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Mathemagenic Activities

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A Technical Memo
Stephen W. Draper
Department of Psychology
University of Glasgow
Glasgow G12 8QQ U.K.


This is a note on the concept of mathemagenic activities.

Link to ITFORUM.

Link to list of ITFORUM participants.

Ian Hart, message to itforum, Sept. 1997

"Mathemagenic" is a term coined by Rothkopf in 1970 to refer to "those activities which give birth to learning", such as "systematic eye fixations while reading".

Since that time the term has been extended: "The concept of mathemagenic activities expresses exactly the idea that there are activities the learner can carry out that will result in their learning." (Laurillard, 1993).

Laurillard went on to coin the term "Anathemagenic", meaning "activities which give birth to loathing" which encompass, in educational technology:

Nada Dabbagh, message to itforum, Sept. 1997

Mathemagenic activities also have two interpretations: generative when initiated by the learner, and supplantive when imposed by the instructional model.

When mathemagenic behaviors are controlled or manipulated by specific design attributes of instruction (such as inserted questions in programmed learning), the result could be a reductive approach to learning (Jonassen, 1984a). Learners become "active performers", responding to activities imposed by the lesson rather than "active learners" where learners impose their own metacognitive activities to produce the desired learnin outcome.

I think there is a continuum of learner generated/instruction generated mathemagenic activities which largely depends on:
- the type of learning outcome;
- learner characteristics and entry skills; and
- the type of instructional strategy used.

Draper: excerpts from message to itforum around July 1996

"Mathemagenic activity" is a term coined by Rothkopf (1970) and means an activity that gives birth to learning. In his words "You can lead a horse to water but only the water that gets into his stomach is what he drinks". In fact a teacher is in an even worse position: not only can a teacher not cause learning directly, they cannot even perceive it happening directly unlike horse minders who can at least see and hear whether a horse drinks and how much. This learner autonomy and the indirectness of teacher power is of course an aspect of constructivism. It is linked here with the idea that learners' actions have a big effect on learning. The impulse behind categorisations of types of CAL, types of learner-computer interaction, or indeed types of educational intervention is that it would be a big help to teachers if types of learner activity could be associated with types of learning. The notion of mathemagenic activity and attempts to classify their types is the essence of this approach to understanding teaching and learning.

Laurillard (1993 p.103) takes up this notion and proposes a model in which there are 12 mathemagenic activities. These apply to all subject areas. The basic approach is that no software to date covers all 12 activities, so any complete approach will combine software (multimedia, etc.) with other activities. This model is not without its problems: to explain how learning occurs without overt support for and observation of some activities, I have to hypothesise that these may be internalised. This leads to further predictions that have not yet been properly tested e.g. that learners will report these internalised activities when interviewed, and that study skill training would equip learners to perform such activities internally without further support. Nevertheless, I find this model to have real predictive power. That is, rather than studying the effects of media or of interactive modes on learning, I vote for studying these mathemagenic activities as primary factors determining learning.

Laurillard's 12 mathemagenic activities

    Conceptual description

  1. Teacher describes the conception.
  2. Student re-expresses the conception.
  3. Teacher redescribes the conception in the light of the student's expression or action.
  4. Student redescribes the conception in the light of the teacher's redescription.

    Personal experience / action

  5. Teacher sets task goal.
  6. Student acts to achieve the task goal.
  7. Teacher's world gives feedback on the action.
  8. Student modifies actions in the light of feedback.

    Reflection (linking description and experience)

  9. Student reflects on action to modify description.
  10. Student adapts action in light of concept.
  11. Teacher adapts task goal in light of student's description.
  12. Teacher reflects on action to modify description.

What defines an M-act?

What defines an M-act: what a teacher does, what a learner does overtly, tacit internal learner actions?

The issue here is: what is the nature and limitations of the notion of M-act (mathemagenic activity), on which the Laurillard model is based? For maximum utility we would like to identify an element that caused learning directly AND could be manipulated by designers directly. Tough.

  1. What affects learners is learner actions, even if these are hidden mental actions. Not external actions by teachers, computers, or learners.
  2. The actions that most directly affect learning are probably things like interpretation and understanding cf. deep and shallow stuff. Not directly observable, although perhaps can get at them by interviews.
  3. Laurillard's version of M-acts is intermediate: they are things like "attend to an exposition". They leave invisible the deep vs. shallow issue of attending to pass a test or attending to understand. But they are activities not materials; and they attempt to enumerate the basic types of action and the basic aspects or types of academic knowledge.

