Enlighten younger students about the concept of "procedural justice" in mathematics?

Question

I am tutoring a 16-year-old student from my home country (in Asia) in, roughly speaking, precalculus. I would like to give him a feeling of procedural justice, so to speak, in modern mathematics, which I wish at some point in my youth a math teacher would somewhat stress. Since my home country's culture possibly overly stresses that sort of engineering mathematics all through the math classes prior to college education, I especially would like to show him a sense of pure mathematics.

However, it seems that perhaps I need to find a way to make this process more smoother; I found that I am losing this student little by little gradually.

I can understand that many people naturally won't appreciate or even care about how one arrives at a result, let alone the discernment of logical rigor. I have no intention to convert any student to become an automatic reasoner; but some experience suggests that many, without a proper sense of procedural justice, would hold instead a business mindset that only learning or seeing important results matters and how a result, important or not, is obtained does not matter. This concerns me in the sense that such a thinking always looks anti-scientific to me; I mean I cannot think of any esteemed field of study that does not stress a sense of procedural justice. Besides, a sense of procedural justice is natural in a sense, since I can hardly find someone who is not curious about what really happened after a magic trick was done for them. On the contrary, many seem surprisingly impatient when being asked to prove $1+1 = 4/2$, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge.

Answer 1

On the contrary, many seem surprisingly impatient when being asked to prove 1+1=4/2, whose proof (with properly delimited deepness) involves nothing beyond and possibly well below most people's working knowledge.

The practice of starting students out with trivial arithmetic proofs like proving 1+1=4/2 seems to be pretty common, but I'm very skeptical of it. In fact, I think the problem is precisely that the proof is so trivial.

1. It doesn't look to students like procedural justice, it looks like a new arbitrary imposition from their teacher. Suddenly their teacher has decided that some facts are now called axioms, and those count as obvious, while other equally obvious facts for some reason don't count, and they now have to do this new pointless, tedious exercise involving some technical rules about how these axioms work.

2. I have yet to see anyone successfully produce a list of rules at this level which is intelligible to students, and also has no hidden assumptions. Getting the technical details about things like equality rules and substitution of terms correct is pretty subtle. So, in practice, students are usually given a list of rules and are told "you can only use these, and no other facts, even if they seem obvious", right up until they run against one of the gaps in the system they've been given, at which point they get told something like "of course you can substitute equivalent terms (or whatever the gap is); that's obvious". Which just highlights for them how utterly arbitrary the entire activity is.

3. Finally, for all that work, the students don't see the value added. The students see no "procedural justice" in the activity, because these new formal arguments are less persuasive to them than their previous reasoning ("both sides are obviously equal to 2!")

I think you're right that some curiosity about where these things come from is natural, but I don't think proofs of trivial facts like $1+1=4/2$ actually triggers that curiosity. I think you need to start with things where students can see the value added, i.e., in places where the answers aren't obvious, and where students can make mistakes, and then see how a more rigorous argument can avoid mistakes.

Once students have internalized the idea that something can seem obvious but be subtly flawed, then you can hope to go back to things they thought were obvious and convince them it's worthwhile to be more rigorous.

Answer 2

"Since my home country's culture possibly overly stresses that sort of engineering mathematics all through the math classes prior to college education, I especially would like to show him a sense of pure mathematics. However, it seems that perhaps I need to find a way to make this process more smoother; I found that I am losing this student little by little gradually."

This statement screams of "I like proofy stuff and want to force people to do more of it, despite them not liking it like I do, and despite it not being the curriculum they are supposed to learn to support chem/physics/engineering classes."

My recommendation is to BACK OFF. Get a win first on the core topic. Then, and only then introduce the extra proofy stuff (that you love). Note, if the kid is a weak student, than you basically never will get to this. [You did not specify if the tutoring was remedial, normal, or enrichment. But remember, the pedagical "equation" has a variable of student in it, not just of material and method.]

Even in the case that the kid is above average, consider using other "lures" to interest him in pure math. It wasn't the picky details of rigorous proof that drew Andrew Wiles into algebraic number theory. It was a cool problem. Consider small amounts of brain teasers related to definitions. For example, "If the second derivative is zero, does that mean it's an inflection point? If no, give me a simple smooth function to show this." [Answer is y=x^4.]

Finally, even if the kid is strong (which is a pre-requisite for any push towards pure math), that many strong kids prefer to do physics or the like. Look at Onsanger, Feynman, etc. Even for people like this, you can probably still interest them in some details of pure math, but it is more likely to be around definitions, suitability of methods, etc. More than procedural justice. Again, I think even the strongest pure mathematicians discovering huge new breakthroughs, don't discover them via procedure. Rather that is what they do to check their work and publish it. Again, Wiles is great example. But many of the writings of Polya* are also applicable.

*Speaking of which, you might also look at his books on the Stanford math problems (pre calculus, but tricky and interesting) or even his volume of analysis problems (but be very selective here...since many are post calculus). Again...for this even to be an option, it only applies if you have a stronger than average student and he is mastering the base materials.

P.s. And it's not like normal curriculum has no proofs. That's the whole point of geometry--lot of proof writing. More emphasis on that, than on the actual calculations.