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Creating a prompt

Prompts are mathematical statements, equations or diagrams stripped back to the bare minimum, while simultaneously loaded with the potential for exploration. In short, a prompt should have "less to it and more in it" as one head of a mathematics department has said.
    
(1) A prompt must promote curiosity and questioning in students of the sort "Is it true that ...?" or "I've noticed ...". Prompts should be intriguing and ripe for speculation or conjecture. 
     
(2) A prompt must be aimed at students' developing mathematical knowledge, challenging them to decide whether new concepts are required to understand it fully. It must be accessible, yet should stand just beyond the recognition of a class. It should not be designed to intimidate a class; rather, students must feel confident enough to be able to manipulate and change the prompt.
    
(3) A prompt must be open enough to offer students the opportunity to regulate their own activity. Ideally, it will offer a number of pathways and incorporate different areas of the curriculum at both abstract and concrete levels.
  
(4) A prompt should provide opportunities for different forms of thinking, including induction (exploration and generalisation) and deduction (logical reasoning and proof). 

The ideal prompt holds seeming opposites in a creative tension, being simultaneously
  • familiar and unfamiliar
  • accessible and inaccessible
  • obvious and intriguing
  • closed and open
  • fixed and changeable.


  
Further guidance on creating a prompt  
im The question of a 'real-life' inquiry prompt
im How to set an inquiry prompt just above the level of a class
im Can a prompt be created for any topic in the maths curriculum?
im When intrigue alone is not enough to make a prompt

Transforming a rich task into an inquiry prompt
  
Jonathan Hall (@StudyMaths) posted the task below on twitter. In combining the concepts of algebra with range and median, Jonathan has created a rich task that invites students to think about patterns. Moreover, his questions, particularly the third and fifth ones, require students to explore in order to find examples that satisfy the constraints.
The layout of the task suggests the means by which the students are to explore. In giving the first three lines of a table of results and leaving the next two blank, Jonathan is encouraging the students into a systematic search. This, of course, is an important mathematical approach. From an inquiry perspective, the task design is rooted in the inductive processes of collecting results and spotting patterns.

  
Emmy Bennett (@msbennett_math) identified the potential of the task as an inquiry prompt, saying
In her design, the prompt keeps the algebraic terms and expression, but replaces the table and teacher's questions with an inequality. When presented 
with the prompt, students posed the questions and made the observations in the picture:
What do we notice about the students’ initial responses to the prompt?
  • There is already secure knowledge in the classroom about median, range and substitution into terms and expressions.
  • Notwithstanding the fact that there is uncertainty over how to calculate the mean, the fact that it has been brought up allows the inquiry teacher to develop different pathways of inquiry. As Jonathon replied on twitter, “I like the fact that you can replace the median and range with any two of M, M, M or R and have a completely new task to play with.”
  • Students have independently attempted to identify patterns.
  • Students have already checked the cases of a = 2, 3, 4 and 8. This has led to an interesting result that the inquiry teacher could draw out: for a = 2, median < range; for a = 3 and a = 4, median > range; and for a = 8, median < range. The results already suggest two points between 2 < a < 3 and 4 < a < 8 at which the inequality changes. A search could focus on those points.
  • Three observations suggest other lines of inquiry: (a) Summing the terms and expressions (a2 + 6a + 8) could lead on to an inquiry into the values of a that give a sum that is even, odd, negative or positive; (b) The comment about a, a2 and 4a being in the a times table could lead to an inquiry about the values of a that also make a + 8 a multiple of a; and (c) The question about finding the median by halving 4a and a2 could lead to an inquiry about when, if ever, 4a and a2 are the middle two terms of an ordered set.
The students’ responses, then, suggest at least five different lines of inquiry. Of course, the inquiry teacher might restrict the students to just one pathway in the first lesson and even structure that pathway if the class is inexperienced in inquiry. However, the key point is that all the lines have come out of the students’ own questions and observation. This is not only hugely motivating, but also empowers students to take responsibility for the direction of their learning.
  
One final point. The inquiry teacher would want students to explain and, if appropriate, prove their results. Teachers who use the rich task would also, no doubt, want students to do the same. However, the task itself directs students into an inductive process. Inquiry keeps the teacher’s and students’ options open. For example, the line of inquiry about when 4a and a2 are the middle two terms of an ordered set is susceptible to a deductive approach from the start:
  
4a and a2 have to be greater than a, which occurs when a > 1
4a and a2 have to be less than a + 8, which occurs when a < 3
 
The prompt has the advantage of initiating an inquiry that starts with students’ questions and observations and also enables students to take an inductive or deductive approach.


Five steps to creating an inquiry prompt
Three maths teachers were involved in creating the inquiry prompt that appears at the end of this sequence. Each step shows the prompt becoming progressively more open - literally being stripped back to the essence of the mathematical diagram. Sometimes it is difficult for just one teacher to see the full potential in a prompt, and successive iterations might take the involvement of other inquiry teachers.
  
Step 1
The diagram was devised by Kier Tipple, a maths teacher in Brighton (UK). Defining A, B, and C as the areas of the three regions, he posed the problem: "At what distance do you need to set the radii apart to have the three sections equal in area?" Kier introduced the problem at a meal for maths educators, drawing the diagram on the back of a train ticket.
Step 2
The diagram remains the same, but the problem has gone. This gets nearer an inquiry prompt because students are expected to pose the questions. However, in retaining the equation A = B = C, it is clear what is on the mind of the teacher: the prompt relates to three areas.
Step 3
This version hints at the original problem by including the centres of the circles. However, the suggestion of finding their areas has gone and the inquiry is potentially more open for it.
Step 4
In the fourth version, we return to the concept of area. The shading of two of the regions suggests they have something in common, but it is not obvious what that might be. For the first time, the prompt could be interpreted as a Venn diagram.
Step 5
The final prompt is stripped back to the minimum - two overlapping circles. This is the most open, offering a number of paths of exploration - including Venn diagrams, labelling parts of a circle, areas and circumferences of circles, and trigonometry and Pythagoras' Theorem (if we go back to the original problem).

   
Colm Sweet (a maths teacher in Horsham, UK) - who was responsible for steps 3, 4, and 5 above - has since developed the prompt further by combining it with triangles and squares to create the overlapping shapes prompt.