Da un progetto dell’università di Cambridge:
Katie Binks, from Long Road Sixth Form College, shares her experience of using Two Way Calculus with a group:
“I displayed the grid on the board from the beginning and asked some questions about the wording to make sure they understood, e.g. a student sketched what they thought it meant to be increasing for x>1. Students suggested some techniques they might try in order to discover things. They then worked for 10 minutes and then I had students feed back what they had found out. At this point we had a suggestion of an even (!) number of stationary points, but got the increasing and minimum filled in and a suggested function for the top row, which was the translated standard quadratic in completed square form. From then on the students went up and filled in things if they thought they had something that worked, and worked in twos or fours, as they liked consulting freely between groups.
The result was quite lively but very productive.
1. One pair of students ended up using Autograph with the constant controller to investigate the effect of removing different bits of a quintic after realizing that there was something which was having the effect of “collapsing” turning points together to create a stationary point of inflection and thinking about both the differential and second differential having a factor of x.
2. There was lots of discussion between different students about the number of roots and how they could prove how many there were. This included a lot of work on comparing algebraic attempts to using their calculator functionality and understanding how they could use the two together to feel more confident in their solutions.
3. One pair of students fairly quickly realized they could reuse functions and filled in the table quickly. They were then challenged to come up with a different row title for the middle row.
4. One group found a link between the two functions on the middle row concerning the form of the differential (which actually was linked to having x=-3 as a stationary point, but they did not recognize that so had a complicated row title) so formed their function by integrating the differential that they felt fitted the pattern.
5. At the end we discussed what they had learned, which ranged from students finally cracking factorizing, to understanding more about shapes of polynomials, links between functions and their differentials and between similar functions. Students were very positive about the confidence they had gained.”