes getting short shrift. On the other hand, electronic structure of atoms and chemical bonding have been moved earlier. I don’t think that’s a necessarily bad choice overall (but do we really need orbitals at the G-Chem level?) and I’m comfortable with this move. That being said, I also do some calculational work (moles/masses, some energy calculations) earlier in the semester.

 

One potential drawback of the current sequence is that G-Chem 2 becomes much more math-heavy (thermodynamics, kinetics, equilibria). Students who are struggling with procedural fluency in doing calculations get mired down in those details and aren’t getting the conceptual material because they’re floundering in the procedural parts. I don’t think loading more conceptual knowledge upfront helps them because the conceptual material (in thermodynamics which is all about keeping quantitative track of energy) is dependent on being able to work the relevant calculations. My experience (in office hours) is that students who can work the calculations also grok the conceptual parts. Those that struggle with the calculations have little grasp of the conceptual material. Even though I always lead with some (although not a lot) of conceptual material for each subtopic in G-Chem 2.

 

Finally, a word about assessment. Rittle-Johnson’s article has a section titled “Evaluation Criteria”. Measuring conceptual knowledge is challenging. Measuring procedural knowledge is easier because you can check this by posing calculational problems. If the procedural task involves “near transfer”, this can actually be a good measure (albeit oblique) of conceptual knowledge. I’m not surprised by this, which is why I think exams are a good assessment tool in G-Chem and P-Chem, despite naysayers (who almost always do not teach chemistry). To some extent, the conceptual pieces of chemistry are acquired gestalt-like, and are not amenable to reductionist breaking-into-pieces. The procedural parts, on the other hand, can be atomized into pieces – and once the student gains fluency (thus moving their acquired procedural knowledge into long-term memory), they now have the bandwidth to synthesize their conceptual knowledge. But we can’t expect them to do so automatically on their own. Hence the need to teach both procedural and conceptual knowledge and keep going back-and-forth between the two. That is the road to understanding (chemistry).