Frequently, I see students asking questions more-or-less along the lines of “What’s the trick for this problem?” with the assumption that for every type of problem, there is a ready-made trick to that allows said students to get the answer quickly – essentially asking for a full solution that is copied for a similar problem that “requires” the same so-called trick. The reason this annoys me is that it fosters an environment where problem-solving and understanding are discouraged and rote memorization is encouraged.
While this rant is primarily aimed at poorly-implemented pedagogy in mathematics, it applies to pretty much every other subject as well. These learning crutches come in many forms. The following are some I simply decided to rant about.
Formulae
Formulae easily make up some of the worst repeat-offenders I have seen so far. In particular, I am referring to when instructors teach math as formulae to be memorized. It is true that having certain formulae memorized is quite useful, but not taking the time to try to understand why said formulae work is detrimental to problem-solving. Through personal experience as a teacher and tutor, I have found that students who have been taught this way tend to get the most upset over “the teacher didn’t show me to use the formula like this!”
Vocabulary
Like formulae, vocabulary is good to know as it’s a way of concisely communicating descriptions to other individuals. I have also found that it’s quite easily abused. In this case, I am referring to when instructors teach math as “This is a(n) __ problem, so you should do this.” This is when vocabulary becomes a problem-solving crutch because many students develop the following solve process:
\[\begin{aligned} \text{Problem}&\implies\text{Vocabulary}\\ &\implies\text{Memorized process} \end{aligned}\]when instead the process should look more like \[\text{Problem}\implies\text{Solve via understanding}.\] As usual, many students who I’ve seen taught via vocabulary also tend to complain that their teacher didn’t show them exactly how to do a problem.
The “Trick”
Knowing the “trick” to a problem is quite useful as it can save a ton of time; however, it is an abomination to teach students by showing them a problem, and showing them the “trick” and having them memorize it blindly. Almost every student I’ve worked with who has been taught like this solves problems in the following manner:
\[\begin{aligned} \text{Problem}&\implies\text{Recognize trick}\\ &\implies\text{Memorized process} \end{aligned}\]Instead the process should instead look like \[\text{Problem}\implies\text{Solve via understanding}.\] I’m not saying it’s bad to know “tricks”, in fact, “tricks” can be really useful. What I am saying is that students should understand how and why these “tricks” work (or at the very least, get an intuition of why they work) instead of blindly memorizing. Or even better, discover the tricks on their own as a result of problem-solving. And again, many students who I’ve seen taught to solve via “tricks” also tend to complain that their teacher didn’t show them exactly how to do a problem. By the way, the abomination known as the acronym F.O.I.L. (a stupid mnemonic for multiplying two binomials together) falls under this category.
In the three main “offending” categories I outlined above, they all contribute negative factors:
- Hampers learning by encouraging memorization instead of problem-solving (hence the teacher-blaming).
- Causes students to use tricks/formulae incorrectly due to a lack of understanding.
- An inability to make solid arguments or reasonings about obtained results.
- Gives the false impression that every kind of problem has been solved implying no need for creative thinking.
The most common counterargument I receive to this is “If teaching in these ways are so bad, then why hasn’t anyone done anything about it?” Unfortunately, education is simply 50 years (or more) behind the present and undoing these trends is really difficult. To get every school to teach problem-solving effectively means teachers need training to accept a new mindset of the present. A generation or two ago, being able to do routine problems by hand was more useful then than it is today. In the modern age where computers are commonplace, things are very different. Computers are by far more efficient than humans when it comes to routine problems. What would take a human several minutes can be done in the blink of an eye (or a few) by a budget Chromebook.
But how would we fix such an issue? Unfortunately, this is not a quick-fix matter. A majority of the education system (even at the university-level) rewards memorization-based work. It would be more realistic to scrap the entire system and redo it from scratch (in my opinion, anyway).
What I do with each of my students is teach them the valuable lesson of struggle and attempt to eradicate the “static mindset” and foster a “growth mindset” as much as possible (check Carol Dweck’s works on mindsets for more information). Struggle means a student is being challenged, now assuming this done appropriately (too much struggle is also a bad thing), then the student is learning. They’re attempting to solve a problem that they can’t simply apply a formula or trick to and call it a day. They have to reason the problem out and try different strategies of tackling the problem – in other words, problem-solving.
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