The difference between ‘inquiry’ and ‘enquiry’ in mathematics classrooms
 
In an Inquiry Maths workshop a few years ago, I was asked what the difference is between enquiry with an ‘e’ and inquiry with an ‘i’. While some people use the terms interchangeably, the dictionary makes a distinction. An enquiry is an informal one-off query; an inquiry is a formal judicial examination of evidence to uncover the truth. I think this distinction is helpful in mathematics education. Enquiry suggests a short, structured and time-limited one-off task; inquiry is more a philosophy of teaching that promotes student agency and aims for open classrooms.
  
I was reminded of the workshop question when I received the latest newsletter from the government-sponsored reSolve (Mathematics by Inquiry) project in Australia. It includes two classroom tasks that exemplify the project’s approach. The tasks focus on how algebra can develop as generalised arithmetic. They encourage children to reason by exploring and expressing mathematical structure, pattern and relationships.
 
The year 4 task is called Number Maze. The teacher sets pupils the task of moving through a number grid in a specified way so that the sum of the numbers in the cells they pass through is odd. The aim is summarised in this way: “Through the course of this task, students are encouraged to look at how many odd and even numbers are in each pathway. They see that an odd number of odds is always required to give an odd total.”
 
The year 9 task Addition Chain follows the same course. The teacher requires a student to choose two numbers from which to start a chain where each term is the sum of the two previous terms. Once the chain has ten numbers, the teacher asks the class to find their total. Using the ‘trick’ that the total is 11 times the seventh number, the teacher announces the answer to the surprise of the class before students have the chance to begin the calculation. The teacher has used a property of the Fibonacci sequence. Starting with two numbers (a, b), the seventh term of the sequence is 5a + 8b and the sum of the first 10 terms is 55a + 88b.
 
The reSolve tasks follow the same model. They combine the two key mathematical processes of inductive exploration and deductive reasoning. Students choose a particular case to explore before being introduced to an explanation of the general structure. In both tasks, the source of inquiry and the teacher’s role are also the same. The teacher starts by giving instructions that allow students little flexibility in how to carry out the task. In year 4 the children choose their own routes through the maze, but the teacher provides the maze used in the task and pupils can only move in prescribed ways. In year 9 students choose the two starting numbers, but again the process they follow is laid out in the teacher’s instructions. Once the class has reached the realisation about odd numbers in year 4 or has understood the trick in year 9, the teacher’s role is to generalise from particular cases. In year 4, the teacher introduces a visualisation through which pupils can ‘see’ why an odd number of odds is required. Similarly, the teacher introduces the algebraic form of the Fibonacci sequence to year 9.
   
Student questioning and regulation
  
Students’ questions, which we at Inquiry Maths hold to be fundamental as a source of inquiry and as a precursor to teacher explanations, seem to have a limited role in reSolve classrooms. The description of the year 9 task states that with the introduction of the algebraic form “the door is opened here to many more mathematical investigations.” There follows a number of questions about how the task could proceed. It is not clear where the questions have come from. Are they examples of questions that students have posed in classrooms or are they suggested extensions from the task designers? In his introduction to the newsletter, Steve Thornton (reSolve Executive Director), says that “at each step of the lesson students learn through the teacher’s active intervention.” This suggests that the teacher poses the questions and students have the choice of which ones to follow.
 
The restricted potential for students’ questions has a serious consequence when students do not or cannot follow the path laid out by the task designers. In the year 4 task, for example, it is not clear how pupils can influence the course of the inquiry if they do not notice what they are required to notice. Steve Thornton says that pupils are not expected to discover results in reSolve classrooms, but in the year 4 task they are encouraged to ‘see’ a specific mathematical property. While the distinction between discovering and ‘seeing’ might seem to rest on semantics, the more important point relates to how students can contribute to resolving the impasse caused by not noticing. The reSolve model lacks a student-driven mechanism (be it questions to the teacher or, as in the Inquiry Maths model, regulatory cards) for overcoming an obstacle to inquiry. Ultimately, the teacher has to tell the class what to ‘see’ in line with the design of the lesson.
 
Agency
  
There seems little scope for students’ agency in the reSolve tasks. The teacher provides the source of the inquiry and its direction and the task designer determines the timing of the explanation. In the tasks we have reviewed the students have the opportunity to decide their own path through the maze or to select a pair of numbers to use, but these are limited responsibilities within closely defined parameters. In contradistinction, Inquiry Maths prompts establish a wider ‘landscape’ or ‘zone’ for exploration in which students have the space to ask questions and participate in directing the inquiry. 
   
From an Inquiry Maths perspective, we might call the reSolve tasks ‘enquiries’. They are restricted to a pre-determined outcome, structured by the designer and directed by the teacher and fit neatly into a predictable time frame. Of course, each task could open up into a wider inquiry by encouraging students’ agency in developing their own pathways. Year 4 pupils could suggest changes to the maze or to the rules for moving between the cells or to the property of the result. Year 9 students could suggest changes to the rule for summing the terms of the sequence. While the task designers say they welcome “alternate representations”, the reSolve model does not make students’ questions and suggestions an integral or essential part of inquiry.

Andrew Blair
August 2017