I agree with the premise that it is destructive for students to merely learn tricks in order to pass classes.
However, the existence of tricks is enormously useful once understanding the underlying mechanism of a particular tool is not the focus of a problem at hand. Abstraction is a fundamental human cognitive faculty. For instance, if a student understands why the cross-multiplication 'trick' works, then they should be free to use it as they please, provided they can actually explain why it works if prompted to do so. The notion that there was a 'right' way to do something (like use common denominators) was stifling and frustrating during my school years. If I can explain and justify the trick - then let me use it. On the other hand, being boxed into doing things the instructor-sanctioned way can lead to equally vacuous understanding: "Teacher says find a common denominator so that's what I'll do even though I don't know why".
Additionally, I will argue that all methods for doing computations with fractions are 'tricks' at some level. After all, they are just theorems on the field of quotients of the integers embedded in the reals. One should not be precluded from using a 'trick' because one of these theorems ('common denominator method good - your trick bad') is more familiar to an instructor. Replacing one 'trick' with what is actually just another does not facilitate understanding.
Of course, this is predicated on actually understanding the tool in the first place.
I take an opposite viewpoint from the author(s). Students should be encouraged to develop and use tools. The utility of hiding complexity [0] with tooling is part of the very essence of what it means to be a hacker. It is also very useful in other fields. For example, a physicist solving for the flow of some fluid does not need to think about why a particular fraction trick works. This would draw precious cognitive resources better served elsewhere.
Once a concept is understood, tricks become useful tools. In a field such as building construction, short-cuts are often expensive in the long run because the benefit of making some compromise (e.g. use cheap plaster) is outweighed by its consequences (e.g. need to re-plaster after short amount of time). This mode of thinking does not apply to mathematics.
Tricks are not 'bad' and should not be nixed. They should be embraced and presented as tools of great utility.
Many of the "tricks" this book indicts aren't good hacks and aren't easily formulated as theorems. They are terrible hacks that are ridiculously inefficient, distracting, and only necessary if you fundamentally don't understand what's actually going on.
See "Butterfly Method, Jesus Fish" or "Backflip and Cartwheel":
They're both super inefficient tricks that are totally unnecessary if you know and understand the theorem they embody.
> If I can explain and justify the trick - then let me use it.
This book is a guide for teachers, not a rule book for students.
For the tricks that are arguably good hacks, the authors provide a simple argument: the time investing in teaching the trick is not worth it, and is better spent somewhere else (see the Jim Doherty quote just before the TOC).
The argument isn't that students should not be allowed to use certain theorems if they understand those theorems. Rather, the argument is that teachers should invest their finite teaching resources explaining other theorems instead.
Often, the authors are arguing that there is an equivalent and equally useful formulation of the same theorem (or a similar one) that's easier to derive and understand. Which is the sort of justification any working mathematician should be on board with (they don't need to agree with the conclusion, but the form of the justification is at least reasonable).
> This mode of thinking does not apply to mathematics.
But this does apply in education writ large. E.g. sacrificing understanding for good performance on a standardized test.
This is not a book telling you how you should do mathematics in your work day. It's a book advocating for certain ways of allocating classroom teaching time.
However, the existence of tricks is enormously useful once understanding the underlying mechanism of a particular tool is not the focus of a problem at hand. Abstraction is a fundamental human cognitive faculty. For instance, if a student understands why the cross-multiplication 'trick' works, then they should be free to use it as they please, provided they can actually explain why it works if prompted to do so. The notion that there was a 'right' way to do something (like use common denominators) was stifling and frustrating during my school years. If I can explain and justify the trick - then let me use it. On the other hand, being boxed into doing things the instructor-sanctioned way can lead to equally vacuous understanding: "Teacher says find a common denominator so that's what I'll do even though I don't know why".
Additionally, I will argue that all methods for doing computations with fractions are 'tricks' at some level. After all, they are just theorems on the field of quotients of the integers embedded in the reals. One should not be precluded from using a 'trick' because one of these theorems ('common denominator method good - your trick bad') is more familiar to an instructor. Replacing one 'trick' with what is actually just another does not facilitate understanding.
Of course, this is predicated on actually understanding the tool in the first place.
I take an opposite viewpoint from the author(s). Students should be encouraged to develop and use tools. The utility of hiding complexity [0] with tooling is part of the very essence of what it means to be a hacker. It is also very useful in other fields. For example, a physicist solving for the flow of some fluid does not need to think about why a particular fraction trick works. This would draw precious cognitive resources better served elsewhere.
Once a concept is understood, tricks become useful tools. In a field such as building construction, short-cuts are often expensive in the long run because the benefit of making some compromise (e.g. use cheap plaster) is outweighed by its consequences (e.g. need to re-plaster after short amount of time). This mode of thinking does not apply to mathematics.
Tricks are not 'bad' and should not be nixed. They should be embraced and presented as tools of great utility.
[0] Such as how fractions work
edit: spelling