Initial Publication Date: March 16, 2018

Fostering Strategy #9: Learners invent a way to represent a data type they have not previously encountered

(most recent update 24jan2018) (return to workshop front page)

Contributors: Elizabeth Joyner, Tiffany Herder, Paul Parsons


  • Curriculum developer / instructor contrives a situation in which groups of learners make a deep exploration of a system, before they are instructed in the formalisms by which professionals represent the system.
  • Learners develop a holistic, experiential understanding of some aspects of the system's structure or behavior, and a commitment to an interpretation or discovery about the system. Along with this comes a desire to communicate their discovery.
  • Students invent a way to represent their discovery graphically. The instructor offers encouragement and guidance to foreground promising approaches and encourage development towards useful representational techniques.
  • If different student groups have developed different representational approaches, the approaches are compared and the affordances of each are articulated, and perhaps a consensus approach is developed.
  • Eventually, the normative representational technique is revealed. Rather than viewing it as an arbitrary convention imposed by authority, learners welcome it as an ingenious solution to a deeply felt representational challenge.
  • Optional extension: videos or readings about the inventors of the now-accepted representational technique (e.g. invention of topographic contours)


  • The literature documents examples of student invention of graphing (diSessa, et al, 1991) and topographic maps (Enyedy, 2005) in classroom settings with young children.
  • Kastens (unpublished data) has used this approach in a research project, where undergraduate participants invented graphic techniques for displaying dip and strike (geologist's conventions for showing the steepness and direction of a slanted layer of rock), see figure.
  • Problem-based learning seems like a fruitful venue for this approach, because it lends itself to deep engagement in unfamiliar systems accompanied by emotional commitment.

Affordances of this strategy/what it is good for:

  • Learners come to see themselves as creative or innovative people, capable of inventing new ways to put their ideas into external visualizations. Such creative/innovative capacity is needed to push science and technology forward into domains for which there are no graphical conventions (yet).
  • Learners must think deeply about the problem at hand, engage in critical thinking, and integrate information that is presented in other modalities.
  • When normative convention for representing the phenomenon is finally shown, learners may be more likely to accept it than they would have been if they had not struggled with the representational challenge themselves.
  • Learners have more autonomy and agency over their own learning/discovery process than in a more directed activity.
  • Discovery and invention emerge from collaborative discourse among peers.
  • Cross-curricular opportunities merging STEM and art.

Potential pitfalls & challenges:

  • Takes a LOT of time.
  • Takes a very skillful and patient instructor, to pick up on teachable moments, and guide groups away from flailing and towards insights. Instructor must also have a deep understanding of the system being represented, to recognize even the most embryonic representation as meaningful.
  • Takes a collaborative classroom culture, accepting of trials and errors.
  • Needs skillful prompts to help promote innovation.
  • Students may follow what they already know about the conventions rather than inventing. If even one student in a group knows the formal conventions for displaying a type of data, the group may go to that convention and by-pass the invention process.
  • Students may think of themselves as "not creative," and thus may reject the opportunity to even engage with the task.
  • Eventually, if learners go on to further study or work in the domain, they must adopt the normative representational approaches and conventions in order to communicate with others.

Emergent insights:

  • This may be the hardest to implement of the fostering strategies considered at this workshop. It takes a lot of time and very skillful facilitation. It is tricky to set up a situation where the learners know enough about the referent system that they have something they really want to say/show, but they haven't encountered the normative form of representation.
  • Every one of the representational conventions that specialists use today had to be invented by somebody or somebodies, and before that invention, even experts in the domain must have struggled to convey their ideas or observations about the phenomenon. Learners can get a feel for this invention process, and value it, though this kind of exercise.
  • Although the literature examples of this strategy incorporate a very ambitious leap of innovation ("inventing mapping"; "inventing graphing"), it may be possible to design instructional sequences where students make a lesser leap by bootstrapping from a familiar visualization to an unfamiliar context. For example, if students have seen isolines (contours) for one kind of spatially distributed attribute (for example, chemical concentrations), they may be able to come up with using contours to represent a different kind of data that they have never seen displayed in this way. This is a form of reasoning from analogy, and the literature on analogical reasoning suggests that the desired insight is more likely to emerge if the reasoner has seen more than one instance of the source analog. In other words, people who have seen contours used to display attributes A and B are more likely to come with the idea of using contours when they are trying to figure out a way to display attribute C than are people who have only seen contours used to display attribute A (or only attribute B).

Researchable questions:

  • What kinds of activities would support innovation in representations of a data type?
  • Examples in the literature are from young children (diSessa, et al, 1991; Enyedy, 2005); how does this strategy differ when used with other audiences?
  • If a curriculum gives learners lots of opportunities for small-scale representational innovations (for example, coming up with the idea of using contours to show a new data type), does that eventually move them towards bolder leaps of innovation (of the scale of inventing an entirely new type of representation)?


  • diSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing graphing: Meta-representational expertise in children. Journal of Mathematical Behavior, 10, 117-160.
  • Enyedy, N. (2005). Inventing mapping: Creating cultural forms to solve collective problems. Cognition and Instruction, 23(4), 427-466.
  • Lehrer, R., & Schauble, L. (2003). Symbolic communication in mathematics and science: Co-constituting inscription and thought. In Language, Literacy, and Cognitive Development: The Development and Consequences of Symbolic Communication (pp. 167–192).
  • Kastens, K. A., Liben, L. S., & Agrawal, S. (2009). How students and field geologists reason in integrating spatial observations from outcrops to visualize a 3-D geological structure. International Journal of Science Education, 31(3), 365-393. [This paper describes the experiment in which the invented dip and strike symbols were collected--although the invented symbols were not the focus of the paper.]

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