Imagine if we would introduce calculus with such a text:
“The sense of intellectual discomfort by which the calculus was provoked into consciousness in the seventeenth century lies deep within memory. It arises from an unsetting contrast, a division of experience. Words and numbers are, like the human beings that employ them, isolated and discrete; but the slow and measured movement of the stars across the night sky, the rising and the setting of the sun, the great ball that arise at the far end of consciousness, linger for moments or for months, and then, like barges moving on some sullen river, silently disappear -these are all of them, continuos and smoothly flowing processes. Their parts are inseparable. How can language account for what is not discrete, and numbers for what is not divisible? ” (p. xi. A tour of the calculus)
And as the index of the course this text:
The overall structure of the calculus is simple. The subject is defined by a
- Fantastic leading idea
- One basic axiom
- A calm and profound intellectual invention
- A deep property
- Two crucial definitions
- One ancillary definition
- One major theorem
- The fundamental theorem of the calculus
The fantastic leading idea: The real world may be understood in terms of the real numbers
The basic axiom: Brings the real numbers into existence
The calm and profound invention: The mathematical function
The deep property: The continuity
The crucial definitions: Instantaneous speed and the area underneath a curve
The ancillary definition: A limit
One major theorem: The mean value theorem
The fundamental theorem of the calculus is the fundamental theorem of the calculus
Imagine that this would be the index or syllabus of the course and the instructions were: Pick one of these topics, the one that makes you wonder more, that makes you awe, develop it until you can integrate it into a meaningful whole with your course.
They have to choose which group takes which topic, that will make them start to get involved in this aspects of the whole in order to see why they want to work on a particular topic. And if there are any groups coinciding in their choice they would have to negotiate (which is a very important skill to learn). They will need to figure out -a little bit- what calculus is about but as they have some mathematical background they will find their way out
Despite my idea, I will name a few studies that had brought to the field -didactics of mathematics- evidence of the advantages of introducing history and cultural context into the teaching of mathematics.
Doorman, M. and van Maanen, J. (2008). A historical perspective on teaching and learning calculus. Australian Senior Mathematics Teacher. Vol. 22(2) pp. 4-14.