Inquiry is NOT discovery learning
If we were to believe the critics, classroom inquiry is just another variation of discovery learning. Citing their favourite article, they conflate teaching models under the umbrella term ‘minimal guidance’. Rather than analyse the specific nature of each model, the critics lazily dismiss one by association with the perceived weaknesses of another. In the learning of mathematics, discovery and inquiry are very different processes.
  
Discovery
In discovery learning, students are expected to derive a procedure or concept from an activity devised by the teacher. For example, a class might be required to work out the areas of squares on the sides of right-angled triangles and then notice that the sum of the areas of the squares on the two short sides equals the area of the square on the hypotenuse. This ‘discovery’ of Pythagoras’ Theorem can be a memorable and exciting experience. The theorem can seem novel, even when students find out later that it is well known.
   
However, the discovery classroom is often an uncomfortable place for the teacher, especially in a subject like mathematics that is built on axioms and proof. The first problem occurs when students do not make the required discovery and ask for direction or clues. Teachers are forced into subterfuges such as pretending not to hear the student or replying that they are “not at liberty to say” or they “don’t know”. Another approach sees teachers assert that it is not in the interest of the students to be told and that finding the concept independently will “help them learn more”.
  
A second problem occurs when the student makes the wrong discovery. In an attempt to tackle the misconception, while simultaneously preserving the potential for a correct discovery, the teacher gives hints such as “it’s not quite right” or asks whether the student has considered an alternative approach. A third problem arises when one student experiences the ‘aha’ moment and wants to share the discovery. The teacher is forced into attempting to quieten that student to avoid ruining the experience for the rest of the class.
  
Inquiry
The 
procedure or concept that appears at the end of the discovery process is incorporated into the course of an inquiry. It is used to answer students’ questions and develop their observations. In the right-angled triangles inquiry (see prompt right), for example, Pythagoras’ Theorem is deliberately introduced to pursue an inquiry pathway. The key issue for the teacher becomes how and when to introduce the theorem.
    
Frequently 
in the initial phase of the inquiry, a student will ask if the length of the hypotenuse forms a linear sequence in the same way as the lengths of the short sides. (The word ‘hypotenuse’ could be introduced by the teacher when she reformulates a question about the ‘longest’ side.) Alternatively, if the question does not arise and the teacher aims to 'cover' the theorem through the inquiry, she might pose the question herself. 
  
In whatever way the question arises, the teacher has a number of options over how to proceed. She could decide to explain Pythagoras’ Theorem immediately; she could use the selection of the regulatory card 'Ask the teacher to explain' to justify an explanation; or, alternatively, she could ask students to research the theorem and report their findings to the class. The decision would depend on her evaluation of the appropriate level of inquiry for the class. An immediate explanation is characteristic of a structured inquiry, the use of the cards would form part of a guided approach and student research might indicate a more open inquiry.
  
In discovery learning, the teacher attempts to preserve the pretence of discovery, even to the extent of withholding knowledge; in inquiry, the teacher, as a participant in the classroom activity, aims to introduce subject-specific knowledge when it is most relevant and meaningful to her students. 

Andrew Blair
June 2017

Continuum
In response to the post, Mike Ollerton (@MichaelOllerton) wrote: I see discovery learning as a complementary subset of enquiry-based learning. I do not see them in terms of a binary divide. At issue is when I choose to tell students something and when I choose not to; the intervention or interference continuum.
Andrew Blair replies: There is a continuum in inquiry, but it relates to the level of control students have in directing the learning process. The aim is to develop their ability to regulate a mathematical inquiry. Rather than the teacher deciding when or when not to intervene, students learn how to overcome an impasse by requesting new knowledge.
  
"I've discovered ..."
In a recent Inquiry Maths workshop at a conference in Birmingham (UK) when participants were feeding back on progress in an inquiry, one teacher said "I've discovered ..." before correcting himself. He reminded the participants of a slide from a presentation at the start of the workshop that said inquiry is not discovery learning. However, inquiry does not preclude the discovery of novel pathways or applications of mathematics to the prompt. The point is that discovery of a concept or procedure is not the aim of inquiry.
  

Leigh Taylor (@leigh_taylor13) inspired this post by asking on twitter for clarification about Inquiry Maths and discovery learning.