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curriculum higraphs

August 12, 2015

I’m a teacher, too, remember? If you’re here for the research, this post might fill a much needed gap in your life.

In a typical university curriculum, we arrange the classes (or modules or courses or whatever you call them) in a dependency graph: to do this class, you should already have done that class, and knowledge from that class will be assumed, etc. I find it useful to push them same sort of structure down a level, the better to organise the broad topics within a class, and even sometimes to structure the process of learning a topic. “Where am I in this picture?” is the question the students should ask themselves: we should make it easy for them to find the answer.

what’s the picture?

Suppose the hierarchichal structure is given like a file system where each node is a directory. In a real file system, we might indeed represent a node as a directory, but not everything which is a directory will necessarily correspond to a node: each node will need a file which maps its internal curriculum, indicating which subdirectories are subnodes and, for each subnode, which other subnodes are immediate prerequisites. What do I mean by “A is a prerequisite of B?”. I mean “Mastery of A is necessary for study of B to be sensible.” There should be no cycles in that graph: mutual relevance does not imply any prerequisite status.

Every such hierarchy can be flattened by giving each node an entry point (a prerequisite of all subnodes) and an exit point (whose prerequisites are the entry point and all subnodes), then linking a node’s entry point from the exit points of its prerequisites. It may also be helpful (but it’s certainly not vital) to indicate for each subnode its external prerequisites, being those nodes elsewhere (neither a sibling nor an ancestor) on which it depends. These should be consistent with the internal curricula, in the sense that the flattening must remain acyclic. Note that we can have D/A/X -> D/B/Y and D/B/W -> D/A/Z,
but if so neither A nor B may be considered a prerequisite of the other.

Colour schemes: if you have mastered a node, it’s safety-blue; if you have mastered the prerequisites for a node but not the node itself, it’s activity-green; if you haven’t yet mastered the prerequisites for a node, it’s danger-red. Red/green distinctions are not ideal for colourblind people, so we should write the captions on green nodes in a more emphatic font: they constitute the frontier where progress can be made.

why bother?

If we plot the curriculum graph, to whatever level of detail we can locally muster, we give some structure to the businesses of teaching, learning and assessment. Learning is what the students do: we can have no direct effect on learning, except perhaps surgery, a fact which sometimes goes astray in discussions of ‘learning enhancement’. Teachers can have an impact on the environment in which learners learn, but ultimately we’re passive in their learning process, and they’ll take whatever they take from the experiences we give them. Learners are going to make a journey through the curriculum. Teaching is how we try to propel them on that journey. Assessment is how we and they determine where they have reached on that journey. If we associate teaching and assessment activities with nodes in the curriculum graph, we make clearer their specific utility in the learning process.

Crucially, by identifying the prerequisition structure, we focus attention in from the whole picture to the student’s frontier of green nodes where they can sensibly study but have not yet achieved mastery. Now, by aligning teaching materials to nodes, the students can identify those which are immediately relevant to their progress, and by aligning assessment to nodes, the students can tell whether progress is happening. My purpose is simply to make it as clear as possible where each student is stuck and what they can do about it.

Teaching takes a variety of forms. We might write notes. We might give lectures. We might offer lab sessions or tutorials. We can arrange ad hoc help sessions. We should expect the delivery of lectures to be an admissible linearisation of the prerequisition structure: that is, lectures should also have a frontier. The students should be able to compare their own frontier with the lecture frontier, and with a fuzzed out version of the cohort’s frontiers. It’s a danger sign if a student’s frontier is strictly behind a lecture: they aren’t best placed to get much from it, at least not at first. If notes are available online in advance of lectures, we acquire the means to cue advance reading and to detect it. The more tightly aligned assessment is to the curriculum, the easier it is to prescribe remedial activities.

Moreover, at the beginning of a lecture, I should very much like to know how the class divides as red/green/blue for that topic. Too much red (or, less likely, overwhelming blue) and I should maybe reconsider giving that lecture. Of course, you only get useful tips like that if you’re doing enough assessment (little and often is better than in overwhelming clumps) to gauge the frontier accurately.

what’s a student’s online view?

When a student visits a node, they should see its hierarchy of ancestors and its immediate subnode structure. The status of all of these nodes should be clear. The rest of the information before them should answer as readily as possible the question ‘What can I usefully do?’. They should see a chronological programme of activities for that node, first into the future and then into the past: lectures may have reading associated; tutorials may have homework handins associated. A node may have a number of assessment items associated, some of which may be active: the list of all assessment items at or below the current node should be readily accessible. Every piece of teaching material should be accompanied by the means to record engagement and comprehension, and to leave a note for the attention of staff.

more later

I want to stop writing for now and publish the story so far. I haven’t said much about assessment mechanisms or what mastery might consist of: these issues I have also been thinking about. But I hope I’ve at least given some reason to believe that there is something to be gained by refining the graph structure of the curriculum when it comes to modelling the progress made by learners and promoting useful activity, for us and for them.

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