> Lack of clarity often comes in the form of ambiguity
This is sometimes useful. Mathematicians have precision ingrained but this often repeals pupils.
When you start with 'f is a continuous function defined in R of x defined in R, and for each x ..." - well, I am lost. This was the introduction of differentials in my son's high school.
I am a physicist, so I started the other way round, by talking about speed, how it is calculated, how one can get more precise by shortening time and that, eventually, we get to the exact momentary speed.
My son started to ask all kind of question such as how to "get closer" on my wavely drawing, to which I told him "good question - this is possible only when we know the function d(t)", etc.
It is only when he understood the general reason for differentials to exist that we went back to the conditions (continuity, planes, etc.). He actually deduced the continuity constraint himself because he understood the "why".
My math told told us once "I will show you a neat trick that you will not understand this year, but when you understand it next year it will be way less useful to you. Just know that it works only when this and that".
So lack of perfection is sometimes useful for people to understand something at all and not wander off after the first two introductory sentences.
Somewhat tangental but I am very jealous of a high school that introduces differentials. My high school didn't have calculus - lack of interest and no one qualified to teach it.
This is interesting - so what (roughly) did you have in high school?
In the few Europeans schools I know of, calculus is brought in though all the 3 or 4 years in high school because it allows, ultimately, to draw functions. (I am not a math teacher, just someone who went though French school and was very interested in the school system in some other European countries).
It comes in pieces: you get to learn differentials and their meaning, usually then used in physics curriculum to denote speed and force.
Then you get limits and how to calculate them for infinity or when approaching a non-continuous function from both sides. And l'Hôpital rule, etc.
Again, it has direct implications in physics, and sometimes in biology (I saw that only once, when the teacher introduced enzyme kinematics).
Then second differentials to analyze convexity.
All this takes a huge chunk of math in high school, and you cannot du much in functional analysis without using differentials - so I am genuinely interested what was the pressure put on in your school (would you mind naming the country?). Trigonometry for instance? Or probability? (these are the two others I can think of that are the other important pieces they learn).
Some of the schools ended by introducing integrals, usually though the concept of calculating surfaces. It was quite useful to have a rough idea of integrals before going to university (where they were introduced anyway)
This is sometimes useful. Mathematicians have precision ingrained but this often repeals pupils.
When you start with 'f is a continuous function defined in R of x defined in R, and for each x ..." - well, I am lost. This was the introduction of differentials in my son's high school.
I am a physicist, so I started the other way round, by talking about speed, how it is calculated, how one can get more precise by shortening time and that, eventually, we get to the exact momentary speed.
My son started to ask all kind of question such as how to "get closer" on my wavely drawing, to which I told him "good question - this is possible only when we know the function d(t)", etc.
It is only when he understood the general reason for differentials to exist that we went back to the conditions (continuity, planes, etc.). He actually deduced the continuity constraint himself because he understood the "why".
My math told told us once "I will show you a neat trick that you will not understand this year, but when you understand it next year it will be way less useful to you. Just know that it works only when this and that".
So lack of perfection is sometimes useful for people to understand something at all and not wander off after the first two introductory sentences.