>These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.
Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case.
When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics.
Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method.
The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.
To make this more explicit, mathematics (even at the primary school level) is arguably more about “proving” things than it is about computation.
Obviously you’re not writing proofs in 3rd grade, but the emphasis should be on students being able to say why their answer is correct rather than just being able to produce the correct answer.
Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case.
When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics.
Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method.
The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.