Creating a prompt
What is a mathematical inquiry 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 Sarah Beaumont (a head of mathematics in Kenya) 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 concrete and abstract levels.
(4) A prompt should provide opportunities for different forms of thinking, including induction (exploration and noticing) 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, fixed and changeable and closed and open.
When intrigue alone is not enough
Daniela Vasile (Head of Mathematics in South Island School, Hong Kong) sent this intriguing equation to Inquiry Maths. It certainly qualifies as a prompt on the basis that it has less to it. However, it proves not to have more in it. There is only one other non-trivial example of the same type.
As the equation does not allow for a successful search phase, it would be hard to maintain students' interest and motivation. An algebraic approach might provide another pathway for older students. Kier Tipple (a teacher of mathematics in Brighton, UK) analyses the general case here.
In mathematics education, inquiry is commonly divided into two types. We characterise them as disciplinary inquiry and modelling.
Each type is generated from a different type of prompt. Internal prompts, such as those on this website, make mathematics itself the object of inquiry. External prompts arise outside the discipline and mathematics is used as a tool in modelling a problem or context.
Two types of prompt elaborates on the distinction and the connections between the two types.
Transforming a rich task into a prompt
Jonathan Hall 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 identified the potential of the task as an inquiry prompt. In her design (see illustration below), 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 deductive reasoning 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.
The question of a 'real-life' inquiry prompt
The flags prompt (see picture) does not feature a real flag – national or otherwise. This is very deliberate. Flags can generate debate that detracts from mathematical inquiry, leading students to argue over the nation represented or, in some cases, drawing them into emotional or political reactions. More widely, the associations that students can attach to real-life contexts might even obstruct the emergence of a mathematical understanding. As Bert van Oers says in an important paper: "It is inconceivable how the higher, abstract levels of mathematical thinking can be based on real-life situations."
There are two separate ways that real life is introduced into classrooms of mathematical inquiry. In the first way, an imaginary real-life context is used as the starting point of inquiry (such as pretending to be responsible for an initiative as a member of a company or institution). This leads to problems when students attempt to fit the hypothetical context into their everyday experience. In one study, for example, children tried to understand a bus journey mentioned in a test question by replacing it with an actual journey they had made. As Alan Schoenfeld says in this paper, real life is used in maths classrooms as a meaningless "cover story for arithmetic."
In the second way, mathematical inquiry originates in the student’s actual life experiences. Dewey argued that the inquiry process has no meaning for children unless they use materials and resources that are familiar everyday objects. However, educators have expressed concerns that mathematics "cannot be learnt directly from the everyday environment"1, particularly abstract concepts. Students might develop arithmetic fluency in concrete settings, but the use of formal algebra derives from de-contextualised situations.
For that reason, the prompts on this website are designed to be as devoid of context as possible - “less to them, more in them” as one Inquiry Maths teacher commented. Each one aims to encourage students to apply their existing and new knowledge to 'fill' the prompt with meaning. In this way, students create a context of inquiry that is far more meaningful because it originates and develops through their own ideas and activity. In the case of the flags prompt, students might move on to investigate actual flags once they have learned to pose mathematical questions unhindered by real-life associations.
1. Skemp, R. R. (1971). The Psychology of Learning Mathematics. Harmondsworth: Penguin Books. p. 32.
Setting a prompt just above the level of the class
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.
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 prior knowledge? 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.
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)
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).