Taking the inquiry out of Inquiry Maths 
A reply to Naveen Rizvi

It is always welcome when a teacher engages with the Inquiry Maths website. Naveen Rizvi has done just that in her post Teaching an Inquiry Maths Problem and, indeed, gone further by recommending the site to her readers. She acknowledges the depth of the mathematical ideas in the prompts, focusing her post on the percentages prompt. From my point of view her post is interesting because it demonstrates how two completely different models of teaching might develop out of the same idea. Although starting with the prompt from the website, Naveen's intention is to take the inquiry out of Inquiry Maths.
  
The difference between our approaches is apparent when Naveen writes, “There is so much mathematics that needs to be communicated.” It quickly becomes clear that the teacher is doing the communicating. Naveen proceeds to break the teaching of the prompt into very small steps. The teaching sequence, which maps out all the calculations students might need to manipulate and extend the prompt, is impressive for its thoroughness and detail. Based on Engelman's idea of Direct Instruction, Naveen uses a carefully constructed script and encourages students to generalise the calculations by listing the procedures they should follow to solve problems with the same structure. The presentation of one concept at a time, Naveen claims, excludes the possibility of students forming a misconception. Furthermore, she contends, this will “increase the probability that all pupils can be successful in learning the subject content.” Such an approach, according to Engelman, constitutes logically faultless communication.
  
The “so much mathematics that needs to be communicated” in Naveen's approach becomes the “so much mathematics that needs to be explored” in my own. The mathematical concepts and procedures might be the same, but the way we want students to interact with them are very different. My view of mathematics follows Polya's in How to Solve It: “Mathematics presented with rigour is a systematic deductive science but mathematics in the making is an experimental inductive science.” The experimental side of mathematics involves questioning, exploring and, as Polya says in Mathematics and Plausible Reasoning, guessing. Yes, mathematics involves small steps, but it also involves great leaps of imagination. 
  
The inquiry classroom aims to incorporate these two sides of the subject by presenting the prompt, which appears midway through Naveen’s teaching sequence, at the start of the inquiry. The prompt functions as a proto-typical mathematical structure that requires students to specialise and generalise if they are to understand and extend that structure.
  
Specialising and generalising from 
Mason et al., Thinking Mathematically
 
The form of generalisation in an Inquiry Maths classroom is different to Naveen's. Rather than the transfer of a set of procedures to a new problem, inquiry involves identifying an underlying structure, creating more examples with the same structure (and consciously transforming that structure), making and testing conjectures and using mathematical language (including algebra) to express generalisations. Mason et al. call this process "the essence of mathematical thinking."
  
Perhaps the biggest difference between our approaches lies in the relationship between the teacher and student. In Naveen's approach, it is one-way - teacher communicating to student. In the inquiry classroom, students are encouraged to participate (under the guidance of the teacher) in the direction of the inquiry. Rather than follow the pre-programmed steps of Direct Instruction, students and teacher explore together what is known and what is not known and, moreover, what needs to be known to specialise and generalise with purpose. In this way, students begin to understand that mathematics is a tool with which to understand different contexts and over which they can develop control.
  
Two definitions of 'mathematics'
When I read Naveen's post, I was reminded of Skemp's seminal paper on relational and instrumental understanding – not, I hasten to add, because I want to characterise Naveen's approach as instrumental and inquiry as relational. Rather, I was reminded of Skemp's main point: “I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’." As in relational and instrumental classrooms, the subject matter of Direct Instruction and inquiry is the same, but, paraphrasing Skemp, the approaches are so different that we must regard them as being based on two different definitions of 'mathematics'. On the one hand, we have a definition of mathematics, at least insofar as the student is concerned, as a logical chain of small steps that limits generalisation to applying a procedure to a new problem; on the other hand, we have a definition of mathematics that interweaves induction and deduction, involves small steps and leaps, and combines specialising and generalising from proto-typical structures.
  
Now, at this point, teachers who advocate Direct Instruction might argue that secondary school students are 'novices' who are incapable of studying the mathematics that I describe. Only once students have understood the atomised content can they be expected to reason mathematically without cognitive overload. The problem I see in this line of argument (aside from the imponderable of how much content is sufficient to reason like an 'expert') is that when the time comes students will not know how mathematicians reason or, more importantly, why they reason in the way they do. Once the teacher's small steps are withdrawn, the danger is that mathematics will cease to have any independent meaning over which students feel they can exercise control.
  
Inquiry aims to replace the over-reliance on the teacher that occurs in the Direct instruction model by giving students agency to act on their questions and observations about the prompt. Inquiry puts students in the position to act like mathematicians (not
nota beneact as mathematicians). In a paper about fostering communities of inquiry, eminent professor Alan Schoenfeld writes 
that parallels can be drawn between the mathematics community and the inquiry classroom because "some of the felt experience [of the mathematician] is the same from the students' point of view." Indeed, he continues, "there are remarkably strong parallels between the two environments - despite the fact that one is 'real' and one (by some measures) 'artificial'. What matters, I think, is that both are communities dedicated to exploration and sense-making."

Andrew Blair
May 2019