We want to pass on understanding of some material.
All people involved should have to spend as little time as possible overall.
Everyone can be a reader at one time and an author at another.
Nowadays in many mathematical texts, it is common for readers to fill some gaps.
Dinstiction between gap sizes:
By "small" I mean: roughly one line of intermediate steps is missing.
By "large" I mean: more than that is missing.
Small Gaps:
Small gaps are, I think, okay. Most readers can easily fill them in mentally. This makes the text shorter and easier to navigate.
Large Gaps:
I would argue, however, that large gaps increase the total time effort:
Qualitative argument
1.: Repeated reading effort for one person:
Even if you are an expert, you are still a human being who forgets. If you filled a gap once, but years later want to look something up, it may happen that you have to fill the gap again - almost as if it were the first time.
2.: Many readers, few authors
A mathematical text can be written once by a few authors, but read by many people, now and in the future.
Quantitative argument
Assumptions:
Consider one argument whose written-out version would take about half a page.
Two cases:
Case 1: The argument is not written out.
Case 2: The argument is written out.
one author;
writing effort in Case 1: 5 minutes;
writing effort in Case 2: 60 minutes;
n readers;
each reader reads the argument twice over their lifetime;
in Case 1, each reading costs 40 minutes;
in Case 2, each reading costs 20 minutes;
General formulas
The total time costs are:
Case 1: T_1 = 5 min + 2n* 40 min
and
Case 2: T_2 = 60 min + 2n * 20 min.
Case 1 takes more time than case 2 exactly when T_1 > T_2.
That is equivalent to
T_1 - T_2 > 0.
The larger this difference, the more time Case 2 saves relative to Case 1.
The difference is
T_1 - T_2 = (5 - 60) min + 2n(40 - 20) min = (-55 + 40n) min.
For n=2 readers this would already be positive meaning that additional writing effort would save time overall.
The larger n is, the more time is saved.
Example - A widely read text
Suppose n = 100 000 readers over 100 years. Then:
(-55 + 40 * 100 000) min = 3 999 945 min
which is about
66 666 hours = 8 333 eight-hour workdays.
If one counts about 260 workdays per year, this is roughly 32 years of work.
Conclusion:
Large gaps should usually be filled by authors rather than by readers.
In mathematical education, I think there is a lot of room for improvement if we shift some work from reading to writing.
More generally:
I suspect that similar opportunities exist in non-mathematical education as well.
I am interested in connecting with people who want to explore and use this potential.
Starting Point:
Dinstiction between gap sizes:
Small Gaps:
Small gaps are, I think, okay. Most readers can easily fill them in mentally. This makes the text shorter and easier to navigate.
Large Gaps:
I would argue, however, that large gaps increase the total time effort:
Qualitative argument
1.: Repeated reading effort for one person:
Even if you are an expert, you are still a human being who forgets. If you filled a gap once, but years later want to look something up, it may happen that you have to fill the gap again - almost as if it were the first time.
2.: Many readers, few authors
A mathematical text can be written once by a few authors, but read by many people, now and in the future.
Quantitative argument
Assumptions:
General formulas
The total time costs are:
and
Case 1 takes more time than case 2 exactly when T_1 > T_2.
That is equivalent to
T_1 - T_2 > 0.
The larger this difference, the more time Case 2 saves relative to Case 1.
The difference is
T_1 - T_2
= (5 - 60) min + 2n(40 - 20) min
= (-55 + 40n) min.
For n=2 readers this would already be positive meaning that additional writing effort would save time overall.
The larger n is, the more time is saved.
Example - A widely read text
Suppose n = 100 000 readers over 100 years. Then:
(-55 + 40 * 100 000) min
= 3 999 945 min
which is about
66 666 hours
= 8 333 eight-hour workdays.
If one counts about 260 workdays per year, this is roughly 32 years of work.
Conclusion:
Large gaps should usually be filled by authors rather than by readers.
In mathematical education, I think there is a lot of room for improvement if we shift some work from reading to writing.
More generally:
I suspect that similar opportunities exist in non-mathematical education as well.
I am interested in connecting with people who want to explore and use this potential.