    However there seem to be four ways to develop the notion of M-act, generated by two two-way choices.

  4. Firstly, as discussed above, is the issue of whether to define mathemagenic activities in terms of the unseen inner cognitive actions that directly determine learning but are not directly observable or controllable, or in terms of observable outer activities (such as answering questions, reading a text, etc.). The former has theoretical power but is much less useful to teachers.
  5. The second issue is whether to define them as necessary to learning, or simply likely on average to promote it. Again, the former would be theoretically appealing, but the latter more practicable. I.e. could just regard the L-acts as likely to increase learning statistically: but neither always necessary nor always effective.


Mathemagenic just means "likely to induce learning". You can't be sure of their importance a) because learners may never get it; b) because able learners may not need an activity to induce the processing that causes learning, but do it by reflection or automatic inference.

For instructional /ped. design, we want it to be a teacher action or at least an overt activity. Then we design courses by designing the M-acts; we can do like Laurillard and categorise CAL by M-acts. But actually external acts don't cause learning reliably. But if M-acts are invisible (internal) then they may be causally correct but are almost useless as you can't know when they happen and they don't help much in designing ped. In other words, there seems a tension between developing the concept for theory (can make it true, but not useful and almost without strong predictions for observations), or make it practical but false.

M-acts must really be defined as internal mental acts; although often with behavioural (observable) aspects e.g. student writing notes or essays or speaking. Whether an M-act occurs depends as much on learner as on the CAL / educational intervention.

The right position is probably (1) that M-acts are internal, but you can get fairly reliable / direct evidence of them simply by asking students (talouds, SSI right after). E.g. Alec standing outside a chemistry lab asking students what the lab was about and finding that they clearly didn't really know (at conceptual level). (2) But we can also classify CAL or other ped. design by whether there is any direct support for each M-act. (3) And furthermore, classify that support as weak or strong. E.g. providing a notepad is weak support for M-act-2: it shows designer has thought about it and provided for it, but it is entirely possible (and empirical studies show it is in fact usual) for students not to use this. In contrast, pre-labs requiring students to calculate parameter values before they can start the lab is strong support for M-act-10.

Are M-acts necessary? sufficient? ...

[This question in part triggered by the Clark/Kosma debate, and some of the critiques of Clark for requiring factors (e.g. media) to be necessary. The trouble is, this may rule out the factors such as M-acts that we do want.]

Are all the 12 M-acts necessary for learning? On the face of it, clearly not because people can learn from hearing something once with no visible re-expression activities. Are they sufficient: forgetting suggests not.

But my position might be:
Learning is directly caused by hidden mental actions / tasks. The M-acts are a bit higher level, and should be thought of as structuring these actions such that it is very likely but not certain that they occur and so learning occurs. Forgetting might then come later. Thus M-acts are approximately sufficient.

M-acts may be internalised, in which case you won't see them unless you get the right data i.e. ask students at the time. Are they necessary? Approximately, in that it is conceivable but unlikely that some other activity would drive the necessary lower level mental actions.
Actually, no they aren't necessary. I think that if you are interested and prepped right, then you would process stuff as it comes in, making the right links and learning. Re-expression is an activity likely to exercise this in most students, but isn't necessary for everyone. On the other hand, it can only help. So I see M-acts as a blunderbuss approach, each activity likely to cause learning, but only partly necessary depending on what other activities that learner has really engaged in.

Furthermore, within the conceptual level there is no binary learned/not-learned. The more connections you have made a) the longer retention is likely, b) the more new/transfer tasks you are likely to be able to do and/or do fast. The M-acts promote these connections, but on the one hand there is no limit to the number of connections you might make (cf. Einstein working out more), on the other, very slender connections can be enough for later processing at test time.

Time ordering of M-activities

You can't take the ordering of the activities (the arrows on arcs in the diagram) too seriously:

Summary issues

  1. Are M-acts to be chosen by the learner, or imposed by the teacher?
  2. Are M-acts those visible as overt behaviour (e.g. a tutorial), or invisible mental events (e.g. understanding)? If overt, then useful for instructional designers, but false or at least unnecessary.
  3. Are M-acts necessary for learning?
    Are all 12 M-acts necessary for learning, or will any one do?
    Are M-acts sufficient to cause learning?


Laurillard, D. (1993) Rethinking university teaching: A framework for the effective use of educational technology (Routledge: London).

Rothkopf,E.Z. (1970) "The concept of mathemagenic activities" Review of educ. research vol.40 pp.325-336