Is inquiry compatible with instruction?
  
In schools, students have to acquire mathematical knowledge. How they acquire knowledge – indeed, what constitutes knowledge – and how they use that knowledge are contested issues. In the discussions around knowledge, inquiry and instruction are often presented as opposite (and, even, contradictory) forms of teaching. If they are used in the same classroom, then they appear in a strict sequence: students receive instruction on a particular topic before applying the new knowledge through inquiry. However, teachers using prompts from this website have suggested that the Inquiry Maths model combines both forms of teaching. And, recently, the Executive Director of the reSolve (Mathematics by Inquiry) project in Australia has argued that inquiry is a form of explicit instruction.

 
The difference between explicit and direct instruction
 
Explicit
instruction aims to achieve specific behavioural and cognitive outcomes, which are communicated to students. Knowledge and skills are clearly ‘framed’ and teacher-directed interaction occurs within the boundary set by the lesson’s aims.
 
Direct instruction (of which Engelmann and Bereiter’s ‘Direct Instruction’ is the best known model) is tightly programmed with teachers following a step-by-step, lesson-by-lesson script. The approach is based on stimulus-response and conditioning psychological models. Lessons are tightly paced, follow a prescribed and pre-determined sequence of skill acquisition, aim to maximise time-on-task and involve positive reinforcement of student behaviours.
 
Explicit instruction gives the teacher more opportunity to respond to students’ prior knowledge and misconceptions than direct instruction, but that response occurs within a restricted boundary or ‘frame’.
 
Professor Steve Thornton (Executive Director of reSolve) makes the case for inquiry as a form of explicit instruction:
The word explicit comes from the Latin words ex (out) and plicare (to fold). To make something explicit therefore literally means ‘to unfold’. This idea of explicitness is completely in line with our view of inquiry, which focuses on unfolding important mathematical ideas by encouraging students to ask questions and seek meaning.
In the reSolve model, the teacher guides students to ‘unfold’ the mathematical ideas behind a classroom task. This involves modelling, the use of enabling prompts to provide access, attending to misconceptions, and the unpacking of alternative strategies. These teacher ‘interventions’ are conceived of at the task design stage and the timing of some, such as modelling a general form, are pre-determined. The structured reSolve tasks might be said to resemble explicit teaching in that the teacher establishes a boundary and aims to achieve a specific outcome. As we argue here, the tasks are better described as one-off ‘enquiries’, rather than as part of a fully-fledged inquiry model of teaching.
  
Similarly, teachers who argue that instruction should precede inquiry also conceive of inquiry in a limited way. If a task is used to apply knowledge or a skill that has been recently learnt, then, by its very nature, the task is restricted. It lies within the boundary set by the instructional phase and the outcome is pre-determined. Teacher-directed interactions help to facilitate and structure the students’ application of the knowledge or skill to a particular context. As the potential for open inquiry is precluded in this sequence, the task might also be called an ‘enquiry’. However, even that label is inappropriate if students do not have any creative input at all. Tasks designed for mechanical application cannot be considered to be a form of inquiry.
  
In Inquiry Maths lessons, teachers have characterised phases of teacher explanation as explicit instruction. In this view, the inquiry itself, rather than the teacher’s intention, acts to ‘frame’ new knowledge. The initial phase of questioning and noticing entices students into the topic area, making them receptive to new knowledge. The teacher then gives the class explicit instruction before students go on to use the knowledge in answering their own questions in the remainder of the inquiry. The benefit of this approach is that students realise why the teacher is explaining; they see the content of the explanation as both meaningful and relevant. In this way, the teacher connects with the students’ intent to answer their own questions. Therefore, I would not characterise this period as ‘explicit instruction’, even if the teacher had pre-planned the explanation and would have given it regardless of the questions. The overall approach is an inquiry because students have autonomy to set and plan their own outcomes (rather than have them communicated by the teacher) within the mathematical field implied by the prompt.
 
Inquiry and explicit instruction are pedagogical approaches that originate in different epistemologies. Explicit instruction sees knowledge as transmitted from teacher to student and teaching as effective transmission; inquiry sees knowledge as constructed by students and teaching as facilitating that construction. Vygotsky was right when he said in Thinking and Speech that explicit instruction is “pedagogically fruitless”, achieving “nothing but a mindless learning of words, an empty verbalism.” He went on, “the formation of a [mathematical] concept only begins at the moment a child learns a verbal definition”, and the full generalisation arises through and is formed by “an extraordinary effort of his own thought.”
  
The difference between explicit or direct instruction and inquiry is neatly summarised by Professor Peter Sullivan in the reSolve newsletter. During explicit and direct instruction, the teacher explains first and then students practice using the new knowledge. The questions are normally graded to go from easier to harder, so by the end of the lesson almost every student encounters a problem they cannot do; they “transition from a state of knowing to not knowing”. In inquiry, students begin with a context or prompt that they do not immediately understand, but one that promotes a desire to know more. As the inquiry develops students come to understand; they “transition from a state of not knowing to knowing.”

Andrew Blair
August 2017