How to set an inquiry prompt just above the level of the class
   
Recently, Michael Fenton (a maths teacher from Fresno, California) posted on his blog the key characteristics of a rich mathematical task. According to Michael, a rich task:
  • Has a low floor and a high ceiling;
  • Has multiple entry points, inviting the use of multiple representations;
  • Has multiple solution paths, providing opportunity for rich discussion;
  • Integrates multiple topics; and
  • Engages student interest; is mathematically/cognitively challenging.
These characteristics would be important elements of an Inquiry Maths lesson, except the first one.
    
Just above
An inquiry prompt does not have a 'low floor'; instead, it should be set just above the students' current level of understanding. It is designed to pique curiosity by being simultaneously familiar and unfamiliar. Curiosity leads to questioning, which, when the questions are framed in mathematically valid ways, initiates inquiry. Such a prompt encourages students to reach above themselves - to seek out new conceptual knowledge in order to answer their own questions.
     
If the prompt is set below the class's current level of understanding, then it is unlikely to provoke curiosity amongst students and is in danger of being considered unworthy of their attention. A 'low floor' might also lead to a lot of unnecessary activity (drawing a series of diagrams, for example) that does not advance a student's understanding of the situation. Similarly, if the prompt is set too far above existing knowledge, then a class will not recognise anything and quickly become disengaged from learning. That is why selecting an appropriate starting point is a key skill of the inquiry teacher, resting, as it does, on an intimate knowledge of students' current levels of learning.
   
   
Vygotsky (1987) described this process as demanding "more than the child is capable of, leading the child to carry out activities that force him to rise above himself" (p. 213). By acting above herself, the child is drawn into her Zone of Proximal Development (ZPD). She brings spontaneously-formed concepts, constructed out of her everyday experiences, into the zone from below and meets scientific concepts that make up the mathematical culture from above (see box below).
    
The student's current level of mental functions is restructured as she starts to think in abstract concepts; in turn, the scientific concepts are filled with concrete contexts from her empirical experience. The two forms of concept develop in a reciprocal relationship: "... while scientific and spontaneous concepts move in opposite directions in development, these processes are internally and profoundly connected with one another" (p. 219, italics in original).
    
Just above for 30 students
A teacher might ask at this point: How is it possible to set a prompt 'just above' the current knowledge of a class of 30 students who have a range of abilities? To answer this, it might be helpful to consider a collective Zone of Proximal Development for the class in which students start at different levels and have overlapping zones of development. 
     
In a simplified version of a collective zone, the diagram shows the ZPDs of five students. The level of the prompt is set just above the highest level of actual development, but within the ZPD of all students. This is a highly idealised picture of the operation of a class. It is impossible in practice for the teacher to be so exact when measuring each student's upper and lower limits. What's more, each student's ZPD will change from concept to concept. However, this diagram can be a useful heuristic to represent the capabilities of a class during an inquiry, particularly in helping the teacher to decide on which student is able to support another.

      
The inquiry prompt should be susceptible to student questioning at different levels and also be responsive to interpretations at different levels of mathematical sophistication. In this way, it can accommodate the various levels of students' actual development. For example, at the start of the inquiry a student with a lower level in the particular conceptual area of the prompt might ask for the definition of a term, another with knowledge of the terms could request instruction in a calculation method, and a third with existing conceptual knowledge might propose a conjecture. Thus, it is possible to set an inquiry prompt just above the current level of the class.

Vygotsky, L. S. (1987). Thinking and Speech. In Rieber, R. W. and Carton, A. S. (Eds.) The Collected Works of L. S. Vygotsky, Vol. 1. New York: Plenum Press (first published 1934).


Andrew Blair
August 2013
Learning concepts in the ZPD
   
When Vygotsky's Zone of Proximal Development is mentioned in educational texts, it is often defined as the distance between the level of a child's actual development (based on independent activity) and the potential activity that she could achieve with assistance. However, Vygotsky's explanation of conceptual development in the zone is referred to much less often: 
   
“The developmental paths taken by the child’s spontaneous and scientific concepts can be schematically represented as two lines moving in opposite directions. One moves from above to below while the other rises from below to above. If we designate the earlier developing, simpler, and more elementary characteristics as lower and the later developing, more complex characteristics (those concerned with conscious awareness and volition) as higher, we can say that the child’s spontaneous concepts develop from below to above, from the more elementary and lower characteristics to the higher, while the scientific concepts develop from above to below, from the more complex and higher characteristics to the more elementary” (1987, pp. 218-219, italics in original).