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.

Problem solving - creating runnable mental models

Game creation is gaining recognition as a valuable learning activity. I have justified it in the past on 3 grounds

• transferable cognitive skills,
• metacogitive skills and
• affective benefits

and my attention for transferable cognitive skills has mainly been on near transfer,

• Cartesian coordinates
• negative number
• position, speed, acceleration
• and many more like this
  

I would like to argue the case for generalised problem solving skills in the visual domains like mathematics and physics through improved competence at creating and using runnable mental models.

Bloom' Taxonomy (note 1) is not particularly helpful for understanding higher order thinking and problem solving in visual thinking so I have already had a try at describing problem solving in visual domains as the creation and running of mental models.

Last year I described how one could solve problems like eg. the forces in structures, from first principles with a bit of prior knowledge by building, validating and running a mental model

There is a bit of literature on mental models, Betrancourt & Chassot refer to a "runnable mental model" in Mayer, R. E. (1989). Models for understanding. Review of Educational Research but I can't download that. Jonassen 1 2 refers to runnable mental models:

He says:
(1) Mental models are internal representations.
(2) Language is the key to understanding mental models; i.e.. they are linguistically mediated.
(3) Mental models can be represented as networks of concepts.
(4) The meanings for the concepts are embedded in their relationships to other concepts.
(5) The social meaning of concepts is derived from the intersection of different individuals' mental models.
These assumptions, we believe, are probably necessary but not sufficient for defining mental models....

Generally, mental models are thought to consist of

• an awareness of the structural components of the system and their descriptions and functions,
• knowledge of the structural interrelatedness of those components,
• a causal model describing and predicting the performance of the system (often formalized by production rules),
• and a runnable model of how the system functions

Jonassen's mental might be a bit different to my concept, mine are visual rather than linguistically mediated. Jonassen also states that mental models are multi-modal so the meaning of linguistically mediated is unclear. Jonnasen also seems to lump together mental models that are a community mental model that is socially negotiated with the construction of a problem space inside a problem solver's head.

I like the words in Dunn quoting Jennifer Wiley: students’ “active construction of a runnable mental model” significantly improves their comprehension of any dynamic system.

To give another example of creating, validating and running mental models, consider my manual transmission car which has a noise. Assume I have a vague recollection of the function of the clutch to break the drive train and remove forces from the gearbox. I produce two visual images of engine-clutch-gearbox-wheels and engine-gearbox-clutch-wheels as possible mental models of the car. To validate them I run them, test their output for what I know of the behaviour of cars. Only the first model is consistent with double declutching (remember having to do that? You are old). I can now use my validated model to diagnose my noise, the noise is present, stationary in neutral , but not with the clutch depressed. Run the model, only the gearbox input shaft meets these conditions.

My belief is that there are generalised problem solving and higher order thinking skills that relate to the ability to build complex and robust mental models and then interrogate or run them. Good problem solvers are good at building and running mental models. These skills can be exercised when programming, particularly in the syntax-free iconic languages such as Scratch, Etoys and GameMaker, also when playing problem solving games. The higher order thinking is the debugging where you compare the behaviour of your program and the mental model of your program. Good learning environments keep learners in a tight cycle of test-implement-debug.

Care should always be exercised when talking about generalised higher order thinking skills, one can make sweeping claims without ever defining what higher order thinking is. Pea & Kurland (ON THE COGNITIVE EFFECTS OF LEARNING COMPUTER PROGRAMMING) criticised similar claims about Logo: whether "spontaneous experience with a powerful symbolic system will have beneficial cognitive consequences, especially for higher order cognitive skills. Similar arguments have been offered in centuries past for mathematics, logic, writing systems, and Latin" That is why I think it is important to have a clear understanding of what higher order thinking is.

I'm thinking about the mental model you have of a computer program as you write the program and the debugging process in the context of cognitive conflict or cognitive dissonance. How helpful is it to view a mental model of a program in the dimensions of the Event Indexing Model - time, space, protagonist, causality, and intentionality? For example, think about event driven programming vs linear, with causality indexing for event driven vs temporal for linear.

Finally, you add the magic ingredient of games: a relevant and authentic challenge, the right tools and a collaborative environment which encourages peer tutoring, flow and the ZPD.


Note 1, Reeves "Kyllonen and Shute (1989) have proposed a taxonomy that represents the spectrum of internal states with which cognitive psychologists are concerned. Their taxonomy begins with simple propositions (e.g., stating that Japan sells more electronic products than any other nation), proceeding through schema, rules, general rules, skills, general skills, automatic skills, and finally, mental models (e.g., analyzing the potential of a trade war between Japan and the United States based on an analysis of balance of trade trends). The latter type of knowledge seems particularly important because mental models are the basis for generalizable problem-solving abilities (Halford, 1993)."


more to come.. work in progress