In algebra, there is a certain order in which operations must be done. If you’re reading this site, there’s a fair chance that you’re so familiar with this concept that it’s basically subconscious. If not, you’ve probably heard of the mnemonic BEDMAS, meaning brackets, exponents, division/multiplication, and addition/subtraction. These rules are formulated so that there is logical consistency in algebra.
I believe there is also an “order of operations” for how we learn science and mathematics. It’s not quite as strict as a set of rules, but there are definitely optimal paths to take.
In my mind, this is usually the best path to take: question, analogy, example, rigorous explanation.
A good way to start learning about a new concept is to ask a question. This forces a student to reflect on what would be the best way to tackle such a question. At the very least, it primes them to be receptive to the strategy. Additionally, relating the new concept to an older concept (when possible) is great because it gives the student a link to what they have already learned.
A good example that I recently went through is in my Electricity and Magnetism (E&M) class. I first learned about electric fields before moving on to potential difference. These two concepts are obviously related, so it made sense for my teacher to relate them. Plus, our knowledge of the former made it easier to ask starting questions about what to do to find the latter (such as the back-and-forth of integrating and taking the gradient of the electric field).
Asking a question as the first step makes the transition from known knowledge to unknown knowledge more seamless.
Depending on the context, an analog can be well suited to familiarizing students with a new concept. An analogy is powerful because it once again links something that a student knows with something a student doesn’t know. Therefore, employing the analogy early on gives the students a way to understand the new concept.
Reaching back again to my E&M class, this was well done with the concept of equipotential maps. These are lines that show the potential of various points in a given region of space (usually with a few charges in the space). The lines on the map represent points on the map where the potential is equal. It’s basically a contour plot if you’re mathematically inclined.
The analogy for this concept was a topographic map, which have the exact same kinds of lines as equipotential maps. The lines represent a certain elevation, and so one can deduce where the high points on the map are.
This is a powerful analogy because it immediately places the student in the context of topography, which people seem to intuitively understand. Therefore, this sort of abstract notion of a potential field can be linked to a very concrete idea of topography.
Much like a question, a good analogy gives students a chance to get to the same “level” of thinking as the teacher.
I’ve found that a lot of concepts tend to be overwhelming at first. Even with a teacher posing great questions and giving a masterful analogy, it can sometimes be difficult to understand what the heck is going on with the mathematics of the situation.
In this scenario, I’ve found it helpful to jump straight into an example. Doing so gives students direct manipulation into the equations or procedures that one needs, instead of going through a bunch of theory first. I like theory as much as anyone else, but it can be debilitating if one spend forty minutes going into deep theory and then suddenly jumps to an example of something looked at long ago. Therefore, jumping into an example early on rids the student of this long stretch of theory.
I’ve found this step to be especially useful in mathematics, when the concepts are nearly always abstract and so make it difficult to follow every little thing that is going on in a lecture. Additionally, many things that are written out in mathematics tend to be more difficult to understand than when actually applying it to an example, so it makes sense to start with an example. I like doing this in order to get myself use to the mathematics, so that when I start taking notes on the theory behind it, I can actually follow along with what I am writing.
Giving students an example near the beginning of the subject allows them to concretely understand what is going on before the theory is presented to them, dodging the whole bit where students become note-taking zombies.
The final piece of the puzzle is the in-depth explanation. I save this for last for a simple reason: students should be able to follow along during these explanations, and the previous steps give them the best chance of following these explanations without issue.
This is the place where formal definitions will be given and equations will be derived. The idea is that once a student has a sense of what is going on or is trying to be achieved, then one can make an effort to formalize the concept using mathematics. By taking the informal approach first, students stand a better chance of remembering what is happening.
The best test I know of to see if students really understood a concept: do they have to flip back through their notes in order to read everything that was done? If so, then there’s a good chance a lot of the explanations went over their heads.
Start with questions, analogies, and examples, and students will understand what is happening more quickly.