Problem visualisation in mathematics and programming
I have been researching into problem visualisation in mathematics and programming following a discussion, originally by Mark Guzdial and continued by Bill Kerr Alan Kay and others and also the parallel discussion by Rob Costello
One paper that has me thinking is the influence of texts' mental images upon problems' resolutions by Giorgio Bagni (2000). It examines the importance of the level of detail of the mental models constructed by students. Bagni asks whether imagining a situation in all its details helps problem solving. Bagni suggests that it can be an obstacle.
Problems were given to 3 Italian school classes, (13-14 years), (14-15 years) and (15-16years).
The first version was briefly stated as an abstract mathematical problem:
The length of the basis AB of an isosceles triangle ABN is 1 000 000 m; the sum of its sides AN, BN is 1 000 001 m; find the length of the height NM.
The second version was embedded in a real world context and included a diagram:
Let us tie a row to two nails very far, say… 1000 km; let us imagine to use a row whose length is exactly 1000 km: so this row will be tight. Then let us tie to the same nails another row, whose length is 1000 km and 1 m; so this second row is a bit longer than the distance between the nails and it will not be tight: in order to stretch it, let us bring the second row in its middle point and let us “raise” such point (see the picture), in order to take it away from the first row, until the second row is completely tight.
a diagram was shown followed by more text
Well, how much must we take away the middle point of the second row? Find the distance between the middle point of the first row, M, and the middle point of the second row, N.
I am hoping that the second version has suffered in the translation from the original Italian to English because it does not read well in English.
Finally the students were shown the correct solution by Pythagoras' theorem which has the result, which may be surprising at first, that by allowing a 1 metre increase in total path length, it is possible to deviate sideways by over 700 metres.
The students in the first group, those given the abstract problem, did significantly better than the second group with the real world problem.
When shown the correct solution, many in the second group, the group that had the problem given in a real world context, had difficulty accepting the correctness of the solution because it was counter intuitive that a such a small increase in total path length would allow such a disproportionately large sideways displacement.
Bagni concludes: “that D’Amore (1997) clearly proved that the full possibility to imagine a situation does not help pupils; now we state that, sometimes, this full possibility can constitute an obstacle to the resolution (or, as in the examined case, to the acceptation of the correct resolution)”.
I hear echoes of Cognitive Load Theory here. The idea that if you are teaching Pythagoras, you should remove any distractors from Pythagoras, so as to minimise the cognitive load on the learner. The learner can then focus their limited processing powers on the material which is to be learnt.
My thought is that the cognitive conflict created by the second problem is valuable for learning and that students should be given time to experience and resolve such cognitive conflict if they are to have more than a superficial understanding of a subject.
This particular cognitive conflict relates to deeper understanding of geometry and physics. It is closely related to the concepts that:
For small x, sin(x) = x.
Over a given distance, the tangent is very close to the circumference if the circle origin is remote.
At x=0, the derivative of sin(x) is maximum but the derivative of cos(x) is zero.
Why is it increasingly hard to pull a rope span, a catenary, the closer to straight it gets? Because your mechanical advantage tends to 1/infinity. Which group of students would you rather have specifying the support structure for electricity cables crossing a busy road?
Jonassen talks of schooling to create real world problem solvers. In school you know that you are doing Pythagoras this week so any problem you are given will be solved in a similar way. Real world problems are poorly specified and multi disciplinary, they require students to have engaged in messy problems.